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+arXiv:2301.00305v1  [math.CT]  31 Dec 2022
+UNIVERSITY OF CALGARY
+The functorial semantics of Lie theory
+by
+Benjamin MacAdam
+A THESIS
+SUBMITTED TO THE FACULTY OF GRADUATE STUDIES
+IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE
+DEGREE OF DOCTOR OF PHILOSOPHY
+GRADUATE PROGRAM IN COMPUTER SCIENCE
+CALGARY, ALBERTA
+JUNE, 2022
+© Benjamin MacAdam 2022
+
+Abstract
+Ehresmann’s introduction of differentiable groupoids in the 1950s may be seen as a starting point for
+two diverging lines of research, many-object Lie theory (the study of Lie algebroids and Lie groupoids)
+and sketch theory. This thesis uses tangent categories to build a bridge between these two lines of
+research, providing a structural account of Lie algebroids and the Lie functor.
+To accomplish this, we develop the theory of involution algebroids, which are a tangent-categorical
+sketch of Lie algebroids. We show that the category of Lie algebroids is precisely the category of invo-
+lution algebroids in smooth manifolds, and that the category of Weil algebras is precisely the classi-
+fying category of an involution algebroid. This exhibits the category of Lie algebroids as a tangent-
+categorical functor category, and the Lie functor via precomposition with a functor
+∂ : Weil1 → �Gpd,
+bringing Lie algebroids and the Lie functor into the realm of functorial semantics.
+ii
+
+Preface
+This thesis is the original work of the author.
+This thesis project arose from attempts to broadly understand differential geometry, and in partic-
+ular the differential geometry of mechanics. First steps were taken with Jonathan Gallagher in under-
+standing how the enriched perspective on category theory introduced in Garner (2018) might relate to
+differential geometric structures. The basic structures in this thesis, however, came out of discussions
+and collaboration with Matthew Burke in the Lie theory context.
+Chapter 1 is an introduction to the theory of tangent categories, and contains no new results. Chap-
+ter 2 began as a joint project with Matthew, who ultimately had to leave the project due to time con-
+straints, although by that time we had already found the basic structure of the proof that differential
+bundles are vector bundles in the category of smooth manifolds (proved in Theorem 2.5.1). The cur-
+rent structure of the chapter, particularly with the emphasis on associative coalgebras of the weak tan-
+gent comonad (T,ℓ) and the tight connection to Grabowski’s previous work on the Euler Vector Field
+construction (Grabowski and Rotkiewicz (2009)), is original work. The basic results in that chapter ap-
+pear in MacAdam (2021); however, the results have been streamlined (there was originally a notion of
+a strong differential bundle, in Section 2.4 it is observed that all differential bundles are strong), and
+there are some new observations about linear connections (Theorem 2.6.6).
+Chapter3 isa rewrite ofa preprintwritten with Matthew on involution algebroids(Burke and MacAdam
+(2019)), while Section 3.4 is due to conversations with Richard Garner (we expect to release a new pa-
+per on involution algebroids based on these results as a joint work). The original idea of augmenting
+an anchored bundle with an involution map is entirely due to Matthew, and the isomorphism on ob-
+jects between involution and Lie algebroids follows calculations shared by Richard Garner (Proposition
+3.4.12). My own contribution in this section comes from connecting this to the work of Martínez (2001),
+as well as giving the bijection on morphisms for involution and Lie algebroids (Theorem 3.5.1).
+The first three chapters provide the background required for the deeper results in Chapter 4. In
+the work of Weinstein (1996); Martínez (2001); de León et al. (2005) on Lie algebroids, it was observed
+that the “prolongation” of a Lie algebroid acted like a tangent bundle. Proposition 4.5.1 makes this
+intuition precise by showing the prolongation is a second tangent structure on the category of Lie al-
+gebroids. The main theorem in this chapter (Theorem 4.4.8) shows that involution algebroids in a
+tangent category � are equivalent to tangent functors (A,α) from Weil1 to � so that the functor A pre-
+serves transverse limits and the natural transformation α is T -cartesian (Definition 4.4.1). These are
+entirely new observations about Lie algebroids and are original work of the author.
+Finally, Chapter 5 puts the first four chapters into the language of enriched category theory, us-
+ing Garner’s enriched perspective on tangent categories (Garner (2018)). The first original results in
+this chapter demonstrate that differential bundles and anchored bundles are models of � -sketches
+(Propositions 5.2.6 and 5.2.10), where � is the site of enrichment for tangent categories. Next, the
+enriched theories framework of Bourke and Garner (2019) is used to prove Theorem 5.4.14, that invo-
+lution algebroids are models of a nervous theory, which is the enriched version of Theorem 4.4.8. The
+thesis concludes with the Lie Realization, Theorem 5.5.13, which is a new characterization of Lie differ-
+iii
+
+entiation and introduces an entirely new way to construct adjunctions between categories of “smooth
+groupoids” and categories of “Lie algebroids” using purely enriched-categorical methods (this is in
+contrast to the geometric approach used in Crainic and Fernandes (2003) and the homotopy theoretic
+approach in Sullivan (1977)).
+This thesis touches on relatively advanced topics in two areas of math: differential geometry (Lie
+algebroids) and enriched category theory (enriched nerve constructions). I have striven to keep it as
+self-contained as possible, introducing the category of smooth manifolds and tangent categories and
+including an appendix with the basics of enriched category theory and locally presentable category the-
+ory. Material on foundational category theory (which is to say, anything that can be found in MacLane
+(1988)) and basic differential calculus (see any calculus textbook) is used without citation or introduc-
+tion; this includes limit, adjunction, monads and monadicity theorems, and the calculus of Kan exten-
+sions and coends. Some facts about horizontal composition spans are used in Chapter 4, but nothing
+that goes beyond the basic definition.
+iv
+
+Acknowledgments
+I would first like to thank my supervisor, Robin Cockett, for guiding this project and for his investment
+of time in reading various preprints and walking through proofs with me. I also owe a special debt of
+thankstoMatthew Burke, whowasa close collaboratoron thisprojectand spearheaded the application
+of tangent categories to Lie theory, and to Jonathan Gallagher for extensive discussions about tangent
+categories. I am also grateful to Rory Lucyshyn-Wright for spending a summer teaching me enriched
+categorytheory, and toKristine Bauerforhelpful conversationsoverthe years. I would alsolike tothank
+all of the members of the Peripatetic Seminar and others over the years for friendly discussion and
+feedback: Chad Nester, Prashant Kumar, J.S. Lemay, Priyaa Srinivasan, Cole Comfort, Daniel Satanove,
+Rachel Hardeman, and Geoff Vooys.
+Finally, I am grateful to my parents and family, and to my wife Niloofar for her support as I wrote
+this thesis.
+v
+
+For my wife, Niloofar.
+vi
+
+Notation
+We start with a table of symbols:
+Notation
+�,�,...
+A (usually tangent) category, treated as a general context for mathematics, de-
+noted using mathbb
+� ,� ,...
+A small category treated as a mathematical object, denoted by mathcal
+A,B,...
+Objects in a category and also functors. written using capital letters
+f ,φ
+Morphisms in a category and natural transformations: lower-case Roman and
+Greek letters
+f ◦ g
+The composition of two maps g : A → B, f : B → C (applicative notation)
+πi
+The projection from the i t h component of an n-fold pullback A0q0×q1...qn−1×qnAn
+or a product
+�n Ai
+F.G
+Composition of two functors, F : � → �,G : � → � (applicative notation)
+φ.G
+Whiskering of a natural transformation
+⊗
+Tensor product in a monoidal category
+⊠
+A restricted notion of span composition, introduced in Definition 4.3.3
+T,p,0,+,ℓ,c
+The data for an arbitrary tangent category, introduced in Definition 1.3.2
+D,⊙,0,!,δ
+The data for an infinitesimal object, introduced in Definition 1.3.7
+T n
+Iterated application of an endofunctor T
+Tn
+The n-fold pullback power of p : T → id for a tangent category, T p ×p ...p ×p T
+vii
+
+We also provide a table of categories:
+Notation
+SMan
+The category of smooth manifolds
+�ℓ
+The category of lifts in a tangent category �, introduced in Definition 2.2.3
+NonSing(�)
+The category of non-singular lifts in a tangent category �, introduced in Defini-
+tion 2.3.1
+PDiff(�)
+The category of pre-differential bundles in a tangent category �, introduced in
+Definition 2.4.1(i)
+Diff(�)
+The category of differential bundles in a tangent category �, introduced in Defi-
+nition 2.4.1(ii)
+LieAlgd
+The category of Lie algebroids, introduced in Section 3.1
+InvAlgd(�)
+The category of anchored bundles in a tangent category �, introduced in Defini-
+tion 3.2.1
+Anc(A)
+The category of anchored bundles in a tangent category �, introduced in Defini-
+tion 3.2.1
+Anc� (A)
+The category of involution algebroids with chosen prolongations in a tangent cat-
+egory �
+Weil1
+The category of Weil algebras, introduced in Section 4.1
+W
+Notation for the Weil algebra �[x]/x 2
+�
+The category of transverse-limit-preserving functors Weil1 → Set, introduced in
+Definition 5.1.1
+Weiln
+1
+The category of Weil algebras with width n, used in Definition 5.2.8
+Weil∗
+1
+The full subcategory of Weil1 spanned by {�,W }
+�
+The classifying � -category of an anchored bundles and all of its prolongations,
+introduced in Definition 5.4.12
+viii
+
+Table of Contents
+Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+ii
+Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+iii
+Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+v
+Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+vi
+Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+vii
+Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
+1
+Tangent categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+4
+1.1
+Differential calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+5
+1.2
+The category of smooth manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+8
+1.3
+Tangent structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+15
+1.4
+Local coordinates in a tangent category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+21
+1.5
+Submersions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+25
+2
+Differential bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+29
+2.1
+Vector bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+30
+2.2
+Lifts for the tangent weak comonad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+34
+2.3
+Non-singular lifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+38
+2.4
+Differential bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+41
+2.5
+The isomorphism of categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+44
+2.6
+Connections on a differential bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+45
+3
+Involution algebroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+48
+3.1
+Lie algebroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+50
+3.2
+Anchored bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+55
+3.3
+Involution algebroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+63
+3.4
+Connections on an involution algebroid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+69
+3.5
+The isomorphism of Lie and involution algebroid categories . . . . . . . . . . . . . . . . . . . .
+76
+4
+The Weil nerve of an algebroid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+78
+4.1
+Weil algebras and tangent structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+80
+4.2
+Tangent structures as monoidal actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+83
+4.3
+The Weil nerve of an involution algebroid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+87
+4.4
+Identifying involution algebroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+97
+4.5
+The prolongation tangent structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
+5
+The infinitesimal nerve and its realization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
+5.1
+Tangent categories via enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
+5.2
+Differential and anchored bundles as enriched structures . . . . . . . . . . . . . . . . . . . . . . 113
+5.3
+Enriched nerve constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
+5.4
+Nervous monads and algebroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
+5.5
+The infinitesimal approximation of a groupoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
+6
+Conclusions and future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
+6.1
+Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
+6.2
+Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
+ix
+
+Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
+Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
+x
+
+Introduction
+The study of Lie groupoids and Lie algebroids goes back to Charles Ehresmann and his student Jean
+Pradines in the late 1950s, building upon Sophus Lie’s original research into the application of groups of
+smooth symmetries to solving ordinary differential equations (Lie (1893)). Motivated by partial differ-
+ential equations, Ehresmann (1959) introduced the notion of a differentiable groupoid, which models
+the internal symmetries of a smooth manifold, in contrast to the external symmetries given by a Lie
+group (a group object in the category of smooth manifolds) or Lie group action. Pradines (1967) ex-
+tended the Lie functor (which sends Lie groups and Lie group actions to Lie algebras and Lie algebra
+actions) from external to internal symmetries and introduced the notion of a Lie algebroid. In doing
+so, he identified some shortcomings in Ehresmann’s original definition of differentiable groupoids, in-
+troducing the modern notion of a Lie groupoid.
+Ehresmann’s investigations into differentiable groupoids initiated one of the major differential ge-
+ometry research programmes of the second half of the twentieth century, the study of Lie groupoids
+and Lie algebroids (which one may refer to as many-object Lie theory to distinguish it from the “single-
+object” Lie groups and Lie algebras that had classically been studied). Research into many-object Lie
+theory in the 80s and 90s focused on extending Lie’s second theorem and the Cartan–Lie theorem,
+which in modern terms state that Lie algebras form a coreflective subcategory of Lie groups; that is, the
+Lie functorhasa fullyfaithful leftadjoint(Mackenzie and Xu (2000); Moerdijk and Mrˇcun (2002); Nistor
+(2000)). The leftadjointisoften called Lie integration, and in theirfamouspaperCrainic and Fernandes
+(2003) found the exactconditionsgoverningwhethera Lie algebroid integratestoa Lie groupoid. Weinstein
+(1996) initiated a line of research into classical mechanics on Lie algebroids and groupoids, extending
+Poincarè’s development of mechanics on a space with a Lie group action and the Euler–Poincarè equa-
+tions (Poincaré (1901); see Marle (2013) for a modern treatment); this has been further developed by
+Eduardo Martinez and his collaborators (de León et al. (2005); Martínez (2001); Martínez (2018); Fusca
+(2018)).
+However, Ehresmann’s work in differentiable groupoids signalled a change in focus for his own re-
+search, as he increasingly focused within the then-new area of category theory. Over the course of
+the 60s and 70s Ehresmann published a string of influential papers in the nascent area of functorial
+semantics, developing the formalism of sketch theory. Sketch theory has proven to be highly influen-
+tial in mathematical logic, and has been active for some 40 years, being extended to syntax/semantics
+adjunctions Gabriel and Ulmer(2006), enriched category theory Kelly (1982) and generalized limit doc-
+trines Adámek et al. (2002). The influence of sketch theory can still be seen on Lie theory in the work
+of Kirill Mackenzie and his collaborators to develop the theory of double Lie algebroids, Lie-algebroid
+groupoids, double-vector bundles, and other tensor-product theories (Mackenzie (1992, 2011)).
+This thesis aims to provide a structural account of the Lie functor from Lie groupoids to Lie al-
+gebroids, using tangent categories Cockett and Cruttwell (2014) to unify Ehresmann’s many-object Lie
+theory and sketch theory. Tangent categories provide a syntactic description of tangent structure based
+on Kock and Lawvere’s synthetic differential geometry (Kock (2006), Lawvere (1979)) and the Weil func-
+tor formalism (a comprehensive account may be found in Kolár et al. (1993), while the explicit link to
+1
+
+abstract tangent structure is found in Leung (2017), another line of research in differential geometry
+that has run parallel to modern Lie theory (albeit with some exchange, e.g. Kolár (2007)). Recent work
+has recast tangent categories as a class of enriched categories (Garner (2018)), making it possible for
+modern techniques from sketch theory and functorial semantics to be applied to differential geome-
+try. In doing so, we demonstrate that the language of tangent categories sheds light on the study of
+classical mechanics on Lie algebroids and groupoids, as well as Mackenzie’s investigations into “Ehres-
+mann doubles” of vector bundles and Lie algebroids.
+Overview
+The first three chapters of this thesis build on previous work in the tangent category literature (for
+example, Cockett and Cruttwell (2017, 2018); Lucyshyn-Wright (2018)), providing tangent-categorical
+sketches of differential-geometric structures. These structures follow Ehresmann’s original notion of
+a sketch quite closely; they are specified as graphs (a collection of objects and arrows in the category)
+with a set of diagrams that must commute and cones that must be universal, except that the data may
+now include the tangent functor T and the tangent natural transformations p, 0, +, ℓ, and c . Each
+sketch is accompanied by a proof that its category of models in smooth manifolds (which we shall
+often write SMan) is precisely the category of geometric structures it seeks to model.
+The first chapter reviews the basic theory of tangent categories, paying particular attention to the
+category of smooth manifolds. The first examples of “sketches” from the tangent categories literature
+are covered, namely differential objects and affine connections, which model vector spaces and con-
+nections respectively (Cockett and Cruttwell (2017, 2018)). The chapter concludes with a study of tan-
+gent submersions, which model submersions from classical differential geometry and are a useful ex-
+ample of the sort of work that occurs in Chapters 2 and 3.
+The second and third chapters develop tangent categorical sketches for vector bundles and Lie
+algebroids respectively. Chapter 2 extends the observation due to Grabowski and Rotkiewicz (2009)
+that the category of vector bundles is a full subcategory of multiplicative monoid actions by the non-
+negative reals �+, and then applies the Euler vector field construction (Definition 2.1.8). The tangent
+categorical sketch for a vector bundle, called a differential bundle, is then developed based on a mor-
+phism λ : E → T E , and an isomorphism of categories between differential bundles in SMan and the
+category of smooth vector bundles is proved. Chapter 3 introduces involution algebroids, which re-
+place the bracket of a Lie algebroid with an involution map
+σ : A̺×T πT A → A̺×T πT A
+(where ̺ : A → T M is the anchor of the Lie algebroid). Using Martinez’s presentation of the structure
+equations for a Lie algebroid (Martínez (2001)), we are once again able to prove an isomorphism of
+categories, this time that the category of Lie algebroids is isomorphic to that of involution algebroids
+in SMan. This provides the initial bridge between differential geometric structures and tangent cate-
+gorical sketches, making it possible to apply more sophisticated techniques in Chapters 4 and 5.
+The fourth chapter constructs a syntactic tangent category for Lie algebroids, and demonstrates
+that Lie algebroids are precisely generalized tangent bundles. We call this result the Weil nerve, as it
+follows the same structure as Grothendieck’s original nerve theorem Segal (1974), in this case using the
+categories presentation of tangent categories due to Leung (2017). This result has useful implications
+for the study of generalized mechanics and geometric structures on Lie algebroids, as it introduces a
+novel tangent structure on the category of Lie algebroids. This novel tangent structure corresponds to
+Poincarè/Weinstein/Martinez’s characterization of classical mechanics on a Lie algebroid.
+The fifth and final chapter introduces the enriched categories perspective on tangent categories
+from Garner (2018), so that the work in Chapters 2 and 3 may be rephrased using the enriched sketches
+2
+
+of Kelly (2005). We construct a functor from the syntactic category of involution algebroids to the syn-
+tactic category of a groupoid-in-a-tangent-category, thus giving a presentation of the Lie functor in
+the spirit of Ehresmann’s sketch theory. The syntactic version of the Lie functor is built by construct-
+ing another novel tangent structure on the category of Lie groupoids (or more generally, groupoids in
+a tangent category), which also agrees with previous investigations into classical mechanics on a Lie
+groupoid. As a final result, we demonstrate that in a locally presentable tangent category we may use a
+left Kan extension to construct a left adjoint to the Lie functor, which we call the the Lie realization.
+3
+
+Chapter 1
+Tangent categories
+Tangent categories are an example of convergent evolution in mathematics, in which two unrelated
+lines of research with very different aims have arrived at a common endpoint, in this case the same
+formal setting for abstract differential geometry. The older line of research has its roots in differential
+geometry proper and, in particular, Weil’s algebraic characterization of the tangent bundle of a smooth
+manifold (Weil (1953)). Weil’s work motivated Kock and Lawvere’s development of synthetic differen-
+tial geometry, presented in the book of the same name Lawvere (1979); Kock (2006) as well as the Weil
+functor formalism of Kolár et al. (1993). The second, more recent line of research has its foundations
+in theoretical computer science following the publication of Linear Logic by Girard (1987). Ehrhard
+and Regnier noticed that some models of linear logic have a notion of the “Taylor series approxima-
+tion” of a proof; this led to the development of differential linear logic by Ehrhard and Regnier (2003).
+Blute, Cockett, and Seely studied the categorical semantics of models of linear logic equipped with the
+derivative operation - that is, they identified those categories whose internal language are were models
+of differential linear logic - developing a categorical theory of differentiation in Blute et al. (2006, 2009).
+Tangent categories arise naturally in each line of research: on the first path with the distillation
+of synthetic differential geometry into abstract tangent functors in Rosický (1984), and more recently
+when Cockett and Cruttwell refined abstract tangent functors following their investigations into the
+manifold categoriesofcartesian differential restriction categoriesin Cockett et al.(2011); Cockett and Cruttwell
+(2014). In a sense, they are categories that axiomatize Weil’s characterization of the tangent bundle as
+an endofunctor in a way that captures the combinatorics of higher-order derivatives when looking at
+a certain class of internal commutative monoids (Cockett and Seely (2011)), as will be made precise
+in Chapter 4. Tangent categories also pull Kock and Lawvere’s synthetic differential geometry into the
+framework of enriched category theory, which is explored in Chapter 5.
+Instances of tangent structure abound throughout mathematics and computer science. For exam-
+ple, many categories of geometric spaces have natural tangent structure, such as the category of conve-
+nient manifolds (the categoryofmanifoldsmodelled locallybyconvenient vectorspaces Kriegl and Michor
+(1997)) and the category of schemes (see point (ii) in Example 2 of Garner (2018)). An example from
+mathematical logic is the category of Köthe sequence spaces( Ehrhard (2002)), and categorical models
+ofthe differential lambda-calculus(Cockett and Gallagher(2019)). More recently, tangentand differen-
+tial categorieshave found applicationsin differentiable programmingand machine learning(Wilson and Zanasi
+(2021)), and to understanding Johnson and McCarthy’s functor calculus Bauer et al. (2018).
+This thesis studies differential geometric structures using the language of tangent categories, fol-
+lowing the tradition of synthetic differential geometry. As such, this chapter will develop tangent cate-
+gories with a focus on the category of smooth manifolds. Extending the study of these formal structures
+in the context of novel tangent categories is a significant endeavour and should be treated as a direc-
+4
+
+tion for future research. The first section introduces Cartesian differential categories as the categorical
+axiomatization of multivariable calculus. The second section introduces the category of smooth man-
+ifolds and two characterizations of its tangent bundle (kinematic versus operational), while the third
+section identifies the structure of the kinematic tangent bundle that characterizes abstract tangent
+structures. The fourth section presents a pair of structures that allow for “local-coordinate calcula-
+tions” in the tangent category of differential objects and connections. The final section introduces
+tangent submersions. A submersion is a differentiable map between differentiable manifolds whose
+differential is everywhere surjective; as a preview of the work in Chapters 2 and 3, this section shows
+that in the category of smooth manifolds, a tangent submersion is precisely a submersion. Section 1.5
+first appeared in MacAdam (2021), and is the only original work in this chapter.
+1.1
+Differential calculus
+As with most treatments of synthetic differential geometry, e.g. Kock (2006), it makes sense to begin
+with the differential calculus - in this case, an introduction to the categorical theory of differentiation.
+Categorical differentiation has recently gained quite a bit of attention due to its relationship with ma-
+chine learning Cockett et al. (2020), and applications to homotopy theory Bauer et al. (2018). This sec-
+tion will just consider the basic structures introduced in Blute et al. (2009), and the canonical example
+of a cartesian differential category (the category of finite-dimensional real vector spaces and smooth
+maps between them).
+Definition 1.1.1. [Definition 1.2.1 Blute et al. (2009)] A cartesian left additive category is a cartesian
+category1 � so that:
+(i) Each hom-set �(A,B) is a commutative monoid with addition +AB and zero map 0AB : A → B (the
+subscript AB will be suppressed when the context is clear).
+(ii) The composition operation ◦ preserves addition on the left:
+(g + h) ◦ f = g ◦ f + h ◦ f
+(iii) Projection is an additive map (preserves addition):
+πi ◦ (f + g ) = (πi ◦ g ◦ f ) + (πi ◦ g )
+Where πi denotes the projection from the i t h component of a product or pullback. j
+There are various examples of cartesian left additive categories - they all fit the same pattern of a
+category where each object is equipped with a non-natural, but coherent, choice of linear structure:
+Example 1.1.2.
+(i) Any category with biproducts is a cartesian left additive category where every map is additive.
+(ii) The category of cartesian spacesCartSp, whose objects are finite-dimensional real vector spaces and
+morphisms are smooth maps between them, is a cartesian left additive category." Clearly smooth
+maps from A → B are closed under addition, projection is an additive map, and (g + h) ◦ f =
+g ◦ f + h ◦ f .
+1We use the standard notation where 1 is the terminal object, × is product, and πi is the i t h projection.
+5
+
+(iii) The category of topological vector spaces and continuous morphisms is a cartesian left additive
+category.
+In fact, cartesian left additive categories may equivalently be described as cartesian categories
+where each object has a coherent choice of commutative monoid structure.
+Proposition 1.1.3 (Blute et al. (2009)). The following are equivalent:
+(i) � is a cartesian left additive category
+(ii) � is a cartesian category so that each object has a chosen commutative monoid structure (A,+A,0A)
+where the following coherence holds:
+(A × B)2
+A × B
+A2 × B 2
++A×B
+τ
++A×+B
+(where τ = ((π0 ◦ π0,π0 ◦ π1),(π1 ◦ π0,π1 ◦ π1))).
+(iii) There is a category with biproducts �+ and a bijective-on-objects subcategory inclusion i : �+ �→ �
+that creates products.
+Cartesian left additive categories provide an appropriate to define a differentiation operation. Re-
+call that the usual derivative of a map f : � → � from elementary calculus can be written
+∂ f
+∂ x : � → �.
+More generally, for a map f : �n → �, one writes the Jacobian of f at x:
+J [f ] : �n → (�n ⊸ �m) :=
+
+
+∂ f1
+∂ x1
+...
+∂ f1
+∂ xn
+...
+...
+...
+∂ f1
+∂ x1
+...
+∂ f1
+∂ xn
+
+
+The Jacobian, however, requires some notion of a “matrix”2 representing a linear map from �n to �m—
+not every category that has a notion of differentiation supports that operation. Instead, the directional
+derivative:
+D [f ](x,v) := lim
+d→0
+f (x + t · v)
+t
+gives an appropriately general notion of differentiation that extends to categories where the space of
+linear maps A → B is not representable by an object in the category 3. A cartesian differential category
+axiomatizes the directional derivative as a combinator on a cartesian left-additive category.
+2This would be called an internal hom in the categorical logic literature.
+3In the case of automatic differentiation, it is also worth noticing that computing the directional derivative of a map
+�n → �m has complexity 2� (f ), while forming the Jacobian has complexity n� (f ), so the directional derivative is a more
+appropriate primitive for purely practical computational reasons (see Section 5 of Hoffmann (2016) for a discussion of the
+computational complexity of forward-mode automatic differentiation).
+6
+
+Definition 1.1.4. [Definition 2.1.1 in Blute et al. (2009)] A cartesian differential category is a cartesian
+left additive category equipped with a combinator (e.g. a function on hom-sets)
+A
+f−→ B
+A × A −−→
+D[f ] B
+satisfying the following axioms:
+[CD.1] Additive:
+D [f + g ] = D[f ]+ D[g ]
+D[0] = 0
+[CD.2] Additive in the second variable:
+D[f ]◦ (g ,h + k) = D[f ]◦ (g ,h) + D[f ]◦ (g ,k)
+D[f ]◦ (g ,0) = 0
+[CD.3] Projection is linear:
+D [πi ] = πi ◦ π1
+D[id ] = π1
+[CD.4] Pairing:
+D[(f ,g )] = (D[f ],D[g ])
+[CD.5] Chain rule:
+D[g ◦ f ] = D[g ]◦ (π0,D[f ])
+[CD.6] Linear in the second variable:
+D[D [f ]]◦ ((a,0),(0,d )) = D[f ]◦ (a,d )
+[CD.7] Symmetry of partial differentiation:
+D [D[f ]]◦ ((a,b ),(c ,d )) = D[D [f ]]◦ ((a,c ),(b,d ))
+Example 1.1.5. The category of cartesian spaces (Example 1.1.2(ii)), CartSp, is the canonical cartesian
+differential category. Let f : �n → �m, and consider its Jacobian at x ∈ �n, J [f ](x) ∈ �n×m. Define the
+differential combinator:
+D [f ]◦ (u,v) = J [f ](v) · u = lim
+t →0
+f (x + t · v)
+t
+.
+In Bauer et al. (2018), the authors construct a cartesian differential category based on the Abelian functor
+calculus of Johnson and McCarthy (1998).
+In Wilson and Zanasi (2021), the authors consider a cartesian differential category whose objects are �2-
+modules to apply gradient-based methods to learn the parameters of of Boolean circuits.
+Every cartesian differential category comes with a notion of linearity. This notion of linearity is
+strictly stronger than additivity - there do exist examples of non-linear additive maps.
+Definition 1.1.6 (Definition 2.2.1 of Blute et al. (2009)). A map f : A → B is linear whenever D[f ] ◦
+(0AB,id ) = f .
+We denote the category of linear maps in a cartesian differential category � as Lin(�). The category
+Lin(�) will have biproducts, and will be a cartesian differential subcategory of �.
+7
+
+Lemma1.1.7(Corollary2.2.3in Blute et al. (2009)). Let � be a cartesian differential category, and denote
+its category of linear maps as Lin(�).
+(i) Linear maps preserve addition.
+(ii) The category Lin(�) is a bijective-on-objects subcategory of � with biproducts, and the inclusion
+Lin(�) �→ � creates products.
+(iii) Every category with biproducts is a cartesian differential category, where
+D [f ] = f ◦ π1,
+and this differential structure makes the inclusion Lin(�) �→ � preserve the left additive structure
+and differential combinator (that is, it is a cartesian differential functor).
+1.2
+The category of smooth manifolds
+Tangent categories axiomatize a more general structure than differential calculus, one in which spaces
+are only “locally linear.” The category of smooth manifolds gives the historically canonical example,
+and a good portion of this thesis relates to structures internal to that category, so it seems worthwhile to
+set a working definition for that context. We follow Tu (2011), and allow for disconnected components
+of a manifold to have different dimensions.
+Definition 1.2.1. [Definitions 5.5–5.7 in Tu (2011)] A chart on a topological space M is pair (Ui,φi :
+Ui �→ �n), where Ui is an open subset Ui ⊆ M and φi : Ui → �n is a local homeomorphism. An atlas is
+a collection of charts {(Ui,φi : Ui → �n)|i ∈ I } (where n is fixed for each connected component of M ) so
+that for each i, j ∈ I , the transition function ψi,j that completes the diagram
+φ−1
+i (Ui ∩Uj)
+�n
+Ui ∩Uj
+Uj
+φi
+⊆
+φj
+ψi,j
+is a smooth map. A smooth manifold is a topological space equipped with a (maximal4) atlas. A mor-
+phism of smooth manifolds is a topological map f : M → N that is locally smooth - for each chart pair of
+charts (Ui,φi) on M and (Vj ,θj), see Figure 1.1 for an illustration. The map f is smooth whenever each
+map fi,j that completes the diagram is smooth
+Ui
+Vj
+M
+N
+f
+fi,j
+The category of smooth manifolds, SMan, is the category of smooth manifolds and their morphisms.
+Remark 1.2.2. In Chapter 2, some results will implicitly use partition of unity arguments, which require
+that the underlying topological space for a manifold is Hausdorff and has a countable basis (i.e. it is
+second-countable). We will avoid any direct reference to these properties, and so we omit them from the
+definition of a smooth manifold.
+4With respect to subset inclusion.
+8
+
+φi
+φ−1
+i
+φ−1
+j
+φj
+M
+Ui
+Uj
+φi(Ui)
+�m
+ψi j
+φj(Uj)
+�m
+Figure 1.1:
+Overlapping charts in an atlas (Credit for this tex code belongs to user Cragfelt
+https://tex.stackexchange.com/a/388493/101171.)
+Example 1.2.3.
+(i) Each vector space �n, for n ∈ �, has a canonical smooth manifold structure whose atlas is a single
+chart (the identity map �n → �n).
+(ii) Most geometric shapes that do not have any singularities or sharp edges can be equipped with an
+atlas without any issue. For example, consider the circle: {(cos(x),sin(x)) : x ∈ [−π,π)}
+For any appropriately small ε > 0, there are two charts from Iε = (−ε,π+ ε);
+φ0(t ) = (cos(t ),sin(t )), φ1(t ) = (cos(t − π),sin(t − π))
+making the circle a smooth manifold.
+Category theory has not seen as many applications in differential geometry as it has in topology
+or algebra, likely because the category of smooth manifolds (Definition 1.2.1) is somewhat poorly be-
+haved. The two main reasons that the category of smooth manifolds is “inconvenient” are as follows:
+• The set of smooth maps between two manifolds M ,N fails, in general, to form a smooth mani-
+fold; thus, the category is not cartesian closed.
+• The category of manifolds does not have quotients or arbitrary fibre products.
+9
+
+However, the category of smooth manifolds admits some limits, for example finite products.
+Proposition 1.2.4 (1.12 Kolár et al. (1993)). The category SMan of smooth manifolds has finite products.
+Proof. Given two manifolds M ,N , take the product of their underlying topological spaces together
+with the product charts
+(φi × ψj ) :Ui × Vj → �n × �m.
+The category of smooth manifolds—using our definition where disconnected components of a
+manifold may have different dimensions—does have a class of (co)limits used throughout this paper,
+namely idempotent splittings.
+Definition 1.2.5 (Borceux and Dejean (1986)). An idempotent is an endomorphism e : E → E so that
+e ◦e = e . The splitting of an idempotent is given by a pair of maps e = s ◦r so that r ◦s = id . The existence
+of a splitting of an idempotent e is equivalent to asking that the following pair of parallel arrows has a
+(co)equalizer:
+E
+E
+e
+The idempotent splitting � of a category (also known as the Cauchy completion of the category), is the
+full subcategory of presheaves [�op ,Set] that are retracts of representable functors. Any functor into a
+category with idempotent splittings will factor through this inclusion of categories, so it is the free co-
+completion of � under idempotent splittings.
+Proposition 1.2.6 (Lawvere (1989)). The category of smooth manifolds is the idempotent splitting of
+the category whose objects are open subsets of Cartesian spaces and whose morphisms are smooth maps
+f :U → V .
+An idempotent in the idempotent splitting of a category is also an idempotent in the base category,
+and thus admits a splitting.
+Corollary 1.2.7. The category of smooth manifolds is closed to idempotent splittings: for every map
+e : M → M so that e = e ◦ e there exists a pair of maps r :Q → M ,s : M →Q so that e = s ◦ r,r ◦ s = id .
+The main construction of interest on the category of smooth manifolds, for tangent categories at
+least, is the tangent bundle. Given a smooth map f : M → N , restrict it to a morphism between coor-
+dinate patches, so it may be regarded as a map f |U :U → V , where U ⊆ �m,V ⊆ �n. This gives a local
+derivative operation (remembering that πi denotes the i t h projection from a product
+�n Ai )5
+D[f |U ] :U × �m → �n,
+(f |U ◦ π0,D[f |U ]) :U × �m → V × �n.
+The tangent bundle makes this construction global; that is, there is a functor T : SMan → SMan giving
+an assignment
+T.M
+T.f
+−→ T.N
+that agrees with the local derivative on coordinate patches of M ,N .
+5The wording here was originally muddled, it has since been corrected.
+10
+
+Definition 1.2.8 (Kolár et al. (1993)). Write the algebra of smooth functions on a manifold as C ∞(M ) :=
+SMan(M ,�). The set SMan(�,M )/ ∼= of tangent vectors on a smooth manifold M comprises the curves
+� → M subject to the equivalence relation that for a pair of curves φ,θ : � → M , φ ∼= θ if and only if
+φ(0) = θ(0) and for every f ∈ C ∞(M ),
+∂ f ◦ φ
+∂ x
+(0) = ∂ f ◦ θ
+∂ x
+(0).
+The set SMan(�,M )/ ∼= has a naturally determined smooth manifold structure which we call the tangent
+bundle over M , T M .
+Example 1.2.9.
+(i) For a vector space, the space of linear paths crossing through a point v ∈ V is isomorphic to V , so
+T V ∼= V × V .
+(ii) The tangent bundle above the circle is diffeomorphic to the cylinder. This is follows from the classi-
+cal result that a tangent vector on the circle must be perpendicular to its position vector.
+The tangent bundle lifts the “local derivative” into a globally defined construction, so the tangent
+bundle construction is functorial.
+Proposition 1.2.10. The tangent bundle is a product-preserving endofunctor on the category of smooth
+manifolds.
+Proof. Functoriality follows by showing that a morphism of smooth manifolds preserves the equiva-
+lence relation on curves that defines a tangent vector:
+∀g ∈ C ∞(M ), ∂ g ◦ φ
+∂ x
+(0) = ∂ g ◦ θ
+∂ x
+(0)
+Note that if f : N → M ∈ C ∞(N ), so that g ◦ f ∈ C ∞(M ), the chain rule ensures that
+∀g ∈ C ∞(N ), ∂ f ◦ g ◦ φ
+∂ x
+(0) = ∂ f ◦ g ◦ θ
+∂ x
+(0)
+To show that T is product-preserving, it suffices to show that the equivalence classes of curves are
+stable under pairing. First, note that for any M and φ ∼= θ : � → M ,
+(φ,id ) ∼= (θ,id ) : � → M × �
+Given a pair of curves θM ,ψM : � → M , where θM ∼= ψM and similarly θN ∼= ψN for N , this implies that
+f ◦ (φM ,φN )
+= f ◦ (φM × id ) ◦ (id ,φN )
+= f ◦ (φM × id ) ◦ (id ,θN )
+= f ◦ (id ,θN ) ◦ (φM × id )
+= f ◦ (id ,θN ) ◦ (θM × id )
+= f ◦ (θM × θN )
+so (φM ,φN ) ∼= (θM ,θN ).
+11
+
+The scalar action by � on tangent vectors and a partially defined addition additionally give the
+tangent bundle the structure of a fibered �-module (that is, an �-module in the slice categorySMan/M
+whose objects are morphisms into M , f : X → M , and morphisms are commuting triangles).
+Proposition 1.2.11. The tangent bundle over M is an �-module in SMan/M , as follows: let γ,ω be
+tangent vectors on M , and define
+• p : T M → M ;p(γ) = γ(0).
+• 0 : M → T M ;0(m) = [r �→ m] (the constant map � → M sending all r ∈ � to m ∈ M )
+• ·p : T M × � → T M ;γ ·p r = [x �→ γ(r · x)]
+• + : T M p ×p T M → T M := [γ],[ω] �→ [γ + ω] (where addition around γ(0) = ω(0) is defined using
+local coordinates).
+The second derivative is involved in the more nuanced axioms for a cartesian differential category,
+namely linearity in the vector argument and the symmetry of mixed partial derivatives. First, set f |U
+to be the restriction of f : M → N to a map between local coordinate patches U ⊆ M ,V ⊆ N , and then
+define
+f0 = f |U ◦ π0 ◦ π1,
+,
+and
+f1 = D[f ]◦ (π0 ◦ π0,π1 ◦ π0),
+f2 = D[f ]◦ (π0 ◦ π0,π0 ◦ π1) .
+The axioms [C D C .6],[C D C .7] then give:
+(U × �n) × (�n × �n)
+(V × �m) × (�m × �m)
+T 2M
+T 2N
+T M
+T.N
+(U × �n)
+(V × �m)
+T 2.f
+T.f
+((f0,f1),(f2,D[D[f |U ]]))
+(f ,D[f |U ])
+ℓ
+ℓ
+((π0,0),(0,π1))
+((π0,0),(0,π1))
+(U × �n) × (�n × �n)
+(V × �m) × (�m × �m)
+T 2M
+T 2N
+T 2M
+T 2.N
+(U × �n) × (�n × �n)
+(V × �m) × (�m × �m)
+c
+((π0◦π0,π0◦π1),(π1◦π0,π1◦π1)
+T 2.f
+T 2.f
+c
+((π0◦π0,π0◦π1),(π1◦π0,π1◦π1)
+((f0,f1),(f2,D[D[f |U ]])
+((f0,f2),(f1,D[D[f |U ]])
+The two natural transformations ℓ and c —the vertical lift and canonical flip—capture these coher-
+ences. Locally, ℓ is the map inserting zeros into the second and third coordinates, while c flips the
+second and third arguments, leading to the coherences established in the next proposition. To capture
+12
+
+these coherences on the tangent bundle, first note that a tangent vector on T M is equivalent to an
+equivalence class of surfaces on M ,
+φ ∼= θ : �2 → M ⇐⇒ φ(0,0) = θ(0,0)
+and ∀f ∈ C ∞(M ), ∂ f ◦ φ
+∂ xi
+(0,0) = ∂ f ◦ θ
+∂ xi
+(0,0),i = 0,1
+Proposition 1.2.12 (Cockett and Cruttwell (2014)). There are two natural transformations
+ℓ : T M → T 2M ;ℓ([γ]) = [γ ◦ (π0 ·� π1)]
+c : T 2M → T 2M ;c ([γ]) = [γ ◦ (π1,π0)]
+satisfying the following coherences:
+(i) ℓ.T ◦ ℓ = T.ℓ◦ ℓ6
+(ii) The following maps are morphisms of fibred �-modules.
+T M
+T 2M
+M
+T M
+ℓ
+p
+p.T
+0
+T M
+T 2M
+M
+T M
+ℓ
+p
+T.p
+0
+(iii) c ◦ c = id
+(iv) T.c ◦ c .T ◦ T.c = c .T ◦ T.c ◦ c .T
+(v) c ◦ ℓ = ℓ
+(vi) T.c ◦ c ◦ T.ℓ = ℓ.T ◦ c .
+Observation 1.2.13. Equation (iv) is known as the Yang–Baxter equation. It is one of the coherences for
+a symmetric monoidal category, and states that the twisting operation between two variables is coherent.
+We may regard the category of endofunctors on a category as a strict monoidal category and use string
+diagram notation (see e.g. Selinger (2010)). Interpreting the map c as twisting two strings, the coherence
+becomes
+=
+The projection p : T M → M is locally trivial: for each connected component of M that is modeled
+on �n, each point m lies in an open subset Um so that p −1(Um) ∼=Um ×�n. This local triviality property
+leads to the following universality condition.
+Proposition 1.2.14. The following diagram is an equalizer:
+T M
+T 2M
+T M
+p
+T.p
+0◦p◦p
+ℓ
+6Recall that we are using the 2-categorical notation described in the front-matter
+13
+
+Therefore the diagram in the following corollary is a pullback:
+Corollary 1.2.15. Write the map µ : T M p ×p T M → T 2M to be T. + ◦(ℓ× 0). The following diagram is a
+pullback:
+T M p ×p T M
+T 2M
+M
+T M
+0
+T.p
+µ
+⌟
+The five maps (p,0,+,c ,ℓ), along with their coherences and universal properties, characterize the
+kinematic tangent bundle, axiomatized as a tangent structure in the next section. However, there is an
+equivalent characterization of the tangent bundle for a finite-dimensional smooth manifold that will
+be important throughout this thesis: the operational tangent bundle. We first need to define the mod-
+ule of vector fields on a manifold, where a vector field is essentially an ordinary differential equation
+defined on a manifold rather than a cartesian space.
+Definition 1.2.16 (3.1,3.3of Kolár et al. (1993)). A vector field on a manifold M is a section X : M → T M
+of the projection p : T M → M so that p ◦ X = idM . The set of vector fields on a manifold M is written
+χ(M ) and carries a C ∞(M )-module structure using the fibered �-module structure on p : T M → M :
+X +χ(M ) Y := +.M ◦ (X ,Y ),
+0χ(M ) := 0.M ,
+f ·χ(M ) X (m) := f (m) ·T M X (m)
+where X ,Y ∈ χ(M ), f ∈ C ∞(M ).
+The module χ(M ) has an important universal property as a C ∞(M )-module—it is precisely the
+module of derivations of C ∞(M ):
+χ(M ) = {X : C ∞(M ) → C ∞(M ) : ∀f ,g ∈ C ∞(M ),X (f ·g ) = X (f ) ·g + f · X (g )}
+Proposition 1.2.17 (3.4 of Kolár et al. (1993)). There is an isomorphism of C ∞(M ) modules:
+Der(C ∞(M )) ∼= χ(M ).
+Finally, we observe that there is a C ∞(M )-Lie algebra structure on χ(M ), with two equivalent def-
+initions. First, there is the kinematic definition of the bracket, which is induced using the universality
+of the vertical lift. Given vector fields X ,Y on M , note that
+X = p.T ◦ (T.Y ◦ X −T M c ◦ T.X ◦ Y ),
+0 = T.p ◦ (T.Y ◦ X −T M c ◦ T.X ◦ Y )
+so by Corollary 1.2.15, there is a unique map [X ,Y ] : M → T M so that
+(T.Y ◦ X −T M c ◦ T.X ◦ Y ) = µ([X ,Y ],X ).
+(1.1)
+Similarly, there is a bracket defined on Der(M ) using the anticommutator of derivations:
+[X ,Y ](f ) = X (Y (f )) − Y (X (f )).
+(1.2)
+These brackets are equivalent for finite-dimensional smooth manifolds.
+Proposition1.2.18. [Mackenzie(2013)]Recall that a Lie algebra overa ring R isan R-module A equipped
+with a bilinear map
+[−.−] : A ⊗ A → A
+that is alternating and satisfies the Jacobi identity:
+[X ,[Y ,Z ]]+ [Z ,[X ,Y ]]+ [Y ,[Z ,X ]] = 0.
+The two brackets on χ(M ) (viewed as a Lie algebra) from Equations 1.1 and 1.2 coincide.
+14
+
+1.3
+Tangent structures
+This section develops the categorical framework to study more general categories of smooth manifolds
+by axiomatizing the tangent bundle, tangent categories, which first appeared in Rosický (1984). An ar-
+bitrary tangent category is significantly more general than smooth manifolds and captures examplesof
+categorieswith “tangentbundle” from computerscience and logic, asdeveloped in Cockett and Cruttwell
+(2014). First, observe that tangent categories forget the base ring from the previous section and only
+consider the fibered commutative monoid structure of the tangent bundle.
+Definition 1.3.1. An additive bundle in a category � is a triple E
+q−→ M ,+ : E q×qE → E ,ξ : M → E which
+gives (q,+,ξ) the structure of a commutative monoid in the slice category �/M . If (q,ξ,+),(q ′,ξ′,+′) are
+both additive bundles, a bundle morphism
+E
+E ′
+M
+M ′
+q
+f
+q
+f0
+is additive if f ◦+ = +′ ◦(f ◦π0, f ◦π1) and f ◦ξ = ξ′ ◦ f0. The category of additive bundles in a a category
+� is given by additive bundles in � and additive bundle morphisms.
+We will often write pullback powers of an additive bundle E q ×q ...q ×q E as En, and use infix no-
+tation to write addition so + ◦ (a,b ) becomes a + b . In the category of smooth manifolds, the tangent
+bundle functor gives a well-behaved functorial vector bundle. A tangent category has a functorial ad-
+ditive bundle and axiomatizes the coherences and universal properties of the tangent bundle from the
+category of smooth manifolds.
+Definition 1.3.2 (Rosický(1984); Cockett and Cruttwell (2014)). A tangent structure consists of a functor
+T : � → � equipped with natural transformations
+p : T ⇒ id , 0 : id ⇒ T,+ : T p ×p T ⇒ T
+ℓ : T ⇒ T.T
+c : T.T ⇒ T.T
+satisfying the following axioms; we call a category equipped with a tangent structure a tangent category.
+[TC.1] Additive bundle axioms:
+(i) Pullback powers of p exist and are preserved by T ; write these Tn.
+(ii) Each triple (p.M : T M → M ,0.M : M → T M ,+.M : T2M → T M ) is an additive bundle.
+T
+T
+T2
+T
+id
+id
+id
++
+p◦πi
+p
+p
+0
+(1.3)
+[TC.2] Symmetry axioms:
+(i) Involution:
+T.T
+T.T
+T.T
+c
+c
+(1.4)
+15
+
+(ii) Yang–Baxter: c .T ◦ T.c ◦ c .T = T.c ◦ c .T ◦ T.c
+T.T.T
+T.T.T
+T.T.T
+T.T.T
+T.T.T
+T.T.T
+T.c
+c .T
+T.c
+c .T
+c .T
+T.c
+(1.5)
+(iii) Naturality equations:
+T.T
+T.T
+T
+T
+T.T.T
+T.T.T
+T.T.T
+T.T
+T.T
+T.T2
+T2T
+T.T
+T.T
+T.T
+T.T
+T
+T
+T.ℓ
+c
+c .T
+T.c
+ℓ.T
+(c ◦T.π0,c ◦T.π1)
+T.+
++.T
+c
+c
+p.T
+T.p
+0.T
+c
+T.0
+(1.6)
+[TC.3] Lift axioms:
+(i) Naturality with addition:
+T
+T.T
+T.T
+T.T
+T2
+T.T2
+id
+T
+T
+T
+T
+T.T
+(ℓ◦π0,ℓ◦π1)
++
++
+ℓ
+T.ℓ
+0
+0
+0.T
+ℓ
+p
+0
+T.p
+(1.7)
+(ii) Coassociativity:
+T
+T.T
+T.T
+T.T.T
+ℓ
+ℓ
+T.ℓ
+ℓ.T
+(1.8)
+(iii) Symmetric co-multiplication:
+T
+T.T
+T.T
+ℓ
+c
+ℓ
+(1.9)
+(iv) Universality: for µ := T. + ◦(0◦ π0,ℓ◦ π1), the following diagram is a pullback for all X :
+T2X
+T 2X
+X
+T X
+µ
+p◦πi
+T.p
+0
+16
+
+Definition1.3.3(Cockett and Cruttwell (2014)). Atangent category ismonoidal whenever� isa monoidal
+category, T is a monoidal functor, and +,p,c ,ℓ are monoidal natural transformations. A tangent cate-
+gory is cartesian whenever � is cartesian and is a strict monoidal tangent category for products.
+Notation 1.3.4. Throughout this thesis, two key pieces of notation apply:
+• Tn denotes pullback powers of p : T ⇒ id (and more generally En for q : E → M ); iterated powers
+of T are written T n.
+• There will often be long strings of Tn pullbacks and functors F : A → B, so a 2-categorical notation
+where functor composition is written with a period, Tn.F.T ′
+m, will often be adopted. While the
+natural transformation c at Tn.Tm.M under the image of the functor T would often be written
+T (cTn.Tm.M ), in this thesis it will be written as
+T.c .Tn.Tm.M : T.T 2.Tn.Tm ⇒ T.T 2.Tn.Tm
+while the composition of 2-cells will be written using ◦ in applicative order, rather than the dia-
+grammatic order typically used in the tangent category literature. That is, the composition
+A
+g−→ B
+f−→ C
+would be written f ◦ g rather than g f .
+Example 1.3.5. Applying the results from section 1.2, the category of smooth manifolds is a tangent
+category. Recall that if M is a smooth manifold, there is a coordinate patch m ∈ U �→ M around each
+point m ∈ M so that U ∼=U ′ ⊆ �n. fibre above U , p −1(U ), is locally isomorphic to U ′ ×�n and similarly
+(p ◦ p)−1(U ) ∼=U ′ × (�n)3, so that:
+p :U × �n → U
+(m, x) �→ m
+0 :U →U × �n
+m �→ (m,0)
++ :U × �n × R n →U × �n
+(m, x, y ) �→ (m, x + y )
+ℓ :U × �n →U × �n × �n × �n
+(m, x) �→ (m,0,0, x)
+c :U × �n × �n × �n →U × �n × �n × �n
+(m, x, y,z) �→ (m, y, x,z)
+The study of tangent categories is closely related to Lawvere’ssynthetic differential geometry, first in-
+troduced in Lawvere (1979) and laterdeveloped in Kock (2006); Lavendhomme(1996); Moerdijk and Reyes
+(1991). The setting of synthetic differential geometry is a topos � (for our purposes, we need only a com-
+plete, cartesian closed category) equipped with a chosen ring object R that satisfies the Kock-Lawvere
+axiom: given the object of nilpotent elements in R, D = [d : R|d 2 = 0], the following map is an isomor-
+phism:
+α : R × R → [D,R]; α(a,b ) = (d �→ a + d b ).
+One can find a class of objects that form a tangent category: the infinitesimally linear objects.
+Definition 1.3.6. Let � be a model of synthetic differential geometry where the ring object is (R,·,1,+,0).
+An object M in a model of synthetic differential geometry is infinitesimally linear when it satisfies the
+following axioms.
+(i) The following natural morphism must be an isomorphism:
+[D (2),M ] ∼= [D,M ]0∗×0∗[D,M ]; where D(2) := [(d0,d1) ∈ D 2 : di ·d j = 0].
+17
+
+(ii) M satisfies property W (credited by Kock (2006) to Gavin Wraith), namely that the following dia-
+gram is a ternary equalizer:
+[D,M ]
+[D × D,M ]
+[D,M ]
+[0◦!×D,M ]
+[(0×0)◦!,M ]
+[D×0◦!,M ]
+[·,M ]
+(We will often use M as shorthand for idM in diagrams).
+An infinitesimal object generalizes the object D in the category of infinitesimally linear objects in
+a model of synthetic differential geometry. The definition adopted in this thesis is a strict generaliza-
+tion of the definition given in Cockett and Cruttwell (2014): here, we work with a symmetric monoidal
+closed category rather than a cartesian closed category in order to capture some examples from logic.
+Definition 1.3.7 (Definition 5.6 Cockett and Cruttwell (2014)). An infinitesimal object in a symmetric
+monoidal category
+(�,⊗,I ,α,ρ,σ)
+is a tuple (⊙ : D ⊗ D → D,0 : I → D,ε : D → I ,δ : D → D(2)) (where D(n) denotes pushout powers of 0)
+so that:
+[IO.1] Pushout powers D(n) of 0 : I → D exist, and ε ◦ 0 = idI .
+[IO.2] ⊙ is a commutative semigroup with zero, so that the following diagrams commute:
+D ⊗ D
+D ⊗ D
+D ⊗ D ⊗ D
+D ⊗ D
+D
+D ⊗ D
+D
+D
+D
+D
+⊙
+σ
+⊙
+⊙⊗D
+⊙
+D⊗⊙
+⊙
+(D,0)
+⊙
+0D ◦ε
+The third diagram shows that 0 is an absorbing element rather than a unit (think of 0 as being in
+the commutative semigroup on the set [0,1) given by multiplication).
+[IO.3] The map δ : D → D(2) makes (0 : I → D,δ,ε) into a commutative comonoid in the coslice I /�:
+D
+D(2)
+D
+D(2)
+D
+D (2)
+D
+D (2)
+D(3)
+D(2)
+δ
+δ
+D+I δ
+δ+I D
+δ
+δ
+(ι1|ι0)
+(0◦ε+I D)
+δ
+[IO.4] The following diagram commutes (⊙ is coadditive):
+D ⊗ D
+D
+D ⊗ D(2)
+D(2)
+1×δ
+(ι0◦⊙|ι1◦⊙)
+δ
+⊙
+The notation (f |g ) : A + B → C for pushouts/coproducts is dual to pairing (f ′,g ′) : C → A × B for
+pullbacks/products, just as ιi is dual to projection. Therefore, (f |g ) ◦ ι0 = f just as π0 ◦ (f ,g ) = f .
+18
+
+[IO.5] The following diagram is a coequalizer:
+D ⊗ I
+D ⊗ D
+D(2)
+0◦ε⊗0
+D⊗0
+(⊙|I ε⊗D )◦(δ⊗id)
+There are two tangent structures associated to an infinitesimal object in a symmetric monoidal
+category �. The first relies on the exponentiability of the infinitesimal object, while the second tangent
+structure is on the opposite category of �. The enriched perspective on tangent structure in Section
+5.1 will clarify the relationship between these two tangent structures.
+Proposition 1.3.8. Let (�,⊗,I ,α,ρ,σ) be a symmetric monoidal category, and
+⊙ : D ⊗ D → D,0 : I → D,ε : D → I ,δ : D → D(2)
+define an infinitesimal object in �.
+(i) There is a tangent structure on �op , where T = D ⊗ (−).
+(ii) If � is also a symmetric monoidal closed category, then it is a tangent category with T = [D,−].
+Proof.
+(i) The opposite category of � is a symmetric monoidal category:
+(�op,⊗,I ,α−1,ρ−1,σ).
+The tangent functor is D ⊗ (−), and the projection is
+p = D ⊗ (−)
+0op
+−→ I ⊗ (−)
+ρ−1
+−→ (−).
+The zero map is given by
+(−)
+ρ−→ I ⊗ (−)
+εop
+−→ D ⊗ (−).
+Addition in �op is given by co-addition in �:
+D(2) ⊗ (−)
+δop ⊗(−)
+−−−−→ D ⊗ (−).
+The lift is given by the semigroup structure (along with the monoidal coherences):
+D ⊗ (−)
+⊙⊗(−)
+−−−→ (D ⊗ D) ⊗ (−)
+α−1
+−→ D ⊗ (D ⊗ (−)).
+The flip is also given by monoidal coherences, combined with the symmetry on the monoidal
+category:
+D ⊗ (D ⊗ (−))
+α−→ (D ⊗ D) ◦ (−)
+σ⊗(−)
+−−−→ (D ⊗ D ) ⊗ (−)
+α−1
+−→ D ⊗ (D ⊗ (−)).
+(ii) First, we identify the following natural isomorphisms:
+u : id ⇒ [I ,−],
+b : [A,[B,−]] ⇒ [A ⊗ B,−].
+19
+
+• The tangent functor is [D,−],
+and the triple (0
+:
+I
+→
+D,δ
+:
+D
+→
+D(2),
+ε : D → I ) gives the additive bundle structure
+p : [D,−]
+0∗
+−→ [I ,−]
+u−→ id ,
+0 : id
+u−→ [I ,−]
+[ε,−]
+−−→ [D,−]
++ : [D (2),−]
+δ∗
+−→ [D,−].
+Note that by the continuity of [−,M ] : �op → �, we have [D(2),M ] ∼= [D,M ]p ×p [D,M ].
+• The lift is given by
+ℓ : [D,−]
+⊙∗
+−→ [D ⊗ D,−]
+b −1
+−→ [D,[D,−]].
+• The canonical flip is given by
+c : [D,[D,−]]
+b−→ [D ⊗ D,−]
+σ∗
+−→ [D ⊗ D,−]
+b −1
+−→ [D,[D,−]].
+The coherences and couniversality properties of an infinitesimal object, along with the continuity
+of
+[−,M ] : �op → �
+then induce the coherences and universality properties for the tangent bundle. This completes
+the proof.
+The tangent structure on �op induced by the infinitesimal object is the dual tangent structure on
+�. This will be revisited in Chapters 4 and 5 when looking at tangent categories from the enriched
+perspective.
+Returning to synthetic differential geometry, the co-universality conditions on an infinitesimal ob-
+ject corresponds to infinitesimal linearity (Definition 1.3.6). In some sense, the category of infinitesi-
+mally linear objects is the largest subcategory of � for which D is an infinitesimal object.
+Corollary 1.3.9. In a model of synthetic differential geometry (�,R), the object D = [d ∈ R|d 2 = 0] is an
+infinitesimal object in the category of infinitesimally linear objects in �.
+This thesis makes use of the 2-category of tangent categories (Section 2.3 of Cockett and Cruttwell
+(2014)), which formalizes the notion of a morphism of tangent structure and 2-cells between them.
+This 2-categorical framework is a departure from the classical theory of synthetic differential geometry,
+where the literature only really addresses morphisms of tangent structure in the form of fully faithful
+embeddings SMan �→ Microl(�).
+Definition 1.3.10. Let (�,�),(�,�′) be a pair of tangent categories. A pair (F : � → �,α : F.T ⇒ T ′.F ) is
+a tangent functor if the following diagrams commute:
+F
+F.T
+F.T2
+F.T
+F.T
+F
+T.F
+T2.F
+T.F
+T.F
+F.T
+T.F
+F.T 2
+T.F.T
+T 2.F
+F.T 2
+T.F.T
+T 2.F
+F.T 2
+T.F.T
+T 2.F
+F.+
+α2
++.F
+α
+α
+F.p
+p.F
+0.F
+F.0
+α
+ℓ.F
+F.ℓ
+α
+α.T
+T.α
+α.T
+T.α
+α.T
+T.α
+F.c
+c .F
+20
+
+A tangent functor is strong whenever α is a natural isomorphism; for a sub-(tangent category) inclusion,
+α = id . A tangent functor (F,α) between cartesian tangent categories is a cartesian tangent functor if F
+is an isomonoidal functor and α is a monoidal natural transformation.
+Example 1.3.11.
+(i) The coherences on the canonical flip c guarantee that (T : � → �,c : T.T ⇒ T.T ) is a strong tangent
+endofunctor on any tangent category.
+(ii) Given a pair of tangent functors (A,α) : � → �,(B,β) : � → �, the composition
+(B.A : � → �,β.A ◦ B.α : B.A.T C
+B.T D .A
+T E .B.A
+B.α
+β.A
+)
+is a tangent functor.
+(iii) A model of synthetic differential geometry (�,R) is well-adapted whenever there is a fully faithful,
+strict tangent functor fromSMan to the category of microlinear spaces of �,SMan �→ Microl(�). The
+original development of well-adapted models for synthetic differential geometry may be found in
+Dubuc (1981), and the reader may check Bunge et al. (2018) for a recent account of the construction
+of such models, or section 3 of Kock (2006).
+Definition1.3.12. Atangentnatural transformation γ between tangent functors(F,α),(G ,β) isa natural
+transformation so that the following diagram commutes:
+F.T (A)
+G .T (A)
+T.F (A)
+T.G (A)
+γT A
+αA
+βT A
+T γA
+If F,G are cartesian tangent functors, then γ is cartesian whenever it is an isomonoidal natural transfor-
+mation.
+Definition 1.3.13. We will call the 2-category of tangent categories, tangent functors, and tangent nat-
+ural transformations TangCat. The 2-category of cartesian tangent categories is CartTangCat.
+1.4
+Local coordinates in a tangent category
+This section develops some structures to facilitate reasoning about higher tangent bundles, using “lo-
+cal coordinates.” In the case of a cartesian differential category, T n(A) =
+�
+2n A, whereas for an open
+subset U ⊆ �m the tangent bundle decomposes as T nU ∼= U × (�m)2n−1. This section introduces two
+structures that allow for these arguments in an arbitrary tangent category: differential objects and con-
+nections.
+Definition 1.4.1 (Cockett and Cruttwell (2018)). A differential object7 in a cartesian tangent category
+is a commutative monoid (A,+A,0A) such that there is a section-retract pair A
+λ−→ T A
+ˆp−→ A which exhibits
+T (A) as a biproduct in the category of commutative monoids:
+A ⊕ A ∼= T A.
+Concretely, a differential object is a commutative monoid equipped with λ : A → T A, ˆp : T A → A, ˆp ◦λ =
+id so that the following axioms hold:
+7Not to be confused with 4.1 from Barr (2002).
+21
+
+DO.1 Coherence between + and λ, ˆp:
+T (A × A)
+T A
+A × A
+A
+T A × T A
+T A × T A
+A × A
+A
+T (A × A)
+T A
+ˆp
+T.+A
+(T π0,T π1)
+ˆp× ˆp
++A
++A
+λ×λ
+(T π0,T π1)−1
+T.+A
+λ
+DO.2 Coherence between 0A and 0.A:
+A
+1
+A
+1
+T A
+A
+T A
+A
+0
+!
+0A
+ˆp
+λ
+p
+!
+0A
+DO.3 Coherence between +A and +.A:
+T2A
+A × A
+A × A
+T2A
+T A
+A
+A
+T A
+ˆp× ˆp
++A
++
+ˆp
+λ×λ
++A
+λ
++
+DO.4 Coherence between λ and ˆp with ℓ:
+T A
+T 2A
+A
+T A
+A
+T A
+T A
+T 2A
+ℓ
+T. ˆp
+ˆp
+ˆp
+λ
+λ
+ℓ
+T.λ
+A map f : A → B between differential objects (A,λA, ˆpA,+A,0B) → (B,λB, ˆpB,+B,0B) is linear whenever
+f preserves the lifts and projections
+(T.f ◦ λA = λB ◦ f ) and ( ˆpB ◦ T.f = f ◦ ˆpA).
+Following the work in Section 3 of Cockett and Cruttwell (2018), it is sufficient to check that f pre-
+serves λ or ˆp (each condition implies the other).
+Example 1.4.2.
+(i) In the category of smooth manifolds, the real numbers are a differential object, as T.� = �[x]/x 2,
+so the lift map in this case is
+λ(a) = 0+ a · x,
+ˆp(a + b · x) = b.
+More generally, finite-dimensional real vector spaces, with their canonical smooth manifold struc-
+ture, are exactly differential objects in the category of smooth manifolds. The lift map is defined
+using T � ∼= �[x]/x 2, so that
+V
+(0,1�)
+−−−→ T (V × �)
+T ·V
+−→ T V.
+This is equivalent to the isomorphism T.V = R[x]/x 2 ⊗ V .
+22
+
+(ii) Every object in a cartesian differential category has a canonical differential object structure, as
+T A := A × A.
+Classically, the tangent space above each point of a smooth manifold is a vector space. We have a
+similar result for differential objects from Cockett and Cruttwell (2018).
+Lemma 1.4.3. Suppose we have the following pullback in a cartesian tangent category �, and all powers
+of T preserve it.
+E
+T M
+1
+M
+ι
+!
+p
+m
+There is a unique differential object structure (E ,λ, ˆp) so that ℓ◦ ι = T.ι ◦ λ.8
+There are two natural classes of morphisms between differential objects, linear and “smooth” (that
+is, arbitrary morphisms).
+Definition 1.4.4 (Cockett and Cruttwell (2018)). Let � be a cartesian tangent category. We define the
+following categories:
+(i) Diff(�) is the category of differential objects and arbitrary morphisms, so for any differential objects
+A,B, we have Diff(�)(A,B) := �(A,B).
+(ii) DLin(�) is the category of differential objects and linear morphisms.
+The category of differential objects and smooth maps is a cartesian differential category, exhibiting
+differential calculus as a specialized logic in a tangent category.
+Proposition 1.4.5 (Section 3.5 of Cockett and Cruttwell (2018)). Let � be a cartesian tangent category.
+Then:
+(i) Diff(�) is a cartesian differential category, where we define the differential combinator D to be
+A
+f−→ B
+A × A
+λ×0
+−−→ T (A × A)
+T.+A
+−−→ T (A)
+T (f )
+−−→ T (B)
+ˆpB
+−→ B
+(where λA,+A are the lift and addition for the differential object structure on A, and ˆpB is the pro-
+jection map on the differential object structure on B).
+(ii) There is an equality of categories, Lin(Diff(�)) = DLin(�), meaning that a morphism between differ-
+ential objects in the cartesian tangent category � is linear if and only if it is linear in the cartesian
+differential category Diff(�).
+Recall that if M is an open subset of �n, the second tangent bundle of an open subsetU ⊆ �n splits
+as
+T 2(U ) = T (U × �n) = (U × �n) × (�n × �n) = T3U .
+Every smooth manifold admits such a decomposition on its second tangent bundle; these are known
+as a affine connections9 and provide a way to reason about an object as though it has local coordinates
+in an arbitrary tangent category.
+8Some diagrammatic notation had creeped into the original draft here, I have fixed it.
+9The prefix “affine” differentiates these from more general connections on differential bundles, which are introduced in
+Section 2.6.
+23
+
+Definition 1.4.6 (Cockett and Cruttwell (2017)). In a tangent category �, define the following:
+(i) An affine vertical connection is a map κ : T 2M → T M so that
+a) κ is a vertical descent, namely a section of the vertical lift ℓ : T M → T 2M , so κ ◦ ℓ = id ;
+b) κ is compatible with both lifts on T 2M : T.κ ◦ ℓ.T = T.κ ◦ T.ℓ = ℓ◦ κ.
+(ii) An affine horizontal connection is a map ∇ : T2 → T 2M so that
+a) ∇ is a horizontal lift, namely a section to the horizontal descent (p.T,T.p) : T 2 ⇒ T2, so that
+(p.T,T.p) ◦ ∇ = id ;
+b) ∇
+is
+compatible
+with
+the
+linear
+structures
+on
+T 2,T2:T.∇
+◦
+(ℓ
+×
+0)
+= ℓ◦ ∇and T.∇(0× ℓ) = T.ℓ◦ ∇.
+(iii) An affine connection is a pair (κ,∇) comprising a vertical and horizontal connection on M satisfy-
+ing the compatibility conditions:
+a) T. + ◦(+.T ◦ (ℓ◦ κ,T.0◦ p.T ),∇(p.T,T.p)) = idT 2M ,
+b) κ ◦ ∇ = 0◦ p ◦ πi.
+An affine connection is torsion-free if κ ◦ c = κ.
+The data ofa vertical connection issufficienttodefine a full connection, asobserved in Lucyshyn-Wright
+(2018).
+Lemma1.4.7(Lucyshyn-Wright (2018)). Afull connection isequivalent to a vertical connection in which
+the following diagram is a fiber product:
+T M
+T 2M
+T M
+M
+T M
+p
+p
+p
+p.T
+T.p
+κ
+Example 1.4.8.
+(i) Every differential object in a tangent category has a canonical vertical connection given by (p ◦
+p.T, ˆp ◦ T. ˆp). A morphism of differential objects will preserve this vertical connection.
+(ii) Every smooth manifold has a non-natural choice of Riemannian metric. By the fundamental the-
+orem of Riemannian geometry, there is a torsion-free connection associated with the metric. (See
+any standard reference on Riemannian geometry, e.g. do Carmo (1992).)
+Connections allow for arguments on higher powers of the tangent bundle to be pushed down to
+pullback powers of T .
+Observation 1.4.9. Suppose M ,N each have connections (κ−,∇−). The map T 2.f : T 2M → T 2N can be
+written using local coordinates �
+T 2.f : T3M → T3N as
+T 2M
+T 2N
+T3M
+T3N
+T.+◦(+.T ◦(ℓ◦π2,T.0◦π0),∇(π0,π1))
+T 2.f
+(p.T,T.p,κN )
+24
+
+where �
+T 2.f is given by
+(T.f ◦ π0,T.f ◦ π1,T.f ◦ π2 +N ∇[f ](π0,π1))
+with ∇[f ] := κN ◦ T 2.f ◦ ∇M .
+Note that in the case that f preserves the connections, ∇[f ] = 0, as
+κN ◦ T 2.f ◦ ∇M = T f ◦ κM ◦ ∇M = T f ◦ 0◦ p = 0◦ f ◦ p.
+1.5
+Submersions
+The category of smooth manifolds is incomplete: there are cospans10 X
+f−→ M
+g←− Y for which the
+pullback fails to exist. Following Thom (1954), the pullback of a cospan exists and is preserved by T
+(i.e. it is a T-limit) whenever for each point f (x) = g (y ), the direct sum of the images of Tx f and Ty g is
+the full vector space Tf (x)M , such cospans are called transverse. Submersions, then, form a convenient
+class of maps, as any cospan where one map is a submersion will be transverse. More precisely:
+Definition
+1.5.1.
+If
+A
+and
+B
+are
+smooth
+manifolds,
+a
+smooth
+function
+f
+:
+A → B is a submersion if and only if the derivative D f |a of f at every point a ∈ A is a surjective lin-
+ear map.
+With this definition we have the following result:
+Proposition 1.5.2. In the category of smooth manifolds, let the class of submersions be denoted by � .
+(i) Submersions are closed under the tangent functor: f ∈ � ⇒ T.f ∈ � .
+(ii) Submersions are closed to pullback along arbitrary maps:
+X
+g ∗X
+X
+N
+M
+N
+M
+g
+f ∈�
+g ∗ f ∈�
+g
+f
+¯g
+⌟
+This will often be referred to as T -stability under reindexing, as it induces a functor between slice
+categories:
+g ∗ : Submersions/M → Submersions/N .
+The properties of the class of submersions in the category of smooth manifolds were studied in
+Cockett and Cruttwell (2018) and axiomatized as a tangent display system.
+Definition 1.5.3. A tangent display system in a tangent category � is a class of maps � in � that is
+• stable under the tangent functor, d ∈ � ⇒ T.d ∈ �,
+• T -stable under reindexing (as in Proposition 1.5.2).
+We call any tangent display system that is closed to retracts in the arrow category a retractive display
+system. If for all M , pM ∈ �, we call � a proper (retractive) display system.
+10Following a general conventionincategory theory,where the prefix "co"-X means anX inthe opposite category,a cospan
+in � is a span in the dual category of �.
+25
+
+This section will show that the submersions in the category of smooth manifolds give a retractive
+display system, yielding a general construction of retractive display systems from display systems.
+The definition of a submersion may be rephrased as follows: f is a submersion if and only if for all
+a ∈ A and all v ∈ T (B) such that f a = pv, there exists a w ∈ T (A) such that T.f ◦w = v. This is a weakly
+universal cone over A
+f−→ B
+p←− T B: there exists at least one morphism into it for any other cone over
+the diagram.
+Definition 1.5.4. A commuting square is a weak pullback if for any x : X → A and y : X → B so that
+f x = g y , there exists a map X → W making the following diagram commute:
+X
+W
+A
+B
+C
+x
+y
+∃
+a
+b
+f
+g
+If the above diagram is a weak pullback for each T n, then it is a weak T -pullback.
+Lemma 1.5.5. Should the pullback of A
+f−→ C
+g←− B exist, Definition 1.5.4 is equivalent to asking that the
+induced map (a,b ) : W → A f ×g B be a split epimorphism.
+Proof. Let r be a retract of (a,b ) : W → A f ×g B. For any X
+(x,y )
+−−→ A f ×g B, the map r ◦ (x, y ) exhibits the
+diagram as a weak pullback. For the converse, the unique map (a,b ) : W → A f ×g B will be a section of
+any map A f ×g B → W induced by weak univerality.
+We now restate the submersion property for a map f using global elements (for all a ∈ A and all
+v ∈ T (B) such that f a = pv, there exists a w ∈ T (A) such that T (f )w = v) using generalized elements.
+Definition 1.5.6. An arrow f : A → B in a tangent category is a tangent submersion if and only if the
+naturality diagram
+T A
+T B
+A
+B
+T f
+p
+p
+f
+is a weak T -pullback.
+Following Lemma 1.5.5, in the case that the pullback exists this is equivalent to asking for a section
+h : A f ×p T B → T A of the horizontal descent (p,T f ) : T A → A f ×p T B (this section is sometimes called
+a horizontal lift in differential geometry literature Cordero et al. (1989)). In smooth manifolds, the T -
+pullback along the projection p : T ⇒ id always exists, so to prove that every submersion is a tangent
+submersion it suffices to show the existence of a horizontal lift.
+Proposition 1.5.7. In the category of smooth manifolds, the tangent submersions are precisely the sub-
+mersions (Definition 1.5.1)
+Proof. There is an explicit construction of a horizontal lift for a classical smooth submersion in VII.1
+of Cushman and Bates (2015).
+26
+
+It is possible to show that the T -stability properties for submersions in the category of smooth
+manifolds follow from the general theory of weak pullbacks. We begin by showing that weak pullbacks
+satisfy a weakened version of the pullback lemma and then show that the retract of a weak pullback is
+a weak pullback (the second lemma is Lemma 2.1 of Adámek et al. (2010)).
+Lemma 1.5.8 (Pullback lemma). Consider the diagram
+•
+•
+•
+•
+•
+•
+(A)
+f
+g (B)
+(i) If f ,g are jointly monic and (A) + (B) is a weak pullback, then (A) is a weak pullback.
+(ii) If (A),(B) are weak pullbacks then (A) + (B) is a weak pullback.
+Proof.
+(i) If (A)+(B) is a weak pullback, a map can be induced for a cone over (A) by concatenating it with (B);
+the jointly monic condition on f ,g guarantees that the map induced for (A) + (B) will commute
+for (A).
+(ii) Given a cone for (A) + (B) induce a map for (B), which then induces a cone for (A).
+Lemma 1.5.9. (Weak) pullbacks are closed to retracts.
+Proof. Suppose that S ′ is a weak pullback, and S is a retract of it in the category of commuting squares.
+Consider the following diagram (suppressing the subscripts for s,r ):
+Z
+A
+A′
+A
+B
+B ′
+B
+C
+C ′
+C
+D
+D ′
+D
+x
+y
+s
+x ′
+y ′
+r
+x
+y
+w
+s
+w ′
+r
+w
+z
+s
+z ′
+r
+z
+s
+r
+Given a cone for S, there is a corresponding cone for S ′ which induces a map Z → A′ and postcompo-
+sition with rA gives the desired map into A.
+Using these lemmas, it is straightforward to prove that the following T -stability properties hold for
+tangent submersions.
+Lemma 1.5.10. In any tangent category �,
+(a) tangent submersions are closed to composition;
+(b) tangent submersions are closed to retracts;
+(c) any T -pullback of a tangent submersion is a tangent submersion.
+27
+
+Proof. (a) follows from Lemma 1.5.8 while (b) follows from Lemma 1.5.9. It remains to prove (c). Con-
+sider a T -pullback, where u is a tangent submersion:
+A
+M
+B
+N
+f
+v
+u
+g
+By naturality, the outer paths of the following two diagrams are equal:
+T A
+T M
+M
+T B
+T N
+N
+T f
+T v
+p
+T u
+u
+T g
+p
+=
+T A
+A
+M
+T B
+B
+N
+p
+T v
+f
+v
+u
+p
+g
+Note that the left diagram is a weak pullback by composition. Therefore the outer perimeter of the right
+diagram is a weak pullback, and the right square is a pullback, so the left square is a weak pullback by
+Lemma 1.5.8, as desired.
+The class of tangent submersions is closed to retracts in the arrow category and is conditionally T -
+stable under reindexing (if the T -pullback of a tangent submersion exists, it is a tangent submersion).
+This stability property leads to the following result:
+Proposition 1.5.11. Let � be a tangent category that allows for reindexing of the class of tangent sub-
+mersions �. Then the class of tangent submersions is a display system.
+Proof. Any class of maps that is closed to reindexing is a tangent display system, and the class of sub-
+mersions is closed to retracts in the arrow category.
+Corollary 1.5.12. The class of submersions in the category of smooth manifolds is a proper retractive
+display system.
+28
+
+Chapter 2
+Differential bundles
+Our principal aim in this thesis is to provide an abstract tangent-categorical axiomatization for Lie
+algebroids. To accomplish this, we must provide an axiomatization for Lie algebroids which is essen-
+tially algebraic (in the sense of Freyd and Kelly (1972)). However, in the category of smooth manifolds,
+Lie algebroids are defined in terms of vector bundles and these are prima facie a highly non-algebraic
+notion.
+In addition to algebraic axioms which make it an �-module in the slice over its base M , a vector
+bundle q : E → M satisfies a crucial topological requirement: it must be locally trivial. This means
+that the projection q : E → M must be locally isomorphic to a projection π0 : U × �n → U for some
+open subset U of M and natural number n. It is this property that permits calculations using local
+coordinates, an approach deeply enshrined in the culture of differential geometry.
+Cockett and Cruttwell (2017) introduced the algebraicnotion ofa differential bundle. Evidence that
+differential bundles are the appropriate generalization of vector bundles was provided by showing how
+classical results for vector bundles could be generalized to differential bundles in any tangent category
+Cockett and Cruttwell (2017, 2018). However, the precise relationship in the category of smooth man-
+ifolds between vector bundles and differential bundles was left open. The main result of this chap-
+ter (see MacAdam (2021)) is that vector bundles and differential bundles coincide in the category of
+smooth manifolds.
+The axiomatization of differential bundles focuses on another important property of vector bun-
+dles: given a vector bundle q : E → M and a vector v in the fibre Ex above x ∈ M , the tangent space
+Tv(Ex) can be naturally identified with Ex . This gives a lift map λ : E → T E which can be axiomatized.
+While the lift map had long been noted in the differential geometry literature in the guise of the Euler
+vector field (see 6.11 of Kolár et al. (1993) and also Section 1 of Michor (1996) which explicitly uses the
+term "lift"), it had not been adopted as the basis of an abstract axiomatization.
+More recently, in the differential geometryliterature, Grabowski and Rotkiewicz (2009) and Bursztyn et al.
+(2016) realized that the multiplicative �+-action on the total space E determines the vector bundle
+structure of q : E → M , and conversely such a multiplicative action determines a vector bundle pre-
+cisely when its Euler vector field (Definition 2.1.8) satisfies an additional “non-singular” property. This
+chapter extends these more recent observations on vector bundles to differential bundles.
+The chapter begins by reviewing vector bundles, describing the Euler vector field construction that
+sends a vector bundle q : E → M to a "lift" map λ : E → T E from Grabowski and Rotkiewicz (2009),
+whereas the rest of the chapter contains new results developed in collaboration with Matthew Burke.
+The second section establishes that these lifts are associative coalgebras for the weak comonad (T,ℓ),
+and that there is a fully faithful functor from vector bundles into the category of lifts for smooth mani-
+folds. The third section identifies the universal property satisfied by the lift (or equivalently Euler vector
+29
+
+field), while the fourth section shows that a non-singular lift corresponds precisely to a differential bun-
+dle. The fifth section proves the main theorem of the chapter: vector bundles are precisely differential
+bundles for smooth manifolds. The final section contains some remarks on extending affine connec-
+tions to arbitrary differential bundles, which will be useful in Chapter 3.
+2.1
+Vector bundles
+A vector bundle over a manifold M axiomatizes the notion of a smoothly varying family of vector spaces
+indexed by the points m ∈ M . The driving example is that of the tangent bundle over a smooth man-
+ifold M , where the fibre above each point m ∈ M is the tangent space TmM . The manifold structure
+guarantees that the projection is locally trivial: given a chart U �→ M , the next pullback splits as a
+product:
+U × �n
+T M
+U
+M
+p
+⌟
+The local triviality of the tangent bundle is essential for various constructions and is part of the defini-
+tion of a vector bundle.
+Definition 2.1.1. A vector bundle is a tuple
+(q : E → M ,ξ : M → E ,+ : E q ×q E → E ,· : � × E → E )
+of morphisms in SMan so that
+(i) the tuple (q,ξ,+,·) defines an �-module in SMan/M ;
+(ii) the map q : E → M is locally trivial.
+The fibred �-module structure means E is a family of vector spaces indexed by M , {Em|m ∈ M }.
+It is important to note that the local triviality axiom guarantees that the projection of a vector bun-
+dle is a submersion (Definition 1.5.1); thus pullback powers of q : E → M exist and are preserved by
+the tangent functor.
+Example 2.1.2. Consider the cylinder, defined as the subset of �3 spanned by C = {(x, y,z)|x 2 + y 2 =
+1,z ∈ �}:
+�
+Above each point i ∈ S 1 = {(x, y )|x 2 + y 2 = 1} the fibre over i is �. For each point i, we can choose a
+sufficiently small ε and take the open set
+Ui = {(x, y ) ∈ S 1|(ix − x)2 + (iy − y )2 ≤ ε},
+which may be flattened to (−(1+ ε),1+ ε) × �.
+30
+
+The sections of a vector bundle also give rise to a C ∞(M )-module, generalizing that aspect of the
+tangent bundle’s fibred �-module structure.
+Lemma 2.1.3. Given a vector bundle q : E → M , write the set of sections of q as Γ(q); as, for example,
+Γ(p.M ) = χ(M ) (recall the notation from Definition 1.2.16). The set Γ(q) has a C ∞(M )-module structure
+in much the same way as χ(M ):
+X +Γ(q) Y := + ◦ (X ,Y ), 0Γ(q) := ξ, (f ·Γ(q) X )(m) := f (m) · X (m)
+There are also a variety of general constructions that yield vector bundles.
+Example 2.1.4.
+(i) The tangent bundle is a vector bundle: the construction in Section 1.2 makes it clear that the pro-
+jection p : T M → M is a locally trivial, fibred �-module over the base space M .
+(ii) A trivial vector bundle over M with fibres in V is the product M × V . In particular, every vector
+space is a trivial vector bundle above the one-point space {∗}.
+(iii) Each TkM will be locally trivial; locally it lookslike the k-fold product of the tangent space p −1
+k (U ) ∼=
+U × (�n)k for an n-dimensional manifold M . More generally, one can take the fibrewise pullback
+Ek = E q ×q E q ×q ...q ×q E and discover a vector bundle over M .
+(iv) The cotangent bundle of M , T ∗M , has the dual vector space of TmM above each point M : T ∗
+m(M ) =
+(TmM )∗. This space can be appropriately topologized to be smooth, and a set of sections of Γ(T ∗M )
+is isomorphic to the set of morphisms T M → � that are linear in each fibre. This construction may
+be applied to any vector bundle and is called the dual vector bundle.
+(v) Consider the space Λn(E ), the alternating tensor product of E ∗. The set of sections of this vector
+bundle is equivalent to the alternating n-linear morphism En → �; when restricted to the tangent
+bundle, this is the space of differential n-forms.
+There are two constructions on vector bundles that will be necessary to prove the main theorem of
+this section.
+Proposition 2.1.5. Let (q : E → M ,ξ,+q,·q) be a vector bundle.
+(i) For any map f : N → M , the T -reindexing of q by f is a vector bundle:
+f ∗E
+E
+N
+M
+q
+f
+f ∗q
+¯f
+⌟
+(2.1)
+(ii) Any retract of q in the space of arrows is a vector bundle; that is, given
+F
+E
+F
+N
+M
+N
+q
+r ′
+r
+π
+s
+s ′
+π
+(2.2)
+if there is a vector bundle structure on q, then there is a vector bundle structure on π.
+31
+
+The category of vector bundles has “locally linear” bundle morphisms as its maps.
+Definition 2.1.6. A morphism of vector bundles between q : E → M and π : F → N is a commuting
+square
+E
+F
+M
+N
+q
+f
+v
+π
+that is fibrewise linear, so that above each fibre
+f |m : Em → Fv(m)
+isa linearmorphism of vectorspaces. Thismay equivalently be stated asa morphism of fibred �-modules,
+so that the following diagrams commute:
+E
+F
+E
+F
+M
+N
+M
+N
+E2
+F2
+� × E
+� × F
+E
+F
+E
+F
+q
+f
+v
+π
+ξ
+ζ
+f
+v
++
++
+f
+f2
+·E
+�×f
+·F
+f
+Example 2.1.7.
+(i) For the pullback vector bundle in Diagram 2.1, the pair ( ¯f , f ) is a linear bundle morphism.
+(ii) For the section/retract vector bundle structure from Diagram 2.2, the section and retract are linear
+morphisms. Note that this is exactly the splitting of a linear idempotent on q : E → M .
+The lift on the tangent bundle was defined in Section 1.2 as
+[γ]∼ �→ [γ ◦ ·�]∼.
+Instead, consider the action of � on a tangent vector:
+([γ]∼,r ) �→ [γ ◦ (r · x)]∼.
+Note that T.· gives the equation
+T. ·◦([ω ◦ (x, y )],[(a,b ) �→ a + b · x]) = (ω ◦ (a · x,a · b · y ));
+so the lift map ℓ can be rederived as follows:
+T M
+T M × �
+T (T M × �)
+T 2M
+[γ]
+([γ],1)
+([γ ◦ π0],[r �→ 1• r ])
+[γ ◦ •�]
+(id,1�)
+0×λ
+This general construction is known as the Euler vector field of a multiplicative action by �+.
+32
+
+Definition 2.1.8. Consider a multiplicative monoid action a : �+ × E → E . The Euler vector field1 of
+the action is the morphism λ : E → T E constructed as follows:
+λ := E
+(id,1�◦!)
+−−−−→ E × �
+0×λ
+−−→ T E × T � ∼= T (E × �)
+T.a
+−→ T E .
+Local triviality for the tangent bundle is encoded by the universality of the vertical lift condition. A
+similar universality condition holds for vector bundles.
+Proposition 2.1.9. Let q : E → M be a vector bundle with corresponding Euler vector field λ. Then the
+following diagram is a T -pullback:
+E
+T E
+M
+E × T M
+(ξ,0)
+q
+λ
+(p,T.q)
+Exploiting the fact that fibred �-modules have subtraction, the following result holds.
+Corollary 2.1.10. The following two diagrams are T-equalizers:
+E2
+T E
+T M
+T M p ×q E
+T E
+E
+µE :=0◦π0+T q λ◦π1
+νE :=T.ξ◦π0+p λ◦π1
+T.q
+T.q◦0◦p
+p
+p◦T.ξ◦T.q
+(recall that E2 is the pullback of a submersion along a submersion and is therefore guaranteed to exist
+and be preserved by the tangent functor).
+Proof. Given v : X → T E so that
+T.q ◦ v = T.q ◦ 0◦ p ◦ v
+then
+p ◦ (v −T.q 0◦ p ◦ v) = p ◦ v −q p ◦ v = ξ ◦ q ◦ v.
+So there is a unique v ′ so that
+λ ◦ v ′ = (v −T.q 0◦ p ◦ v)
+meaning that
+v = 0◦ p ◦ v +T.q λ ◦ v ′ = µ(0◦ p ◦ v,v ′)
+as required. The projection q is a submersion, so the pullback E2 is preserved by the tangent functor,
+as is the pullback in Proposition 2.1.9, and the same calculation may be applied for each T n. The proof
+for ν follows by the same argument.
+Recall that the class of submersions forms a retractive display system in the category of smooth
+manifolds (Definition 1.5.3), so they are stable under reindexing and closed to retracts. We may now
+infer the following:
+Corollary 2.1.11. The projection for a vector bundle is a submersion.
+Proof. This follows from the fact that π : T E → E is a submersion, so that q ◦ π0 : E q ×p T M → M is a
+submersion, so the map q : E → M is a retract of the projection p.E : T E → E in the arrow category.
+1Somewhat confusingly, the Euler vector field is almost never a vector field.
+33
+
+Preservation of the Euler vector field is also sufficient to guarantee that a morphism f : E → F
+determines a vector bundle morphism.
+Proposition 2.1.12. Let q : E → M ,π : F → N be a pair of vector bundles with Euler vector fields λE ,λF .
+Then a bundle morphism (f ,v) : q → π is a vector bundle morphism if and only if
+λF ◦ f = T.f ◦ λE
+Proof. Note that the νF map from Corollary 2.1.10 is monic, and if f preserves the lift, it preserves ν:
+νF ◦ (T.v, f ) = + ◦ (T.ζ ◦ T.v,λF ◦ f ) = + ◦ (T.f ◦ T.xi,T.f ◦ λE ) = T.f ◦ νE .
+Next, observe that T.v × f is the unique map making the following diagram commute:
+M
+E
+T M p ×q E
+T E
+T N p ×q F
+T F
+N
+F
+∃!
+ν
+ν
+T.f
+p
+p
+f
+q◦π1
+π◦π1
+ζ
+ξ
+⌟
+⌟
+Now ν is a vector bundle morphism and monic, and T.f is a vector bundle morphism, so it follows
+that T.v × f is a vector bundle morphism and hence f is also a vector bundle morphism. The reverse
+implication is immediate.
+2.2
+Lifts for the tangent weak comonad
+The lift ℓ : T ⇒ T 2 gives rise to a weak comonad. Weak comonads were introduced in Wisbauer (2013)
+and have a natural notion of an associative algebra2. An associative coalgebra of the weak comonad
+(T,ℓ) is called a lift; this chapter will demonstrate that lifts provide the essential structure necessary to
+formulate vector bundles.
+Definition 2.2.1. A weak comonad on a category � is an endofunctor S : � → � equipped with a coas-
+sociative map δ : S ⇒ S.S:
+S
+S.S
+S.S
+S.S.S
+δ
+δ
+S.δ
+δ.S
+An associative algebra of a weak comonad is an object E equipped with a map λ : E → SE so that
+E
+S.E
+S.E
+S.S.E
+λ
+λ
+δ.E
+S.λ
+2This is not strictly true. Wisbauer has a more nuanced hierarchy of almost-monads, and in his language (T,ℓ) would be
+an endofunctor with an associative product.
+34
+
+A morphism of these algebras is a map f : (E ,λ) ⇒ (D,γ) so that
+E
+D
+S.E
+S.D
+f
+S.f
+λ
+γ
+Recall that for a full (co)monad, there is an adjunction between the base category and the category
+of (co)algebras. That result is weakened in this case:
+Lemma 2.2.2. For every weak comonad on a category �, there is a free coalgebra functor
+F : � → CoAlg(�);E �→ (S.E ,δ : S.E → S.S.E )
+an underlying object functor
+U : CoAlg(�) → �;(E ,λ) �→ E
+and a natural transformation
+λ : id ⇒ F.U ;
+E
+S.E
+S.E
+S.S.E
+λ
+S.λ
+λ
+δ
+Definition 2.2.3. A lift in a tangent category � is an associative coalgebra of (T,ℓ), namely, a pair (E ,λ :
+E → T E ) so that the following diagram commutes:
+E
+T E
+T E
+T 2E
+λ
+λ
+ℓ.E
+T.λ
+A morphism of lifts is a coalgebra morphism. The category of lifts and lift morphisms in a tangent cate-
+gory � is written Lift(�).
+Note that the tangent bundle is not, in general, a comonad: while the tangent projection has the
+correct type for a counit, p : T ⇒ id , it does not satisfy p ◦ℓ = id . In fact, if p were a counit, this would
+force id = p ◦ ℓ = 0 ◦ p so that 0 = p −1, thus if (T,ℓ,p) is a comonad then T is naturally isomorphic to
+the identity functor.
+Example 2.2.4.
+(i) For every object M in a tangent category �, the pair (T M ,ℓ : T M → T 2M ) is a lift, called the free
+lift on M .
+(ii) Every object M in a tangent category has a trivial lift, 0 : M → T M , where T.0◦ 0 = ℓ◦ 0.
+(iii) Every differential object has a lift λ; the coherence is equivalent to axiom [D0.3] in Definition 1.4.1.
+(iv) The Euler vector field of a multiplicative �+-action h : R + ×E → E in SMan is a lift. Recall that the
+Euler vector field over the scalar action sM : T M × � → � induces the vertical lift on a manifold:
+ℓ = T.sM ◦ (0,λ� ◦ 1�◦!).
+35
+
+In the category of smooth manifolds, T � ∼= �[x]/x 2, where λ(r ) = [x �→ r ·x] corresponds to the map
+λ′(r ) = 0+r ·x. Similarly, there is an isomorphism �[x, y ]/(x 2, y 2) so that for the maps 0.Tand T.0,
+0.T (a + r · x) ∼= a + r · x + 0y + 0x y,
+T.0(a + r · x) ∼= a + 0x + r · y + 0x y.
+Since we know that (0 + x)(0 + y ) = (0 + x y ) in �[x, y ]/(x 2, y 2) (following 1.4.2), we can use these
+isomorphisms to see that
+ℓ◦ λ� ◦ 1�◦! = (0.T ◦ λ� ◦ 1�◦!) ·T 2.� (T.0◦ λ� ◦ 1�◦!).
+Consider a monoid action (�+,h) on a manifold E . The Euler vector field of this action,
+λ : E
+(id,1�◦!)
+−−−−→ E × �
+(0,λ�)
+−−−→ T E × T �
+T.h
+−→ T E
+will define an algebra if the induced scalar action on T E commutes with the natural scalar action:
+T.λ ◦ λ = T 2.h ◦ (T.0◦ λ,T.λ ◦ 0◦ 1�◦!)
+= T 2.h ◦ (T.0◦ T.h ◦ (0,λ ◦ 1�◦!),0◦ λ ◦ 1�◦!)
+= T 2.h ◦ (T 2.h ◦ (T 0◦ 0,T 0◦ λ ◦ 1�◦!),0◦ λ ◦ 1�◦!)
+= T 2.h ◦ (T.0◦ 0,(T.0◦ λ ◦ 1�◦!) ·T 2.� (0.T ◦ λ ◦ 1�◦!))
+= T 2.h ◦ (ℓ◦ 0,ℓ◦ λ ◦ 1�◦!)
+= ℓ◦ T.h ◦ (0,λ ◦ 1�◦!)
+= ℓ◦ λ.
+Observation 2.2.5. Recall that by Proposition 2.1.12, morphisms preserve a monoid action if and only
+if they preserve the associated Euler vector field of the action. This means the Euler vector field construc-
+tion gives a fully faithful functor from monoid actions to lifts in the category of smooth manifolds, and
+therefore from the category of vector bundles to the category of lifts in SMan.
+The following proposition gives a pair of constructions on lifts—closure under the tangent functor
+and finite T -limits—that will be useful in this section.
+Lemma 2.2.6. Let � be a tangent category.
+(i) The tangent functor lifts to an endofunctor on the category of lifts in �.
+(ii) Given a diagram
+D : � → Lift(�)
+in the category of lifts of �, if the T -limit ofU .D exists in �, then limU .D has a natural lift λ′ asso-
+ciated to it so that (limU .D,λ′) is the limit of D in Lifts(�). (That is, T -limits of lifts are computed
+pointwise in the base category.)
+Proof.
+(i) Simply check that
+T.(c ◦ T.λ) ◦ c ◦ T.λ = T.c ◦ T 2.λ ◦ c ◦ T.λ = T.c ◦ c .T ◦ T 2.λ
+= T.c ◦ c .T ◦ T.ℓ◦ T.λ = ℓ.T ◦ c ◦ T.λ.
+36
+
+(ii) Concretely, a tangent terminal object will have a lift:
+(1,1
+0−→ T.1 ∼= 1).
+Given (E ,λ) and (F,l ), if the tangent product E × F exists there is a lift
+(E × F,E × F
+λ×l
+−−→ T E × T F ∼= T (E × F )).
+Given the T -equalizer of a fork f ,g : (E ,λ) → (F,l ), the equalizer has a lift induced as follows:
+C
+E
+F
+T.C
+T.E
+T.F
+f
+g
+k
+λ
+k
+λ
+T.f
+T.g
+l
+Proposition 2.2.7. The category of lifts is a tangent category.
+Proof. The tangent functor sends
+f : (E ,λ) → (F,l )
+T.f : (T E ,c ◦ T.λ) → (T F,c ◦ T.l ).
+To see that this is still an algebra morphism, compute
+c ◦ T.l ◦ T.f = c ◦ T 2.f ◦ T.λ = T 2.f ◦ c ◦ T.λ
+The structure maps are the structure maps on the underlying object of the lift; the universality condi-
+tions follow by Proposition 2.2.6.
+The following idempotent is key in the theory of lifts and will be used in defining non-singular
+lifts (Definition 2.3.1), and its splitting will present the projection and zero-section of a vector bundle
+(Definition 2.4.1).
+Proposition 2.2.8. The category of lifts in a tangent category � has a natural idempotent:
+e : id ⇒ id ;e(E ,λ) : (E ,λ)
+p◦λ
+−−→ (E ,λ).
+Proof. First, we see that e = p ◦ λ is an idempotent:
+p ◦ λ ◦ p ◦ λ = p ◦ p.T ◦ T.λ ◦ λ = p ◦ p.T ◦ ℓ◦ λ = p ◦ 0◦ p ◦ λ = p ◦ λ.
+Moreover, every f : (E ,λ) → (F,l ) preserves the idempotent:
+f ◦ p ◦ λ = p ◦ T.f ◦ λ = p ◦ l ◦ f .
+Finally, note that the idempotent is a lift morphism.:
+T.λ ◦ λ = ℓ◦ λ = c ◦ ℓ◦ λ = c ◦ T.λ ◦ λ
+which implies that
+λ ◦ e = λ ◦ p ◦ λ = p ◦ T.λ ◦ λ = p ◦ c ◦ T.λ ◦ λ = T.p ◦ T.λ ◦ λ = T.e ◦ λ.
+37
+
+2.3
+Non-singular lifts
+Grabowski and Rotkiewicz (2009) introduced the notion of a non-singular lift as a means to axioma-
+tize the Euler vector field of a vector bundle’s multiplicative �+-action. While it is not immediately
+clear that our definition is the same as Grabowski’s, the results of Section 2.5 will justify the use of this
+language as they are necessarily the same.
+Definition 2.3.1. A lift (E ,λ) in a tangent category � is non-singular whenever the following diagram is
+a T -equalizer:
+E
+T E
+T E
+λ
+T.e
+e .E
+where e .E = p ◦ ℓ (the idempotent associated to the free lift on E ) and T.e = T.p ◦ T.λ (the image of the
+idempotent associated to (E ,λ) under the tangent functor). The category of non-singular lifts is written
+NonSing(�).
+The most prominent class of examples is given by the Euler vector field of the �-action on a vector
+bundle.
+Proposition 2.3.2. The Euler vector field of a vector bundle is a non-singular lift.
+Proof. Let (q : E → M ,+,ξ,·) be a vector bundle with Euler vector field λ. By Proposition 2.1.9, the
+diagram
+T M
+E
+T E
+E
+λ
+T.q
+p◦T.ξ◦T.q
+p
+T.q◦0◦p
+is a T -limit. The T -universality of this diagram will hold if and only if the diagram is universal after
+each parallel pair of arrows is post-composed by a T -monic. A section is a T -monic, so the previous
+diagram is T -universal if and only if the following diagram is T -universal:
+E
+T E
+E
+T E
+T M
+T E
+p
+ξ◦q◦p
+T.q
+0◦q◦p
+λ
+0
+T.ξ
+Now simplify this diagram using the fact that T.(ξ ◦ q) = T.e ,0◦ p = e .E :
+T E
+E
+T E
+T E
+e .T
+e .E ◦T.e
+T.e
+e .E ◦T.e
+λ
+38
+
+Note that the the two pairs of parallel arrows have a common arrow, implying that they may be pulled
+together into a single ternary equalizer. All that remains to check, then, is that for any x : X → T E ,
+(e .e ◦ x = T.e ◦ x = e .E ◦ x) ⇐⇒ (T.e ◦ x = e .E ◦ x).
+The forward implication is trivial, so it remains to prove the reverse. Suppose T.e ◦ x = e .E ◦ x; then
+e .e ◦ x = e .E ◦ T.e ◦ x = e .E ◦ e .E ◦ x = e .E ◦ x
+giving the result, namely that the diagram
+E
+T E
+T E
+λ
+T.e
+e .E
+is a T -equalizer.
+Every map f : E → F gives a map of free coalgebras T.f : (T E ,ℓ) → (T F,ℓ) and the idempotent e is
+a coalgebra morphism by Proposition 2.2.8, so the following is immediate:
+Proposition 2.3.3. A non-singular lift is an equalizer in the category of lifts:
+(E ,λ)
+(T E ,ℓ)
+(T E ,ℓ)
+λ
+T.e
+e .E
+Observe that the category of non-singular lifts is closed under finite limits in the category of lifts.
+Proposition 2.3.4. The category of non-singular lifts in � is closed under T -limits:
+(i) The tangent functoron liftspreservesnon-singularlifts, so that if (E ,λ) isnon-singularthen (T E ,c ◦
+T.λ) is non-singular.
+(ii) The trivial lift on an object, 0 : M → T M , is non-singular.
+(iii) T -products of non-singular lifts are non-singular lifts.
+(iv) T -equalizers of non-singular lifts are non-singular lifts.
+Proof.
+(i) This follows from the fact that the non-singularity condition is a T -limit.
+(ii) The zero map splits the idempotent 0◦ p, so it is the equalizer of 0◦ p,p ◦ 0 = id .
+(iii) This follows by stability of limits under products.
+(iv) The following diagram commutes by naturality:
+C
+E
+F
+T.C
+T.E
+T.F
+T.C
+T.E
+T.F
+λ
+λ
+f
+T.f
+T.g
+e .E
+T.e
+e .F
+T.e
+T.f
+T.g
+g
+λ
+e .C
+T.e
+Each horizontal diagram is a T -equalizer, and the two columns on the right are T -equalizers, so
+the column on the left is a T -equalizer.
+39
+
+Finally, when a tangent category has certain T -equalizers, there is an idempotent monad on the
+category of lifts, whose algebras are non-singular lifts:
+Theorem2.3.5. Let � be a tangent category with chosen T -equalizersof idempotents. Then the following
+equalizer determines a left-exact idempotent monad on the category of lifts, whose algebras are non-
+singular lifts.
+(F,l )
+(T.E ,ℓ)
+(T.E ,ℓ)
+e .E
+T.e
+Proof. First, take the equalizer in Lift(�); the functor sends (E ,λ) to the chosen limit (F,l ).The unit of
+the monad is the unique morphism from (E ,λ) to the equalizer (F,l ) induced by universality:
+(F,l )
+(T E ,ℓ)
+(T E ,ℓ)
+(E ,λ)
+λ
+T.e
+e .E
+∃!
+Note that non-singular lifts are closed under finite limits, so (F,l ) is a non-singular lift. If (E ,λ) is a
+nonsingular lift then λ : (E ,λ) → (T E ,ℓ) equalizes the diagram, so there is a unique isomorphism
+(F,l ) ∼= (E ,λ), making the multiplication of the monad a natural isomorphism (and thus yielding an
+idempotent monad). The functor is defined as a T -limit and therefore preserves all T -limits of lifts, so
+it is left-exact.
+In the category of smooth manifolds, ℓ is the Euler vector field of an �+-action, this guarantees that
+every λ is the Euler vector field of a multiplicative �+-action. We note the following corollary.
+Corollary 2.3.6. In the category of multiplicative �+ actions in SMan, multiplication by 0 is equivalent
+to the natural idempotent e in the fully faithful functor sending an �+-action to its Euler vector field.
+By non-singularity, the following diagram is an equalizer, giving E a multiplicative action by �+ whose
+Euler vector field is λ:
+(E ,·E )
+(T E ,·T )
+(T E ,·T )
+T.e
+e .T
+λ
+Moreover, λ is the Euler vector field of this lift, and the following diagram commutes:
+E
+T E
+T E
+�+ × E
+�+ × T E
+�+ × T E
+T (�+ × E )
+T (�+ × T E )
+T (�+ × T E )
+T E
+T T E
+T T E
+T (�+×e .E )
+T (�+×T.e )
+�+×λ
+·p
+T ·p
+e .E
+T.e
+T λ
+T ·E
+(1�,id)
+(1�,id)
+(1�,id)
+λ�×0
+λ�×0
+λ�×0
+e .E
+T.e
+�+×e .E
+�+×T.e
+λ
+�+×λ
+40
+
+2.4
+Differential bundles
+This section introduces (pre-)differential bundles, which provided the rest of the data for a vector bun-
+dle: namely the projection, the zero section, and the addition map. The zero section and projection
+data arise by splitting the natural idempotent e : id ⇒ id , and non-singularity will induce the addi-
+tion map. Every differential bundle satisfies a pair of universality diagrams, linking this presentation
+of differential bundles to the original definition in Cockett and Cruttwell (2018).
+Definition 2.4.1.
+(i) A pre-differential bundle is a lift λ : E → T E equipped with a chosen splitting of the natural idem-
+potent e = p ◦ λ from Proposition 2.2.8. Pre-differential bundles are formally written (q : E →
+M ,ξ,λ), where q : E → M is the retract, ξ : M → E the section, and λ : E → T E the lift (the types
+are only necessary for the projection as the rest may be inferred, and will generally be suppressed to
+save space).
+(ii) A differential bundle is a pre-differential bundle (q : E → M ,ξ,λ) with the properties that λ is
+non-singular and T -pullback powers of q exist.
+Morphisms of (pre-)differential bundles are exactly morphisms of their underlying lifts. The categories
+of (pre-)differential bundles are exactly (pre-)differential bundles and lift morphisms, and are written
+Pre(�),DBun(�) respectively.
+Recall that as e is a natural idempotent in the category of lifts, any lift morphism will preserve e
+and will consequently preserve its idempotent splitting. Preserving the idempotent means that every
+differential bundle morphism is a bundle morphism, where the base map is given by
+E
+F
+M
+N
+q
+f
+π
+m:=π◦f ◦ξ
+We now look at the limits of (pre-)differential bundles.
+Observation 2.4.2.
+(i) The limit for a diagram of pre-differential bundles is the limit of the underlying lifts equipped with
+a chosen splitting of p ◦ λ due to basic properties about idempotent splittings.
+(ii) The limit for a diagram of differential bundles is the limit in the category of pre-differential bun-
+dles (the lift will be universal by Proposition 2.3.4), so long as T -pullback powers of the resulting
+projection exist.
+Because there is a projection associated to a (pre-)differential bundle, the T -reindexing operation
+described in Proposition 1.5.2 can now be applied. This gives a pullback differential bundle in a similar
+way to Proposition 2.1.5.
+Lemma 2.4.3 (Cockett and Cruttwell (2018)). Let (q : E → M ,ξ,λ) be a (pre-) differential bundle in a
+tangent category �, and consider a T -pullback in �:
+u∗E
+E
+N
+M
+u
+q
+u∗q
+¯u
+⌟
+41
+
+Then the induced triple maps
+T.u∗E
+T E
+N
+u∗E
+E
+u∗E
+E
+N
+M
+N
+M
+T N
+T M
+u
+q
+u∗q
+¯u
+⌟
+T. ¯u
+λ
+0
+T.q
+T.u
+0
+T.u∗q
+u∗λ
+q
+u
+⌟
+ξ◦u
+u∗ξ
+induce a (pre-)differential bundle (u∗q : u∗E → N ,u∗ξ,u∗λ). If λ is non-singular and T -pullback pow-
+ers of u∗q exist, then (u∗q,u∗ξ,u∗λ) is a differential bundle.
+Proof. Note thata pre-differential bundle (q : E → M ,ξ,λ) in � mayalsobe regarded asa pre-differential
+bundle (q : (E ,λ) → (M ,0),ξ,λ) in Lift(�). Take the following pullback in Lift(�):
+(u∗E ,u∗λ)
+(E ,λ)
+(N ,0)
+(M ,0)
+u
+q
+⌟
+It follows by construction that ι ◦ u∗λ = λ ◦ ι. Thus the result holds.
+Proposition 2.4.4. Let (q : E → M ,ξ,λ) be a non-singular pre-differential bundle in a tangent category
+�, so that T -pullback powers of q exist. Then there is an additive bundle structure (q,ξ,+q) so that the
+differential bundle morphisms are additive:
+(λ,ξ) : (q,ξ,+q) → (p,0,+)
+(λ,0) : (q,ξ,+q) → (T.q,T.ξ,T.+q).
+Furthermore, every differential bundle morphism preserves addition.
+Proof. Non-singularity forces the existence of an addition map:
+E2
+T2E
+T2E
+E
+T E
+T E
+∃!+q
++
+λ×λ
+λ
+e .E
+T.e
++
+(e ×e ).E
+T2.e
+Note that this diagram commutes because T2E is a pre-differential bundle whose lift is ℓ×ℓ, and also +
+is a linear morphism, so it commutes with the addition map e ◦a +e ◦b = e ◦(a ×b ). Post-composition
+with λ ensuresthatξ isthe unitand thatassociativityholds. Adifferential bundle morphism will induce
+a morphism of the equalizer diagrams that induce each addition map to preserve addition.
+Recall that by Proposition 2.3.2, the Euler vector field for every vector bundle is a non-singular lift.
+If T M q ×q E exists, the map
+νE : T M p ×q E
+T.ξ×λ
+−−−→ T2E
++.E
+−→ T E
+may be formed. Similarly, using the additive bundle structure from Proposition 2.4.4, the µ map may
+be formed:
+µE : E q ×q E
+0×λ
+−−→ T (E q ×q E )
+T.+q
+−−→ T E
+Note that any differential bundle morphism will preserve µ and ν.
+42
+
+Lemma 2.4.5. Differential bundle maps preserve µ(x, y ) := 0◦ x +T.q λ◦ y and ν(v, y ) := T.ξ◦v +p λ◦ y .
+Proof. The following diagram demonstrates that lift maps preserve ν:
+E q ×p T M
+F q ′×p T N
+T.E
+T.F
+T.E
+T.F
+ν
+ν
+f ×T.m
+T.f
+T.g
+e .e
+T.e
+e .e
+T.e
+T.f
+T.g
+g ×T.n
+Similarly, this diagram demonstrates that lift maps preserve µ:
+T E
+T F
+T E2
+T F2
+E2
+F2
+(T.f ×T.f )
+(λ×0)
+(λ′×0)
+(f ×f )
+T.+
+T.+′
+T f
+Lemma 2.4.6. Consider the full subcategory of differential bundles in � whose objects are differential
+bundles (E ,λ) so that the forks
+E2
+T E
+T M
+T M p ×q E
+T E
+E
+µE :=0◦π0+T q λ◦π1
+νE :=T.ξ◦π0+p λ◦π1
+T.q
+T.q◦0◦p
+p
+p◦T.ξ◦T.q
+(2.3)
+are T -equalizers. This subcategory is closed under T -equalizers.
+Proof. Start with the T -equalizer of lifts:
+C
+E
+F
+f
+g
+k
+Observe that k is a lift map, so by Lemma 2.4.5, the following diagram commutes:
+C q c ×p T P
+E q ×p T M
+F q ′×p T N
+T.C
+T.E
+T.F
+T.C
+T.E
+T.F
+ν
+ν
+f ×T.m
+T.f
+T.g
+e .e
+T.e
+e .e
+T.e
+T.f
+T.g
+k×T.k ′
+ν
+e .e
+T.e
+T.k
+T.k
+g ×T.n
+43
+
+Next, since k is a morphism of pre-differential bundles, by Lemma 2.4.5 the following diagram com-
+mutes:
+C2
+E2
+F2
+T.C
+T.E
+T.F
+T.C
+T.E
+T.F
+µ
+µ
+f2
+T.f
+T.g
+e .e
+T.e
+e .e
+T.e
+T.f
+T.g
+g2
+µ
+e .e
+T.e
+The top row follows because maps satisfying Rosicky’s universality condition preserve µ, and the bot-
+tom row by the naturality of e .C and the fact that linear maps preserve the natural idempotent. Thus, if
+E ,F satisfy the universality diagrams in Diagram 2.3, the equalizer C will as well, because T -equalizers
+are closed to T -limits in the category of fork diagrams.
+Theorem 2.4.7. For every differential bundle (q : E → M ,ξ,λ) in a tangent category, the diagram
+E2
+T E
+T M
+µE
+T.q
+T.q◦0◦p
+is a T -equalizer, and if T M p ×q E exists, then
+E q ×p T M
+T E
+E
+νE :=λ◦π0+p T.ξ◦π0
+p
+p◦T.ξ◦T.q
+is a T -equalizer.
+Proof. The tangent bundle satisfies both universality conditions, and by Lemma 2.4.6every differential
+bundle will satisfy these conditions by the non-singularity of the lift.
+Corollary 2.4.8. In a complete tangent category, the category of differential bundles is precisely the cat-
+egory of algebras of the monad in Theorem 2.3.5 on pre-differential bundles.
+Remark 2.4.9. The universality conditions in Theorem 2.4.7 demonstrate that this definition of differ-
+ential bundle agrees with that of Cockett and Cruttwell (2018).
+2.5
+The isomorphism of categories
+It is now straightforward to show that the main theorem of this chapter holds. First, observe that for
+every differential bundle (q : E → M ,ξ,λ) in the category of smooth manifolds, the T -pullback E p ×q
+T M exists , as p is a submersion. Note that the universality of the two diagrams is equivalent:
+E q ×p T M
+T E
+E
+νE :=λ◦π0+p T.ξ◦π0
+p
+p◦T.ξ◦T.q
+E q ×p T M
+T E
+M
+E
+νE
+λ◦π0+p T.ξ◦π1
+p
+ξ
+⌟
+Thus the following holds.
+44
+
+Theorem 2.5.1. There is an isomorphism of categories between vector bundles and differential bundles
+in smooth manifolds.
+Proof. Note that Proposition 2.3.2 gives a fully faithful functor from the category of vector bundles to
+differential bundles of smooth manifolds, as the lift associated to the vector bundle is non-singular and
+the projection and zero section give the rest of the structure of a differential bundle: as remarked after
+Definition 2.1.1, local triviality guarantees that the projection is a submersion, so pullback powers of
+the projection exist, yielding a differential bundle.
+To see there is an isomorphism on objects, recall that every differential bundle is a fibred �-module
+by Corollary 2.3.6, and that this identification is a bijective mapping (it recovers the original �-action
+from the Euler vector field of the �-action, and vice versa). Last, note that the universality condition
+E q ×p T M
+T E
+M
+E
+νE
+λ◦π0+p T.ξ◦π1
+p
+ξ
+⌟
+along with Proposition 2.1.5 ensures the local triviality of q, so that the unique fibred �-module struc-
+ture associated to a differential bundle is indeed a vector bundle.
+2.6
+Connections on a differential bundle
+The connections discussed in this section generalize the notion of an affine connection to a differential
+bundle, giving a “local coordinates” presentation for T E similar to the presentation of T 2M as T3M
+induced by an affine connection. Chapter 3 makes extensive use of connections on vector bundles, so
+it is useful to set out the basic definitions before embarking on algebroid theory.
+Definition 2.6.1 (Cockett and Cruttwell (2017)). Let (q : E → M ,ξ,λ) be a differential bundle in a tan-
+gent category �.
+• A vertical connection is a map κ : T E → E so that
+(i) κ is a vertical descent, and hence a retract of the lift; thus, κ ◦ λ = id ;
+(ii) κ is compatible with both differential bundle structures on T E , so that the maps
+κ : (T E ,ℓ) → (E ,λ)
+κ : (T E ,c ◦ T.λ) → (E ,λ)
+are lift morphisms.
+• A horizontal connection is a map ∇ : E q ×p T M → T E so that
+(i) ∇ is a horizontal lift, and so is a retract of (p.E ,T.q) : T E → E q×pT M 3; thus, (p,T.q)◦∇ = id ;
+(ii) ∇ is compatible with each pair of lifts, so that the maps
+∇ : (E q ×p T M ,λ × ξ) → (T E ,c ◦ T.λ)
+∇ : (E q ×p T M ,0× ℓ) → (T E ,ℓ)
+are lift morphisms.
+3That (p.E ,T.q) land in the pullback E q ×p T M is a consequence of naturality, as p ◦ T.q = q ◦ p.
+45
+
+• A full connection is a pair (κ,∇) that satisfies the following compatibility relations:
+(i) κ ◦ ∇ = ξ ◦ q ◦ π0,
+(ii) ∇(p,T.q) +T.q µ(p,κ) = id , so that there is an isomorphism E q ×p T M p ×q E ∼= T E .
+A notion that will be useful when dealing with classical differential geometry is that of a covariant
+derivative, whose definition is equivalent to that of a vertical connection in the category of smooth
+manifolds.
+Definition 2.6.2. Let (q : E → M ,ξ,λ,κ) be a vertical connection.4 The covariant derivative associated
+to (κ,∇) is the map
+∇(−)[=] : Γ(π) × Γ(p) → Γ(π);(A,X ) �→ (κ ◦ T A ◦ X ).
+Lucyshyn-Wright (2018) drastically simplified the notion of a full connection by showing that it is
+exactly a vertical connection satisfying a universal property.
+Proposition 2.6.3 (Lucyshyn-Wright (2018)). A connection on a differential bundle is equivalently spec-
+ified by a vertical connection
+κ : T E → E
+so that the following diagram exhibits T E as a biproduct in the category of differential bundles over M :
+E
+T E
+T M
+M
+E
+p
+T q
+κ
+q
+q
+p
+There is also a notion of flatness for connections that extends to vertical connections on a differen-
+tial bundle.
+Definition 2.6.4. A connection κ on a differential bundle (q : E → M ,ξ,λ) is flat whenever
+κ ◦ T.κ ◦ c = κ ◦ T.κ.
+We can see that connections are closed under similar constructions to differential bundles, in par-
+ticular idempotent splittings and the reindexing construction from Lemma 2.4.3.
+Lemma 2.6.5. Let (q : E → M ,ξ,λ) be a differential bundle equipped with a vertical connection.
+(i) Any linear retract of (q,ξ,λ) will have a vertical connection.
+(ii) The pullback differential bundle induced by pulling back q along f : N → M will have a vertical
+connection.
+If the vertical connection is flat or is part of a full connection (that is, it satisfies the universality condition
+in Proposition 2.6.3), then the induced connection will be as well.
+Proof. The universal property induces the vertical connection in each case. The construction will pre-
+serve flatness as it is an equational condition, and it preserves effectiveness by the commutativity of
+limits.
+4The definition of a covariant derivative only uses a vertical connection.
+46
+
+There is no reason for every differential bundle in a tangent category to have a connection (for
+example, the tangent bundle in the free tangent category Weil1 in Chapter 4 is a differential bundle
+that does not have a connection). However, if the total space of a differential bundle has an affine
+connection, this induces a compatible connection on the differential bundle and its base space.
+Theorem 2.6.6. Let (q : E → M ,ξ,λ) be a differential bundle in a tangent category in �, where E has a
+(flat) vertical connection. Then the total space M and the differential bundle (q,ξ,λ) each have a (flat)
+vertical connection. If the connection is full (so there is a compatible horizontal connection), then the
+induced connections are likewise full.
+Proof. By the strong universality condition for differential bundles, the differential bundle (q ◦ π0 :
+E q×pT M → M ,(ξ,0),λ×ℓ) is the pullback differential bundle of p : T E → E along ξ : M → E . This gives
+(q◦π0,(ξ,0),(λ,ℓ)),a (flat, effective) vertical connection byLemma 2.6.5, and soityields(q : E → M ,ξ,λ)
+and (p : T M → M ,0,ℓ) as (flat, effective) vertical connections by the idempotent splitting property.
+Every smooth manifold has an affine connection, thus inducing a connection on any vector bundle.
+Corollary 2.6.7. Every vector bundle has a connection.
+47
+
+Chapter 3
+Involution algebroids
+This chapter accomplishes the first major goal of this thesis by providing a tangent-categorical axiom-
+atization of Lie algebroids, namely involution algebroids. Much like vector bundles, Lie algebroids are
+a highly non-algebraic notion (in the sense of Freyd and Kelly (1972)), being vector bundles equipped
+with a Lie algebra structure on the set of sections of the projection. Furthermore, the bracket on sec-
+tions must satisfy a product rule with respect to �-valued functions on the base space (the Leibniz
+law), introducing another piece of non-algebraic structure to the definition. The tangent-categorical
+definition of Lie algebroids will treat the tangent bundle as the “prototypical Lie algebroid” in which
+the vertical lift ℓ : T ⇒ T 2 identifies the vector bundle structure and the canonical flip c : T 2 ⇒ T 2 plays
+the role of the Lie bracket.
+Lie algebroidsare the natural many-objectanalogue toLie algebras, in the same waythatLie groupoids
+are the many-object analogue of Lie groups. In the single-object case, a Lie group is classically thought
+of as a space of symmetries for some smooth manifold (one often identifies a group action G ×M → M ),
+and a Lie algebra may similarly be thought of as a space of derivations (often identified as a sub-Lie-
+algebra of χ(M ) for a manifold M ). The extension of groups to groupoids is natural; in fact, Brandt’s
+introduction of groupoids in Brandt (1927) predates MacLane and Eilenberg’s invention of category
+theory in Eilenberg and MacLane (1945) by nearly two decades. The translation of Lie algebras to the
+many-object case is not as straightforward. The first step is to replace the vector space underlying a Lie
+algebra with a vector bundle (π : A → M ,ξ,λ). The idea is to axiomatize this vector bundle so that each
+section in Γ(π) corresponds to a derivation on C ∞(M ). The anti-commutator operation on derivations
+from Proposition 1.2.18 suggests there should be a Lie bracket [−,−] : Γ(π) ⊗ Γ(π) → Γ(π) (similar to the
+partially-defined multiplication for a groupoid), while the correspondence with derivations on C ∞(M )
+suggests there be a vector bundle morphism ̺ : A → T M satisfying the Leibniz law:
+[X , f · Y ] = f ·[X ,Y ]+ [X , f ]· Y ;
+[X , f ] := ˆp ◦ T.f ◦ ̺ ◦ X 1
+(3.1)
+(the full definition of Lie algebroids may be found in 3.1.1). This “operational” definition of Lie alge-
+broids makes it difficult to describe their morphisms, and furthermore it essentially fails to be an alge-
+braic structure in the classical sense, as it axiomatizes structure on the set of sections of a map rather
+than a morphism in the category itself.
+Involution algebroids were introduced to provide a tangent-categorical presentation of Lie alge-
+broids, similar to the relationship between differential bundles and vector bundles. Chapter 2 focused
+on the Euler vector field construction on a vector bundle, showing that this induced a fully-faithful
+functor from vector bundles to associative coalgebras (lifts) of the weak comonad (T,ℓ), and identified
+vector bundles with a subcategory of Lift(SMan) satisfying a universal property. The corresponding
+1Recall the notation from Lemma 2.1.3.
+48
+
+construction for Lie algebroids, then, is the canonical involution, which was identified by Eduardo
+Martinez and his collaborators (a clearly written exposition may be found in Section 4 of de León et al.
+(2005)). Given a Lie algebroid (π : A → M ,̺ : A → T M ,[−,−] : Γ(π) ⊗ Γ(π) → Γ(π)), its canonical involu-
+tion is a map
+σ : A̺×T πT A → A̺×T πT A.
+Using this σ map, there is a straightforward characterization of Lie algebroid morphisms: a Lie alge-
+broid morphism is precisely a vector bundle morphism (f ,m) : A → B that preserves the anchor and
+involution maps:
+A
+B
+A̺×T πT A
+B ̺B ×T.πB T B
+T M
+T N
+A̺×T πT A
+B ̺B ×T.πB T B
+̺A
+̺B
+f
+T.m
+σB
+f ×T.f
+σA
+f ×T.f
+Furthermore, it is implicit in Martinez’s work (Martínez (2001)) that σ satisfies axioms corresponding
+to the Lie algebroid axioms. Thus, involutivity corresponds to antisymmetry of the Lie bracket
+σ ◦ σ = id ⇐⇒ [X ,Y ]+ [Y ,X ] = 0,
+while the Leibniz law holds if and only if the ̺ map sends the algebroid involution to the canonical flip
+on M ,
+T.̺ ◦ π1 ◦ σ = c ◦ T.π◦ ̺ ⇐⇒ ∀f ∈ C ∞(M ),[X , f · Y ] = f ·[X ,Y ]+ [X , f ]· Y
+(using the same definition as before for [X , f ]).
+The idea of an involution algebroid, then, is to axiomatize the canonical involution directly, just as
+differential bundles axiomatize the Euler vector field of a differential bundle. An involution algebroid
+is a differential bundle equipped with a pair of structure maps
+̺ : A → T M , σ : A̺×T πT A → A̺×T πT A
+satisfying a collection of axioms. Some of them are straightforward translations of the structure equa-
+tions for Lie algebroids given in Martínez (2001), for instance
+T.̺ ◦ λ = ℓ◦ ̺, σ ◦ σ = id , T.̺ ◦ π1 ◦ σ = c ◦ T.̺ ◦ π1.
+However, this requires a new coherence between the Euler vector field of the underlying vector bundle
+and the involution map:
+σ ◦ (ξ ◦ π,λ) = (ξ ◦ π,λ).
+The most striking new fact about this coherence is that the Jacobi identity on the bracket [−,−] corre-
+sponds to the Yang–Baxter equation on σ:
+A̺×T πT AT.̺×T 2.πT 2A
+A̺×T πT AT.̺×T 2.πT 2A
+A̺×T πT AT.̺×T 2.πT 2A
+A̺×T πT AT.̺×T 2.πT 2A
+A̺×T πT AT.̺×T 2.πT 2A
+A̺×T πT AT.̺×T 2.πT 2A
+σ×c .A
+σ×c .A
+1×T.σ
+σ×c .A
+1×T.σ
+This is both surprising (it is a new characterization of a central object of study in differential geom-
+etry and mathematical physics) and yet in a way expected (the work in Cockett and Cruttwell (2015),
+49
+
+Mackenzie (2013) indicates that the Lie algebra structure on the set of vector fields over a manifold
+follows from the Yang–Baxter equation on c ). This vector-field-free presentation of the Jacobi identity
+allows for a structural approach to Lie algebroids that drives the work presented in Chapters 4 and 5.
+As with Chapter 2, the first section is expository, and is concerned with introducing the category of
+Lie algebroids. The second section introduces anchored bundles, together with the space of prolonga-
+tions of an anchored bundle. The relationship between anchored bundles and involution algebroids
+is equivalent to that between reflexive graphs and groupoids (the subject of Chapter 5), the space of
+prolongations of an anchored bundle being equivalent to the set of composable arrows for a groupoid.
+This section is mostly a translation of Martinez’s prolongation construction to a general tangent cate-
+gory. The rest of the chapter contains new results, developed in collaboration with Matthew Burke and
+Richard Garner.
+Section 3 introduces involution algebroids, which are anchored bundles equipped with an involu-
+tion map on their space of prolongations. Section 4 considers an anchored bundle in a tangent cate-
+gory with negatives that is equipped with a connection. The connection gives an involution algebroid
+a “local coordinates” presentation (in the sense of Section 1.4) that is equivalent to the local character-
+ization of Lie algebroids from Section 1. The final section of this chapter establishes the main result:
+the category of Lie algebroids is isomorphic to that of involution algebroids in smooth manifolds.
+3.1
+Lie algebroids
+This section reviews the basic theory of Lie algebroids: their definition and that of their morphisms,
+along with some introductory examples. The classical definition will not appear elsewhere in this
+chapter, however, as we quickly introduce Martinez’s structure equations for a Lie algebroid (Martínez
+(2001)), then translate them into tangent-categorical terms using a connection.
+Definition 3.1.1. A Lie algebroid is a vector bundle π : A → M equipped with an anchor ̺ : A → T M
+and a bracket [−,−] : Γ(π) ⊗ Γ(π) → Γ(π) satisfying the following axioms:
+• bilinear: [a X1 + b X2,Y ] = a[X1,Y ]+ b [X2,Y ] and [X ,a Y1 + b Y2] = a[X ,Y1]+ b [X ,Y2]
+• anti-symmetric: [X ,Y ]+ [Y ,X ] = 0
+• Jacobi: [X ,[Y ,Z ]] = [[X ,Y ],Z ]+ [Y ,[X ,Z ]]
+• Leibniz: [X , f · Y ] = f ·[X ,Y ]+ [X , f ]· Y
+(where [X , f ] is defined as in Equation 3.1).
+Example 3.1.2.
+(i) The canonical example of a Lie algebroid is, of course, the tangent bundle using the operational
+tangent bundle from Definition 1.2.16.
+(ii) ALie algebra isa Lie algebroid overthe terminal object: fora groupG , the bundle of source-constant
+tangent vectors is the usual Lie functor from Lie groups to Lie algebras, because a groupoid is a one-
+object group.
+50
+
+(iii) The bundle of source-constant tangent vectors s,t : G → M of a Lie groupoid forms a Lie algebroid.
+This bundle is defined by the pullback
+A
+T G
+T M
+M
+T M ×G
+(T.s,p)
+(0,e )
+π
+T.t
+̺
+⌟
+where the projection is π and the target is given by ̺ in the diagram. There is an injective �-module
+morphism from sections of π, Γ(π) to vector fields on G , χ(G ), and the Lie bracket on G is closed
+over the image of this lift, putting a Lie bracket on Γ(π). In particular, we can see that T M is the
+bundle of source-constant tangent vectors for the pair groupoid on a manifold M :
+T M
+T (M × M )
+M
+(M × M ) × T M
+(p,T.π0)
+(∆,0)
+p
+(id,0◦p)
+⌟
+Given (u,v) : X → T (M × M ) above (m,m) : X → T M with u = 0 ◦ m, it follows that u = 0 ◦ p ◦ u,
+so v is the unique map induced into T M .
+(iv) Every group is a groupoid over a single object. The Lie algebroid associated with a group G , then, is
+the usual Lie algebra.
+(v) From the Hamiltonian formalism of mechanics, every Poisson manifold has an associated Lie alge-
+broid. A Poisson manifold is a manifold M equipped with a Poisson algebra structure on C ∞(M ),
+namely a Lie algebra that is also a derivation:
+[f ·g ,h] = f ·[g ,h]+ [f ,h]·g
+where · is the multiplication in the algebra C ∞(M ) as in Lemma 2.1.3. The cotangent bundle over
+a Poisson manifold M is canonically a Lie algebroid, called a Poisson Lie algebroid Courant (1994).
+(vi) Any Lie algebra bundle—that is, a vectorbundle equipped with a Lie bracket on itsspace of sections—
+is a Lie algebroid, with anchor map ξ ◦ π.
+Morphisms of Lie algebroids are notoriously difficult to work with, and have an involved definition.
+Definition 3.1.3. Let A,B be a pair of anchored bundles over M ,N , and Φ : A → B an anchored bundle
+morphism over a map φ : M → N . A Φ-decomposition of X ∈ Γ(πA) is a set of Xi ∈ Γ(πB) and fi ∈ C ∞(M )
+so that
+Φ ◦ X =
+�
+i
+fi · Xi ◦ φ.
+An anchor-preserving vector bundle morphism Φ is a Lie algebroid morphism if and only if for any X ,Y ∈
+Γ(πA) and Φ-decompositions {Xi , fi},{Yj,g j} of X ,Y , the following equation holds:
+Φ ◦ [X ,Y ] =
+�
+fi ·g j ·([Xi ,Yj]◦ φ) +
+�
+[X , fi ]·(Xi ◦ φ) −
+�
+[Y ,g j](Yj ◦ φ).
+51
+
+The equation defininga Lie algebroid morphism holdsindependentlyofthe choice ofΦ-decomposition
+(see Higgins and Mackenzie (1990) for a proof).
+Example 3.1.4.
+(i) If A,B are Lie algebroids over a base manifold M , Φ : A → B is a Lie algebroid morphism if and only
+if it preserves the anchor and Lie bracket.
+(ii) Given two Lie algebroid morphisms f : A → B,g : B → C , their composition g ◦ f is also a Lie
+algebroid morphism.
+(iii) A Poisson Sigma model (see Bojowald et al. (2005)) is a morphism of Lie algebroids
+φ : T Σ → T ∗M
+for which Σ is a 2-dimensional manifold and T Σ denotes its tangent Lie algebroid, while M is a
+Poisson manifold and T ∗M denotes the Lie algebroid structure on its cotangent bundle .
+Lie algebroids are a natural generalization of Lie algebras to the “multi-object” setting, but they
+are ill-suited for a functorial presentation of the theory. A step in this direction is to consider the
+coordinate-based presentation of the Lie bracket and its coherences due to Martínez (2001). Let A
+be an anchored bundle over M equipped with a bilinear bracket on its space of sections, and choose a
+pair of bases for Γ(π) and χ(M ): for Γ(π) write {eα}, and for χ(M ) write { ∂
+∂ x i }. The anchor and bracket
+then have a presentation in local coordinates:
+̺(eα) =
+�
+i
+̺i
+α
+∂
+∂ x i
+[eα,eβ] =
+�
+γ
+C γ
+αβ eγ
+(from here on out, we use Einstein summation notation to simplify our calculations, so instead write
+̺(eα) = ̺i
+α
+∂
+∂ x i
+[eα,eβ] = C γ
+αβ eγ
+with
+�
+suppressed). The following characterization of the Lie algebroid axioms uses Martinez’s struc-
+ture equations.
+Proposition 3.1.5. [Martínez (2001)] An anchored bundle A over M equipped with a bracket is a Lie
+algebroid if and only if ̺ and [−,−] satisfy the following structure equations:
+(i) Alternating:
+C ν
+αβ + C ν
+βα = 0
+(ii) Leibniz:
+̺ j
+α
+∂ ̺i
+β
+∂ x j = ̺i
+γC γ
+αβ + ̺ j
+β
+∂ ̺i
+α
+∂ x j
+(iii) Bianchi:
+0 = ̺i
+α
+∂ C ν
+βγ
+∂ x i + ̺i
+β
+∂ C ν
+γα
+∂ x i + ̺i
+γ
+∂ C ν
+αβ
+∂ x i + C µ
+βγC ν
+αµ + C µ
+γαC ν
+βµ + C µ
+αβC ν
+γµ
+52
+
+This proposition is a straightforward translation of the Lie algebroid axioms into local coordinates
+using a covariant derivative. First, recall that by the smooth Serre–Swan theorem (11.33 Nestruev
+(2003)), the bilinearity of the bracket
+[−,−] : Γ(π) × Γ(π) → Γ(π)
+as a morphism of C ∞(M )-modules guarantees that it corresponds to a bilinear morphism A2 → A of
+vector bundles, meaning that there exists a globally defined bilinear map A2 → A that is equal to the
+Lie bracket when applied to sections of the projection. We record this as a lemma:
+Lemma 3.1.6. For every Lie algebroid � , there is a bilinear morphism
+〈−,−〉 : A2 → A
+so that for any sections X ,Y ∈ Γ(π), 〈X ,Y 〉 = [X ,Y ].
+There are two structure maps derived from 〈−,−〉 that encode the coherences of a Lie algebroid.
+The first map measures the extent to which the anchor maps fail to preserve the chosen connections
+on the vector bundle π : A → M and the tangent bundle p : T M → M .
+Definition 3.1.7. Let A be a Lie algebroid, and for a chosen horizontal connection ∇ on A and vertical
+connection κ′ on T M , set
+{v, x}κ′,∇ := T M p ×q A
+∇
+−→ T A
+T.̺
+−→ T 2M
+κ′
+−→ T M .
+When the choice of connection is evident by context, we will suppress the subscript.
+Observation 3.1.8. The above parentheses bracket corresponds to the symbol
+{−,−} = ̺ j
+β
+∂ ̺i
+α
+∂ x j .
+The Leibniz coherence may be rewritten as follows:
+Lemma 3.1.9. Let A be an anchored bundle with a bilinear bracket (inducing an involution σ). Choose
+connections (κ,∇),(κ′,∇′) on A and T M respectively. The bracket and anchor map satisfy the Leibniz
+law if and only if
+̺ ◦ 〈x, y 〉(κ,∇) + {̺x, y }(κ′,∇) = {̺y, x}(κ′,∇).
+Proof. The condition is equivalent to the identity
+̺ j
+α
+∂ ̺i
+β
+∂ x j = ̺i
+γC γ
+αβ + ̺ j
+β
+∂ ̺i
+α
+∂ x j .
+The Bianchi axiom measures the failure of the Jacobi identity in local coordinates, and states that
+it must be corrected for by the curvature of the brackets. We see that
+C ν
+αµC µ
+βγ = [eα,[eβ,eγ]]
+while
+∂ C ν
+βγ
+∂ x i eν = κ ◦ T (〈−,−〉(κ,∇)) ◦ ∇A2 ◦ ( ∂
+∂ x i ,eβ,eγ)
+53
+
+determines a trilinear map
+{ ∂
+∂ x i ,eβ,eγ}(κ,∇) := κ ◦ T (〈−,−〉(κ,∇)) ◦ ∇A2 ◦ ( ∂
+∂ x i ,eβ,eγ).
+This is the second derived map used in the structure equations for a Lie algebroid.
+Definition 3.1.10. Let (π : A → M ,ξ,λ,̺) be an anchored bundle equipped with a bilinear map
+〈−,−〉 : A2 → A.
+The derived ternary bracket {−,−,−} : T M p ×πAπ×πA → A is defined as
+{v, x, y }(κ,∇) := T M p ×πAπ×πA
+∇[A2]
+−−−→ T A2
+T.〈−,−〉
+−−−−→ T A
+κ−→ A
+where ∇[A2] is the pairing (∇(π0,π1),∇(π0,π2)).
+Observation 3.1.11. The ternary bracket corresponds to the following symbol:
+{−,−,−}(κ,∇) := ̺i
+α
+∂ C ν
+βγ
+∂ x i .
+Lemma 3.1.12. Let A be an anchored bundle over M with a bilinear bracket and denote the induced
+involution by σ. Choose a pair of connections and write the derived maps 〈−,−〉,{−,−,−}. Then
+(i) the bracket is antisymmetric if and only if its globalization is; that is, 〈eα,eβ〉 + 〈eβ,eα〉 = 0;
+(ii) the bracket satisfies the Jacobi identity if and only if it is alternating in the last two arguments and
+0 =
+�
+γ∈Cy(3)
+〈xγ0,〈xγ1, xγ2〉〉 +
+�
+γ∈Cy(3)
+{̺xγ0, xγ1, xγ2}.
+Proof.
+(i) This is equivalent to C ν
+αβ + C ν
+βα = 0.
+(ii) This is equivalent to
+0 = C µ
+βγC ν
+αµ + C µ
+γαC ν
+βµ + C µ
+αβC ν
+γµ + ̺i
+α
+∂ C ν
+βγ
+∂ x i + ̺i
+β
+∂ C ν
+γα
+∂ x i + ̺i
+γ
+∂ C ν
+αβ
+∂ x i .
+Proposition 3.1.13. A Lie algebroid is exactly an anchored vector bundle (π : A → M ,ξ,λ,̺) with a
+bilinear, alternating map
+〈−,−〉 : A2 → A;
+〈x, y 〉 + 〈y, x〉 = 0
+so that for any connection (∇,κ) on A, the maps
+{v, x}(κ,∇) := κ′ ◦ T.̺ ◦ ∇(v, x),
+{v, x, y }(κ,∇) := κ ◦ T (〈−,−〉(κ,∇)) ◦ ∇A2 ◦ (v, x, y )
+(3.2)
+satisfy the equations
+(i) ̺ ◦ 〈x, y 〉 + {̺x, y }(κ′,∇) = {̺y, x}(κ′,∇),
+54
+
+(ii)
+�
+γ∈Cy(3)〈xγ0,〈xγ1, xγ2〉〉 +
+�
+γ∈Cy(3){̺xγ0, xγ1, xγ2}(κ,∇) = 0.
+There is also a local coordinates presentation of morphisms as in Section 2 of Martínez (2018). An
+anchored bundle morphism A → B is a Lie algebroid morphism whenever
+f β
+γ Aγ
+αδ + ̺i
+δ
+∂ f β
+α
+∂ x i = B β
+θσ f θ
+α f σ
+δ + ̺i
+α
+∂ f β
+δ
+∂ x i .
+The A and B arguments are understood as the brackets, so this condition can be rewritten as
+f β
+γ Aγ
+αδ = f ◦ 〈α,δ〉,
+B β
+θσ f θ
+α f σ
+δ = 〈f ◦ α, f ◦ δ〉.
+Set the following notation for maps between vector bundles with connection:
+f : A → B
+(κA,∇A,A,λA)
+(κB,∇B,B,λB)
+∇[f ] : A2 → B := κB ◦ T.f ◦ ∇A
+(3.3)
+The ̺ terms are understood to be the torsion, so that
+̺i
+δ
+∂ f β
+α
+∂ x i = κ ◦ T.f ◦ ∇(̺eδ,eα) = ∇[f ](̺eδ,eα),
+̺i
+α
+∂ f β
+δ
+∂ x i = κ ◦ T.f ◦ ∇(̺α,δ) = ∇[f ](̺eα,eδ),
+using the notation set up in Equation 3.3. The notion of a Lie algebroid morphism, then, has the fol-
+lowing presentation:
+Proposition 3.1.14 (Martínez (2018)). Let (π : A → M ,̺A,〈−,−〉A),(q : B → N ,̺B,〈−,−〉B) be a pair of
+Lie algebroids with chosen connections (κ−,∇−). An anchor-preserving vector bundle morphism f : A →
+B is a Lie algebroid morphism if and only if
+f ◦ 〈eα,eδ〉 + ∇[f ]◦ (̺eδ,eα) = 〈f ◦ α, f ◦ δ〉 + ∇[f ]◦ (̺eα,eδ).
+3.2
+Anchored bundles
+Anchored bundlesare toLie algebroidswhatreflexive graphsare togroupoids. Each theoryhas(mostly)
+the same structure, but while a reflexive graph is missing a groupoid’s composition operation, an an-
+chored bundle lacks a Lie algebroid’s bracket operation. This section reviews the basic theory of an-
+chored bundles and their prolongations (see Mackenzie (2005) for more details).
+Definition3.2.1. An anchored bundle in a tangent category isa differential bundle (A
+π−→ M ,ξ,λ) equipped
+with a linear morphism
+A
+T M
+M
+̺
+π
+p
+A morphism of anchored bundles is a linear bundle morphism (f ,v) that preserves the anchors
+T A
+T B
+A
+B
+A
+B
+T M
+T N
+λ
+l
+f
+T.f
+ρ
+̺
+f
+T.v
+The category of anchored bundles and anchored bundle morphisms in a tangent category � is written
+Anc(�), and a generic anchored bundle is written (A
+π−→ M ,ξ,λ,̺).
+55
+
+There are two pullbacks that are associated with every anchored bundle. These play the role of the
+spaces of composable arrows G2 := G t ×s G ,G3 := G t ×sG t ×sG for a reflexive graph s,t : G → M .
+Definition 3.2.2. Let (A
+π−→ M ,ξ,λ,̺) be an anchored bundle. Its first and second prolongations are
+given by the limits
+� (A)
+T A
+� 2(A)
+T 2A
+A
+T M
+T A
+T 2M
+A
+T M
+̺
+T.π
+⌟
+T 2.π
+T.̺
+T.π
+̺
+⌟
+(The notation for the fibre product is slightly non-standard, as it is not technically a pullback.) Through-
+out this chapter, it will always be assumed that the first and second prolongations of an anchored bundle
+exist (although no choice of prolongation is explicitly made).
+Remark 3.2.3. It is not strictly necessary that the prolongations of an anchored bundle exist; this con-
+dition is primarily a matter of convenience when discussing involution algebroids. Every result in this
+section, that does not explicitly mention prolongations, holds for an anchored bundle independently of
+their existence.
+Example 3.2.4.
+(i) For any object M , id : T M → T M is an anchor for the tangent differential bundle, and every
+f : M → N yields a morphism of anchored bundles. Moreover, for any anchored bundle over M the
+anchor is itself a morphism of anchored bundles (A,π,ξ,λ,̺) → (T M ,p,0,ℓ,id ). This induces a
+fully faithful functor:
+� �→ Anc(�).
+This inclusion has a left adjoint, which sends an anchored bundle (π : A → M ,ξ,λ,̺) to its base
+space M (the unit is the anchor map ̺), so that � is a reflective subcategory of Anc(�).
+(ii) For any differential bundle (A,π,ξ,λ), the map 0 ◦ π : A → T M is an anchor and every morphism
+f : A → B of differential bundles again yields a morphism of anchored bundles. The naturality of
+0 ensures that every differential bundle morphism will preserve this trivial anchor map, giving a
+fully faithful functor
+DBun(�) �→ Anc(�).
+This functor has a right adjoint that replaces the anchor map with the trivial anchor map
+(π : A → M ,ξ,λ,̺) �→ (π : A → M ,ξ,λ,0◦ π)
+where the counit is given by the natural idempotent e = ξ ◦ π : (A,λ) → (A,λ). It is trivial to check
+that the anchor map is preserved by the bundle morphism (e ,id ):
+̺ ◦ ξ ◦ π = 0◦ T.id ◦ π
+and so the following diagram commutes:
+T M
+T M
+A
+A
+0◦π
+p◦λ
+̺
+56
+
+This means that differential bundles are a coreflective subcategory of anchored bundles.
+(iii) Any reflexive graph (s,t : C → M ,e : M → C ) in a tangent category has an anchored bundle (when
+sufficient limits exist), constructed as
+C ∂
+T.C
+T.C
+e .T1
+T.e s
+(where e s : C → C = e ◦ s). Construct a lift on C ∂ :
+T.C ∂
+T 2C1
+T 2C1
+C ∂
+T.C1
+T.C1
+e .C1
+T.e s
+ℓ
+ℓ
+T.e .C1
+T.T.e s
+λ
+This lift will be non-singular by the commutativity of T -limits. The pre-differential bundle data is
+given by the projection
+C ∂ �→ T.C
+p.e s
+−−→ M .
+The section is induced by
+M
+(0.e )
+−−→ T.C
+while post-composition with T.t gives the anchor map:
+C ∂ �→ T.C
+T.t
+−→ T.C .
+The diagram is a pullback by composition, and the outer perimeter defines the pullback � (A). Note
+that any reflexive graph morphism will give rise to an anchored bundle morphism by naturality,
+making the construction of an anchored bundle from a reflexive graph functorial.
+(iv) For any object M in a tangent category, cM : T 2M → T 2M is an anchor on (T.p,T.0,c ◦ T.ℓ), and
+every map f : M → N gives a morphism of anchored bundles.
+(v) For any anchored bundle (q : E → M ,ξ,λ,̺), the differential bundle (T.q,T.ξ,
+c ◦ T.λ) has an anchor
+̺T : T E
+T.̺
+−→ T 2M
+c−→ T 2M .
+The first prolongation of an anchored bundle � (A) behaves similarly to the second tangent bundle,
+except that it does not have a canonical flip. In the definition of an affine connection, the tangent
+bundle played a similar role to the “arities” of a theory. There is a lift map that makes this connection
+stronger:
+Definition 3.2.5. Let (π : A → M ,ξ,λ,̺) be an anchored bundle in a tangent category �. We define a
+generalized lift:
+ˆλ : A → A̺×T πT A := (ξ ◦ π,λ).
+This generalized lift satisfies the same coherences as the lift on the second tangent bundle:
+Proposition 3.2.6. Let (π : A → M ,ξ,λ,̺) be an anchored bundle in a tangent category �. It follows that
+(i) (Coassociativity of ˆλ) (ˆλ × ℓ) ◦ ˆλ = (id × T.ˆλ) ◦ ˆλ;
+57
+
+(ii) (Universality) the following diagram is a T -pullback:
+A2
+A̺×T πT A
+M
+A
+ˆµ
+π0
+π◦πi
+ξ
+⌟
+where ˆµ := (ξ ◦ π◦ π0,µ).
+Proof.
+(i) Compute
+(ˆλ × ℓ) ◦ ˆλ = (ξ ◦ π◦ ξ ◦ π,λ ◦ ξ ◦ π,ℓ◦ λ)
+= (ξ ◦ π,T.(ξ ◦ π) ◦ λ,T.λ ◦ λ)
+= (π0,T.(ξ ◦ π) ◦ π1,T.λ ◦ π1) ◦ (ξ ◦ π,λ)
+= (id × T (ˆλ)) ◦ ˆλ.
+(ii) Use the pullback lemma to observe that the following diagram is universal for any anchored bun-
+dle:
+A2
+� (A)
+T A
+M
+A
+T M
+ξ
+̺
+0
+T π
+π0
+π◦π1
+ˆµ
+π1
+µ
+The top triangle of the diagram commutes by definition. The right square and outer perimeter
+are pullbacks by definition, and the bottom triangle also commutes by definition. The pullback
+lemma ensures that the left square is a pullback, so for every anchored bundle, the general lift is
+universal for � (A). Now post-compose with the involution:
+A2
+� (A)
+� (A)
+M
+A
+A
+ξ
+π0
+π◦π1
+ˆµ
+id
+pπ1
+σ
+ˆν
+It suffices to check that the top triangle commutes, so σ ◦ ˆµ = ν:
+σ ◦ ˆµ ◦ (a,b ) = σ ◦ ((ξ ◦ π,0) ◦ a +π0 (ξ ◦ π,λ) ◦ b ) = (id ,T.ξ ◦ ̺ ◦ a) +pπ1 (ξ ◦ π,λ) ◦ b.
+Thus, the lift (ξπ,λ) involution algebroid is universal for � (A).
+Example 3.2.7.
+(i) For T (M ) = T (M ), the space of prolongations is T (M )id ×T p T 2M = T 2M , and the second prolon-
+gation is given by T 2M .
+58
+
+(ii) In a tangent category with a tangent display system, if π ∈ � then the prolongations for (π,ξ,λ,̺)
+automatically exist. In particular, for every anchored bundle in the category of smooth manifolds,
+all prolongations exist because the projection is a submersion (see Section 1.5).
+(iii) For any differential bundle with the anchor 0 ◦ π, it follows that � (A) ∼= Aπ×π◦πi (A2) ∼= A3. The
+universality of the vertical lift factors b into (a1,a2), as the following diagram is a pullback:
+A3
+Aπ×πA
+T A
+A
+M
+T M
+π0
+(π1,π2)
+µ
+ππi
+T π
+π
+0
+(iv) Returning to the anchored bundle constructed from a graph, the space of prolongations � (A) em-
+beds into the second tangent bundle of the space of composable arrows:
+A̺×T πT A �→ T 2(G t ×sG ).
+The category of anchored bundles is, in a sense, “tangent monadic” over the category of differential
+bundles: the forgetful functor from anchored bundles to differential bundles “creates” T -limits and
+the tangent structure (this is all made precise in Chapter 5 using an enriched perspective on tangent
+categories).
+Observation 3.2.8. A limit of anchored bundles is the limit of the underlying differential bundles. Be-
+cause the anchor is preserved by every map in the diagram, this induces a natural transformation for
+any D : � → Anc(C ):
+�
+Anc(�)
+Anc
+DBun(�)
+D
+S
+U
+̺
+Thus, limU .S.D computes the limit of the underlying objects, while limU .D computes the limit of the
+underlying differential bundles in the diagram. The anchor/unit map, then, induces a differential bun-
+dle map:
+lim̺i : (limAi ,limλi ) → (T.(lim
+i Mi),ℓ.(lim
+i Mi)).
+This is the limit in the category of anchored bundles (so long as pullback powers of the limit projection
+and the two prolongations of the limit anchor bundle exist).
+The tangent structure on lifts defined in Proposition 2.2.7 lifts to a tangent structure on anchored
+bundles.
+Lemma 3.2.9. The category of anchored bundles in a tangent category has a tangent structure that maps
+objects as follows:
+(π : A → M ,ξ,λ,̺) �→ (T.π : T A → T M ,T.ξ,c ◦ T.λ,c ◦ T.̺)
+where the structure maps are all defined using the pointwise structure maps in �.
+59
+
+Proof. Given an anchored bundle (q : A → M ,ξ,λ,̺), there is an anchor on the differential bundle
+(T q,T ξ,c ◦ T λ) given by c ◦ T ̺; the diagram commutes by naturality and the coherences on c ,ℓ.
+T 2A
+T 3M
+T 3M
+T A
+T 2M
+T 2M
+T M
+T M
+T M
+T q
+T p
+c ◦T λ
+c ◦T ℓ
+ℓ
+c
+p
+T ̺
+T 2̺
+T c
+The tangent structure maps and universality properties all follow from the forgetful property of the
+functor from anchored bundles to differential bundles as a consequence of Observation 3.2.8.
+For any anchored bundle A, there are two differential bundles associated to � (A). The first is the
+usual pullback differential bundle given by pulling back T.π along ̺ as in Lemma 2.4.3:
+A̺×T πT A
+T A
+A
+T M
+π1
+π0
+T π
+̺
+This gives the differential bundle structure
+(A̺×T πT A
+π0
+−→ A,A
+(id,T.ξ◦̺)
+−−−−−−→ A̺×T πT A,A̺×T πT A
+(0,c ◦T.λ)
+−−−−−→ T (A̺×T πT A)).
+Taking the fibre product in the category of anchored bundles as in Observation 3.2.8 yields the second
+differential bundle structure:
+(A,π,ξ,λ)
+(̺,id)
+−−−→ (T M ,p,0,ℓ)
+(T π,π)
+←−−− (T A,p,0,ℓ).
+The two lifts behave similarly to the pair (T.ℓ,ℓ.T ) on the second tangent bundle, and (ℓ,λT ) on the
+tangent bundle of a pre-differential bundle.
+Lemma 3.2.10. Let � be a tangent category and Anc(�) the category of anchored bundles in �. If T -
+pullback powers of p ◦π0,π1 : A̺×T πT A → A exist, then there are two differential bundles on � (A), with
+lifts induced as
+(� (A),λ × ℓ)
+(T A,ℓ)
+(� (A),0× (c ◦ T.λ))
+(T A,c ◦ T.λ)
+(A,λ)
+(T M ,ℓ)
+(A,0)
+(T M ,0)
+̺
+T.π
+⌟
+T.π
+̺
+⌟
+with structure maps
+1. (� (A)
+p◦π1
+−−→ A,A
+(ξπ,0)
+−−−→ � (A),� (A)
+λ×ℓ
+−−→ T ◦ � (A)),
+2. (� (A)
+π0
+−→ A,A
+(id,T.ξ◦̺)
+−−−−−−→ � (A),� (A)
+0×c ◦T.λ
+−−−−→ T.� (A)).
+Furthermore, the two lifts λ� and λ̺ commute:
+c ◦ T.(λ × ℓ) ◦ (0× c ◦ T.λ) = T.(0× c ◦ T.λ) ◦ (λ × ℓ).
+60
+
+Proof. The two differential bundles exist as a consequence of Observation 2.4.2 and Lemma 2.4.3, re-
+spectively. The commutativity of limits follows by postcomposition, as both c ◦T.λ,ℓ and λ,0 commute
+by the differential bundle axioms.
+The above lemma determines a functor, which we denote � , that sends an anchored bundle in �
+to an anchored bundle in Lift(�) (equipped with the tangent structure from Proposition 2.2.7).
+Proposition 3.2.11. There is a functor � from anchored bundles in � to anchored bundles in the cate-
+gory of lifts Lift(�), that sends an anchored bundle (π : A → M ,ξ,λ,̺) to the tuple in Lift(�)
+π� := (� (A),0× c ◦ T.λ)
+p◦π1
+−−→ (A,0),
+ξ� := (A,0)
+(ξ◦π,0)
+−−−→ (� (A),0× c ◦ T.λ)
+λ� := (� (A),0× c ◦ T.λ)
+(λ×ℓ)
+−−→ (T (� (A)),c ◦ T (0× c ◦ T.λ),
+̺� := (� (A),0× c ◦ T.λ)
+π1
+−→ (T A,c ◦ T.λ)
+(note that this functor lands in anchored bundles of non-singular lifts; see Definition 2.3.1). Morphisms
+of anchored bundles
+(f ,m) : (π : A → M ,ξ,λ,̺) → (q : E → N ,ζ,l ,δ)
+are sent to
+� (f ,m) := f Tm ×Tm T f : A̺×T πT A → E δ×T q T E
+Proof. First, note that the tuple
+(π� : � (A) → A,ξ� ,λ� ,̺� )
+is an anchored bundle, and that each morphism is a lift morphism:
+• π� : (� (A),0× c ◦ T.λ) → (A,0) follows because
+T.p ◦ T.π1 ◦ (λ × ℓ) = T.p ◦ ℓ◦ π1 = 0◦ p ◦ π1
+• ξ� : (A,0) → (� (A),0× c ◦ T.λ) follows since
+(T.ξ ◦ T.π,T.0) ◦ 0 = (0◦ ξ ◦ π,T.0◦ 0) = (0◦ ξ ◦ π,c ◦ T.λ ◦ 0) = (0× c ◦ T.λ) ◦ (ξ ◦ π,0)
+• λ� : (� (A),0× c ◦ T.λ) → (T.� (A),0× c ◦ T.(c ◦ T.λ)) follows by the commutativity of 0× c ◦ T.λ)
+and λ × ℓ, and
+• ̺� : (� (A),0× c ◦ T.λ) → (T A,c ◦ T.λ) is a lift by definition of ̺� = π1.
+This gives an anchored bundle in the category of non-singular lifts in �.
+Next, check that the mapping is functorial. To see that � (f ,m) preserves the lifts, note that (f ,m)
+gives a morphism of diagrams for each pullback in Lift(�) defining the two lifts, so the induced map
+� (f ,m) preserves each lift. To see that � (f ,m) preserves the anchor, check that
+̺� B ◦ � (f ,m) = π1 ◦ (f × T.f ) = T.f ◦ π1 = T.f ◦ ̺� A.
+61
+
+Corollary 3.2.12. In a tangent category � where pullbacks along differential bundle projections exist,
+such as a tangent category equipped with a proper retractive display system (Definition 1.5.3) like the
+category of smooth manifolds, there is an endofunctor on the category of anchored bundles in �,
+� ′ : Anc(�) → Anc(�) =U Lift.� .
+Observation 3.2.13. T -Limits in Anc(Lift(�)) are computed as pointwise limits in � by Observations
+3.2.8 and 2.4.2, and � is constructed as a limit, so it follows that � preserves T -limits.
+Proposition 3.2.14. The prolongation endofunctor � ′ : Anc(�) → Anc(�) has natural transformations
+• p ′ : � ′ ⇒ id
+• 0′ : id ⇒ � ′
+• +′ : � ′p ′×p ′� ′ ⇒ � ′
+• ℓ′ : � ′ ⇒ � ′.� ′
+satisfying all of the axioms of a tangent category that do not incvolve the canonical flip (Definition 1.3.2).
+(Sketch). Note that the full argument is given in Section 4.3, but it is not difficult to sketch it out here.
+For the projection, zero map, and addition, use the differential bundle structure induced by Corollary
+3.2.11. To see the lift axiom, note that up to a choice of pullback, we have:
+� ′.� ′(π : A → M ,ξ,λ,̺) =
+
+
+
+
+
+
+
+
+
+(p ◦ π1,p ◦ π2) :
+� 2(A) → � (A)
+(ξ ◦ π◦ π0,0◦ π1,0◦ π2) :
+� (A) → � 2(A)
+(λ × ℓ× ℓ) :
+� 2(A) → T.� 2(A)
+(π1,π2) :
+� 2(A) → T.� (A)
+The “lift map” is then
+ℓ′ : � ′ ⇒ � ′.� ′; � (A)
+A×T.ˆλ
+−−−→ � 2(A)
+(where we recall that ˆλ = (ξ ◦ π,λ) : A → � (A)).
+Remark 3.2.15. The structure described in Proposition 3.2.11 leads to the theory of double vector bun-
+dles, developed by MacKenzie and his collaborators F. and Mackenzie (2019); Mackenzie (1992). A dou-
+ble vector bundle is a commuting square
+D
+B
+A
+M
+q D
+A
+q D
+B
+q A
+q B
+where each projection is a vector bundle projection, and “vertical” and “horizontal” orientations of the
+square are each vector bundle homomorphisms. It was observed by Grabowski and Rotkiewicz (2009)
+that this is equivalent to a pair of commuting �+-actions on the total space E ; following the development
+in Chapter 2, this is a commuting pair of non-singular lifts on E ,
+λA,λB : E → T E , T.λB ◦ λA = c ◦ T.λA ◦ λB.
+A proper exposition of the so-called “Ehresmann doubles” (Mackenzie (2011)) for the structures in Lie
+theory would substantially expand the scope of this thesis, and so it has been relegated to the margins.
+62
+
+Finally, observe that a connection (see Section 2.6) on an anchored bundle’s underlying differential
+bundle behaves similarly to an affine connection.
+Lemma 3.2.16. Let (π : A → M ,ξ,λ,̺) be an anchored bundle with � (A) existing in a tangent category
+�. If (π,ξ,λ) has a connection, then define
+ˆκ := κπ1,
+ˆ∇ := ∇(π0,̺π1)
+and note that
+(i) ˆκ : � (A) → A is a retract of (ξπ,λ), and ˆκ : (� (A),l ) → (A,λ) is linear for both l = (λ×ℓ),(0×c ◦T.λ);
+(ii) ˆ∇ : A2 → � (A) is a section of (π0,p ◦ π1) and is bilinear;
+(iii) there is an isomorphism � (A) ∼= A3:
+∇(p ◦ π1,̺ ◦ π0) +(π,T.π) ˆµ(p ◦ π1, ˆκ) = id .
+Example 3.2.17. Every vector bundle in the category of smooth manifolds has a connection; it follows
+that Lemma 3.2.16 holds for every anchored bundle in the category of smooth manifolds.
+Remark 3.2.18. The category of anchored bundles in a tangent category is almost a tangent category,
+except that it lacks a symmetry map. The “differential objects” in such a category will act like a cartesian
+differential category, except that the symmetry of mixed partial derivatives fails. There has been some
+interest in settings for differentiable programming where the symmetry of mixed partial derivatives need
+not hold (see Definition 3.4 along with the discussion at the end of Section 6 in Cruttwell et al. (2021));
+the category of anchored bundles in a tangent category appears to be a source of examples.
+3.3
+Involution algebroids
+Involution algebroids are a tangent-categorical axiomatization of Lie algebroids. Recall that a Lie al-
+gebroid has almost all of the structure of the operational tangent bundle, in particular a Lie bracket
+on sections that satisfies the Leibniz law so that it gives a directional derivative. In Proposition 3.2.14,
+it was demonstrated that the category of anchored bundles have almost all of the same maps as the
+tangent bundle, the only structure map missing being the canonical flip
+c : T 2M ⇒ T 2M
+which, we may recall from the chapter introduction (and Cockett and Cruttwell (2015), Mackenzie
+(2013)), is used in constructing the Lie bracket of vector fields in a tangent category with negatives.
+Thus, a natural next step is to add an involution map to an anchored bundle and require that it satis-
+fies the same coherences as c from the tangent bundle.
+Definition 3.3.1. An involution algebroid is an anchored bundle (A,π,ξ,λ,̺) equipped with a map
+σ : � (A) → � (A) satisfying the following axioms:
+(i) Involution:
+� (A)
+� (A)
+� (A)
+σ
+σ
+63
+
+(ii) Double linearity:
+T.� (A)
+T.� (A)
+� (A)
+� (A)
+σ
+0×c ◦T.λ
+T.σ
+λ×ℓ
+(iii) Symmetry of lift:
+A
+� (A)
+� (A)
+ˆλ=(ξ◦π,λ)
+σ
+ˆλ
+(iv) Target:
+� (A)
+T A
+T 2M
+� (A)
+T A
+T 2M
+σ
+π1
+T.̺
+π1
+T.̺
+c
+(v) Yang–Baxter:
+� 2(A)
+� 2(A)
+� 2(A)
+� 2(A)
+� 2(A)
+� 2(A)
+σ×c
+σ×c
+σ×c
+A×T.σ
+A×T.σ
+A×T.σ
+A is an almost-involution algebroid if the involution does not satisfy the Yang–Baxter equation. A mor-
+phism of involution algebroids is a morphism of anchored bundles, so that � (f )◦σA = σB ◦� (f ). (Note
+that because σ is an isomorphism and σ = σ−1,
+σ : (� (A),0× c ◦ T.λ) → (� (A),λ × ℓ)
+is linear as well.) Write the category of involution algebroids and involution algebroid morphisms in �
+is written Inv(�).
+Observation 3.3.2. It is not immediately clear that the Yang–Baxter equation is well-typed. This follows
+from the target axiom (iv) and the double linearity axiom (ii). Starting with (u,v,w ) : A̺×T πT AT.̺×T 2.π
+T 2A, we see that σ × c is well-typed if and only if
+T.̺ ◦ π1 ◦ σ(u,v) = T 2.π◦ c ◦ w = c ◦ T 2.π◦ w = c ◦ T.̺ ◦ v.
+Similarly, 1 × T.σ is well-typed if T.̺ ◦ u = T.π ◦ π0 ◦ T.σ(v,w ); then use the double linearity axiom to
+compute
+T.π◦ π0 ◦ T.σ(v,w ) = T.π◦ T.p ◦ T.π1(v,w ) = T.π◦ T.p ◦ w
+= T.p ◦ T 2.π◦ w = T.p ◦ T.̺ ◦ v = T.π◦ v.
+64
+
+This perspective on Lie algebroids has already appearad in the work of Martinez and his collabora-
+tors in de León et al. (2005), where a “canonical involution” was derived on space of prolongations of a
+Lie algebroid using the formula
+σ : � (A) → � (A);σ(x, y,z) = σ(y, x,z + 〈x, y 〉).
+The structure of this map has been largely unexplored; helpfully, involution algebroids succeed in
+reverse-engineering axioms for an involution map that will induce a Lie bracket on the sections of
+the projection map. The bracket from the original Lie algebroid is induced using the same formula as
+for the bracket of vector fields on the tangent bundle:
+λ ◦ [X ,Y ]∗ =
+�
+(π1 ◦ σ ◦ (id ,T.X ◦ ̺) ◦ Y −p T.Y ◦ X ◦ ̺) −T.π 0Y
+�
+.
+Furthermore, a morphism of anchored bundles is a Lie algebroid morphism if and only if it preserves
+the derived involution map.
+Example 3.3.3.
+(i) For any M in �, (T M ,p,0,id ,c ) is an involution algebroid. Furthermore, for any involution alge-
+broid anchored on M , (̺,id ) is a morphism of involution algebroids (by the target axiom). This
+defines a fully faithful functor � �→ Inv(�). The same construction as the anchored bundle case in
+Example 3.2.4(i) exhibits � as a reflective subcategory of Inv(�).
+(ii) For any differential bundle, the trivial anchored bundle (π : A → M ,ξ,λ,0◦ π) has an involution
+using the isomorphism � (A) ∼= A3, given by σ := (π1,π0,π2) (proving this map satisfies the involu-
+tion algebroid axioms is just an exercise in combinatorics). It follows that every differential bundle
+morphism gives rise to a morphism of these trivial involution algebroids. The same construction
+from the anchored bundle case (Example 3.2.4(ii)) exhibits Diff(�) as a coreflective subcategory of
+involution algebroids.
+(iii) Consider a groupoid
+s,t : G → M , e : M → G , (−)−1 : G → G , m : G2 → G .
+The underlying reflexive graph has an associated anchored bundle, constructed in Example 3.2.4,
+and the space of prolongations of this anchored bundle includes into (T 2.G2). Note that there is a
+well-formed involution map:
+σ(u,v) = c ◦ ((0◦ p ◦ v)−1;(T.0◦ u);v)
+The direct proof of this involves a more conceptual construction, which is the focus of Section 5.5.
+An involution algebroid resembles a generalized tangent bundle, and so the lift (ξ◦π,λ) : A → � (A)
+satisfies the same universality conditions as ℓ : T ⇒ T 2. The double linearity condition is equivalent to
+the naturality condition for c ,ℓ in Definition 1.3.2:
+� (A)
+� (A)
+T 2
+T 2
+� 2(A)
+� 2(A)
+� 2(A)
+T 3
+T 3
+T 3
+T.ℓ
+c .T
+T.c
+c
+ℓ.T
+id×T.ˆλ
+σ
+σ×c
+id×T.σ
+ˆλ×ℓ
+(3.4)
+65
+
+which, using string diagrams for monoidal categories (Selinger (2010)), is the equation
+=
+where the circle denotes the lift and c is the crossing of two lines.
+Proposition 3.3.4. Let (π : A → M ,ξ,λ,̺) be an anchored bundle, and suppose that
+σ : � (A) → � (A)
+satisfies the involution axiom (i) and the symmetry of lift axiom (iii),and furthermore that the involution
+“exchanges” the idempotents associated to the two lifts (λ×ℓ) and (0×c ◦T.λ) on � (A) (Proposition 2.2.8)
+σ ◦ ((p ◦ λ) × (p ◦ ℓ)) = (p ◦ 0) × (p ◦ c ◦ T.λ) = id × (T.p ◦ T.λ)
+Then the double linearity axiom is equivalent to the left-hand commuting diagram in Diagram 3.4:
+(ˆλ × ℓ) ◦ σ = (id × T.σ) ◦ (σ × c ) ◦ (id × T.ˆλ).
+Proof. Starting with the left-hand side of the equation,
+(id × T.σ) ◦ (σ × c ) ◦ (id × T.ˆλ) ◦ (u,v)
+= (id × T.σ) ◦ (σ × c ) ◦ (u,T.ξ ◦ T.π◦ v,T.λv)
+= (id × T.σ) ◦ (σ ◦ (u,T.ξ ◦ ̺ ◦ u),T.λv)
+= (id × T.σ) ◦ (ξ ◦ π◦ u,0◦ ̺ ◦ u,T.λv)
+= (id × T.σ) ◦ (ξ ◦ π◦ u,0◦ T.π◦ v,T.λv)
+= (ξ ◦ π◦ u,T.σ ◦ ˆλ◦ v).
+For the right-hand side:
+(ˆλ,ℓ) ◦ σ ◦ (u,v)
+= (ξ ◦ π◦ π0 ◦ σ ◦ (u,v),(λ × ℓ) ◦ σ ◦ (u,v))
+= (ξ ◦ π◦ p ◦ v,(λ × ℓ) ◦ σ ◦ (u,v))
+= (ξ ◦ p ◦ T.π◦ v,(λ × ℓ) ◦ σ ◦ (u,v))
+= (ξ ◦ p ◦ ̺ ◦ u,(λ × ℓ) ◦ σ ◦ (u,v))
+= (ξ ◦ π◦ u,(λ × ℓ) ◦ σ ◦ (u,v))
+= (ξ ◦ π◦ u,T.σ ◦ (0× c ◦ T.λ) ◦ (u,v)).
+So the naturality equation 3.4 is equivalent to
+T.σ ◦ (0× c ◦ T.λ) = (λ × ℓ) ◦ σ.
+The category of involution algebroids is “tangent monadic” over the category of anchored bundles,
+in the same sense that anchored bundles are tangent monadic over the category of differential bundles,
+or internal categories over reflexive graphs in a category. The “tangent monadicity” leads to a similar
+observation about T -limits of involution algebroids as in Observation 3.2.8.
+66
+
+Observation 3.3.5. The forgetful functor from involution algebroids to anchored bundles creates limits;
+that is to say, the T -limits of the underlying anchored bundles give the limits of involution algebroids.
+Recall that by Observation 3.2.13, the limit � (limAi ) = lim� (Ai ) in the category of anchored bundles,
+and this induces a map between objects in �
+limσi : lim� (Ai ) → lim� (Ai ).
+The axioms for an involution algebroid are induced by universality.
+Note that a point-wise tangent structure may be defined following the anchored bundles example
+from Lemma 3.2.9:
+Proposition 3.3.6. For a tangent category �, the category of involution algebroids has a “point-wise”
+tangent structure that maps objects as follows:
+(π : A → M ,ξ,λ,̺,σ) �→ (T.π,T.ξ,c ◦ T.λ,T.̺,σT )
+where σT is defined as
+σT := (1×c c ) ◦ T.σ ◦ (1×c c ) : � (AT ) → � (AT ).
+(3.5)
+Proof. The tangent structure on involution algebroids is inherited from the functor Inv(�) → Anc(�).
+Thus, it suffices to give the involution map for the tangent involution algebroid.
+Note that given (π : A → M ,ξ,λ,̺,σ), the space of prolongations on (T.π,T.ξ,
+c ◦T.λ,c ◦T.̺) is T Ac ◦T.̺×T 2πT 2A. We can construct an isomorphism between the objects � (T A) and
+T (� (A)) in � using the cospan isomorphism
+T A
+T 2M
+T 2A
+T A
+T 2M
+T 2A
+c ◦T.̺
+T 2.π
+T.̺
+T 2.π
+id
+c
+c
+thus inducing a map
+T Ac ◦T.̺×T 2πT 2A
+id×c
+−−→ T (A̺×T πT A)
+T.σ
+−→ T (A̺×T πT A)
+id×c
+−−→ T Ac ◦T.̺×T 2πT 2A
+which we call σT . The linearity and involution axioms follow by construction.
+Now check the rest of the axioms. For the unit:
+(1×c c ) ◦ T.σ ◦ (1×c c ) ◦ (T.(ξ ◦ π),c ◦ T.λ)
+= (1×c c ) ◦ T.σ ◦ (T (ξπ),T λ) = (1×c c ) ◦ (T.(ξ ◦ π),T.λ) = (T.(ξ ◦ π),c ◦ T.λ).
+For the anchor:
+T.̺T ◦ π1 ◦ σT = T (c ◦ T ̺) ◦ π1 ◦ (1×c c ) ◦ T.σ ◦ (1×c c )
+= T.c ◦ c ◦ T 2.̺ ◦ T.π1 ◦ T.σ ◦ (1×c c ) = T.c ◦ c ◦ T.c ◦ T 2.̺ ◦ T.π1 ◦ (1×c c )
+= c ◦ T.c ◦ c ◦ T 2̺ ◦ c ◦ π1 = c ◦ T c ◦ T 2.̺ ◦ π1 = c ◦ T.̺T ◦ π1.
+The Yang–Baxter equation is straightforward to check.
+The tangent bundle is a canonical involution algebroid on every object in a tangent category, and
+the anchor induces a morphism from an involution algebroid to the tangent involution algebroid on
+its base space. The anchor acts as a reflector from involution algebroids in � to � itself.
+67
+
+Proposition 3.3.7. Any tangent category � is a reflective subcategory of the category of involution alge-
+broids in �.
+Proof. First, observe that the inclusion of � into Inv(�) (Definition 3.3.1) is fully faithful because the
+anchor on the tangent involution algebroid is id : T M → T M . This means that the only involution
+algebroid morphisms T A → T B are pairs (T f , f ), f : A → B. Now consider the functor Inv(�) → � that
+sends (π : A → M ,ξ,λ,̺,σ) to M : this gives an endofunctor S : Inv(�) → Inv(�) along with a natural
+transformation ̺ : id ⇒ S, so that S.̺ = ̺.S = id , given by the anchor map. Thus, the category � is
+the category of algebras for an idempotent monad on Inv(�).
+Corollary 3.3.8. Let
+�A = (π : A → M ,ξ,λ,̺,σ)
+be an involution algebroid in a tangent category �.
+(i) The morphism
+(T.π,π) : T A → T M
+is an involution algebroid morphism from the tangent involution algebroid on A to the tangent
+involution algebroid on M .
+(ii) The morphism
+(̺,id ) : �A → T M
+is an involution algebroid morphism.
+If pullback powers of p ◦ π1 : � (A) → A exist, then
+(p ◦ π1 : � (A) → A,(ξ ◦ π,0),(λ × ℓ))
+is a differential bundle; note that π0 acts as an anchor. If the prolongations exist, then
+(p ◦ π1 : � (A) → A,(ξ ◦ π,0),(λ × ℓ),π0,σ′)
+is an involution algebroid; this follows from computing the pullback in the category of involution alge-
+broids (σ′ is induced as in Observation 3.2.8).
+The above corollary puts an involution algebroid structure on the differential bundle (� (A),λ × ℓ).
+Note that the map σ gives an isomorphism of differential bundles
+(� (A),λ × ℓ) → (� (A),0× c ◦ T.λ).
+Martinez observed that the canonical involution σ puts a unique Lie algebroid structure on (� (A),0×
+c ◦T.λ), and σ isa uniquelydetermined isomorphism ofinvolution algebroids(see de León et al. (2005)):
+Corollary 3.3.9. For an involution algebroid (π : A → M ,ξ,λ,̺,σ), the isomorphism of differential
+bundles
+σ : (� (A),0× c ◦ T.λ) → (� (A),λ × ℓ)
+induces a second involution algebroid structure on � (A).
+68
+
+Recall that Proposition 3.2.14 sketched out a proof that the category of anchored bundles in � has
+an endofunctor � ′ and natural transformations p ′,0′,+′,ℓ′ satisfying the axioms of a tangent structure;
+this endofunctor and the natural transformations all lift to Inv(�). The involution map σ is the missing
+piece that gives a tangent structure on Inv(�). The construction may be spelled out here at a big-picture
+level, but the actual proof brings up tricky coherence issues that make up the bulk of Chapter 4.
+Proposition 3.3.10. The category of involution algebroids in a tangent category � has a second tangent
+structure, where the tangent functor is given by
+� ′ : Inv(�) → Inv(�)
+and the tangent natural transformations are given as in Proposition 3.2.14, with the canonical flip
+σ′ : � ′.� ′ → � ′.� ′ := � 2(A)
+1×T.σ
+−−−→ � 2(A).
+Starting with an involution algebroid, � ′(A) and � ′.� ′(A) are given by
+� ′(A)
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+π′ :
+� (A)
+p◦π1
+−−→ A
+ξ′ :
+A
+(ξ◦π,0)
+−−−→ � (A)
+λ′ :
+� (A)
+λ×ℓ
+−−→ T.� (A)
+̺′ :
+� (A)
+π1
+−→ T A
+σ′ :
+� 2(A)
+σ×c
+−−→ � 2(A)
+� ′.� ′(A)
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+π′′
+� 2(A)
+(p◦π1,p◦π2):
+−−−−−−−→ � (A)
+ξ′′ :
+� (A)
+(ξ◦π◦π0,0◦π1,0◦π2)
+−−−−−−−−−−−→ � 2(A)
+λ′′ :
+� 2(A)
+(λ×ℓ×ℓ)
+−−−−→ T.� 2(A)
+̺′′ :
+� 2(A)
+(π1,π2)
+−−−→ T.� (A)
+σ′′ :
+� 3(A)
+(σ×c ×c )
+−−−−−→ � 3(A)
+(where � 3(A) = A̺×T πT AT.̺×T 2.πT.(A̺×T πT A)). The tangent natural transformations are given by
+• p : � ′ ⇒ id ; � (A)
+π0
+−→ A
+• 0 : id ⇒ � ′; A
+(id,T.ξ◦T.π)
+−−−−−−−→ � ′(A)
+• + : � ′
+2 ⇒ � ′; A̺×T.π◦πi T2A
+id×+
+−−−→ � (A)
+• ℓ : � ′(A) ⇒ � ′.� ′; � (A)
+1×T.(ξ◦π,λ)
+−−−−−−→ � 2(A)
+• ̺′ : � ′ ⇒ T
+• c : � ′.� ′(A) ⇒ � ′.� ′; � 2(A)
+1×T.σ
+−−−→ � 2(A).
+Proof. Deferred to Section 4.5.
+3.4
+Connections on an involution algebroid
+In this section we take an involution algebroid with a chosen linear connection on its underlying an-
+chored bundle (Definition 3.3.1, 2.6.1), and rederive Martinez’s structure equations for a Lie algebroid
+(Proposition 3.1.5).
+In a tangent category � with negatives, there is a natural transformation
+n : T ⇒ T
+69
+
+making each fibred commutative monoid (p : T M ⇒ M ,0,+,n) a fibred abelian group. Because the
+additive bundle structure on differential bundles is induced via universality (Proposition 2.4.4), in a
+tangent category with negatives the additive bundle structure for differential bundles will likewise have
+negatives. We adopt the following notation for the “fibred linear algebra” used in this section, as there
+are a significant number of computations done on bundles with multiple choices of additive bundle
+structure (e.g. the second tangent bundle of a differential bundle has three additive bundles structures).
+Notation 3.4.1. Let E be an object with multiple differential bundle structures (πi : E → M i ,ξi,λi)
+in a tangent category with negatives. Addition over a specific differential bundle is written using infix
+notation, where a subscript is added to the symbol denoting the projection of the differential bundle.
+Letting x, y : X → E denote a pair of generalized elements for which the πi -addition operation is well-
+defined; we set
+x +π[i] y = X
+(x,y )
+−−→ E π[i]×π[i]E
++i
+−→ E .
+Similar notation is used for subtraction:
+x −π[i] y = X
+(x,y )
+−−→ E π[i]×π[i]E
+id×n[i]
+−−−−→ E π[i]×π[i]E
++i
+−→ E .
+Throughout this section, we work in a tangent category � with negatives and a chosen anchored
+bundle (π : A → M ,ξ,λ,̺), equipped with a connection (κ,∇), whose base object has a torsion-free
+connection (κ′,∇′) and a morphism
+σ : � (A) → � (A)
+that exchanges the two projection maps p ◦ π1,π0, meaning that the following diagram commutes:
+� (A)
+A
+A
+� (A)
+π0
+p◦π1
+σ
+p◦π1
+π0
+The following notation will be useful when working with local coordinates.
+Notation 3.4.2. First, recall the ∇-notation for morphisms of differential bundle where each morphism
+has a choice of connection from Equation 3.3:
+f : A → B
+(κA,∇A,A,λA)
+(κB,∇B,B,λB)
+∇[f ] := κB ◦ T.f ◦ ∇A : A2 → B
+Let
+(π : A → M ,ξ,λ,̺),(q : B → N ,ζ,l ,ρ)
+be a pair of anchored bundles equipped with connections. “Hatting” a map f : � (A) → � (B) refers to
+the map:
+�f : A3
+ˆν(π0,π2)+ ˆ∇(π0,π1)
+−−−−−−−−−−→ � (A)
+f−→ � (B)
+(π0,p◦π1,κ◦π1)
+−−−−−−−−→ B3.
+Similarly, for f : T A → T B,
+�f : T M p ×πAπ×πA
+ν(π0,π2)+∇(π0,π1)
+−−−−−−−−−−→ T A
+f−→ T B
+(π0,p◦π1,κ◦π1)
+−−−−−−−−→ T M p ×q B π×q B
+70
+
+Clearly, �
+f ◦ g = �f ◦ �g . Similarly, a map A3 → B3 may be “barred” to form a map � (A) → � (B):
+g : � (A)
+(π0,p◦π1,κ◦π1)
+−−−−−−−−→ A3
+g−→ B3
+ˆν(π0,π2)+ ˆ∇(π0,π1)
+−−−−−−−−−−→ � (B).
+It is straightforward to see that �f = f , �g = g .
+Lemma 3.4.3. σ : � (A) → � (A) induces a bracket on Γ(π):
+λ ◦ [X ,Y ] = (σ ◦ (id ,T X ◦ ̺) ◦ Y −T π (id ,T Y ◦ ̺) ◦ X ) − 0◦ Y.
+Proof. Let X ,Y ∈ Γ(π) and compute:
+p ◦ π1 ◦ (σ ◦ (id ,T X ◦ ̺) ◦ Y −π0 (id ,T Y ◦ ̺) ◦ X )
+= p ◦ (π1 ◦ σ ◦ (id ,T X ◦ ̺) ◦ Y −T π T Y ◦ ̺ ◦ X )
+= p ◦ π1 ◦ σ ◦ (id ,T X ◦ ̺) ◦ Y −π p ◦ T Y ◦ ̺ ◦ X
+= π0 ◦ (id ,T X ◦ ̺) ◦ Y − Y ◦ p ◦ ̺ ◦ X
+= Y − Y = ξ,
+π0 ◦ (σ ◦ (id ,T X ◦ ̺) ◦ Y −π0 (id ,T Y ◦ ̺) ◦ X )
+= π0 ◦ σ ◦ (id ,T X ◦ ̺) ◦ Y −π π0 ◦ (id ,T Y ◦ ̺) ◦ X
+= p ◦ π1 ◦ (id ,T X ◦ ̺) ◦ Y −π X
+= p ◦ T X ◦ ̺ ◦ Y −π X = X −π X = ξ.
+The universality of the lift induces a new section [X ,Y ] so that
+λ ◦ [X ,Y ] = (σ ◦ (id ,T X ◦ ̺) ◦ Y −T π (id ,T Y ◦ ̺) ◦ X ) − 0Y .
+Definition 3.4.4. The map σ : � (A) → � (A) is linear whenever the two bundle morphisms
+σ : (� (A),λ × ℓ) → (� (A),0× c ◦ T λ) σ : (� (A),0× c ◦ T λ) → (� (A),λ × ℓ)
+are linear, and cosymmetric if
+σ ◦ �λ = �λ = (ξπ,λ).
+Note that whenever σ is linear and cosymmetric, σ ◦ ˆµ(u,v) = ˆν(u,v).
+Linearity and unit axioms, along with the connection on the differential bundle, force the existence
+of a bilinear bracket 〈−,−〉 as in the definition of a Lie algebroid in Proposition 3.1.13.
+Proposition 3.4.5. For an anchored bundle (π : A → M ,ξ,λ,̺) with connection (κ,∇), a cosymmetric
+and bilinear σ is equivalent to a bilinear 〈−,−〉 : A2 → A, with the correspondence given by
+〈−,−〉 : κ ◦ σ ◦ ∇−π κ ◦ ∇
+σ : � (A) −−−−−−−−−−→
+(π1,π0,π2+〈π0,π1〉)
+� (A)
+71
+
+Proof. Derive
+�σ ◦ (x, y,z) = (y, x,κ ◦ σ ◦ (∇(̺x, y ) +p◦π1 µ(y,z)))
+= (y, x,κ ◦ σ ◦ ∇(̺x, y ) +π κ ◦ σ ◦ µ(y,z))
+= (y, x,κ ◦ σ ◦ ∇(̺x, y ) +π κ ◦ ν ◦ (y,z))
+= (y, x,κ ◦ σ ◦ ∇(̺x, y ) +π z)
+=: (y, x,〈x, y 〉 +π z)
+where 〈−,−〉 is certainly bilinear. The converse is immediate: take
+�σ(u,v,w ) := (v,u,w + 〈u,v〉).
+It is easy to see that σ is linear and cosymmetric.
+The linear bracket must be involutive for the bilinear bracket to be alternating.
+Proposition 3.4.6. If σ is cosymmetric and linear, then the bilinear bracket 〈−,−〉 is alternating if and
+only if σ ◦ σ = id .
+Proof. First, note that σ ◦ σ = id if and only if �σ ◦ �σ = id . Then check that
+�σ ◦ �σ(u,v,w ) = �σ(v,u,w + 〈u,v〉) = (u,v,w + 〈u,v〉 + 〈v,u〉).
+By the cancellativity of addition on A,
+w = w + 〈u,v〉 + 〈v,u〉 ⇐⇒ 0 = 〈u,v〉 + 〈v,u〉.
+Observation 3.4.7. A bilinear 〈−,−〉 on an anchored bundle with a connection induces the maps
+{v, x}(κ,∇) := κ′ ◦ T.̺ ◦ ∇(v, x),
+{v, x, y }(κ,∇) := κ ◦ T (〈−,−〉(κ,∇)) ◦ ∇A2 ◦ (v, x, y )
+from Equation (3.2) in Proposition 3.1.13.
+Proposition 3.4.8. Let σ be cosymmetric and bilinear, with the associated bracket 〈−,−〉. Then
+T ̺ ◦ π1 ◦ σ = c ◦ T ̺ ◦ π1
+if and only if the Leibniz equation is satisfied:
+̺ ◦ 〈u,v〉 + {̺v,u} = {̺u,v.}
+(3.6)
+Proof. Following the given notation and using the hypothesis that the connection is torsion-free on
+M ,
+�
+T ̺ ◦ π1(u,v,w ) = (̺u,̺v,w + {u,v})
+�c (x, y,z) = (y, x,z).
+Computing each side,
+�
+T ̺ ◦ �
+π1 ◦ �σ ◦ (u,v,w ) = �
+T ̺ ◦ �
+π1 ◦ (v,u,w + 〈u,v〉)
+= �
+T ̺ ◦ (̺v,u,w + 〈u,v〉)
+= (̺v,̺u,̺w + ̺〈u,v〉 + D[̺](̺v,u))
+= (̺v,̺u,̺w + ̺〈u,v〉 + {v,u}),
+�c ◦ �
+T ̺ ◦ �
+π1(u,v,w ) = �c (̺u,̺v,̺w + {u,v})
+= (̺v,̺u,̺w + {u,v}),
+so it follows that the two terms are equal if and only if the desired equality holds.
+72
+
+Lemma 3.4.9. Let σ be linear and cosymmetric. Then
+(i) T (〈−,−〉) : T A2 → T A satisfies
+�
+T (〈−,−〉)(ax,uy ,uz,ux y ,ux z)
+= (ax ,〈uy ,uz〉,{ax,uy , yz} + 〈uy ,ux z〉 + 〈ux y ,uz〉);
+(ii) T.σ satisfies
+�
+(id × T (�σ))(ux ,uy ,ux y ,uz,ux z,uy z,ux y z)
+= (ux ,uz,ux z,uy ,ux y ,uy z + 〈uy ,uz〉,
+ux y z + {ax,uy , yz} + 〈uy ,ux z〉 + 〈ux y ,uz〉).
+Proof.
+(i) By the universality of the vertical lift and bilinearity of 〈−,−〉, the outer squares below are pull-
+backs:
+A4
+A2
+T (A)
+T (A)
+T M
+T M
+M
+M
+µA2
+(〈π0,π1〉,〈π0,π3〉+〈π1,π2〉)
+µA
+T 〈,〉
+T π◦T πi
+T π
+0
+0
+so that
+T (〈,〉)(0,uy ,uz,ux y ,ux z) = µA(〈uy ,uz〉,〈uy ,ux z〉 + 〈ux y ,uz〉).
+Now we compute
+T (〈,〉)(µA2((uy , yz),(ux y ,ux z)) +p ∇A2(ax,(uy ,uz)))
+= T (〈,〉)µA2((uy ,uz),(ux y ,ux z)) +p T (〈,〉) ◦ ∇(ax,(uy ,uz))
+= µA(〈uy ,uz〉,〈uy ,ux z〉 + 〈ux y ,uz〉) +p T (〈,〉) ◦ ∇(ax,(uy ,uz))
+and then postcompose this with (T π,p,κ) to obtain
+(0,〈uy ,uz〉, 〈uy ,ux z〉 + 〈ux y ,uz〉) +p (ax,〈uy ,uz〉,{ax,uy , yz})
+= (ax,〈uy ,uz〉,{ax,uy , yz} + 〈uy ,ux z〉 + 〈ux y ,uz〉).
+(ii) Consider the following diagram:
+T M p ×πA3π×πA3
+T M p ×πA3π×πA3
+T (A3)
+T (A3)
+T ◦ � (A)
+T ◦ � (A)
+�
+T �σ
+∇A3+p µA3
+T �σ
+T (π1,π0,π2+〈π0,π1〉)
+T (∇′+π0µ′))
+(T (π3),p,κA3)
+T σ
+T (π0,p◦π1,κ◦π1)
+73
+
+We want to find �
+T �σ =
+�
+T (π1,π0,π2 + 〈π0,π1〉). Note that
+T (π1,π0,π2 +π 〈π0,π1〉) = T (π1,π0,π2) +T π T (ξππi ,π0,〈π0,π1〉).
+The left term is straightforward:
+�
+T (π0,π1,π2)(ax ,(uy ,ux y ),(uz,ux z),(uy z,ux y z))
+= (ax,(uz,ux z),(uy ,ux y ),(uy z,ux y z))
+and for the right term, use part (i) of this lemma:
+�
+T 〈−,−〉◦ �
+T (π0,π1))(ax,(uy ,ux y ),(uz,ux z),(uy z,ux y z))
+= �
+T 〈−,−〉(ax,uy ,ux y ,uz,ux y z)
+= (ax ,〈uy ,uz〉,{ax,uy , yz} + 〈uy ,ux z〉 + 〈ux y ,uz〉)
+Then compute
+�
+T �σ(ax ,uy ,ux y ,uz,ux z,uy z,ux y z)
+= (ax ,uz,ux z,uy ,ux y ,uy z + 〈uy ,uz〉,ux y z + q)
+where q = {ax ,uy , yz} + 〈uy ,ux z〉 + 〈ux y ,uz〉, giving the desired equation.
+Proposition 3.4.10. Let σ be cosymmetric, doubly linear, and involutive, and satisfy the target axiom.
+Then σ satisfies the Yang–Baxter equation if and only if 〈−,−〉 and {−,−,−} satisfy the Bianchi identity:
+0 =
+�
+γ∈Cy(3)
+〈xγ0,〈xγ1, xγ2〉〉 +
+�
+γ∈Cy(3)
+{̺xγ0, xγ1, xγ2.}
+(3.7)
+Proof. We expand �
+id × T σ and �
+σ × c . Start with T (id × σ), which was derived in Lemma 3.4.9:
+σ1(u) =
+�
+(id × T (�σ))(ux,uy ,ux y ,uz,ux z,uy z,ux y z)
+= (ux ,uz,ux z,uy ,ux y ,uy z + 〈uy ,uz〉,
+ux y z + 〈uy ,ux z〉 + 〈ux y ,uz〉 + {̺ ◦ ux,uy ,uz}).
+Using the fact that κ′ is torsion free, so ˆc (ux z,uy z,ux z y ) = (uy z,ux z,ux y z), it follows that
+σ2(ux,uy ,ux y ,uz,ux z,uy z,ux y z) = (uy ,ux,ux y + 〈ux,uy 〉,uz,uy z,ux z,ux y z).
+Then compute
+σ2σ1σ2(u)
+=
+�
+uz,uy ,uy z + 〈uy ,uz〉,ux,ux z + 〈ux,uz〉,ux y + 〈ux ,uy 〉,z1
+�
+,
+σ1σ2σ1(u)
+=
+�
+uz,uy ,uy z + 〈uy ,uz〉,ux,ux z + 〈ux,uz〉,ux y + 〈ux ,uy 〉,z2
+�
+.
+74
+
+Note that the first five terms are equal, so it suffices to check z1 = z2 for z1 = π6σ1σ2σ1(u),z2 =
+π6σ2σ1σ2(u).
+z1 = ux y z + 〈uy ,ux z〉 + 〈ux y ,uz〉 + {̺ ◦ ux,uy ,uz} + {̺ ◦ uz,ux,uy }
++ 〈ux,uy z + 〈uy ,uz〉〉 + 〈ux z + 〈ux ,uz〉,uy 〉
+= ux y z + 〈uy ,ux z〉 + 〈ux y ,uz〉 + {̺ ◦ ux,uy ,uz} + {̺ ◦ uz,ux,uy }
++ 〈ux,uy z〉 + 〈ux,〈uy ,uz〉〉 + 〈ux z,uy 〉 + 〈〈ux ,uz〉,uy 〉
+= ux y z + 〈ux y ,uz〉 + {̺ ◦ ux,uy ,uz} + {̺ ◦ uz,ux,uy }
++ 〈ux,uy z〉 + 〈ux,〈uy ,uz〉〉 + 〈〈ux,uz〉,uy 〉,
+z2 = ux y z + 〈ux,uy z〉 + {̺ ◦ uy ,ux,uz} + 〈ux y + 〈ux ,uy 〉,uz〉
+= ux y z + 〈ux,uy z〉 + {̺ ◦ uy ,ux,uz} + 〈ux y ,uz〉 + 〈〈ux,uy 〉,uz〉.
+So z1 = z2 is equivalent to requiring
+ux y z + 〈ux y ,uz〉 + {̺ ◦ ux,uy ,uz} + {̺ ◦ uz,ux,uy }
++ 〈ux,uy z〉 + 〈ux,〈uy ,uz〉〉 + 〈〈ux,uz〉,uy 〉
+= ux y z + 〈ux,uy z〉 + {̺ ◦ uy ,ux,uz} + 〈ux y ,uz〉 + 〈〈ux,uy 〉,uz〉;
+cancelling alike terms, this is equivalent to
+〈ux,〈uy ,uz〉〉 + 〈〈ux,uz〉,uy 〉 + {̺ ◦ ux,uy ,uz} + {̺ ◦ uz,ux,uy }
+= {̺ ◦ uy ,ux,uz} + 〈〈ux,uy 〉,uz〉.
+Using the fact that 〈−,−〉 and {−,−,−} are alternating in the last two arguments, this is equivalent to
+0 = 〈ux,〈uy ,uz〉〉 + 〈〈ux,uz〉,uy 〉 + 〈uz,〈ux,uy 〉〉
++ {̺ ◦ ux,uy ,uz} + {̺ ◦ uz,ux,uy } + {̺ ◦ uy ,uz,ux}
+= 〈ux,〈uy ,uz〉〉 + 〈uy ,〈uz,ux〉〉 + 〈uz,〈ux,uy 〉〉
++ {̺ ◦ ux,uy ,uz} + {̺ ◦ uz,ux,uy } + {̺ ◦ uy ,uz,ux}
+giving the desired identity.
+Corollary 3.4.11. Let (π : A → M ,ξ,λ,̺) be an anchored bundle in a tangent category with negatives,
+with anchored connection (∇,κ) on A and torsion-free affine connection (∇′,κ′) on M . An involution
+algebroid structure on A is equivalent to a bilinear map
+〈−,−〉 : A2 → A
+with derived maps
+{−,−} : Aπ×p T M → T M := κ′ ◦ T ̺ ◦ (π0,π1),
+{−,−,−} : T M p ×π◦πi A2 → A;{a,u1,u2} := κ ◦ T (〈−,−〉) ◦ (∇(a,u1),∇(a,u2))
+satisfying
+(i) 〈−,−〉 is linear and cosymmetric,
+(ii) 〈−,−〉 is alternating,
+75
+
+(iii) 〈−,−〉 and {−,−} satisfy the Leibniz equation, Equation 3.6.
+(iv) 〈−,−〉,{−,−}, and {−,−,−} satisfy the Bianchi identity, Equation (3.7).
+Morphisms of involution algebroids may also be characterized by preservation of the tensor.
+Proposition 3.4.12. Let A,B be a pair of involution algebroids with chosen connections in a tangent
+category with negatives. Then an anchored bundle morphism f : A → B is an involution algebroid
+morphism if and only if (recalling the notation from Equation 3.3)
+∇[f ](x, y ) + 〈f ◦ x, f ◦ y 〉 = ∇[f ](y, x) + f ◦ 〈x, y 〉.
+Proof. Note that
+(σ ◦ � (f ) = � (f ) ◦ σ)
+⇐⇒ (π1,π0,π2 + 〈π0,π1〉) ◦ (f ◦ x, f ◦ y, f ◦ z + ∇[f ](x, y ))
+= (f , f , f ◦ π2 + ∇[f ](π0,π1)) ◦ (y, x,z + 〈x, y 〉)
+while the second condition reduces to
+∇[f ](x, y ) + 〈f ◦ x, f ◦ y 〉 = ∇[f ](y, x) + f ◦ 〈x, y 〉.
+3.5
+The isomorphism of Lie and involution algebroid categories
+Sections 3.1 and 3.4 have made the relationship between involution algebroids and Lie algebroids clear.
+It is important to note that while the proofs used connections as a tool to identify the local coherences
+satisfied by involution and Lie algebroids, the construction of a Lie algebroid from an involution alge-
+broid (or vice versa) is independent of the choice of connection.
+Theorem 3.5.1. There is an isomorphism of categories between Lie algebroids and involution algebroids
+in smooth manifolds.
+Proof. For the equivalence of categories, note that by Propositions 3.1.13 and 3.1.14, Corollary 3.4.11,
+and Proposition 3.4.12 there is an isomorphism of categories between involution algebroids with a
+choice of connection and Lie algebroids with a choice of connection (morphisms are not restricted to
+connection preserving morphisms). This allows us to chain together isomorphisms
+Inv(SMan) ∼= Inv(SMan)ChosenConn ∼= LieAlgdChosenConn ∼= LieAlgd.
+To complete the proof, we must show that the assignment that sends an involution algebroid to a
+Lie algebroid whose bracket is given by
+λ ◦ [X ,Y ]∗ =
+�
+(π1 ◦ σ ◦ (id ,T.X ◦ ̺) ◦ Y −p T.Y ◦ ̺◦) −T.π 0◦ Y
+�
+,
+(3.8)
+is a bijection on objects, which brings up some subtleties. First, while an involution map
+σ : A̺×T πT A → A̺×T πT A
+76
+
+is defined with respect to a particular choice of pullback A̺ ×T πT A, the category of involution alge-
+broids does not distinguish between different choices of this pullback (and therefore different repre-
+sentations of the map σ), and it is not part of the data of an involution algebroid. It is immediate by
+universality that the left-hand-side of Equation 3.8 is independent of the choice of pullback A̺×T πT A.
+Now, recall thatthe canonical involution ofa Lie algebroid isuniquelybyTheorem 4.7ofde León et al.
+(2005) (this was also mentioned in Corollary 3.3.9). Once we make a choice of prolongation A̺×T πT A,
+we have made a choice of pullback � (A) in LieAlgd, which uniquely determines the canonical involu-
+tion
+σ : A̺×T πT A → A̺×T πT A.
+While the exact map σ depends on the choice of pullback A̺ ×T π T A, they all determine the same
+involution algebroid, thus proving the bijection correspondence of objects.
+77
+
+Chapter 4
+The Weil nerve of an algebroid
+The first three chapters of this thesis demonstrated that tangent categories allow for an essentially alge-
+braic description of Lie algebroids by axiomatizing the behaviour of the tangent bundle, and showing
+that a Lie algebroid over a manifold M is a “generalized tangent bundle”, namely an involution al-
+gebroid. This chapter will make precise the sense in which an involution algebroid is a generalized
+tangent bundle, by showing that the category of involution algebroids in a tangent category � is equiv-
+alent to a certain tangent-functor category from the free tangent category over a single object to �, or
+more generally that there is a fully faithful functor
+Inv(�) �→ TangLax(FreeTangCat(∗),�).
+This functor, the Weil nerve of an involution algebroid, builds a functor from the free tangent category
+over a single object to � using a span construction. This chapter primarily builds on two pieces of work:
+Leung’s construction of the free tangent category Weil1 (Leung (2017)) , and Grothendieck’s original
+nerve construction (first published in Segal (1974)).
+To understand Leung’s construction of the free tangent category, and more generally his actegory-
+theoretic presentation of tangent categories (Section 4.2), we first look at Weil’s original insight relating
+the kinematic and operational descriptions of the tangent bundle in SMan. The definition of a tangent
+vector on a manifold M as an equivalence class of curves (Definition 1.2.8) puts a bijective correspon-
+dence between tangent vectors and �-algebra homomorphisms from the ring of smooth functions
+C ∞(M ) to the ring of dual numbers:
+C ∞(M ) → �[x]/x 2.
+The hom-set �Alg(C ∞(M ),�[x]/x 2) is precisely the set of derivations on C ∞(M ), which defines the
+operational tangent bundle discussed in Definition 1.2.16: there is a natural smooth manifold structure
+on this set. The Weil functor formalism, most notably developed in Kolár et al. (1993), extends this ob-
+servation to a general class of endofunctors on SMan. For example, the fibre product T2M corresponds
+to �-algebra morphisms,
+C ∞(M ) → R[x, y ]/(x 2, y 2, x y ),
+while the second tangent bundle corresponds to �-algebra morphisms into the tensor product,
+C ∞(M ) → R[x]/(x 2) ⊗ R[y ]/(y 2) = R[x, y ]/(x 2, y 2).
+By applying Milnor’s exercise (Problem 1-C Milnor and Stasheff (1974)), which states that the C ∞ func-
+tor
+SMan → �Algop; M �→ C ∞(M ) = SMan(M ,�)
+78
+
+is fully faithful, the structure maps occur as natural transformations. For example, the tangent projec-
+tion is induced by the morphism
+p : �[x]/x 2 a+b x�→a
+−−−−−→ �,
+so that
+T M
+p−→ M = [C ∞(M ),�[x]/x 2]
+(p)∗
+−→ [C ∞(M ),�].
+The zero map and addition are similarly induced by
+0 : �
+a�→a+0x
+−−−−−→ R[x]/x 2 and + : R[x, y ]/(x 2, y 2, x y )
+a+b x+c y
+−−−−−−−→
+�→a+(b+c )x R[x]/x 2,
+respectively, while the lift and flip are induced by the morphisms
+ℓ : �[x]/x 2
+a+b x
+−−−−−→
+�→a+b x y �[x, y ]/(x 2, y 2)
+and
+c : �[x, y ]/(x 2, y 2)
+a+b x+c y +d x y
+−−−−−−−−−−−→
+�→a+c x+b y +d x y �[x, y ]/(x 2, y 2).
+More generally, there is a monoidal category of Weil algebras (Definition 4.1.1) which has a monoidal
+action on the category of smooth manifolds. The Weil functor formalism, then, studies differential ge-
+ometric structures from the perspective of the endofunctors and natural transformations induced by
+this action. Leung’s insight is that there is an analogous category of commutative rigs1 built by replac-
+ing �[x]/x 2 with �[x]/x 2, called Weil1 (Definition 4.1.3); a tangent structure is precisely a monoidal
+action by Weil1 satisfying some universal properties. In particular, this category Weil1 is precisely the
+free tangent category over a point, FreeTang(∗), so that every object A in a tangent category � deter-
+mines a strict tangent functor
+T (−)A : Weil1 → �;V �→ T V A
+and morphisms f : A → B are in bijective correspondence with tangent-natural transformationsT (−)A ⇒
+T (−)(B).
+The axioms of an involution algebroid in a tangent category � correspond bijectively with those
+of the tangent bundle - this suggests that an involution algebroid should determine a tangent functor
+from Weil1 to �. A first guess would lead one to think that p : �[x]/x 2 → � is sent to π : A → M , 0 to
+ξ, and + to +A. As the space of prolongations � (A) = A̺ ×T πT A plays the role of the second tangent
+bundle, we can see that
+ℓ : N [x]/x 2 → N [x, y ]/(x 2, y 2) �→ (ξ ◦ π,λ) : A → � (A),
+and
+c : N [x, y ]/(x 2, y 2) → N [x, y ]/(x 2, y 2) �→ σ� (A) → � (A).
+This pattern may be neatly summed up using span composition - we will construct a functor that sends
+�[x]/x 2 to the span
+A
+M
+T M
+̺
+π
+1A rig is a ring without negatives, i.e. a commutative monoid equipped with a bilinear multiplication.
+79
+
+and the tensor product �[x]/x 2 ⊗ �[x]/x 2 to the span composition (e.g. the pullback)
+� (A)
+A
+T A
+M
+T M
+T 2M
+̺
+π
+T.π
+T.̺
+⌟
+which is the space of prolongations. This leads to the first major result of this chapter, the Weil Nerve
+(Theorem 4.3.9), which states there is a fully faithful embedding
+NWeil : Inv(�) �→ [Weil1,�].
+This bears a strong similarity to Grothendieck’s original nerve theorem, which takes an internal cate-
+gory s,t : C → M and constructs a functor ∆op → � (where ∆op is the monoidal theory of an internal
+monoid) by sending tensor to span composition, and the composition and unit maps given span mor-
+phisms
+C2
+M
+M
+M
+M
+M
+C
+C
+s◦π0
+t ◦π1
+s
+t
+m
+s
+t
+e
+while the unit and associativity axioms for a category are exactly the unit and associativity laws for
+a monoid in this setting. The Segal conditions identify exactly the functors C : ∆op → � that lie in
+the image of the nerve functor N as those whose [n]t h object is sent to the wide pullback C ([n]) =
+C [2]t ×s C [2]... t ×s C [2]. The corresponding result for involution algebroids is found in Theorem 4.4.8,
+which states that a tangent functor (A,α) : Weil1 → � is the nerve of an involution algebroid if and
+only if A preserves tangent limits and α is a T -cartesian natural transformation (this forces A.(W ⊗n) =
+A̺×T.πT A ...T n−1̺×T nπT nA). The similarity between the Weil nerve and Grothendieck/Segal’s nerve
+runs deep, and in Chapter 5 we demonstrate that the enriched perspective on tangent categories puts
+these both into the same formal framework.
+Sections4.1 and 4.2give a detailed introduction tothe Weil functorformalism (Kolár et al.(1993),Bertram and Souvay
+(2014)) and Leung’s unification of Weil functors with tangent categories Leung (2017). The rest of the
+chapter contains contains new results developed by the author. Section 4.3 proves the embedding
+part of the Weil nerve theorem, that the category of involution algebroids embeds into the category
+of tangent functors and tangent natural transformations [Weil1,�]. Section 4.4 identifies exactly those
+tangent functors (A,α) : Weil1 → � that are the nerve of an involution algebroid, completing the proof
+of the Weil nerve theorem. Section 4.5 uses the Weil nerve to develop a novel tangent structure on the
+category of involution algebroids in a tangent category (in particular, the category of Lie algebroids will
+have this novel tangent structure).
+4.1
+Weil algebras and tangent structure
+This section gives a more thorough introduction to the Weil functor formalism of Kolár et al. (1993),
+and in particular how the structure maps of a tangent category may be teased out of it. We begin by
+80
+
+introducing Weil algebras, and the prolongation of a smooth manifold by a Weil algebra. (The relation-
+ship with prolongations of involution algebroids from Definition 3.2.1 will be made clear in Section
+4.3.)
+Definition 4.1.1. An �-Weil algebra2 is a finite-dimensional �-algebra V so that V = � ⊕ ˙V as �-
+modules and ˙V is a nilpotent ideal. The category �Weil is the full subcategory of �Alg spanned by the
+�-Weil algebras. The prolongation of a manifold by a Weil algebra V is given by the manifold
+T V M := �Alg(C ∞(M ,�),V ).
+Eck (1986) showed that every product-preserving endofunctor on the category of smooth mani-
+folds is constructed as the Weil prolongation by some Weil algebra. Consequently, �-algebra homo-
+morphisms induce natural transformations between these product-preserving endofunctors on the
+category of smooth manifolds.
+Example 4.1.2. Consider the following �-Weil algebras and their associated prolongation functors.
+(i) Prolongation by � induces the identity functor, and the tangent bundle is given by �[x]/x 2. The
+tangent projection, then, is equivalent to the �-algebra morphism
+p : �[x]/x 2 → �; p(a + b x) = a
+while the 0-map induces the zero vector field:
+0 : � → �[x]/x 2; 0(a) = a + 0x.
+(ii) The algebra �[xi ]1≤i≤n/(xi x j )1≤i≤j ≤n = (�[x]/x 2)n is the wide pullback TnM = T M p×pT M ...p×p
+T M . In particular, prolongation by �[x, y ]/(x 2, y 2, x y ) gives the bundle T2M = T M p×pT M . The
+�-algebra morphism
++ : �[x, y ]/(x 2, y 2, x y ) → R[x]/x 2; +(a0 + a1x + a2y ) = a0 + (a1 + a2)x
+corresponds to the addition of tangent vectors.
+(iii) The algebra �[x, y ]/(x 2, y 2) = (�[x]/x 2) ⊗ (�[x]/x 2) is the second tangent bundle T 2M = T T M .
+The vertical lift T → T 2 is induced by the morphism
+ℓ : �[x]/x 2 → �[x, y ]/(x 2, y 2); ℓ(a + b x) = a + b x y.
+(iv) The monoidal symmetry map induces c : T 2 ⇒ T 2, as follows:
+c : (�[x]/x 2) ⊗ (�[y ]/y 2) → (�[y ]/y 2) ⊗ (�[x]/x 2);
+c (a + b1x + b2y + b3x y ) = a + b2x + b1y + b3x y.
+(v) For n ≥ 2, the algebra �[x]/x n gives the n-jet bundle. Note that this is the equalizer of ⊗n�[x]/x 2
+by the symmetry actions of Sn.
+2Not to be confused with the normal usage of “Weil algebra” in Lie theory, e.g. Meinrenken and Pike (2021).
+81
+
+Further examples may be found in the monograph Kolár et al. (1993). Tangent categories bridge
+the gap between the Weil functor approach to studying the differential geometry of smooth manifolds
+and the synthetic differential geometry approach of axiomatizing a tangent bundle using nilpotent in-
+finitesimals. The main structure axiomatized here is that of monoidal action of a symmetric monoidal
+category on a category M × � → �, or equivalently, a lift from a category to the category of complexes
+� → [M ,�], which involves translating a bit of classical category theory to the 2-categorical setting.
+Example 4.1.2 leads to the classical theorem that the category of smooth manifolds has an action by
+the category of �-Weil algebras that preserves all connected limits that exist. These “natural” universal
+properties (in the sense of Kolár et al. (1993)) is foundational to synthetic differential geometry; see,
+for example, Chapter Two of Lavendhomme (1996). Unfortunately, Weil algebras are not an ideal syn-
+tactic presentation: they are not a finitely presentable category, and it is not immediately clear when
+a diagram is a connected limit.3 Moving from �-algebras to commutative rigs and restricting to an
+appropriate subcategory solves this problem.
+Definition 4.1.3 (Definition 3.1 Leung (2017)). The category Weil1 is defined to be the full subcategory
+of commutative rigs, CRig, generated by the rig of dual numbers W := �[x]/x 2, constructed as follows:
+1. Start with finite product powers of W in CRig, and make a strict choice of presentation:
+W0 = �, Wn := �[xi ]/(xi x j )i≤j ,0 ≤ i < n.
+2. Then take the closure of Wn under coproduct of commutative rigs, written ⊗. Again, make a strict
+choice of presentation:
+Wn(0) ⊗ ...⊗ Wn(m−1) := �[xi,j ]/(xi j xik)j ≤k,0 < i < m,0 < j < n(i).
+Note that we will often suppress the tensor product ⊗ and simply write
+U V :=U ⊗ V.
+Proposition 4.1.4 (Definition 3.3 Leung (2017) ).
+(i) The category Weil1 is a symmetric strict monoidal category with unit � and coproduct ⊗.
+(ii) � is a terminal object in Weil1.
+Note that there is a forgetful functor
+Weil1 → (CMon/�) → CMon
+that reflects connected limits. This gives the following class of limits, identified in Leung (2017).
+Definition 4.1.5. We say the following pullback diagrams in Weil1 are transverse:
+W
+W
+W 2
+W
+W 2
+W W
+W
+W
+W
+�
+�
+W
+id
+id
+id
+id
+⌟
+π0
+π1
+p
+p
+⌟
+p◦πi
+µ
+0
+pW
+where µ(a +a1x +a2y ) = a +a1x +a2x y . The ⊗-closure of these three pullbacks is the set of transverse
+squares, and they are also pullback squares by Leung (2017).
+3It should be noted that Nishimura and Osoekawa (2007) made progress applying techniques from computer algebra to
+latter problem.
+82
+
+To see that each transverse square in the ⊗-closure is a pullback diagram, take the two non-identity
+squares and rewrite them in CMon:
+� × (� × �)
+� × �
+� × (� × �)
+� × (� × � × �)
+� × �
+�
+�
+� × �
+�×π0
+�×π1
+π0
+π0
+⌟
+π0
+�×(π0,0◦!,π1)
+(id,0◦!)
+�×π1
+⌟
+The coproduct of Weil algebras is the tensor product of the underlying commutative monoids, which
+are finite-dimensional and free, so these limits are closed under ⊗.
+Proposition 4.1.6 (Leung (2017) Proposition 4.1). The category Weil1 is a tangent category, where the
+tangent functor is
+T := W ⊗ (_) : Weil1 → Weil1
+and the natural transformations are given by
+p : W ⊗ (_)
+p⊗(_)
+−−→ (_), 0 : (_)
+0⊗(_)
+−−→ W ⊗ (_), + : W2 ⊗ (_)
++⊗(_)
+−−→ W ⊗ (_),
+ℓ : W ⊗ (_)
+ℓ⊗(_)
+−−→ W ⊗ W ⊗ (_),
+c : W ⊗ W ⊗ (_)
+p⊗(_)
+−−→ W ⊗ W ⊗ (_)).
+The category Weil1 is, in some sense, a finitely presented theory. It is precisely the free tangent
+category on a single object:
+Proposition4.1.7(Proposition 9.5, Leung(2017)). The categoryWeil1 isgenerated by the maps{p,0,+,ℓ,c }
+from Example 4.1.2, closed under composition, tensor, and maps induced by transverse limits.
+Corollary 4.1.8. The category Weil1 is the free tangent category over a single object: every object C in a
+tangent category � determines a strict tangent functor T−.C : Weil1 → �, mapping
+V = W n1 ⊗ ...⊗ W n(k) �→ Tn(1).(...).Tn(k).C = T V C
+so that there isan isomorphism of categoriesbetween � and the category of strict tangent functorsWeil1 →
+� with tangent natural transformations as morphisms.
+Notation 4.1.9. Throughout this section, the notation T V C will refer to the action of the Weil algebra
+V on an object C in a tangent category. In particular, we will make use of the isomorphism T U .T V C =
+T U V C .
+4.2
+Tangent structures as monoidal actions
+The presentation of Weil1 as the free tangent category situates the formal theory of tangent categories
+as an instance of more general categorical machinery, namely monoidal actions. Recall that in a sym-
+metric monoidal category (�,⊗,I ), an internal monoid (C ,•,i) determines a monad:
+(C ⊗ _ : � → �,µ : � ⊗ (� ⊗ _)
+•⊗_
+−→ � ⊗ _,η : _
+ρ−→ I ⊗ _
+e ⊗_
+−−→ C ⊗ _).
+83
+
+The category of algebras for this monad is exactly the category of C -modules, objects with an associa-
+tive and unital action by C . A morphism will be a map on the base object that preserves the action:
+C
+M ⊗ C
+M ⊗ M ⊗ C
+M ⊗ C
+C
+M ⊗ C
+C
+M ⊗ C
+M ⊗ D
+C
+D
+M ⊗∝
+•⊗M
+∝
+∝
+(u,C )
+∝
+f
+∝C
+∝D
+M ⊗f
+Strict actegories are the 2-categorical generalization of modules over a monoid. The coherences for a 2-
+monad follow from the coherences from a strict monoidal category in the 2-category of categories. The
+following proposition relies on a few facts from enriched category theory (treating the cartesian closed
+category Cat as a Cat-enriched category, per Kelly (2005)) but a more general treatment of non-strict
+actegories may be found in Janelidze and Kelly (2001):
+• A 2-functor and 2-natural transformations are exactly a functor and natural transformations
+that satisfy extra coherences. These coherences follow for free by constructing the monad and
+comonad � × _,[� ,_].
+• An algebra of the underlying 1-monad is exactly an algebra of the 2-monad (the same result holds
+for comonads).
+When working with algebras of a 2-monad, four different notions of morphisms can come into play
+(Lack and Power (2009)). These arise through using the 2-categorical data to weaken the notion of a
+morphism:
+(i) Strict: this is exactly a morphism of the underlying algebras. Write the 2-category of strict � -
+actegories.
+(ii) Strong: the morphisms preserve the action to an isomorphism:
+� × �
+� × �
+�
+�
+� ×F
+∝C
+∝D
+F
+α
+(iii) Lax: the 2-cell is no longer an isomorphism:
+� × �
+� × �
+�
+�
+� ×F
+∝C
+∝D
+F
+α
+(iv) Oplax: the 2-cell travels in the opposite direction (these will not figure into this account).
+2-cells between actegory morphisms must satisfy a coherence between the natural transformation
+parts of the actegory morphisms.
+84
+
+Definition 4.2.1. In the case of strict, strong, and lax tangent functors, the same notion of a 2-cell applies:
+a natural transformation γ : F ⇒ G satisfying the following coherences with the natural transformations
+α and β:
+� × �
+� × �
+� × �
+� × �
+�
+�
+�
+�
+� ×F
+∝�
+∝�
+F
+α
+G
+� ×G
+� ×F
+∝�
+∝�
+G
+β
+γ
+� ×γ
+=
+We call these actegory natural transformations.
+Note that for strict actegory morphisms, this condition holds for any natural transformation γ : F ⇒
+G . Now consider the following three 2-categories.
+Definition 4.2.2. Let (� ,⊗,I ) be a strict monoidal category. Define the following three 2-categories.
+1. �Actstrict: the 2-category of strict � -actegories, strict actegory morphisms, and natural transfor-
+mations.
+2. �Actstrong: the 2-category of strict � -actegories, strong actegory morphisms, and actegory nat-
+ural transformations.
+3. �Actlax: the 2-category of strict � -actegories, lax actegory morphisms, and actegory natural
+transformations.
+Note that the inclusions of these 2-categories are locally fully faithful, so only the 1-cells differ.
+The case where the action preserves certain limits in the monoidal category is of particular interest.
+A small category equipped with a class of chosen limits is known as a sketch. The previous correspon-
+dence restricts to the class of limit-preserving actions in this case.
+Definition 4.2.3. A sketch is a small category with a class of chosen limits, and a sketch morphism is a
+functor sending chosen limits to the chosen limits in the domain (up to isomorphism). The category of
+models of a sketch � in a category �, Mod(� ,�), is the full subcategory [� ,�] whose functors preserve
+the chosen limits. A monoidal sketch, then, is a sketch (� ,� ) equipped with a symmetric monoidal
+category structure on � so that _ ⊗ _ preserves limits in each argument.
+Now use the fact that the category Weil1 is a monoidal sketch, since it is a small, strict monoidal
+category equipped with a class of limits stable under the tensor product.
+Theorem 4.2.4 (Theorem 14.1, Leung (2017)). Let � be a category. The following are equivalent:
+(i) A tangent structure on �,
+(ii) A sketch action ∝: Weil1 × � → �.
+Observation 4.2.5. There is a coalgebraic perspective on tangent categories, coming from the equiva-
+lence between algebrasof the 2-monad (Weil1×(_),⊗,I : 1 → Weil1) and the 2-comonad ([Weil1,_],[⊗,_],[I ,_]).
+For any category �, there is a free tangent category given by
+Weil1 × �
+85
+
+and this agrees with the free Weil1-actegory. However, for the cofree tangent category, take
+Mod(Weil1,�),
+the category of transverse-limit-preserving functors Weil1 → �.
+We can use Leung’s theorem to induce a monoidal functor Weil1 → � when a tangent structure is
+induced by a single object.
+Corollary 4.2.6. Let (�,⊗,I ) be a strict monoidal category. If an additive bundle (p : A → I ,+ : A2 →
+A,0 : I → A) equipped with morphisms
+A ⊗ A
+c−→ A ⊗ A
+A
+ℓ−→ A ⊗ A
+determines a tangent structure on � using the endofunctor A⊗(−), then A determines a strict, transverse-
+limit-preserving, monoidal functor
+A(−) : Weil1 → �;Wn[1] ⊗ ...⊗ Wn[k] �→ An[1] ⊗ ...⊗ An[k] = Tn[1]...Tn[k].I
+Note that this allows for a more conceptual description of representable tangent structure.
+Proposition 4.2.7. In a symmetric monoidal closed category, an infinitesimal object is exactly a strict
+symmetric monoidal functor D : Weil1 → �.
+This presentation of an infinitesimal object makes it tautological that �op has a tangent structure.
+Corollary 4.2.8. Given a strict symmetric monoidal functor
+D : Weil1 → �op
+there is a strict action of Weil1 on �op given by
+Weil1 × �op D op ⊗�
+−−−−→ �op × �op
+⊗−→ �op .
+There is a clear correspondence between the notions of a (strict, strong, lax) tangent functor and
+a (strict, strong, lax) actegory morphism. This proposition extends to the following equivalence of 2-
+categories.
+Corollary 4.2.9 (Theorem 14.1 Leung (2017)). The following pairs of 2-categories are equivalent.
+(i) The 2-category of tangent categoriesand strict tangent functorsisthe full sub-2-category ofWeil1ActStrict
+spanned by sketch actions.
+(ii) The 2-category of tangent categoriesand strong tangent functorsisthe full sub-2-category ofWeil1Actstrong
+spanned by sketch actions.
+(iii) The 2-category of tangent categoriesand lax tangent functorsisthe full sub-2-category ofWeil1ActLax
+spanned by sketch actions.
+86
+
+4.3
+The Weil nerve of an involution algebroid
+The construction in this section is analagous to the nerve of an internal category—hence the “Weil
+nerve” construction—and deals with similar technical issues. In particular, the construction in this
+section will mimic the nerve construction for internal categories by replacing the tensor product of
+Weil1 with span composition in the domain category. Recall that every anchored bundle or internal
+category has a canonical span associated with it:
+A
+C
+M
+T M
+M
+M
+π
+̺
+s
+t
+In any category �, there is a category of spans in � as well as span composition.
+Definition 4.3.1. A span from A to B in a category � is a diagram of the form
+X
+A
+B
+l
+r
+There is a notion of span composition, so given a span X : A → B and Y : B → C , then the composition
+of X and Y is the pullback (if it exists):
+Z
+X
+Y
+A
+B
+C
+l
+r
+l ′
+r ′
+⌟
+l ′′
+r ′′
+A morphism of spans is a commuting diagram of the form
+A
+X
+B
+C
+Y
+D
+l
+r
+l ′
+fl
+fr
+r ′
+fc
+Note that if f and g are span morphisms with fr = gl , then the horizontal composition may be formed
+if each respective span composition exists:
+•
+X
+W
+A
+B
+E
+C
+D
+F
+Y
+Z
+•
+l
+r
+l ′
+fl
+fr =gl
+r ′
+fc
+gr
+gc
+⌟
+⌟
+87
+
+When discussing span composition in a tangent category, it is assumed that the pullback is a T -pullback.
+Observation 4.3.2. Note that the category of spans in � is a functor category, so that limits are computed
+pointwise in �. This also means that the horizontal composition operation, when it exists, preserves
+limits in either argument.
+These span constructions can be helpful in constructing functors from a monoidal category into
+a non-monoidal category �, by forming a monoidal category from � using spans. In the case of an
+internal category over M , one takes the slice category �/(M × M ) where the tensor product is span
+composition. An internal category s,t : C → M is a monoid in this category of spans over M , so that it
+determines a monoidal functor
+C : ∆op → �/(M × M ),
+remembering that ∆op is the monoidal theory for monoid (every monoid in a monoidal category �
+determines a monoidal functor ∆ → �). The construction of the corresponding monoidal category for
+spans is more nuanced, as the category Weil1 is not �-indexed. Observe that the prolongation of an
+anchored bundle is constructed as a span composition:
+� (A)
+A
+T A
+M
+T M
+T 2M
+π
+̺
+T π
+T ̺
+π0
+π1
+⌟
+The third prolongation is given by span composition as well:
+� 2(A)
+A
+T.� (A)
+M
+T M
+T 2M
+π
+̺
+T.π◦π0
+T.̺◦π1
+π0
+π1
+⌟
+This horizontal composition will play the same role as the tensor product in �/(M × M ).
+Definition 4.3.3. In a tangent category �, consider a pair of spans
+X : M → T U M , Y : M → T V M .
+Define X ⊠ Y to be the horizontal composition (when it exists):
+Z
+X
+T U Y
+M
+T U M
+T U V M
+lX
+rY
+T U .lY
+T U .r
+⌟
+88
+
+(recall that we will often suppress the ⊗ in Weil1 to save space). So the span composition is
+M
+X−→ T U M
+T U .Y
+−−−→ T U .T V M
+M
+X
+T U M
+M
+Y
+T U M
+M
+A
+T V
+M
+B
+T V ′
+lX
+lY
+θ.M
+rX
+rY
+f
+rA
+rB
+ψ.M
+g
+lB
+lA
+(4.1)
+The horizontal composition f ⊠ g is defined as f ×θ.M θ.g :
+Z
+X
+Y
+M
+T U M
+T U .T V M
+M
+T U ′M
+T U ′.T V ′M
+A
+T U ′B
+C
+lX
+lA
+θ.M
+rX
+rA
+f
+T U .rY
+T U ′.rB
+θ.ψ.M
+θ.g
+T U .lY
+T U ′.lB
+⌟
+⌟
+In any tangent category with a tangent display system (Definition 1.5.3), the category of spans on
+M whose maps are of the form given by Equation 4.1 with l ∈ � is a monoidal category. Any anchored
+bundle in a tangent category gives rise to a monoidal category after a strict choice of T -pullbacks (as-
+suming those T -pullbacks exist).
+Definition 4.3.4. Let (π : A → M ,ξ,λ,̺) be an anchored bundle in a tangent category �. Write the span
+�A.Wn := (M
+π◦πi
+←−− An
+̺n
+−→ TnM ),
+�A.� := (M = M = M ).
+A choice of prolongations for (π,ξ,λ,̺) is a strict choice of horizontal composition for each V ∈ Weil1:
+�A.V = �A.(Wn[1]...Wn[k]) := �A.Wn[1] ⊠ ··· ⊠ �A.Wn[k].
+We will write the span as follows:
+A.V
+M
+T V .M
+πV
+̺V
+(notice that the apex is not hatted). Given a choice of prolongations for an anchored bundle (π,ξ,λ,̺),
+the category Span(π,ξ,λ,̺) is defined as follows:
+89
+
+• Objects are �A.V for V ∈ Weil1.
+• Morphisms are given by pairs
+(f ,φ) : �A.V → �A.U
+where f : A.V → A.U and φ : V →U determine a span morphism of the form
+M
+A.V
+T V .M
+M
+A.U
+T U .M
+πV
+̺V
+φ.M
+πV
+̺U
+f
+as discussed in Definition 4.3.1.
+• Tensor structure: The tensor product is defined using the horizontal composition ⊠ as defined in
+Definition 4.3.1.
+The idea is to show that an involution algebroid structure on an anchored bundle induces a tangent
+structure on the monoidal categoryofprolongations, and then toapplyLeung’stheorem. The following
+two lemmas will simplify this proof.
+Lemma 4.3.5. Let (π : A → M ,ξ,λ,̺) be an anchored bundle with chosen prolongations in a tangent
+category �, and identify the monoidal category Span(π,ξ,λ,̺).
+(i) There is a functor
+U ̺ : Span(π,ξ,λ,̺) → �
+constructed by sending a span morphism to the morphism between the objects at its apex:
+M
+A.V
+T V .M
+A.V
+M
+A.U
+T U .M
+A.U
+πV
+̺V
+φ.M
+πV
+̺U
+f
+f
+(ii) Suppose we have a square
+�A.U
+�A.Y
+�A.X
+�A.Z
+(f ,φ)
+(g ,ψ)
+(l ,α)
+(r,β)
+whose image under U ̺ is a T -pullback in �, and so that the square in Weil1 is a transverse T -
+pullback:
+A.U
+A.Y
+U
+Y
+A.X
+A.Z
+X
+Z
+f
+g
+l
+r
+φ
+ψ
+β
+α
+⌟
+⌟
+Then U ̺ reflects the limit; that is, the original square in Span(π,ξ,λ,̺) is a T -pullback.
+(iii) T -pullbacks of the form described in (ii) are closed under ⊠.
+90
+
+Proof. The functor in in (i) is straightforward to construct, as it simply forgets the left and right legs
+of the spans. For (ii), note that because the Weil1 part of the diagram is a transverse T -pullback, then
+given a pair of maps
+�A.V
+�A.U
+�A.Y
+�A.X
+�A.Z
+(f ,φ)
+(g ,ψ)
+(l ,α)
+(r,β)
+(x,ω)
+(y,γ)
+a unique span morphism �A.V → �A.U may be induced using the apex map from �, and the unique
+map induced in Weil1 by the universality of transverse squares (this square is also universal in �), so
+the span morphism diagram will commute by universality.
+For (iii), T -pullback squares of the form in (ii) are closed under ⊠ as transverse squares in Weil1
+are closed under ⊗, so the result follows by the commutativity of limits and by applying part (ii) of this
+lemma.
+Observation 4.3.6. It will be useful to have a “flat” presentation of the prolongation A.Wn(1) ...Wn(k).
+The higher prolongations of an anchored bundle may be concretely described as the T -pullback of the
+zig-zag below:
+ˆA.Wn(1)...Wn(k)
+ˆA.Wn(1)
+Tn[1]. ˆA.Wn(2)
+Tn(1)...n(k−1) ˆA.Wn(k)
+Tn(1)M
+...
+Tn(1).̺n(2)
+Tn(1)...n(k).(π◦π0)
+̺n(1)
+Tn[1].(π◦π0)
+so that the prolongation A.W n[1]...W n[k] may be written concretely as
+(u1,...,uk) : An[1]̺′×T.π′Tn[1].An[2]T.̺×T 2.π...̺′×T.π′Tn[1]...n[k−1]An[k].
+Furthermore, the choice of prolongation identifies the following limits:
+ˆA.U V
+T U . ˆA.V
+ˆA.U
+T U M
+̺U
+T U .πV
+⌟
+ˆA.U V
+T U ˆA.V
+T U M
+T U M
+ˆA.U
+T U M
+⌟
+̺U
+T U .πV
+so that
+A.U V = A.U ⊠ A.V = A.U ⊠ idM ⊠ A.V
+where idM is the span M = M = M .
+Note that a ̺ sends the involution algebroid structure map to its corresponding tangent structure
+map. Each of the structure maps, then, gives a span morphism where ⊠ is well defined:
+91
+
+Definition 4.3.7. Let (π : A → M ,ξ,+q,λ,̺,σ) be an involution algebroid in � with chosen prolonga-
+tions. Then define the following maps in Span(π,ξ,λ,̺):
+• The projection p : ˆA.W → ˆA.�,
+M
+A
+T M
+M
+M
+M
+̺
+π
+p
+π
+• The zero map 0 : ˆA.� → ˆA.W ,
+M
+M
+M
+M
+A
+T M
+0
+π
+̺
+ξ
+• The addition map + : ˆA.W2 → ˆA.W ,
+M
+A2
+T2M
+M
+A
+T M
+̺2
+π◦πi
++.M
+π
+̺
++
+• The lift map ℓ : ˆA.W → ˆA.W W ,
+M
+A
+T M
+M
+A̺×T πT A
+T 2M
+π
+̺
+ℓ.M
+π◦π0
+T.̺◦π1
+(ξπ,λ)
+• The flip map c : ˆA.W W → ˆA.W W ,
+M
+A̺×T πT A
+T 2M
+M
+A̺×T πT A
+T 2M
+c .M
+π◦π0
+̺◦π1
+π◦π0
+̺◦π1
+σ
+The idea is to show that the monoidal category of chosen prolongations Span(π,ξ,λ,̺) for an in-
+volution algebroid has a tangent structure generated by the structure maps in Definition 4.3.7 and the
+endofunctor �A.W ⊠(−). Using the flat presentation, we can then show thatU ̺ will determine a tangent
+functor in �. The following lemma about ̺ will be useful in constructing the natural transformation
+part of a tangent functor.
+Definition 4.3.8. Let (π : A → M ,ξ,λ,̺) be an anchored bundle with chosen prolongations. Recall that
+by Definition 4.3.4, the right leg of A.U is written ̺, so it induces a span map:
+M
+A.U
+T U M
+M
+T U M
+T U M
+̺U
+̺U
+πU
+pU
+92
+
+This map has a flat presentation as
+A.U V
+A.U ̺U ×T U .πV T U .A.V
+T U .A.V
+T U M id ×T U .πV T U .A.V
+̺U ⊠A.V
+πU ×T U .A.V
+We write the map
+̺U .V := ̺U ⊠ ( �A.V )
+which corresponds to the following span morphism:
+A.U V
+A.U
+T U .A.V
+M
+T U M
+T U V M
+T U M
+T U .A.V
+T U .A.V
+̺U
+T U .̺V
+T U .πV
+̺U
+πU
+⌟
+pU .M
+T U .πV
+Theorem 4.3.9 (The Weil Nerve). There is a fully faithful functor
+NWeil : Inv(�) → [Weil1,�]
+that sends an involution algebroid to the transverse-limit-preserving tangent functor:
+( �A,α) : Weil1 → �
+Proof. For the first step of this proof, we show that an involution algebroid structure on an anchored
+bundle (π : A → M ,ξ,λ,̺) determines a tangent category structure on the monoidal category Span(π :
+A → M ,ξ,λ,̺).
+We check that the endofunctor ˆA ⊠ (−) determines a tangent structure, with the structure maps
+given by Definition 4.3.7:
+[TC.1] Additive bundle axioms:
+(i) Use Lemma 4.3.5 to see that
+ˆA.W2 = ˆA.W p ×p ˆA.W Aπ×πA;
+this is preserved by ˆA.V ⊠ (−).
+(ii) The triple ( ˆA.+, ˆA.p, ˆA.0) = (+q ,π,ξ) is an additive bundle induced by Proposition 2.4.4,
+and ⊠ preserves pullbacks (and therefore additive bundles), so the additive bundle axioms
+hold.
+93
+
+[TC.2] Symmetry axioms:
+(i) ˆA.c ◦ ˆA.c = id follows from the involution axiom σ ◦ σ = id .
+(ii) For Yang–Baxter, note that
+( ˆA ⊠ c ) ◦ (c ⊠ ˆA) ◦ ( ˆA ⊠ c ) = (c ⊠ ˆA) ◦ ( ˆA ⊠ c ) ◦ (c ⊠ ˆA)
+follows from the Yang–Baxter equation on an involution algebroid
+(σ × c ) ◦ (id × T.σ) ◦ (σ × c ) = (id × T.σ) ◦ (σ × c ) ◦ (id × T.σ),
+since
+σ × c .A = ( ˆA.c ) ⊠ ( ˆA.W ) and id × T.σ = ( ˆA.W ) ⊠ c .
+(iii) For the naturality conditions:
+(a) The interchanges of +,0,p all follow from the fact that
+σ : (A.W W,λ × ℓ) → (A.W W,id × c ◦ T.λ)
+is linear, and so is an additive bundle morphism.
+(b) The axiom
+ℓ.T ◦ c = T.c ◦ c .T ◦ T.ℓ
+is equivalent to the equation
+(σ × c ) ◦ (1× T.σ) ◦ (ˆλ× ℓ) = (1× T ˆλ) ◦ σ
+which is equivalent to the double linearity axiom on σ by Proposition 3.3.4.
+[TC.3] The lift axioms:
+(i) The additive bundle equations are a consequence of λ being a lift and + being the addition
+induced by the non-singularity of λ.
+(ii) The coassociativity axiom
+ℓ.T ◦ ℓ = T.ℓ◦ ℓ
+is equivalent to
+(ˆλ× ℓ) ◦ ˆλ = (id × T.ˆλ) ◦ ˆλ
+proved in (i) of Proposition 3.2.6.
+(iii) The symmetry of comultiplication, c ◦ℓ = ℓ, is given by the unique equation for an involu-
+tion algebroid, so that σ ◦ (ξ ◦ π,λ) = (ξ ◦ π,λ).
+(iv) The universality of the lift follows from part (ii) of Proposition 3.2.6; Lemma 4.3.5 ensures
+that for any V ∈ Weil1, �A.V ⊠ µ and µ ⊠ �A.V are universal.
+This lemma puts a tangent structure on Span(π,ξ,λ,̺). Now consider the functor sending spans to
+the apex map,
+U ̺ : Span(π,ξ,λ,̺) → �.
+The family of maps
+{̺U .V : ̺U ⊠ �A.V |U ,V ∈ Weil1}
+94
+
+gives a family of natural transformations
+̺U : A.T U ⇒ T U .A,
+so that the following pair constitute a tangent functor
+(U ̺,̺) : Span(π,ξ,λ,̺) → �.
+Because the universality conditions on Span(π,ξ,λ,̺) followed by reflecting limits in � using Lemma
+4.3.5, it follows that (U ̺,̺) will preserve the tangent-natural limits in Span(π,ξ,λ,̺) corresponding to
+transverse limits in Weil1.
+By Leung’s Theorem 4.1.7 (by way of Corollary 4.2.6), the tangent structure on Span(π,ξ,λ,̺) in-
+duces a strict, monoidal, transverse-limit-preserving functor
+¯A : Weil1 → Span(π,ξ,λ,̺)
+that sends the tensor product ⊗ to the span composition ⊠. By composing the strict tangent functor
+( ¯A,id ) and (U ̺,̺), we have a lax, transverse-limit-preserving, tangent functor:
+(A,̺) : Weil1 → �;V �→ A.V
+Now, check the bijection on morphisms. Starting with an involution algebroid morphism (f ,m) :
+A → B, note that this gives a span morphism ˆf :
+M
+A
+T M
+N
+B
+T N
+T.m
+m
+f
+This gives a natural definition of ˆf .V using the horizontal composition of span morphism, so that
+ˆf .(U V ) = ˆf .U ⊠ ˆf .V and ˆf .� = m,
+(4.2)
+giving a family of maps { ˆfU : U ∈ objects(Weil1)}. Because f will commute with the structure maps
+{π,ξ,+,(ξ ◦ π,λ),σ}, it follow immediately that ˆf is a natural transformation, because the following
+calculation holds for each θ : X → Y ∈ {p,0,+,ℓ,c }:
+ˆf .U Y V ◦ ( ˆA.U ⊠ θ ⊠ ˆA.V )
+= ( ˆf .U ⊠ ˆf .Y ⊠ ˆf .V ) ◦ ( ˆA.U ⊠ θ ⊠ ˆA.V )
+= ˆf .U ⊠ ( ˆf .Y ◦ θ) ⊠ ˆf .V
+= ˆf .U ⊠ (θ ◦ ˆf .X ) ⊠ ˆf .V
+= ( ˆA.U ⊠ θ ⊠ ˆA.V ) ◦ ˆf .U X V
+Tangent naturality will follow by the preservation of the anchor map by f . The equality, for any Weil
+algebra U , of the diagrams
+M
+A.U
+T U M
+M
+A.U
+T U M
+N
+B.U
+T U N
+M
+T U M
+T U M
+N
+T U N
+T U N
+N
+T V N
+T V N
+T U .m
+m
+πU
+̺U
+̺U
+ˆf .U
+̺U
+pU
+πU
+pU
+̺U
+pU
+T U .m
+T U .m
+m
+̺U
+=
+95
+
+is precisely the tangent-naturality condition from Definitions 1.3.12, 4.2.1.
+For the inverse of this mapping, consider a tangent natural transformation (Definition 1.3.12)
+γ : (A,α) → (B,β),
+A ◦ T (A)
+B ◦ T (A)
+T ◦ A(A)
+T ◦ B(A)
+γT A
+αA
+βT A
+T γA
+where (A,α) and (B,β) are tangent functors Weil1 → � built out of involution algebroids with chosen
+prolongations. For any U ,V , the map γ.U V decomposes as γ.U ⊠ γ.V :
+�A.T.T
+T. �A.T
+�B.T.T
+T. �B.T
+�A.T
+T. �A
+�B.T
+T. �B
+�A.p.T =π0
+α.T =π1
+�A.p
+T. �A.p
+�B.p
+T. �B.p
+�B.p.T =π0
+β.T =π1
+γ.T
+T.γ
+T.γ.T
+γ.T T
+Applying this relationship inductively, it is clear that the base maps γ.W and γ.� determine the entire
+morphism γ.V :
+M
+A.U
+T U M
+N
+B.U
+T U N
+M
+An[1]
+Tn[1].M
+...
+Tn[1]...Tn[k].M
+N
+Bn[1]
+Tn[1].M
+...
+Tn[1]...Tn[k].N
+Tn[1]...Tn[k].γ.�
+γ.�
+̺n[1]
+π◦πi
+π◦πi
+̺n[1]
+γ.Wn[1]
+Tn[1].γ.�
+Tn[1].(π◦πi)
+Tn[1].(π◦πi)
+...
+T U .γ.�
+πU
+πU
+̺U
+̺U
+γ.�
+γ.U
+=
+Thus, every tangent-natural transformation is constructed out of a pair
+(γ.� : M → N ,γ.W : A → B)
+using the ⊠ construction from Equation 4.2. All that remains to show is that this pair is an involution
+algebroid morphism.
+Tangent naturality gives the following two coherences:
+̺B ◦ γ.W = T.γ.� ◦ ̺A and σB ◦ γ.W W = γ.W W ◦ σA
+since ̺B = β.W,̺A = α.W,σB = B.c , and σA = A.c by construction. The following diagram proves
+that γ.W preserves the lifts, so that (γ.W,γ.�) is an involution algebroid morphism:
+T.A.T
+T.A.T
+T.A.T
+T.B.T
+T.B.T
+T.B.T
+A.T.T
+A.T.T
+B.T.T
+B.T.T
+A.T
+A.T
+A.T
+B.T
+B.T
+B.T
+A.ℓ
+α.T
+(ξπ,λA)
+π1
+T.γ
+γ.T
+m
+B.ℓ
+β.T
+(ξπ,λB )
+β.T
+λA
+λB
+96
+
+Thus, a tangent natural transformation ( �A,α) → ( �B,β) is exactly a morphism of involution algebroids
+A → B, proving the theorem.
+Now, the projection for a Lie algebroid is a submersion, as we may make a choice of prolongations
+for each U ∈ Weil1. These prolongations lead to a new observation about Lie algebroids: they embed
+into a category of functors into smooth manifolds.
+Corollary 4.3.10. Using the Weil nerve construction, the category of Lie algebroids embeds into the
+tangent-functor category:
+LieAlgd �→ [Weil1,SMan].
+4.4
+Identifying involution algebroids
+This section identifies those tangent functors
+(A,α) : Weil1 → �
+that are involution algebroids as precisely those where A preserves transverse limits and α is a T -
+cartesian natural transformation (Definition 4.4.1). These conditions will force each A.V to be the
+V -prolongation of the underlying anchored pre-differential bundle:
+(A.p : A.T → A, A.0 : A → A.T, A.T
+A.ℓ
+−→ A.T.T
+α.T
+−→ T.A.T, α : A.T → T.A)
+(these conditions also ensure that this tuple is an anchored differential bundle).
+Initially, it is only clear that α is T -cartesian for the projection p. Indeed, recall that the prolonga-
+tion A.U V is defined to be the T -pullback of the cospan:
+�A.U
+αU
+−→ T U .A.�
+T U .A.p V
+←−−−−− T U . �A.V
+Then consider the following diagram:
+�A.U W V
+T U . �A.T.T V
+�A.U .V
+T U . �A.V
+�A.T U
+T U . �A.�
+T U . ˆA.p.V
+�A.T U .p.T V
+�A.T U .p V
+T U . �A.p V
+⌟
+⌟
+This means that every naturality square of α for p is a T -pullback; natural transformations satisfying
+this property for every map in the domain category are called T -cartesian.
+Definition 4.4.1. A natural transformation γ : F ⇒ G is cartesian whenever each naturality square
+F C
+G C
+F D
+G D
+F.f
+γ
+γ
+G .f
+⌟
+is a pullback. A natural transformation between functors into a tangent category is T -cartesian when-
+ever each component square is a T -pullback (we will generally suppress the T when the context is clear).
+97
+
+Now, recall that the Weil complex determined by an involution algebroid has A.U .V determined
+by the following T -pullback squares:
+�A.U .V
+T U . �A.V
+�A.T U
+T U . �A.�
+�A.T U .p V
+T U . �A.p V
+⌟
+Then it is not difficult to show that the T -cartesian condition on a Weil complex forces it to be an
+involution algebroid. We first need:
+Definition 4.4.2. A T -cartesian Weil complex in � is a tangent functor
+(A,α) : Weil1 → �
+for which A sends transverse limits to T -limits and α is a T -cartesian natural transformation.
+The first condition to check is that a T -cartesian Weil complex gives a natural anchored bundle ˆA
+whose Weil prolongations coincide with the functor assignments on objects.
+Proposition 4.4.3. Let (A,α) be a T -cartesian Weil complex. Then we have an anchored bundle
+(M := A.�,
+ˆA := A.W, π := A.π, ξ := A.ξ, λ := α.T ◦ A.ℓ).
+Furthermore,
+� ( ˆA) = A.W W,
+� 2( ˆA) = A.W W W.
+Proof. Suppose we have a tangent functor (F,α) : � → � and a differential bundle (π,ξ,λ) in �. If F pre-
+serves T -pullbacks of π, it preserves the additive bundle structure on (π,ξ,+), so to show (F.π,F.ξ,α ◦
+F.λ) is universal it suffices to show that the following diagram is a T -pullback in �:
+F.A2
+T.F.A
+F.M
+T.F.M
+F.π2
+µα◦F.λ
+F.0
+F.T.π
+Expand this to
+F.A2
+F.T.A
+T.F.A
+F.M
+F.T.M
+T.F.M
+F.π2
+F.µλ
+F.0
+T.F.π
+F.T.π
+α.A
+α.M
+In this case, it restricts to the diagram
+A.T2
+A.T 2
+T.A.T
+A
+A.T
+T.A
+A.µ
+α.T
+A.T.p
+α
+T.A.p
+A.0
+⌟
+98
+
+Each square is a T -pullback by hypothesis, so the universality of the lift follows by the T -pullback
+lemma. Because the complex is T -cartesian, the assignment A.V gives a coherent choice of prolonga-
+tions by the T -pullback
+A.U .V
+T U .A.V
+A.U
+T U .A
+A.U .p V
+αU
+T U .A.p V
+αU
+There is, of course, a natural candidate for the involution map.
+Corollary 4.4.4. Let (π : A → M ,ξ,λ,̺) be the anchored bundle induced by a T -cartesian Weil complex
+in a tangent category �. Then we have an involution map
+σ : � (A)
+A.c
+−→ � (A).
+The equations for an involution algebroid should follow immediately by functoriality; one need
+only ensure that the maps take the correct form.
+Lemma 4.4.5. Let A be a T -cartesian Weil complex in a tangent category �, with (π : A → M ,ξ,λ,σ) its
+underlying anchored bundle. Then we have:
+(i) A.c .T = σ × c ,
+(ii) A.T.c = 1× T.σ,
+(iii) A.ℓ.T = ˆλ × ℓ.A,
+(iv) A.T.ℓ = id × T.ˆλ.
+Proof.
+(i) Consider the diagram
+A.T.T.T
+A.T.T.T
+T.T.A.T
+T.T.A.T
+T.T.A
+T.T.A
+A.T.T
+A.T.T
+A.T.T.p
+αT T
+αT T
+T.T.A.p
+A.T.T.p
+T.T.A.p
+c .A.T
+c .A
+A.c
+A.c .T
+⌟
+⌟
+Observe that this forces A.c .T = A.c × c .A.T = σ × c .
+99
+
+(ii) Likewise, the diagram
+A.T.T.T
+A.T.T.T
+T.T.A.T
+T.T.A.T
+T.A
+T.T.A
+A.T
+A.T
+A.T.p.p
+α.T T
+αT
+T.T.A.p
+A.T.p.p
+αT
+T.T.A.p
+α.T T
+T.A.c
+A.T.c
+⌟
+⌟
+forces A.T.c = id × T.A.c = id × T.σ.
+(iii) The diagram
+A.T.T
+A.T.T.T
+T.A.T
+T.T.A.T
+T.A
+T.T.A
+A.T
+A.T.T
+A.ℓ.T
+α.T
+αT T .T
+ℓ.A.T
+A.T.p
+T.A.p
+T.T.A.p
+ℓ.A
+α
+A.ℓ
+A.T.T.p
+αT T
+⌟
+⌟
+⌟
+⌟
+forces A.ℓ.T = A.ℓ× ℓ.A = ˆλ × ℓ.
+(iv) As α is T -cartesian, the following diagram is a T -pullback:
+A.T.T
+A.T.T.T
+T.A.T
+T.A.T.T
+α.T
+A.T.ℓ
+α.T.T
+T.A.ℓ
+⌟
+Using previous results, this means that A.ℓ.T is the unique map making the following diagram
+commute:
+A̺×T.πT A
+A̺×T.πT AT.̺×T 2.πT 2A
+T A
+T AT.̺×T 2.πT 2A
+(π1,π2)
+T.ˆλ
+π1
+which we can see is id × ˆλ.
+Pulling together this lemma and the previous proposition, the following is now clear:
+Proposition 4.4.6. A T -cartesian Weil complex determines an involution algebroid.
+However, we have not yet exhibited an isomorphism of categories between the image of the Weil
+nerve functor and T -cartesian Weil complexes. At first glance, the Weil nerve construction only gives
+a Weil complex that is T -cartesian for the tangent projection p ∈ Weil1. Being T -cartesian for p is,
+100
+
+however, sufficient: a Weil complex that is T -cartesian for tangent projections will be T -cartesian for
+every map in Weil1 (a similar result appears in the context of differentiable programming languages;
+see Cruttwell et al. (2019)).
+Proposition 4.4.7. A lax tranverse-limit-preserving tangent functor (F,α) : Weil1 → � for which F pre-
+serves pullback powers of each T U .p is T -cartesian if and only if each
+F.T.T
+T.F.T
+F.T
+T.F
+F.T.p
+αA
+T.F.p
+αB
+is a T -pullback.
+Proof. We only check the converse since the forward implication is trivial. We make use of the T -
+pullback lemma.
+(i) c is an isomorphism, so its naturality square is a T -pullback.
+(ii) For projections T2 → T , the retract of a T -pullback diagram is a T -pullback, so the following
+diagram is universal:
+F.T.T2
+F.T.T2
+F.T.T2
+F.T.T
+T.F.T2
+T.F.T
+F.T.T2
+F.T.T2
+α
+α
+F.T.πi
+F.T.πi
+α
+α
+T.F.πi
+F.T.πi
+(iii) For 0, observe that the following two diagrams are equal:
+F.T
+F.T.T
+F.T
+F.T
+F.T
+T.F
+T.F.T
+T.F
+T.F
+T.F
+α
+F.T.p
+T.F.p
+F.T.0
+T.F.0
+α
+α
+α
+α
+⌟
+⌟
+The right diagram is a T -pullback, and the right square of the left diagram is a T -pullback by
+hypothesis. By the T -pullback lemma, the left square of the left diagram is a T -pullback.
+(iv) For ℓ, observe that
+F.T.T
+F.T.T.T
+F.T.T
+F.T.T
+F.T
+F.T.T
+T.F.T
+T.F.T.T
+T.F.T
+T.F.T
+T.F
+T.F.T
+F.T.ℓ
+α.T
+T.F.ℓ
+α.T.T
+F.T.p.T
+T.F.p.T
+α.T
+⌟
+F.T.p
+T.F.p
+α.T
+α
+⌟
+F.T.0
+T.F.0
+α.T
+⌟
+The outer perimeter of the right diagram is a T -pullback (left square by hypothesis, right square
+by (ii)), as is the right square of the left diagram (by hypothesis). By the T -pullback lemma, the
+left square of the left diagram is a T -pullback.
+101
+
+(v) For +, observe that
+F.T.T2
+F.T.T
+F.T.T
+F.T.T2
+F.T.T
+F.T
+T.F.T2
+T.F.T
+T.F.T
+T.F.T2
+T.F.T
+T.F
+F.T.+
+α.T
+T.F.+
+α.T
+F.T.p
+T.F.p
+α.T
+⌟
+F.T.πi
+T.F.πi
+α.T2
+α.T
+⌟
+F.T.p
+T.F.p
+α
+⌟
+The outer diagram on the right is a T -pullback by composition, and the right square on the left
+diagram is a T -pullback by hypothesis, so the result follows.
+To check that the naturality square is a T -pullback for every map in Weil1, we once again use Leung’s
+characterization of maps in Weil1 from Proposition 4.1.7. Inductively, the set of maps generated by
+{p,0,+,ℓ,c } closed under ⊗ and ◦ follows as T -pullback squares are closed to composition. For maps
+induced by a tranverse limit in Weil1, F preserves transverse limits so this follows by the commutativity
+of limits.
+Theorem 4.4.8. For any tangent category �, the replete image of the Weil nerve functor
+Inv(�) �→ [Weil1,�]
+is precisely the category of T -cartesian Weil complexes.
+Corollary 4.4.9. That α : A.T ⇒ T.A is T -cartesian is equivalent to requiring that the tangent functor
+(A,α) : Weil1 → �
+restricts to an anchored bundle
+(π : A.T
+A.p
+−→ A.�, ξ : A
+A.0
+−→ A.T, λ : A.T
+A.ℓ
+−→ A.T T
+α.T
+−→ T.A.T, ̺ : A.T
+α−→ T.A)
+and each A.T V is the V -prolongation of this anchor bundle.
+Remark 4.4.10. The condition in Corollary 4.4.9 is analogous to the Segal conditions identifying those
+simplicial complexes
+∆ → �
+that are internal categories. Note that every simplicial object has an underlying reflexive graph
+tr1(X ) := (s,t : X ([1]) → X ([0]),i : X ([0]) → X ([1]))
+where X ([n]) is isomorphic to the object of n-composable arrows for the underlying reflexive graph.
+Remark 4.4.11. Notably, being T -cartesian for p is enough to force that a natural transformation is
+T -cartesian for the other tangent-structural natural transformations. This has consequences when one
+uses partial maps to combine topological notions with tangent categories. In this context, a partial map
+N → X with domain M �→ N is a span
+M
+N
+X
+f
+m
+whose right leg is monic. The intuition is that the map f is defined on the subobject M of N , which
+introduces a new problem: what is the proper notion of a subobject in a tangent category? Such a notion
+102
+
+should give rise to a stable class of monics: one that is closed under horizontal span composition. One
+answer is the notion of etale monics: a morphism is etale whenever the naturality square for p is a T -
+pullbacks:
+T M
+T N
+M
+T N
+T.f
+T.f
+p
+p
+⌟
+Geometrically, this means that the morphism is a local diffeomorphism; for example, an etale subobject
+of �n in the Dubuc topos is precisely an open subset in the usual sense. An endofunctor lifts to the partial
+map category whenever it preserves the class of monics. A natural transformation lifts to endofunctors
+on the partial map category whenever it is T -cartesian for the class of monics, and the same proof will
+show that this property holds for etale monics Cruttwell et al. (2019).
+4.5
+The prolongation tangent structure
+One of the most important consequences of the Weil Nerve Theorem 4.3.9 is that the category of in-
+volution algebroids (with chosen prolongations) may be equipped with two tangent structures. The
+first tangent structure is the pointwise tangent structure described in Proposition 3.3.6. The tangent
+functor sends
+(A,α) �→ (T.A : Weil1 → �,c .A ◦ T.α : T.A.T ⇒ T.T.A)
+(recall the composition of tangent functors given in Example 1.3.11 (ii)). The structure morphisms will
+be given by whiskering, so in this case θ.A,θ ∈ {p,0,+,ℓ,c }. The restriction to tangent functors that
+preserve transverse limits along with the fact that the natural part α is T -cartesian, however, ensures
+that precomposition with the tangent functor
+(A,α) �→ (A.T : Weil1 → �,T.α ◦ A.c : A.T.T ⇒ T.A.T )
+returns an involution algebroid. The structure maps are once again given by whiskering, with the pre-
+composition tangent structure A.θ,θ ∈ {p,0,+,ℓ,c }. Preservation of transverse limits guarantees that
+this tangent structure will satisfy the necessary universality conditions.
+Proposition 4.5.1 (Proposition 3.3.10). The category of involution algebroids with chosen prolongations
+in a tangent category � has a second tangent structure, where the action by Weil1 is given by
+(A,α) �→ (A.T : Weil1 → �,α.T ◦ ˆA.c : ˆA.T.T ⇒ T. ˆA.T ).
+Proof. The proposition statement means that the structure morphisms for this new involution alge-
+broid are given by
+(A.T,α.T ◦ A.c ) ∼=
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+α.T ◦ A.c = ̺′ :
+� (A)
+π1
+−→ T A
+A.T.p = π′ :
+� (A)
+p◦π1
+−−→ A
+A.T.0 = ξ′ :
+A
+(ξ◦π,0)
+−−−→ � (A)
+̺′ ◦ A.T.ℓ = λ′ :
+� (A)
+λ×ℓ
+−−→ T.� (A)
+A.T.c = σ′ :
+� 2(A)
+σ×c
+−−→ � 2(A)
+103
+
+Similarly, we can see that
+(A.T.T,α.T.T ◦ A.c .T ◦ A.T.c )
+=
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+α.T.T ◦ A.c .T ◦ A.T.c = ̺′′ :
+� 2(A)
+(π1,π2)
+−−−→ T.� (A)
+A.T.p = π′′ :
+� 2(A)
+(p◦π1,p◦π2):
+−−−−−−−→ � (A)
+A.T.0 = ξ′′ :
+� (A)
+(ξ◦π◦π0,0◦π1,0◦π2)
+−−−−−−−−−−−→ � 2(A)
+̺′′ ◦ A.T.ℓ = λ′′ :
+� 2(A)
+(λ×ℓ×ℓ)
+−−−−→ T.� 2(A)
+A.T.T.c = σ′′ :
+� 3(A)
+(σ×c ×c )
+−−−−−→ � 3(A)
+These coincide with the involution algebroids � ′(A),� ′.� ′(A) in Proposition 3.3.10: the second tan-
+gent structure follows from the fact that the natural transformations for the tangent structure there are
+given by
+A.φ : A.U ⇒ A.V,φ :U → V ∈ {p,0,+,ℓ,c }.
+The result follows as a corollary of Theorem 4.3.9.
+The Jacobi identity for involution algebroids
+Classically, the theory of Lie algebroids uses the algebra of sections Γ(π). One key observation is that
+when using the Lie tangent structure (Inv(�),� ), sections of π are in bijective correspondence with
+χ� (A). This observation allows for different statements about Lie algebroids to be translated into for-
+mal statements about the tangent bundle in (Inv(�),� ).
+Proposition 4.5.2. Let A be an involution algebroid in �. There is a bijection between the sections of π
+in � and the vector fields on A in Inv(�):
+X ∈ Γ(π) �→ ((id ,T X ◦ ̺),X ) : A → TL(A);
+ˆX ∈ χ� (A) �→ ˆXR : A.R → A.T.
+Proof. Recall the coherence for tangent natural transformations γ : (H ,φ) ⇒ (G ,ψ):
+H .T
+K .T
+T.H
+T.K
+γ.T
+φ
+ψ
+T.γ
+We specify this to a morphism ˆX : (A,α) ⇒ TL(A,α) = (A.T,α.T ) at �,W :
+A
+A̺×T πT A
+T M
+T A
+X .T
+̺
+π1
+T.X
+and so infer that, if we set X := XR , we have π1 ◦ X .T = T (X ) ◦ ̺. Furthermore, the condition that
+pL ◦ X = id forces id = π0 ◦ X .T ; thus, we can see that every section X of pL is given by a morphism of
+the form ((id ,T X ◦ ̺),X )) on the underlying involution algebroids, where π◦ X = id .
+We now show that every X ∈ Γ(π) gives rise to a section of πL. Observe that the following is a mor-
+phism of involution algebroids:
+A
+A̺×T πT A
+M
+A
+π
+(id,T X ◦̺)
+p◦π1
+X
+104
+
+Note that it is well typed, as T.π◦ T.X ◦ ̺ = ̺ ◦ id . Check that it is a bundle morphism:
+p ◦ π1 ◦ (id ,T.X ◦ ̺) = p ◦ T.X ◦ ̺ = X ◦ p ◦ ̺ = X ◦ π
+and that it is linear:
+(λ ◦ π0,ℓ◦ π1) ◦ (id ,T.X ◦ ̺) = (λ,ℓ◦ T.X ◦ ̺)
+= (λ,T 2.X ◦ ℓ◦ ̺) = (λ,T 2.X ◦ λ) = T (id ,T X ) ◦ λ.
+Then check that it preserves the anchor:
+π1 ◦ (id ,T X ◦ ̺) = T X ◦ ̺
+and the involution:
+(π0,π1,T 2X ◦ T ̺π1) ◦ σ = (σ,T 2X ◦ T ̺ ◦ π1σ)
+= (σ,T 2X ◦ c T ̺ ◦ π1) = (σ(π0,π1),c π3) ◦ (id ,T 2X ◦ T ̺ ◦ π1).
+Thus we have that (id ,T X ◦̺π1) is a morphism of involution algebroids, inducing a morphism of
+T -cartesian Weil complexes. Lastly, we check that it is a section of p L
+A , but this is clear, since
+π0 ◦ (id ,T X ◦ ̺) = id ;
+thus we have the desired bijection.
+Recall that given an involution on an anchored bundle, there is a bracket on its set of sections (see
+the explicit construction in Section 3.4). Given an X ,Y ∈ Γ(π), there is a bracket defined as follows:
+ˆλ ◦ [X ,Y ]A +1 (ξπ,0) ◦ Y = ((σ ◦ (id ,T Y ◦ ̺) ◦ X −2 (id ,T X ◦ ̺) ◦ Y )).
+A direct proof of the Jacobi identity is a detailed calculation (see the original preprint on involution al-
+gebroids Burke and MacAdam (2019)) and still relies on Cockett and Cruttwell’s result for an arbitrary
+tangent category with negatives. As a result of Proposition 4.5.2, we can instead use Cockett and Crut-
+twell’s result directly:
+Corollary 4.5.3. Let A be a complete involution algebroid in a tangent category � with negatives. There
+is a Lie bracket defined on Γ(π), [−,−] induced by
+ˆλ ◦ [X ,Y ]A +1 (ξπ,0) ◦ Y = ((σ ◦ (id ,T Y ◦ ̺) ◦ X −2 (id ,T X ◦ ̺) ◦ Y )).
+Proof. The bracket is induced by Rosicky’s universality diagram, as
+0 = p ◦ ((σ ◦ (id ,T Y ◦ ̺) ◦ X − (id ,T X ◦ ̺) ◦ Y )) − 0Y
+= T p ◦ ((σ ◦ (id ,T Y ◦ ̺) ◦ X − (id ,T X ◦ ̺) ◦ Y )) − 0Y .
+We look at the Lie tangent structure for Inv∗(A); this is precisely the vector field induced by
+e vR([ ˆX , ˆY ]).
+We complete the proof by using the result that for any A in a tangent category with negatives, the
+bracket on χ(A) that is defined by
+ℓ◦ [X ,Y ] = (c ◦ T.X ◦ Y − T.Y ◦ X ) − 0X
+satisfies the Jacobi identity.
+105
+
+Identifying categories of involution algebroids
+Section 4.4 identified whenever a functor Weil1 → � is an involution algebroid, whereas this section
+identifies tangent categories � that embed into the category of involution algebroids in some tangent
+category �. We call this structure an abstract category of involution algebroids. This notion involves
+some 2-category theory, using a modified notion of codescent (see Bourke (2010) for a development of
+codescent).
+Recall that for any tangent category �, the category of involution algebroids has � as a reflective
+subcategory. Furthermore, because limits of involution algebroids are computed pointwise, this reflec-
+tor is left-exact. This left-exact reflection is the main structure we axiomatize.
+Definition 4.5.4. An abstract category of involution algebroids is a tangent category � with a left-exact
+T -cartesian tangent localization (Z ,̺) : � → �, where L satisfies a codescent condition:
+TangCatStrict(Weil1,�) �→ TangCatLax(Weil1,�)
+L∗
+−→ TangCatLax(Weil1,�)
+(where L∗ denotes post-composition by L) is fully faithful.
+Example 4.5.5. The category of involution algebroids in any tangent category � is an abstract category
+of involution algebroids using (Inv(C ),� ). The reflector is the functor sending an involution algebroid
+to its base space; the T -cartesian natural transformation is the anchor map. Any tangent subcategory of
+Inv(�) that contains � as a full subcategory will give rise to an abstract category of involution algebroids.
+Proposition 4.5.6. Let � �→ � be an abstract category of involution algebroids. Then there is an embed-
+ding � �→ Inv(�).
+Proof. The proof follows by treating objects in � as strict tangent functors Weil1 → � and morphisms
+as tangent natural transformations.
+Weil1
+�
+�
+Z
+� (−,A)
+� (−,B)
+� (−,f )
+The natural part of Z is T -cartesian, and the functor part preserves limits, so Z .� (−,A) =: Z [A] de-
+termines an involution algebroid in �, and f a morphism of involution algebroids. The embedding is
+guaranteed by the codescent condition so that the post-composition functor is fully faithful.
+Corollary 4.5.7. An abstract category of involution algebroids � �→ � is exactly a full subcategory � �→
+� �→ Inv(�).
+106
+
+Chapter 5
+The infinitesimal nerve and its realization
+The main thrust of Chapters 2, 3, and 4 has been that the tangent categories framework allows for Lie
+algebroids to be regarded as tangent functors
+Weil1 → SMan
+which satisfy certain universality conditions. This chapter, which is more experimental than the pre-
+vious four chapters and represents work still in progress, puts Lie algebroids into the framework of
+enriched functorial semantics. This new perspective on algebroids uses Garner’s enriched perspective
+on tangent categories (Garner (2018)) and the enriched theories paradigm from Bourke and Garner
+(2019). The functorial-semantics presentation of the Lie functor will generalize the Cartan–Lie the-
+orem (that the category of Lie algebras is a coreflective subcategory of Lie groups) into a statement
+within the general theory of functorial semantics.
+The goal is to show that the infinitesimal approximation of a groupoid, as discussed in Example
+3.1.2, may be constructed as a nerve, just like Kan’s original simplical approximation of a topological
+space. The nerve of a functor K : � → � approximates objects and morphisms in � by � -presheaves,
+so it sends an object in � to the � -presheaf
+NK : � → �;C �→ �(K −,C ).
+Thus there will be an infinitesimal object,
+∂ : Weilop
+1
+→ Gpd(� )
+where Gpd(� ) denotes groupoids in the category � of Weil spaces (formally defined in Section 5.1).
+The nerve of ∂ has a left adjoint, the Lie realization, given by the left Kan extension (just as Kan’s geo-
+metric approximation of a simplicial set is, in Kan (1958)):
+Weilop
+1
+Gpd(� )
+[Weil1,� ]
+�
+∂
+Lan� ∂
+(5.1)
+Theorem 5.5.13 (The Lie Realization). There is a tangent adjunction between the category of involution
+algebroids and groupoids in � , where each functor preserves products and the base spaces.
+Gpd(� )
+Inv(� )
+N∂
+|−|∂
+⊣
+107
+
+Note, however, that the left Kan extension in Equation 5.1 does not immediately give the desired
+adjunction of Theorem 5.5.13, as there is no guarantee that the nerve functor N∂ lands in the category
+of algebroids. To prove this we must revisit the work in Section 4.3 presenting involution algebroids as
+those functors A : Weil1 → � for which each A(V ) is the prolongation of its underlying anchored bundle;
+this leads naturally to the formalism for enriched theories developed in Bourke and Garner (2019).
+The presentation of algebroids as models of an enriched theory requires situating the categories
+of differential bundles, anchored bundles, and involution algebroids in � as full subcategories of � -
+presheaf categories on small � -categories. Section 5.1 reviews the work in Garner (2018) characteriz-
+ing tangent categories as categories enriched in � = Mod(Weil1,Set) (the cofree tangent category on
+Set, by Observation 4.2.5).
+Section 5.2 reconfigures the content of Chapter 2 using the enriched perspective, so that lifts are � -
+functors from the � -monoid D + 1, while differential bundles are a reflective subcategory of functors
+from its idempotent splitting. Similarly, the category of anchored bundles are a reflective subcategory
+of the category [Weil1
+1,�], where Weil1
+1 is the category of 1-truncated Weil algebras (Definition 5.2.8).
+Section 5.3 reviews the basic idea of an enriched nerve/approximation. A particularly important ex-
+ample is the linear approximation of a reflexive graph, the functor introduced in Example 3.2.7, which
+is the nerve of a functor
+∂ : (Weil1
+1)op → Gph(� ).
+Section 5.4 appliesthe enriched theoriesframework introduced in Bourke and Garner(2019), where
+a dense subcategory of a locally presentable � �→ � forms the “arities”, and a bijective-on-objects func-
+tor � → � is the theory. The category of models is the pullback of (enriched) categories:
+��
+[� ,� ]
+�
+[� op,� ]
+The first step is to freely complete the � -category Weil1
+1 of truncated Weil algebras (Definition 5.2.8)
+so that its base anchored bundle has all prolongations; call this � -category � . Then, in every tangent
+category�, the categoryofanchored bundleswith chosen prolongations(Definition 4.3.4) in � embeds
+fully and faithfully into the functor category:
+Anc� (�) �→ [� ,�]
+(here, Anc� (�) denotes the category of anchored bundles with chosen prolongations). In particular,
+the tangent bundle on � in Weil1 determines a bijective-on-objects functor
+� → Weil1
+so that the category of involution algebroids in any tangent category � is the pullback in � Cat:
+Inv∗(�)
+[Weil1,�]
+Anc∗(�)
+[� op,�]
+⌟
+This means that the category of involution algebroids in � is monadic over the category of anchored
+bundles in � using the monad-theory correspondence from Bourke and Garner (2019).
+108
+
+The final section looks at the category of � -groupoids. Essentially, the free groupoid over the lin-
+ear approximation of a graph will now give an infinitesimal object Weil1 → Gpd(� ). The nerve of
+∂ : Weilop
+1
+→ Gpd(� )—the infinitesimal approximation—has a right adjoint via the realization of the
+nerve functor from Definition 5.3.5. This is used to prove the culminating Theorem 5.5.13.
+The individual pieces of categorical machinery used in this chapter are not new (the enriched per-
+spective on tangent categories, enriched nerve constructions, enriched theories). However, all of the
+results dealing with the application of enriched nerve constructions and enriched theories to tangent
+categories is original work of the author.
+5.1
+Tangent categories via enrichment
+This section gives a quick introduction to Garner’s enriched perspective on tangent categories. The
+enriched approach to tangent categories first appeared in Garner (2018) and builds on the category
+perspective on tangent categories introduced in Leung (2017). Garner was able to exhibit some of the
+major results from synthetic differential geometry as pieces of enriched category theory; for example,
+the Yoneda lemma implies the existence of a well-adapted model of synthetic differential geometry.
+The category of Weil spaces is the site of enrichment for tangent categories and is closely related to
+Dubuc’s Weil topos from his original work on models of synthetic differential geometry Dubuc (1981);
+a deeper study of this topos may be found in Bertram (2014). Recall that the category Weil1 is the free
+tangent category over a single object. The category of Weil spaces is the cofree tangent category over
+Set, which is the category of transverse-limit-preserving functors Weil1 → Set by Observation 4.2.5.
+Call this the category of Weil spaces, and write it � . Just as a simplicial set S : ∆ → Set is a gadget
+recording homotopical data, a Weil space records infinitesimal data.
+Definition 5.1.1. A Weil space is a functor Weil1 → Set that preserves transverse limits (Definition 4.1.5):
+that is, the ⊗-closure of the set of limits
+
+
+
+
+
+
+
+Tn+m
+Tm
+Tn
+�
+⌟
+,
+T2
+T 2
+�
+T
+0
+µ
+⌟
+,
+Tn
+Tn
+Tn
+Tn
+
+
+
+
+
+
+
+A morphism of Weil spaces is a natural transformation. Write the category of Weil spaces as � .
+Example 5.1.2.
+(i) Every commutative monoid may be regarded as a Weil space canonically. Observe that for every V ∈
+Weil1 and commutative monoid M , one hasthe free V -module structure on M given by |V |⊗CMonM
+(here |V | is the underlying commutative monoid of V ). The commutative monoid |V | is exactly
+�dimV, so that
+|V |⊗CMon M ∼= ⊕dimV M .
+This agrees with the usual tangent structure on a category with biproducts.
+(ii) Following (i), any tangent category � that is concrete—that is, admitting a faithful functorU : � →
+Set—will have a natural functor into Weil spaces (copresheaves on Weil1). Every object A will have
+an underlying Weil space V �→ USet(T V (A)), and whenever U preserves connected limits (such as
+the forgetful functor from commutative monoids to sets), each of the underlying copresheaves will
+be a Weil space.
+109
+
+(iii) Consider a symmetric monoidal category with an infinitesimal object, which by Proposition 4.2.7 is
+a transverse-colimit-preserving symmetric monoidal functor D : Weil1 → �. Then for every object
+X , the nerve (Definition 5.3.1) ND (X ) : �(D −,X ) : Weil1 → Set is a Weil space.
+(iv) For any pair of objects A,B in a tangent category, �(A,T (−)B) : Weil1 → Set is a Weil space by the
+continuity of �(B,−) : � → Set.
+Unlike the category of simplicial sets, the category of Weil spaces is not a topos. The category of
+Weil spaces does, however, inherit some nice properties from the topos of copresheaves on Weil1 by
+applying results from Section 5.3, as it is locally presentable. The basics of locally presentable categories
+can be found in the Appendix 6.2. Roughly speaking, a cocomplete category � has a subcategory of
+finitely presentable objects �f p, those C so that
+�(C ,−) : � → Set
+preserves filtered colimits (i.e. those colimit diagrams that commute with all finite limits in Set). � is
+locally finitely presentable whenever every object is given by the coend
+C ∼=
+� X ∈�f p
+�(X ,C ) · X
+where S · X is the (possibly infinite) product X |S| with |S| the cardinality of the set S. This means that
+� = Lex(�op
+f p,Set), where Lex means the category of finite-limit-preserving functors.
+The third pointofCorollary5.1.4 below, that� islocallyfinitelypresentable asa cartesian monoidal
+category, means that the category �f p is closed under products. This implies that � is locally pre-
+sentable as a � -category (this ends up being an important technical condition whereby locally pre-
+sentable � -categories make sense). Whenever we discuss an arbitrary � -category, we will assume
+that � is locally presentable as a monoidal category.
+Proposition 5.1.3 (Garner (2018)). The category of Weil spaces is a cartesian-monoidal reflective sub-
+category of [Weil1,Set].
+Corollary 5.1.4. The category of Weil spaces is
+(i) a cartesian closed category;
+(ii) a representable tangent category, where the infinitesimal object is given by the restricted Yoneda
+embedding � : Weilop
+1
+→ � ;
+(iii) Locally finitely presentable as a cartesian monoidal category.
+The cofree tangent structure on � is given by precomposition, that is:
+T U .M .(V ) = M .(U ⊗ V ) = M .T U .V.
+This tangent coincides with the representable tangent structure induced by the Yoneda embedding.
+The proof is an application of the Yoneda lemma. Observe that
+[D,M ](V ) ∼= [Weil1,Set](D × � (V ),M ) ∼= [Weil1,Set](� (W V ),M ) ∼= M (W V )
+where D � (V ) = � (W V ) follows because the tensor product in Weil1 is cocartesian and the reflector
+is cartesian monoidal.
+110
+
+Notation 5.1.5. The tangent category � is a representable tangent category, where D = � W (Definition
+1.3.7). We will write the Yoneda functor � : Weilop
+1
+→ � as D(−), so it is closer to the usual notation
+used in representable tangent categories or synthetic differential geometry (a single D may be used as
+shorthand for D(W ), D (n) for D(Wn), etc.). Note that
+D(V ) = D (⊗kWn(i)) =
+K
+�
+D(ni)
+and that D(ℓ) = ⊗,D(c ) = (π1,π0), D(+) = δ, and so on.
+At this point, we are ready to move to the enriched perspective on tangent categories. The basics
+of enriched category theory may be found in the Appendix 6.2, but one definition in particular is im-
+portant to include here.
+Definition 5.1.6. Let � be a � -category for � a closed symmetric monoidal category. For J ∈ � , the
+power by J of an object C ∈ � is an object C J so that the following is an isomorphism:
+∀D : � (J ,�(D,C )) ∼= �(D,C J )
+whereas the copower is given by
+∀D : � (J ,�(C ,D)) ∼= �(J • C ,D).
+Let � ←� � be a full monoidal subcategory of � . A � -category � has coherently chosen powers by � if
+there is a choice of � -powers (−)J so that
+(C J )K = C J ⊗K .
+Likewise, coherently chosen copowers are a choice of � -copowers so that
+K • (J • C ) = (K ⊗ J ) • C .
+The sub-2-categories of � -categories equipped with coherently chosen powers and copowers are � Cat�
+and � Cat� , respectively.
+Wood (1978) proved that the 2-category of actegories over a monoidal � is equivalent to the 2-
+category of [�,Set]-enriched categories with powers by representable functors (Definition 5.1.6), us-
+ing the monoidal structure on [�,Set] induced by Day convolution1. Moreover, Garner (2018) showed
+that a monoidal reflective subcategory � �→
+ˆ
+� exhibits the 2-category of � -categories as a reflective
+sub-2-category of � -categories; this proves that tangent categories are equivalent to a particular class
+of enriched category.
+Proposition 5.1.7 (Garner (2018)). A tangent category is exactly a � -category with powers by repre-
+sentables.
+Proof. For every A,B ∈ �0, the Weil space is defined as
+�(A,B) :=U �→ �(A,T U B).
+1The Day convolution tensor product of presheaves X ,Y : �op → Set is given by X �⊗Y := Lan⊗(X ⊠Y ), where (X ×Y )(V ) =
+X (V ) × Y (V ) (Day (1970)).
+111
+
+The functor �(A,−) is continuous and T −B is an infinitesimally linear functor, so this is a Weil space.
+The following diagram gives the composition map :
+�(B,C ) × �(A,B)
+�(A,C )
+�(B,T U C ) × �(A,T V B)
+�(A,T V U C )
+�(T V B,T V .T U C ) × �(A,T V B)
+T V ×id
+m
+Note that it is natural in U and V .
+By the Yoneda lemma (as the internal hom in � is the internal hom of copresheaves on Weil1),
+�(A,T V B) = (U �→ �(A,B)(V ⊗U )) = [D (V ),�(A,B)]
+so this category has coherently chosen powers by representable functors.
+Now the original notions of (lax, strong, strict) tangent functors can be shown to correspond to
+power-preservation properties of � -functors between � -categories with coherently chosen powers:
+Theorem 5.1.8 (Garner (2018)). We have the following equivalences of 2-categories:
+(i) the 2-category of � -categories with coherently chosen powers and TangCatLax;
+(ii) the 2-category of � -categories with coherently chosen powers and power-preserving � functors
+and TangCatStrong;
+(iii) The 2-category of � -categories with coherently chosen powers and chosen-power-preserving �
+functors and TangCatStrict.
+Note that for a lax tangent functor (F,α) : � → �, the map
+α.X : F.T.X → T.F.X
+can be seen as the unique morphism induced by universality. Conversely, for a strong tangent functor,
+the natural isomorphism α is the isomorphism from a power F.T.X to the coherently chosen power
+T.F.X . In contrast, a strict tangent functor preserves the coherent choice of powers.
+The ability to work with � -categories that do not have powers by representables allows for signif-
+icant flexibility. It is useful to observe that there are � -categories which are not tangent categories.
+Example 5.1.9.
+(i) Every monoid (M ,m,e ) in � gives rise to a one-object � -category whose hom-object is M , com-
+position is m, and unit is e .
+(ii) Given a tangent category �, it is possible to take the full � -category over some set of objects �,
+even though D may not be closed under iterated applications of the tangent functor.
+(iii) The dual of a � -category is a � category, where
+�op (A,B) = �(B,A) : Weil1 → Set.
+Dually, in the case that � is a tangent category, �op will have coherent copowers by representables.
+112
+
+(iv) If a cartesian category � hasan infinitesimal object D (Definition 1.3.7), it hasa natural � -category
+structure (with coherent copowers by representables)
+�(A,B) : Weil1 → Set := �(A × D (−),B).
+Using the dual tangent structure on �op from Proposition 1.3.8, the enrichment on � is exactly the
+enrichment found by regarding � as the dual � -category of the tangent category �op .
+As a final remark, note that the Yoneda lemma applies to � -categories, so there is an embedding
+� �→ [�op ,Set].
+The powers and copowers by representables are computed pointwise in a presheaf category, so they
+inherit the coherent choice. Thus the following holds:
+Corollary 5.1.10 (Garner (2018)). Every tangent category embeds into a � -cocomplete representable
+tangent category.
+(In fact, showing that the Yoneda embedding applies to tangent categories is the main theorem of
+Garner (2018).)
+5.2
+Differential and anchored bundles as enriched structures
+This section gives an enriched-categorical reinterpretation of the work in Chapter 2 regarding differen-
+tial bundles and Chapters 3 and 4 regarding anchored bundles.
+Differential bundles as enriched structures
+As a first case study using the enriched perspective for tangent categories, consider differential bundles.
+Most of the work in Chapter 2 uses the intuition that differential bundles are some sort of tangent-
+categorical algebraic theory; this section will make that intuition concrete. Recall that a lift (Definition
+2.2.3) is a map λ : E → T E . Using the enriched perspective and treating T E as a power (recall Defini-
+tion 5.1.6), this gives the following correspondence:
+λ : 1 → �(E ,E D )
+ˆλ : D → �(E ,E )
+The commutativity condition for ˆλ, then, is translated as follows:
+E
+T E
+D × D
+�(E ,E ) × �(E ,E )
+T E
+T 2E
+D
+�(E ,E )
+⊙
+mE ,E ,E
+ˆλ׈λ
+ˆλ
+λ
+λ
+T.λ
+ℓ
+That is, ˆλ is a semigroup morphism D → �(E ,E ). In any cartesian closed category with coproducts,
+a semigroup may be freely lifted to a monoid using the "exception monad" (−) + 1 from functional
+programming (see, for example, Seal (2013)).
+Definition 5.2.1. Regard the following monoid as the one-object � -category Λ:
+D × D + D + D + 1
+(ιL ◦m|ιL|ιL |ιR )
+−−−−−−−−→ D + 1
+m : (D + 1) × (D + 1) → D + 1
+113
+
+Thus, a lift λ is exactly a functor ˆλ : Λ → �. Now check that morphisms are tangent natural transfor-
+mations. Note that the semigroup D is commutative, so Λ = Λop; the choice of using Λop in the next
+lemma is to be consistent with conventions used in Section 5.3.
+Lemma 5.2.2. The category of lifts in a tangent category � is isomorphic to the category of � -functors
+and � -natural transformations Λop → �.
+Proof. Check that a � -natural transformation is exactly a morphism of lifts f : λ → λ′. Start with the
+� -naturality square:
+D + 1
+�(A,A) × �(A,B)
+�(A,B) × �(B,B)
+�(A,B)
+(λ,f ◦!)
+(f ◦!,l )
+mAB B
+mAAB
+Now, rewriting D + 1 → �(A,B) as a semigroup map D → �(A,B), we have:
+D
+(λ,f ◦!)
+−−−→ �(A,A) × �(A,B)
+mAAB
+−−−→ �(A,B)
+1
+(λ′,f )
+−−→ �(A,T A) × �(A,B)
+1×T
+−−→ �(A,T A) × �(T A,T B)
+mA,T A,T B
+−−−−−→ �(A,T B)
+1
+T f ◦λ′
+−−−→ �(A,T B)
+Similarly, the other path is exactly λ′ ◦ f . Thus, a � -natural transformation is exactly a morphism of
+lifts.
+It is a classical result in synthetic differential geometry that the object D has only one point. In the
+case of � , this follows from the Yoneda lemma (regarding � as a Set-category):
+� (1,D) = Weil1(�[x]/x 2,�) = {! : �[x]/x 2 → �}
+Note that the natural idempotent e : id ⇒ id from Proposition 2.2.8, then, must be the point 0 : 1 → D.
+Note that this idempotent is an absorbing element of the monoid D + 1, so for any f : X → D + 1, it
+follows that m(f ,0◦!) = m(0◦!, f ) = 0◦!. A pre-differential bundle is exactly a lift with a chosen splitting
+of the natural idempotent p ◦ λ.
+Lemma 5.2.3. For every lift ¯λ : Λop → �, the natural idempotent e = p ◦ λ is exactly
+1
+0−→ D
+ιL−→ D + 1 → �(E ,E ).
+Now that the natural idempotent is understood as a map in Λ, that idempotent splits to give the
+theory of a pre-differential object.
+Definition 5.2.4. The � -category Λ+ is given by the set of objects {0,1} with hom-Weil-spaces. Specifi-
+cally,
+• The hom-spaces are Λ+(1,1) = D + 1, otherwise Λ+(i, j) = 1.
+• Composition (writing the original composition from Λ as m) is given by
+m111 : (D + 1) × (D + 1)
+m
+−→ (D + 1)
+m101 : 1× 1
+ιR−→ (D + 1)
+otherwise: mi j k =!
+114
+
+Idempotent splittings are absolute (co)limits, and are preserved by all functors; as this is a limit
+completion, we have the following.
+Lemma5.2.5. The category of pre-differential bundles is exactly the category of �-functors Λ+ → � (that
+is, �-valued presheaves).
+It is straightforward to exhibit the category of differential bundles as a reflective subcategory of
+pre-differential bundles in [Λ+,�] (so long as � has equalizers).
+Proposition 5.2.6. The category of differential bundles in a tangent category � with T -equalizers and
+T -pullbacks is a reflective subcategory of [Λ+,�].
+Proof. The category of pre-differential bundles in � is isomorphic to [Λ+,�]. By Corollary 2.4.8, the
+category of differential bundles is the category of algebras for an idempotent monad on the category
+of pre-differential bundles in �. The reflector sends a pre-differential bundle to the T -equalizer:
+A
+T E
+T E
+e .E
+T.e
+This equalizer will always exist if � has equalizers, and pullbacks of the projection will exist if � has
+pullbacks, so the pullback is a differential bundle. This reflection gives a left-exact idempotent monad
+on [Λ+,�] whose algebras are differential bundles.
+Now, in the case that � is a locally presentable tangent category (such as � ), [Λ+,�] is locally pre-
+sentable and so is the reflective subcategory of differential bundles; thus the following holds.
+Corollary 5.2.7. If � is locally presentable, then DBun(�) is a locally presentable category.
+Anchored bundles
+There are two ways to think about anchored bundles:
+(i) an anchored bundle is a differential bundle with an anchor A → T M , or
+(ii) an anchored bundle is an involution algebroid without an involution.
+These two perspectives can be unified by regarding A as a cylinder for the weighted limit T M , so that
+the anchor is induced by the unique map A.T.0 → T.A.0:
+Λ+
+�
+A
+T M
+̺
+That is, the syntactic category for anchored bundles is constructed as a full � -category of Weil1 that
+doesn’t include the map c . This may be found by taking the full subcategory of Weil1 whose objects
+are constructed out of Wn,n ∈ �.
+Definition 5.2.8. A Weil algebra has width k ∈ � if it can be written
+V = ⊗0≤i<kWn(i), n(i) ∈ �
+The category of k-truncated Weil algebras, written Weilk
+1 , is the full � -subcategory of Weil1 of Weil
+algebras of width k or less.
+The full subcategory whose objects are {�,W } will be given the special notation Weil∗
+1.
+115
+
+Note that for each V in Weilk
+1 , the enrichment is given by
+Weilk
+1 (U ,V ) := (X �→ Weil1(U ,X ⊗ V )).
+The maps in Weil1
+1—that is, the full subcategory of Weil1 whose objects are
+{Wn|n ∈ �}
+—have the useful property that they may be written without the flip c . This makes Weil1
+1 a natural
+candidate for the syntactic category of anchored bundles.
+Lemma 5.2.9. Every morphism
+Wn → V ∈ Weil1
+may be written without c ; that is, it is generated by the set of maps {p,+,0,ℓ} closed under tensor, com-
+position, and maps induced by transverse limits.
+Proof. Every map Wn → V in Weil1 is the finite sum
+v �→
+�
+x
+(Ax v) • x
+where x ∈ var(V ), and Ax ∈ �n so that An v is the ordinary dot product. Each term can be written
+without c , and the whole term is then constructed by adding each component using the appropriate
+U . + .V -symbols. 2
+It is possible, then, to show that the category of anchored bundles in � is a full sub-tangent-category
+of functors Weil1
+1 → �; note that W acts as a cone for (−)D , so this induces a map
+̺ : F (W ) → T.F (�).
+Proposition 5.2.10. Every anchored differential bundle in � determines a functor Weil1
+1 → �; an an-
+chored bundle morphism is exactly an enriched natural transformation.
+Proof. Start with an anchored bundle (q : E → M ,ξ,λ,̺); then for hom-objects with domain �,
+Weil1
+1(�,�) = 1 �→ idM ,
+Weil1
+1(�,T ) = 1 �→ ξ.
+Forthe hom-objectswith domain T , the problem isslightlymore difficult, asWeil1
+1(T,�)(T V ) = Weil1(T,T V )
+and Weil1
+1(T,T )(T ) = Weil1(T,T.T V ). This part of the proof amounts to constructing maps
+Weil1(T,T V ) → �(E ,T V .M ),
+Weil1(T,T V .T ) → �(E ,T V .E )/
+The first mapping is straightforward: send θ to θ.M ◦̺. For the second map, observe that the following
+diagrams commute:
+T A
+T 2M
+A
+T M
+A
+T M
+A2
+T2M
+A
+T M
+M
+M
+M
+M
+A
+T M
+̺
+T.̺
+λ
+ℓ
+ξ
+0
+̺
+̺
+π
+p
++q
+̺2
++.M
+̺
+2This may also be regarded as a consequence of the graphical notation for maps in Weil1 in Table 1 on page 308 of Leung
+(2017).
+116
+
+The idea is to take an anchored bundle morphism f and rewrite it as a string of compositions that does
+not include c , switching out every occurrence of
+T V .θ.M ,θ ∈ {p,0,+,ℓ}
+and replacing it with T V .(θ ′), where θ ′ is the corresponding map in {q,ξ,+q,λ}. This induces a map
+Weil1
+1(W,W ) → �(E ,E ) ∈ �
+and the ̺ is exactly the unique α : F.W → T.F induced by universality.
+For morphisms, the inclusion (Λ+)op → Weil1 ensures that any tangent natural transformation will
+be a linear morphism on the underlying differential bundle, and the tangent natural transformations
+coherences ensure that a tangent natural transformation will preserve the anchor. Conversely, an an-
+chored bundle morphism will preserve each of the constructed morphisms E → T V .E ,E → T V .M
+(as it preserves each of {q,+q,ξ,λ}), giving a natural tangent transformation. Thus there is a faithful
+embedding Anc(�) �→ [Weil1
+1,�].
+The converse, identifyingthose functorsWeil1
+1 → � thatdetermine anchored bundles, isimmediate.
+Corollary 5.2.11. The category of anchored bundles comprises precisely the � -functors and � -natural
+transformations A : Weil∗
+1 → � (Definition 5.2.8) so that the precomposition Λ+ → Weil∗
+1 → � determines
+a differential bundle. That is to say, the category of anchored bundles in � is the following pullback in
+� Cat:
+Anc(�)
+[Weil1
+1,�]
+DBun(�)
+[Λ+,�]
+⌟
+5.3
+Enriched nerve constructions
+Nerve constructionspresenta powerful generalization of the Yoneda functor that sends an object C ∈ �
+to the representable presheaf �(−,C ) ∈ [�op ,� ] (for a general reference on nerve constructions and
+their realizations, see Chapter 3 of Loregian (2021)). The Yoneda lemma states that this functor is an
+embedding (that is, it is fully faithful), so no information is lost when embedding a category into its cat-
+egory of presheaves. Nerve constructions, then, move this towards an approximation of the original
+category � by some subcategory � �→ �, or more generally by some functor K : � → �. In Section 5.5,
+the “infinitesimal” approximation of a Lie groupoid as a Lie algebroid will be exhibited as approxima-
+tion by a � -functor ∂ : Weilop
+1
+→ Gpd(� ).
+We work with a � that is locally presentable as a monoidal category, such as � ,Set, or the category
+of commutative monoids.
+Definition 5.3.1. The nerve of a � -functor K : � → � is the functor
+NK : � → [� op,� ];
+C �→ �(K −,C )
+Any presheaf A : � → � that is in the image of NK is a K -nerve.
+117
+
+Remark 5.3.2. The “K -nerve” terminology seems to go back to Grothendieck/Segals’s original intuition
+for the nerve construction of a category3 (Segal (1974)) and appears in, for example, Berger et al. (2012)
+and Bourke and Garner (2019). However, the “approximation” of a category by a functor K : � → �
+originally used in topology seems to be a more intuitive description of the functor NK .
+Example 5.3.3.
+(i) The nerve of the identity functor � = � is the usual Yoneda embedding � �→ [�op ,� ].
+(ii) The first example of a nerve construction in the mathematical literature is the simplicial approxi-
+mation of a topological space by Kan (1958). Recall the original construction of the simplicial nerve
+of a topological space X , where Xn = Top(∆n,X ). This is exactly the nerve of the functor ∆ → Top
+that sends n to the n-simplex
+�
+x ∈ �n|
+�
+xi = 1
+�
+.
+(iii) The following example figures into Segal’s original nerve construction for a category, and is revisited
+in Section 5.4. Define a reflexive graph to be a presheaf over the full subcategory ofCatwhose objects
+are the two preorders
+[0] = 0, [1] = 0 < 1.
+This is equivalent to the free category with two parallel arrows and a common retract,
+[1]
+[0]
+t
+s
+e
+so that a graph in a category has an object-of-vertices V and an object-of-edges E ,along with source
+and target maps s,t : E → V ; morphisms of graphs are maps (fE , fV ) that commute with the source
+and target maps.
+By Corollary 5.4.3, any full subcategory containing the representables will be dense. Define the
+category of paths, Pth, to be full subcategory of graphs generated by
+[0]
+[0]
+[1]
+...
+[1]
+[n]
+[t ]
+[s]
+[s]
+[t ]
+⌟
+,n ∈ �
+so that Gph([n],G ) picks out the set of paths of length n in a graph G . We call this subcategory
+P : Pth → Gph, and see that NP sends a graph to its Pth-presheaf of composable paths:
+En
+E
+...
+E
+V
+V
+t
+s
+t
+s
+⌟
+3Segal published the result, but seems to have credited the theorem to Grothendieck.
+118
+
+The pushout [n] is precisely the graph
+0 −→ 1 −→ ... −→ (n − 1) −→ n
+where the reflexive map e at each vertex is suppressed.
+Recall the functor sending reflexive graphs to anchored bundles from Example 3.2.7(iv). This may
+be restated as a nerve construction.
+Proposition 5.3.4. There is a functor
+∂ : Weil∗
+1 → Gph(� )
+so that N∂ : Gpd(� ) → [Weil∗
+1,� ] is the linear approximation of a reflexive graph as described in Exam-
+ple 3.2.7 (iv).
+Proof. Let s,t : C → M ,e : M → C be a reflexive graph, and recall that the anchored bundle C ∂ → C is
+induced by the equalizer
+C ∂
+T.C
+T.C
+e .T1
+T.e s
+The category Gph is a representable tangent category, where D is the discrete graph D = D. By the
+Yoneda lemma, we have that
+[Gph,� ](� 1,G ) = G1,[Gph,� ](� 0,G ) = G0,[Gph,� ](D × � (i),G ) = T.Gi,
+so the limit diagram defining the infinitesimal approximation of a graph becomes
+C ∂
+Gph(D × I ,C )
+Gph(D × I ,C ).
+(e ×I )∗
+(D×e −)∗
+Now observe thatGph(� )op with the dual tangentstructure from Proposition 1.3.8 hasa reflexive graph
+object
+s,t : 1 → I ,! : I → 1.
+Construct its linear approximation as in Example 3.2.7(iv) (e.g. take the coequalizer of � -graphs):
+D × I
+D × I
+∂ .
+e ×I
+D×e s
+By Proposition 5.2.10, this determines a functor
+∂ : Weil1
+1 → Gph(� )op.
+By the continuity of the hom-functor, the nerve N∂ : Gph(� ) → [Weil∗
+1,� ] lands in the category of
+anchored bundles. The continuity of the hom-functor ensures that this is indeed the linear approxi-
+mation from Example 3.2.7 (iv).
+The simplicial localization in Kan (1958) has a left adjoint, the geometric realization, that constructs
+a topological space using the data of a simplicial set. This realization may be constructed using a left
+Kan extension.
+119
+
+Definition 5.3.5. Let K : � → � be a � -functor for a cocomplete �. The realization of K is the left Kan
+extension
+A
+�A
+�
+K
+�
+|−|:=LanK �
+|C |K :=
+� A
+�(K A,C ) • K A
+A nerve/realization context is a � -functor K : � → � from a small A to a cocomplete �.
+The adjunction between simplicial sets and topological spaces follows from general categorical
+machinery as a nerve/realization context:
+Lemma 5.3.6. For every nerve/realization context K : � → �, the realization of K is left adjoint to the
+nerve of K .
+5.4
+Nervous monads and algebroids
+Transposing the characterization of algebroids from Theorem 4.3.9 to the enriched perspective intro-
+duces the challenge of finding an appropriate framework to describe algebroids as enriched structures.
+Kelly’s theory of enriched sketches (see Chapter 6 of Kelly (2005)), small � -categories equipped with
+a chosen class of limits cones, seems to be a natural candidate. However, when regarding a tangent
+functor as a � -functor, the natural part
+α : F.T ⇒ T.F
+is the unique morphism induced by the universality of T as a weighted limit, so it becomes unclear
+how to translate the condition that α is a cartesian natural transformation (Definition 4.4.1).
+A clue for how to proceed may be found in Kapranov (2007), which proved that Lie algebroids
+are monadic over anchored bundles (when allowing for infinite-dimensional vector bundles). Recent
+work in Bourke and Garner (2019) and Berger et al. (2012) has developed the appropriate notion of (en-
+riched) theories that correspond to monads over general locally presentable categories. A critical in-
+sight is that for a filtered-colimit-preserving monad � on Set, the opposite category of the Lawvere
+theory � is precisely the full subcategory of the Kleisli category whose objects are [n] =
+�
+n 1, and the
+nerve of the inclusion
+� op �→ Set�
+is fully faithful; that is, when the functor K is dense. Formally, a functor is dense whenever its nerve
+behaves like the Yoneda embedding. The theory of dense functors is developed in Chapter 5 of Kelly
+(2005). We continue to assume that an arbitrary site of enrichment � is locally presentable as a � -
+category.
+Definition 5.4.1. A functor K : � → � is dense whenever the nerve of K is fully faithful, and a subcate-
+gory inclusion that is dense will be called a dense subcategory.
+Dense functors are poorly behaved under composition, but there is a useful cancellativity result
+from Section 5.2 of Kelly (2005).
+Proposition 5.4.2. Consider a diagram of � -categories
+X
+Z
+Y
+K
+F
+J
+α
+120
+
+where α is a natural isomorphism and K is dense. If this diagram exhibits (α, J ) as the left Kan extension
+of K along F , then J is also dense.
+Corollary 5.4.3. If K is dense, F is fully faithful, and J .F = K , then F is a dense subcategory.
+Generally speaking, we will often refer to dense subcategories rather than dense functors. This is
+achieved by factoring the dense functor
+K = �
+K ′
+−→ im(K ) �→ �
+where im(K ) is the � -category whose objects are those of � and whose hom-objects are given by
+im(K )(A,B) = �(K A,K B), so that the inclusion of im(K ) into � is fully faithful and therefore dense
+by Corollary 5.4.3.
+Example 5.4.4.
+(i) The category of finite cardinals Σ is the full subcategory of Set whose objects are given by finite
+coproducts of the terminal object, [n] =
+�
+n 1. This is a skeleton of the category of finite sets, and a
+dense subcategory of Set (see e.g. Bourke and Garner (2019)).
+(ii) Recall that the category of � -presheaves on � , [� op,� ] is the free colimit completion of � (see
+e.g. Kelly (2005)). If � is cocomplete, and K : � → � is dense, then the realization |− |K exhibits �
+as a reflective subcategory of [� op,� ], and is this a locally presentable category4. Conversely, if �
+is a reflective subcategory of [� op ,� ] that contains the representable functors, then � is a dense
+subcategory of �.
+(iii) By Corollary 5.2.7, the category of differential bundles in � is a reflective subcategory of [Λ+op,� ],
+so � : Λ+ �→ [Λ+op,� ] is a dense subcategory of differential bundles in � following the argument
+in the above example.
+Nervous theories (Bourke and Garner (2019)) are generalizations of Lawvere theories that extend to
+arbitrary locally finitely presentable � -categories. Recall that a classical Lawvere theory is a bijective-
+on-objects, product-preserving functor
+t : Σ → �
+where � is a cartesian category. Nervous theories replace Σ with a dense subcategory of some locally
+presentable � -category, and the product preservation condition with conditions on the nerve of the
+theory map.
+Definition 5.4.5. Let K : � → � be a dense � -subcategory of a locally finitely presentable �. We call
+the replete image of NK in [� op,� ] the category of K -nerves. An � -theory is a bijective-on-objects
+� -functor J : � → � , where each
+� (J −,a) : � → �
+is a K -nerve. The category of models for an � -theory is the pullback in � CAT:
+��
+�
+�
+�
+�
+�
+J ∗
+NK
+⌟
+(These are called concrete models in Bourke and Garner (2019).)
+4In fact, an equivalent definition of a locally presentable category is as a cocomplete category with a dense subcategory.
+121
+
+Remark 5.4.6. The category of models for a theory is monadic. The core of the argument is due to Weber’s
+nerve theorem, found in Weber (2007), but an exposition on that result is beyond the scope of this thesis.
+Example 5.4.7.
+(i) A functor t : Σ �→ � is a Σ-theory if and only if � is a Lawvere theory, where we use the fact that
+Σ (Example 5.4.4) a skeletal subcategory of finite sets. The nerve conditions in this case identify the
+models of the Lawvere theory, as the nerve of Σ �→ Set sends a set to the strict product-preserving
+functor [n] �→ An.
+(ii) As shown in Berger et al. (2012) and Bourke and Garner (2019), the original nerve construction
+from Segal (1974) may be restated as saying that small categories arise as models of a Pth-theory
+(Example 5.3.3). The set Gph([n],G ) is the set of paths of through the graph G that have n non-
+identity elements, as
+G t ×s G ∼= G t ×id M id ×s G ∼= G t ×s◦e ◦sG t ◦e ◦t ×sG .
+Next, recall that the category ∆ may be regarded as follows:
+• Objects: A strict order [n] = 0 < 1 < ··· < n, for n ≥ 0, regarded as a category.
+• Maps: Functors.
+There is a bijective-on-objects functor from Pth → ∆ that sends the graph [n] to the pre-order [n],
+as every graph homomorphism between paths will be order-preserving. Now observe that a model
+is precisely a simplicial set, where
+X ([0]) = M ,X ([1]) = C ,X ([2]) = C t ×s C ,X ([3]) = C t ×s C t ×s C .
+Furthermore, the map
+0
+0
+1
+1
+2
+in ∆ becomes a composition map:
+X ([2]) → X ([1])
+C t ×s C → C
+Associativity and unitality (for the section e : M → C ) follow by functoriality. Thus, a model of
+Pth → ∆ is a small category, and a morphism is exactly a functor.
+Corollary 5.2.11, then, gives a monadicity result for � -anchored bundles over � -differential bun-
+dles.
+Proposition 5.4.8. � -anchored bundles are models of the theory (Λ+)op → (Weil∗
+1)op (see Section 5.2).
+122
+
+Proof. Note that the inclusion Λ+ → Weil1
+1 is a differential bundle, so it is a nerve. Then the diagram:
+Anc(� )
+[Weil∗
+1,� ]
+DBun(� )
+[Λ+,� ]
+⌟
+exhibits � -anchored bundles as models of a (Λ)op -theory.
+As a corollary, we see that the category of anchored bundles is locally presentable.
+Corollary 5.4.9. The category of anchored bundles is locally presentable, and (Weil∗
+1)op �→ Anc(� ) is
+dense.
+In Bourke and Garner (2019), the authors identify exactly those monads on a locally presentable
+� -category that correspond to the models of a theory. Recall the notation that the category of algebras
+for a monad is �T and the category of free coalgebras is �T . For a dense subcategory � �→ �, use �T
+for the category of free algebras over objects in � . Recall that the Lawvere theory
+K : FinSet → �
+for a filtered-colimit-preserving monad � on Set may be re-derived as the full subcategory of free alge-
+bras over finite sets, KT : FinSet� �→ Set� �→ Set�. Nervous monads abstract this property.
+Definition 5.4.10. Let � be a locally presentable � -category, with K : � → � a dense sub-� -category.
+A � -monad � over � is K -nervous if
+1. the inclusion KT : �T �→ �T is dense;
+2. the following diagram is a pullback in � CAT:
+�T
+[� op
+T
+,� ]
+�
+[� op,� ]
+Theorem 5.4.11 (Bourke and Garner (2019)). Let K : � �→ � be a dense sub-� -category of a cocomplete
+� -category �. There is an equivalence of categories between � -theories and � -nervous monads on �.
+The first step in showing that algebroids are monadic over anchored bundles is to construct a dense
+subcategory of Anc(� ) that plays the role of the V -prolongations from Definition 4.3.4. The enriched
+framework makes this straightforward: prolongations are weighted limits, with which we may freely
+complete Weil∗
+1.
+Definition 5.4.12. Consider the category Anc(� ) as a representable tangent category, and take the dual
+tangent structure on Anc(� )op. The category � is defined as the full subcategory of Anc(� )op whose
+objects are generated by the prolongations of the Yoneda embedding � : Weil∗
+1 → Anc(� ).
+For any small � -category �, the free completion of � is the opposite � -category of the category of
+copresheaves on �.5 Thus, � is the free completion of Weil1
+1 with prolongations. Therefore, a choice
+5This is the dual statement of the classical theorem that the category of presheaves is the free cocompletion of a small
+category �.
+123
+
+of prolongations on an anchored bundle A determines a unique functor � → � given by right Kan
+extension (see e.g. Kelly (1982)). The continuity of
+Anc(� )(−,A) : Anc(� )op → �
+ensures that Anc(� )(LV ,A) is AV .
+By Theorem 4.3.9, the category of involution algebroids in � is a full sub-� -category of [Weil1,�]
+that has a forgetful functor down to the category of anchored bundles. A consequence of Theorem
+4.4.8 is that a functor Weil1 → � is an involution algebroid if and only if the precomposition
+Weil∗
+1 �→ Weil1 → �
+restricts to an anchored bundle, with each V sent to the V -prolongation of this anchored bundle. In
+other words, algebroids are models of the following enriched theorem (as in Definition 5.4.5).
+Definition 5.4.13. Define the � op-theory of algebroids as the functor
+a : � → Weil1.
+Thisfunctorisbijective-on-objectsby definition, and satisfiesthe nerve condition because the V -prolongation
+of the tangent bundle is T V , so each presheaf
+Weil1(a−,V ) : � → �
+is an � -nerve.
+The Weil nerve, then, translates to the following characterization of the category of involution alge-
+broids in a tangent category �.
+Theorem 5.4.14. The category of involution algebroids with chosen prolongations in a tangent category
+� is precisely the pullback in � CAT:
+I nv ∗(�)
+[Weil1,�]
+Anc ∗(�)
+[� ,�]
+⌟
+(5.2)
+Consequently, in � involution algebroids are monadic over anchored bundles.
+Proof. Recall the correspondence
+¯A : Weil1 → � ∈ � Cat
+( ˆA,α) : Weil1 → � ∈ TangCatl a x
+If
+Weil∗
+1 → � → Weil1 → �
+determines an anchored bundle (π : A → M ,ξ,λ,̺) whose V -coprolongation is ˆA.V , so that ˆA is the
+nerve of an involution algebroid by Corollary 4.4.9.
+Thus, we may characterize � -involution algebroids as algebras for a � -nervous monad on the
+category of � -anchored bundles.
+124
+
+Corollary 5.4.15. The category of involution algebroids in � is equivalent to the category of algebras
+for the � -nervous monad on Anc(� ) generated by the theory
+a : � → Weil1
+from Definition 5.4.13, using Theorem 19 from Bourke and Garner (2019).
+Remark 5.4.16. The construction of involution algebroids as the category of models for a � -theory is
+remarkably similar to the original nerve construction for categories in Segal (1974), with the symmetric
+nerve construction for groupoids replacing ∆ with the category of finite sets Σ (see Example 44 (iv) of
+Bourke and Garner (2019)). Each construction truncates the original category to two objects and builds
+a new category with a bijective set of objects by freely adding limits to the two-object truncation.
+5.5
+The infinitesimal approximation of a groupoid
+This section contains the main theorem of the chapter, namely that there is an adjunction
+Gpd(� )
+Inv(� )
+⊣
+This adjunction follows by inducing a nerve/realization context on the category of groupoids in � :
+∂ : Weilop
+1
+→ Gpd(� ).
+Asimplified version ofthisfunctorappeared in Proposition 5.3.4, the linearapproximation ofa reflexive
+graph. In fact, ∂ will be exactly the free groupoid over the graph ∂ from Proposition 5.3.4 (if we had
+worked in the category of reflexive graphs equipped with an involution). Recall that groupoids and
+Weil spaces are both models of a sketch (recall Section 5.1). The symmetry of the theories gives two
+equivalent presentations of groupoids in � :
+Definition 5.5.1. The tangent category of � -groupoids is equivalently
+(i) the tangent category of internal groupoids in � , where the tangent structure is computed point-
+wise;
+(ii) the tangent category of transverse-limit-preserving functors Weil1 → Gpd(Set) with the cofree tan-
+gent structure from Observation 4.2.5.
+Itisimportanttonote thatthe tangentstructure on � -groupoidsisrepresentable, asitisa cartesian
+closed category with an infinitesimal object given by Weil1 �→ � �→ Gpd(� ) (a proof that Gpd(�) is
+cartesian closed for a cartesian closed category � with sufficient limits a may be found in Section B2.3
+of Johnstone (2002)).
+There are two classes of “trivial” � -groupoids that will be useful.
+Example 5.5.2.
+(i) Every small groupoid G may be regarded as a groupoid in � as the constant functor Weil1 →
+Gpd(Set) sending V to the groupoid G preserves transverse limits, we may then apply the symmetry
+of theories to find a groupoid in the category of Weil spaces.
+125
+
+(ii) Every object M in a category has a cofree groupoid given by s,t : M = M , this is the discrete
+groupoid whose only morphisms are the identity maps. Thus, every Weil space M ∈ � has a corre-
+sponding discrete groupoid (this will often be written M = M to denote the discrete groupoid over
+M ).
+Using the cartesian closure of � and the full subcategories of trivial and discrete groupoids, we
+have the following:
+Observation 5.5.3. The category of � -groupoids is both
+(i) a � -category, with powers by � . The power of a � -groupoid � = s,t : G → M by a Weil space E
+is given by the � -groupoid
+[E ,�] = {[E ,s],[E ,t ] : [E ,G ] → [E ,M ], [E ,e ] : [E ,M ] → [E ,G ]};
+(ii) aGpd-enriched category with powersby small groupoids. The powerof a � -groupoid � : Gpd → �
+by a groupoid � is the � -groupoid
+[� ,�](V ) = [� ,�(V )]Gpd.
+The following two � -groupoids form the basic building block for the main result.
+Example 5.5.4.
+(i) Set D := � W in � . The discrete groupoid D = D represents the tangent functor internally, and the
+copresheaf Gpd(D,−) : Gpd(� ) → � sends a groupoid s,t : G → M to T M .
+(ii) The arrow groupoid is the free groupoid generated by the graph:
+• → •
+Write the trivial � -groupoid on this groupoid as I . Note that the power by I will send a groupoid
+to its “arrow groupoid” � →, the groupoid whose objects are arrows in �, with a map u → v being
+a commuting square. It follows that Gpd(I ,�) is the space of arrows of the groupoid, Gpd(I × I ,�)
+the space of commuting squares, and so on.
+The next proposition gives the necessary properties of I to construct the Lie derivative.
+Lemma 5.5.5.
+(i) For every groupoid � = s,t : G → M , there is an isomorphism
+G t ×t G s ×t G ∼= Gpd(� )(I × I ,�);
+this corresponds to the unique filler for the diagram
+•
+•
+•
+•
+u
+v
+w
+∃!w −1◦v◦u
+126
+
+(ii) The arrow groupoid has a semigroup with a zero structure (like an infinitesimal object D), and the
+multiplication is the coequalizer
+1× I
+I × I
+I × I
+I
+I × 1
+!×I
+s×I
+I ×!
+I ×s
+m
+The dual result of part (ii) is somewhat more obvious to see, as the following fork is always an
+equalizer in � for a groupoid G :
+G
+G □
+G □
+G e [s]×I
+G I ×e [s]
+G m
+(where we write G as the object of arrows and Gpd(� )(I × I ,G ) as G □). Note that G e [s]×I and G I ×e [s]
+correspond to the idempotents on G □:
+•
+•
+•
+•
+•
+•
+•
+•
+•
+•
+•
+•
+q
+u
+v
+w
+u
+u
+q
+q
+G I ×e [s]
+G e [s]×I
+Also note that a commuting square is equalized by these two maps if and only if u = q = id , which
+forces w = v; these commuting squares are exactly the image of G m.
+The semigroup structure on I represents the map sending
+Y
+X
+Y
+X
+X
+X
+u
+u
+u
+Now look at the ∂ defined in Proposition 5.3.4, and take the same colimit in Gpd(� ). Groupoids are
+monadic over reflexive graphs with an involution, so this is essentially the free groupoid over that graph
+(as the free functor is a left adjoint and therefore preserves colimits).
+Definition 5.5.6. Define the � -groupoid ∂ to be the � -coequalizer
+D × I
+D × I
+∂
+e ×I
+D×e [s]
+where e = 0◦!,e [s] = i ◦ s. Note that just as in Proposition 5.3.4, ∂ is an anchored bundle.
+The properties of the arrow groupoid make it possible to prove the following theorem.
+Theorem 5.5.7. The object ∂ determines a cartesian � -functor
+∂ (−) : Weilop
+1
+→ Gpd(� )
+that is both an infinitesimal object in � -groupoids and an involution algebroid in Gpd(� )op.
+127
+
+Proof. Write the pushout powers of ξ : 1 → ∂ as ∂ (n), and for any Weil algebra V = Wn(1) ⊗ ... ⊗ Wn(k),
+write ∂ (V ) = ∂ (n1)×...×∂ (nk). Composition for internal categories will be written in the diagramattic
+order, writing composition as an infix semicolon “;” to keep it distinct from composition of morphisms
+in � .
+The proof has two main steps:
+1. For every Weil algebra V , there is an isomorphism
+∂V ∼= ∂ (V )
+where ∂V is the V -prolongation of the anchored bundle in Gpd(� )op.
+2. The universal lift for the anchored bundle, given as a coequalizer
+∂ × ∂
+∂ × ∂
+∂
+e ∂ ×∂
+e ∂ ×∂
+⊡
+is a semigroup.
+From (1) and (2), we may infer that the object ∂ is an infinitesimal object. The semigroup map ⊡ is
+commutative and has a zero by (2), and satisfies all of the couniversality axioms by (1).
+1. The proof of this step follows by induction on the width (Definition 5.2.8) of the Weil algebra V ,
+where the cases 0,1 both hold by definition. In the case n = 2, we need only check this holds for
+∂ × ∂ ∼= ∂W W :
+D × ∂
+D
+L2
+∂ × ∂
+∂
+(id,0)
+δ
+⌟
+(δ,id)
+(id,0)
+For any G , a map X : ∂ × ∂ → G corresponds to a commuting square in T 2G of the form
+•
+•
+•
+•
+T.e ◦u
+v
+e .T ◦v
+u
+so that T.e ◦ v = id and e .T ◦ u = id . It follows that u = (e .T ◦ v)−1;(T.e ◦ u);v (Lemma 5.5.5 (i)),
+and that precomposition with the uniquely induced map determines ( ¯u,v) : X → G L(2), where
+T.0◦ ¯u = u.
+Observe that any pair ( ¯u,v) : X → G L(2) determines a square
+•
+•
+•
+•
+T.0◦ ¯u
+v
+e .T ◦v
+128
+
+where T.e ◦ v = id and T.e ◦ ¯u = id . Note that T.0 ◦ T.p ◦ T.0 ◦ ¯u = T.0 ◦ ¯u, and that the bottom
+horn is u = (e .T ◦ v)−1;(T.0◦ ¯u);v; now check
+T.e ◦ u = (T.e ◦ e .T ◦ v)−1;(T.e ◦ T.0◦ ¯u);(T.e ◦ v)
+= id ;T.0◦ ¯u;T.e ◦ v = T.0◦ ¯u
+and
+e .T ◦ u = e .T ◦ (e .T ◦ v)−1;(T.0◦ ¯u);v
+= (e .T ◦ e .T ◦ v)−1;(e .T ◦ T.0◦ ¯u);(e .T ◦ v)
+= (e .T ◦ v)−1;id ;(e .T ◦ v) = id
+thus determining a map X → G ∂ ×∂ . The two maps are inverse to each other, giving an isomor-
+phism.
+For the inductive case, look at the prolongations of anchored bundles and recall that
+AU V ⊠M AZ = AU V ⊠� (U ,A) AU Z
+Where AU V and AU Z are treated as spans over AU , so that
+AU
+id⊠πV
+←−−− AU V
+an c U ⊠AV
+−−−−−−→ T U .AV
+AU
+id⊠πZ
+←−−− AU Z
+an c U ⊠AZ
+−−−−−−→ T U .AZ
+and observe that their span composition is
+AU V Z
+AU V
+T V .AU Z
+AU
+T V .AU
+T Z .T V .AU
+̺U ⊠AV
+T U .(̺U ⊠AZ )
+⌟
+So it now suffices to prove that the diagram
+∂U × DV
+∂U × (DV × ∂Z )
+∂U × ∂V
+∂U × (∂V × ∂Z )
+⌟
+is a pushout. But by the inductive hypothesis,
+DV
+DV × ∂Z
+∂V
+∂V × ∂Z
+⌟
+so the result follows by the cocontinuity of ∂U × (−), and Gpd(� ) is a cartesian closed category
+(so X × (−) is cocontinuous).
+129
+
+2. Recall that the fork
+I × I
+I × I
+I
+m
+e [s]×I
+I ×e [s]
+is a coequalizer. Also note that e ∂ is the coequalizer of e ,e [s]. The commutativity of colimits
+then ensures that there is a multiplication map induced by the following diagram:
+(D × I )2
+(D × I )2
+∂ 2
+(D × I )2
+(D × I )2
+∂ 2
+D × I
+D × I
+∂
+Thus the multplication is associative, is commutative, and has a zero given by 1
+0×s
+−→ D × I .
+Aan immediate corollary of Theorem 5.5.7 is that the ∂ determines a nerve/ realization where the
+nerve factorsthrough the categoryofinvolution algebroids. Thisputsthe Lie functorintoa nerve/realization
+context (recall Definition 5.3.5).
+Definition 5.5.8. The Lie realization is the left Kan extension
+|− |∂ = Lan∂ : Inv(� ) → Gpd(� ).
+This functor is well behaved. First, ote that it preserved products:
+Lemma 5.5.9. The realization functor preserves products.
+Proof. Product preservation is a consequence of |−|∂ being the left Kan extension of a cartesian functor
+along a cartesian functor (Day and Street (1995)). Note that this implies
+� v ∂ (v) = 1.
+Next, we see that the realization of an involution algebroid has the same base space.
+Lemma 5.5.10. The base space of the groupoid ∂ is D v.
+Proof. When constructing the colimit of groupoids, the colimit’s base space is the ordinary colimit for
+the diagram of the base spaces. The reflector from simplicial objects to groupoids preserves products,
+so it suffices to check that the base space of ∂ (n) is D(n).
+The base space of I is 1+1, and the map e s
+0 is given by δ−◦!. Since D is a discrete cubical object, its
+base space is D × 1. Thus the coequalizer defining ∂ is
+D + D
+D + D
+∂0
+0◦!+0◦!
+id+0◦!
+.
+A map γ : D + D → M is a pair of maps γ0,γ1 : D → M . We can see that γ0 and γ1 agree at 0 (they are
+both γ0(0)), and γ(0) is a constant tangent vector. It follows that γ1 ◦ (id |id ) = γ.
+130
+
+Now recall the co-Yoneda lemma: for any � -presheaf A : �op → � ,
+F (C ) =
+� C ′∈�
+�(C ,C ′) ⊗ F (C ′).
+In particular, for an involution algebroid in � (or any � -presheaf on Weilop
+1 ),
+A(U ) =
+� V ∈Weil1
+Weilop
+1 (U ,V ) × A(V ) =
+� V ∈Weil1
+Weil1(V,U ) × A(V ).
+Lemma 5.5.11. For any involution algebroid A,
+A(R) =
+� V ∈Weil1
+A(V ) × D V .
+Proof. Use the tangent structure on Weil1 to regard it as a � -enriched category:
+D V = � (V ) = Weil1(V,−) = (U �→ Weil1(V,U ))
+= (U �→ Weil1(V,U ⊗ R)) = Weil1(V,R).
+The following computation gives the result:
+A(R) =
+� V
+Weil1(V,R) × A(V ) =
+� V
+D V × A(V ).
+We can now see that the base space of an involution algebroid is isomorphic to the base space of
+its realization. That is to say, the realization sends an involution algebroid over a Weil space M to a
+groupoid over the Weil space M (up to isomorphism).
+Proposition 5.5.12. Let A be an involution algebroid in � . Then |A|([0]) = A(R).
+Proof. Use the Yoneda lemma, and the fact that � [0] = 1 is a small projective so that Gpd(1,−) is � -
+cocontinuous. Now apply Lemma 5.5.10 and Lemma 5.5.11:
+|A|([0]) = Gpd(1,|A|)
+= Gpd
+�
+1,
+� v∈Weil1
+A(v) • ∂ v
+�
+=
+� v
+A(v) ×Gpd(1,∂ v)
+=
+� v
+A(v) × D v = A(R).
+Thus, as a final result, we have achieved an adjunction between the category of involution alge-
+broids and groupoids in � that is product-preserving and stable over the base spaces.
+131
+
+Theorem 5.5.13 (The Lie Realization). There is a tangent adjunction between the category of involution
+algebroids and groupoids in � , where each functor preserves products and the base spaces.
+Gpd(� )
+Inv(� )
+N∂
+|−|∂
+⊣
+Remark 5.5.14. Just as the introduction to this thesis begins with the work of Charles Ehresmann, we
+should take a moment to see how the Lie realization relates to his original research into sketch theory.
+Rather than sketches, we use their closely related cousins, essentially algebraic theories, which in this
+case are small, finitely complete � -categories. We may regard Gpd(� ) and Inv(� ) as � -functor cate-
+gories
+Lex(�Gpd,� ),
+Inv(�Gpd,� ),
+respectively (where Lex means finite-limit-preserving � -functors). The functor ∂ : Weilop
+1
+→ Gpd(� )
+induces a left-exact � -functor
+ˆ∂ : �Inv → �Gpd.
+This means the functor from Lie groupoids to Lie algebroids is induced by a morphisms of essentially
+algebraic theories, and may thus be presented as a morphism of sketches.
+132
+
+Chapter 6
+Conclusions and future work
+6.1
+Conclusions
+We now take stock of what this thesis has accomplished.
+Enrichedessentiallyalgebraicpresentationsofgeometricstructures
+Thisthesishasbeen concerned
+with categories of vector bundles and Lie algebroids in smooth manifolds. While these are not models
+of a limit sketch in the category of smooth manifolds, classical results such as the Serre–Swann theo-
+rem or Vaintrob’s presentation of Lie algebroids indicate that these categories do have some algebraic
+description. By combining the results in Chapters 2 and 3, we can see that vector bundles and Lie
+algebroids are characterized by tangent categorical gadgets, differential bundles (Theorem 2.5.1) and
+involution algebroids (Theorem 3.5.1). By combining these observations with the enriched perspec-
+tive on tangent categories, the results in Section 5.2 exhibit vector bundles and Lie algebroids as models
+of enriched sketches (Proposition 5.2.6 and Theorem 5.4.14), as can also be found in Chapter 6 of Kelly
+(2005). Note that from this perspective, the construction of the Weil nerve of an involution algebroid
+is mostly important as an intermediate step.
+New tangent structures for Lie algebroids and groupoids
+It is well known that the categories of
+Lie algebroids and groupoids have tangent structures induced by post-composition with the tangent
+functor, yielding the tangent algebroid and tangent groupoid, respectively. The key result in Chapter
+4 exhibiting the category of Lie algebroids as transverse-limit-preserving cartesian-tangent functors
+Weil1 → SMan, then, gives the category of Lie algebroids a second tangent structure that corresponds
+to Martinez’s prolongation Lie algebroid. Similarly, the construction of the infinitesimal nerve of a lo-
+cal groupoid restricts to a new tangent structure on the category of Lie groupoids; in this case, the
+base space is the Lie algebroid of the groupoid G , and the total space is the Lie algebroid of the arrow
+groupoid G 2.
+While this may seem like a piece of formal category theory, it is closely related to symmetry re-
+duction in classical mechanics using Lie theory. This goes back to Poincaré’s celebrated note Poincaré
+(1901), which introduced the Euler-Poincare formulation ofLagrangian mechanicson a manifold equipped
+with a Lie group action, where the tangent bundle is replaced with a Lie algebra action (see Marle
+(2013) for a modern exposition). Poincaré’s formalism has been extended to Lie groupoids and Lie
+algebroids in Weinstein (1996) and Martínez (2001), and the thesis Fusca (2018) investigates fluid me-
+chanics through this lens. This work may be interpreted as using the novel tangent structures on Lie
+groupoids and Lie algebroids given in Chapters 4 and 5, and warrants further investigation.
+133
+
+Functorial semantics of Lie theory
+The work in Section 5.5 puts the Lie derivative on a new formal
+grounding:
+(i) In the enriching category � , the Lie derivative functor now arises as a nerve/realization context.
+This guarantees the existence of a left adjoint—the realization—that constructs a Lie groupoid
+from an algebroid. The realization preserves products and is stable over the base space.
+(ii) It is only a small extension of the work in Section 5.5 to show that the Lie derivative of a groupoid
+in a tangent category may generally be regarded as a weighted limit, where AV = {∂ (V ),G } for
+each Weil prolongation of the involution algebroid of the groupoid.
+These appear to be new results, although in a nerve/realization approach they had previously appeared
+in Lie theory in the form of Sullivan’s construction (Sullivan (1977)), which takes the differential graded
+algebra of a Lie algebroid and constructs a simplicial set:
+DGAop(Ω(∆n),A)
+where Ω(∆n) is the de Rham cohomology of the n-simplex in cartesian space.
+6.2
+Future work
+This section outlines various lines of research that were either cut from the thesis while writing due to
+time and/or space constraints, or new lines of research that the thesis-writing process has motivated
+but which have not yet been pursued.
+Enriched sketches and Mackenzie theory
+Following Voronov (2012), Mackenzie theory refers to the
+body of research developed by Kirill Mackenzie and his collaborators into Lie theory, particularly struc-
+tures like double Lie algebroids (Mackenzie (1992)), VB-Lie algebroids (Bursztyn et al. (2016)), double
+Lie groupoids and so on. Intuitively, a double Lie algebroid is a Lie algebroid in the category of Lie alge-
+broids, while a VB-Lie algebroid is a vector bundle in the category of Lie algebroids. These structures
+have an intuitive relationship with tensor products of sketches; that is, for limit sketches A,B there is
+a chain of isomorphisms of categories:
+Mod(A,Mod(B,Set)) ∼= Mod(A ⊗ B,Set) ∼= Mod(B,Mod(A,Set)).
+We have already demonstrated that the Lie algebroids and vector bundles are � -sketchable, so the
+natural next step is to revisit Mackenzie theory via this lens. Certain results such as the symmetry of
+partial Lie derivatives from Mackenzie (1992) (given a double Lie groupoid, the order in which the Lie
+functor is applied doesn’t matter) should be immediate.
+A tangent categorical formalization of mechanics
+Several papers in synthetic differential geometry
+have translated aspectsofLagrangian and Hamiltonian mechanicsintothe syntheticsetting(Bunge and Heggie
+(1984); Nishimura (1997)), and to a degree this has made it difficult to build enthusiasm for a tangent
+categorical presentation of mechanics. These approaches generally emphasize the ability to construct
+function spaces, or the use of a topos-theoretic internal language. The novel tangent structures for
+Lie algebroids and groupoids presented here, however, provide a new line of inquiry: to develop a uni-
+fied framework for Lagrangian or Hamiltonian mechanics in a tangent category that agrees with the
+algebroid and groupoid approaches to mechanics.
+134
+
+Appendix: Background on locally
+presentable � -categories
+Synthetic differential geometry uses a topos-theoretic framework, such as can be found in the final
+chapters of Lavendhomme (1996). The internal language of a topos is a key tool, and models of syn-
+thetic differential geometry are constructed using sheaf constructions. Tangent categories have an
+analagous toolbox, drawing from enriched locally presentable categories. We begin with locally pre-
+sentable categories; basic material on these may be found in Adámek and Rosický (1994), and on en-
+riched categories in Kelly (2005). The following class of colimits is foundational in the theory of locally
+presentable categories.
+Definition .0.1. A category is filtered whenever any finite diagram in � has a cocone. A colimit whose
+diagram category is filtered is called a filtered colimit. Note that a category � is filtered if and only if
+every colimit diagram � → Set commutes with every finite limit in Set.
+Example .0.2.
+(i) Any finite category � with a terminal object is filtered: for any diagram D : � → � , the natural
+transformation ! : id ⇒ K1 is a cocone via whiskering.
+�
+�
+�
+D
+K1
+!
+(ii) Any category with finite colimits is filtered, because each diagram has a colimit (and therefore a
+cocone).
+Definition .0.3. An object in a cocomplete � is finitely presentable whenever �(C ,−) preserves filtered
+colimits. A cocomplete category � is locally finitely presentable whenever it has a small subcategory
+�f p of finitely presentable objects so that every object in � is given by the coend
+C ∼=
+� X ∈�f p
+�(X ,C ) · X .
+Example .0.4. In the category Set, the subcategory of finitely presentable objects is precisely the category
+of finite sets. Every set is a filtered colimit of finite sets, so that Set is locally finitely presentable.
+Intuitively, this means that every object in � is generated by gluing together finitely presentable
+objects in a canonical way; hence the name locally finitely presentable categories.
+The subcategory of finitely presentable objects in a cocomplete category is itself finitely cocom-
+plete. This follows from the fact that �(−,X ) : �op → Set is a continuous functor (it sends colimits in �
+135
+
+to limits in Set) and also that filtered colimits commute with finite limits in Set. Thus, for any filtered
+colimit of representables,
+�(colD,limF ) ∼= limX limY �(D(X ),F (Y )) ∼= limY limX �(D(X ),F (Y )).
+There are several ways to characterize locally finitely presentable categories; these may be found in
+Adámek and Rosický (1994).
+Proposition .0.5. For a category � the following are equivalent:
+(i) � is a locally finitely presentable category.
+(ii) For a small cocomplete category � , � = Lex(� ,Set) (where Lex means finite-limit-preserving) and
+� = �op
+f p.
+(iii) � is the category of models for some limit sketch (e.g. a small category � equipped with a class of
+cones � , where a model is a functor into Set sending cones to limit diagrams).
+(iv) � is a reflective subcategory of a presheaf topos for some small category � (this holds tautologically
+for presheaf topoi).
+(v) � is a full subcategory of a locally presentable category that is closed under limits (this relies on a
+large cardinal axiom known as Vopenka’s principle).
+One of the remarkable aspects of the theory of locally finitely presentable categories is that its re-
+sults translate over to enrichment in a monoidal category � without any significant changes.
+Definition .0.6. Let � be a monoidal category. A � -graph � is given by a (large) set of objects �0, and a
+map
+� : �0 × �0 → � .
+Given a collection of maps
+mABC : �(B,C ) × �(A,B) → �(A,C ),
+jA : I → �(A,A)
+we say � is a � -category whenever the following associativity and unitality diagrams commute:
+�(B,D) ⊗ �(A,B)
+(�(C ,D) ⊗ �(B,C )) ⊗ �(A,B)
+�(A,D )
+�(C ,D) ⊗ (�(B,C ) ⊗ �(A,B))
+�(C ,D ) ⊗ �(A,C )
+m
+m⊗id
+m
+α
+id⊗m
+I ⊗ �(B,C )
+�(B,B) ⊗ �(B,C )
+�(B,C )
+�(B,C )
+�(B,C ) ⊗ I
+�(B,C ) ⊗ �(C ,C )
+λ
+j ⊗id
+m
+ρ
+id⊗j
+m
+136
+
+Examples of enrichment abound throughout category theory.
+Example .0.7.
+(i) Any category with biproducts is enriched in the category of commutative monoids.
+(ii) A symmetric monoidal closed category is self-enriched using the internal hom [−,−].
+(iii) A 2-category is a category enriched in Cat.
+(iv) The unit � category is the 1-object � category whose hom-object is I .
+(v) When � is symmetric monoidal, we can construct the opposite � -category of � , where �op (A,B) =
+�(B,A). The new composition is defined by
+mop : �op (B,C ) ⊗ �op(A,B)
+σ
+−→ �(B,A) ⊗ �(C ,B)
+m
+−→ �(C ,A) = �op (A,C )
+There is a 2-category of � -categories for any � .
+Definition .0.8. A � -functor F : � → � is an assignment on objects F0 : �0 → �0 and a collection of
+maps FA,B : �(A,B) → �(F0A,F0B) so that the following diagrams are satisfied:
+�(B,C ) ⊗ �(A,B)
+�(F0B,F0C ) ⊗ �(F0A,F0B)
+I
+�(A,A)
+�(A,C )
+�(F0A,F0C )
+�(F0A,F0A)
+m C
+F ⊗F
+m D
+F
+F
+j C
+A
+j D
+F0A
+A � -natural transformation F ⇒ G is a collection of maps γA : I → �(F0A,G0A) so that the following
+diagram commutes:
+�(F0B,F0C )
+�(F0B,F0C ) ⊗ I
+�(F0B,F0C ) ⊗ �(G0B,F0B)
+�(B,C )
+�(G0B,F0C )
+�(G0B,G0C )
+I ⊗ �(G0B,G0C )
+�(G0B,F0B) ⊗ �(G0B,G0C )
+G
+F
+ρ
+id⊗γ.B
+m
+λ
+γ.C ⊗id
+m
+� Cat is the 2-category of � -categories, � -functors and � -natural transformations.
+Enriched categories have a richer notion of (co)limit than ordinary categories. A limit for a functor
+is defined as a universal cone; that is, a right Kan extension along !:
+1
+�
+�
+!
+KA
+F
+β
+so that the universal property
+∀C : �(C ,limF ) ∼= [�,Set](KC ,F −)
+137
+
+is satisfied. Dually, a colimit is a universal cocone, and so a left Kan extension:
+�
+�
+1
+F
+!
+KA
+β
+where the couniversal property
+∀C : [�op ,Set](F −,KC ) ∼= �(colimF,C )
+is satisfied. This does not translate over to � -categories, for a few reasons. An obvious one is that �
+need not be cartesian monoidal, so there is no reason a � -category should have a functor � → 1. The
+notion of a weighted colimit is more appropriate:
+Definition .0.9. Let � be a � -category, and call W : � → � the weight and D : � → � the diagram. The
+limit of D weighted by W , or the weighted limit {W,�}, is an object satisfying the following universal
+property:
+∀C : � (W,[�,� ](F −,KC )) ∼= �(F,W ,C ).
+Dually, given a diagram P : �op → �, the colimit of P weighted by W , W ∗P, is an object in � satisfying
+the universal property
+∀C : � (W,[�,� ](KC ,F −)) ∼= �(C ,F ∗ W ).
+Example .0.10.
+(i) A conical (co)limit is weighted by the constant functor into the unit I , KI : � → w . This coincides
+with the definition of ordinary (co)limits.
+(ii) The power of C by an object V ∈ � is an object C V satisfying the universal property that
+�(B,C V ) ∼= � (V,�(B,C )).
+Dually, the copower by V is given by
+�(B • V,C ) ∼= � (V,�(B,C )).
+(iii) In ordinary categories, the power of an object C by V is given by the (possibly infinitary) V -indexed
+coproduct:
+C V ∼=
+�
+v∈V
+C
+whereas the copower is the possibly infinitary V -index product of C :
+V • C ∼=
+�
+v∈V
+C .
+(iv) In a symmetric monoidal closed category � , the power by � is given by [V,−] and the copower by
+V ⊗ _.
+Following Kelly’s work, we observe that every finite weighted limit is composed of powers and con-
+ical limits.
+Proposition .0.11. Every weighted (co)limit is the composition of conical (co)limits and (co)powers.
+138
+
+Corollary .0.12. Weighted limits are exactly conical/ordinary limits in ordinary categories.
+The previous definitions relating to locally presentable categories transfer over mostly unchanged.
+Definition .0.13.
+(i) A monoidal closed � is presentable as a closed category if �f p is a monoidal subcategory.
+(ii) Given � as in (i), a weight W : � → � is filtered whenever W ⋆ (−) : �
+� → � preserves finite limits.
+(iii) An object in a cocomplete � category is finitelypresentable when �(C ,−) preservesfiltered weighted
+colimits.
+(iv) A cocomplete � -category � is locally finitely presentable whenever every object C is the coend
+C ∼=
+� X ∈�f p
+�(X ,C ) · X .
+Proposition .0.14 (Freyd and Kelly (1972); Kelly (2005)). All of the previous results about locally pre-
+sentable categories lift up to locally presentable � -categories, provided that � is presentable as a closed
+category.
+139
+
+Bibliography
+J. Adámek and J. Rosický. Locally presentable and accessible categories, volume 189. Cambridge: Cam-
+bridge University Press, 1994. ISBN 0-521-42261-2/pbk.
+J. Adámek, F. Borceux, S. Lack, and J. Rosický. A classification of accessible categories. J. Pure Appl.
+Algebra, 175(1):7–30, 2002. ISSN 0022-4049. doi: https://doi.org/10.1016/S0022-4049(02)00126-3.
+URL https://www.sciencedirect.com/science/article/pii/S0022404902001263. Spe-
+cial Volume celebrating the 70th birthday of Professor Max Kelly.
+J. Adámek, H. P. Gumm, and V. Trnková. Presentation of set functors: a coalgebraic perspective. J. Logic
+Comput., 20(5):991–1015, 2010. ISSN 0955-792X. doi: 10.1093/logcom/exn090.
+M. Barr. Acyclic models, volume 17. American Mathematical Society (AMS), Providence, RI, 2002. ISBN
+0-8218-2877-0/hbk.
+K. Bauer, B. Johnson, C. Osborne, E. Riehl, and A. Tebbe. Directional derivatives and higher-order chain
+rules for abelian functor calculus. Topology Appl., 235, 2018.
+C. Berger, P.-A. Melliès, and M. Weber. Monads with arities and their associated theories. J. Pure Appl.
+Algebra, 216(8-9):2029–2048, 2012. ISSN 0022-4049. doi: 10.1016/j.jpaa.2012.02.039.
+W. Bertram. Weil spaces and weil-lie groups, 2014. URL https://arxiv.org/abs/1402.2619v1.
+[math.GR](unpublished preprint).
+W. Bertram and A. Souvay. A general construction of Weil functors. Cahiers Topologie Géom. Différen-
+tielle Catég., 55(4):267–313, 2014. ISSN 1245-530X.
+R. F. Blute, J. R. B. Cockett, and R. A. G. Seely. Differential categories. Math. Structures Comput. Sci., 16
+(6):1049–1083, 2006.
+R. F. Blute, J. R. B. Cockett, and R. A. G. Seely. Cartesian differential categories. Theory Appl. Categ., 22:
+622–672, 2009.
+M. Bojowald, A. Kotov, and T. Strobl. Lie algebroid morphisms, Poisson sigma models, and off-shell
+closed gauge symmetries.
+J. Geom. Phys., 54(4):400–426, 2005.
+ISSN 0393-0440.
+doi: 10.1016/j.
+geomphys.2004.11.002.
+F. Borceux and D. Dejean. Cauchycompletion in categorytheory. CahiersTopologie Géom. Différentielle
+Catég., 27(2):133–146, 1986. ISSN 0008-0004.
+J. Bourke. Codescent objects in 2-dimensional universal algebra. PhD thesis, University of Sydney, 2010.
+J. Bourke and R. Garner. Monads and theories. Adv. Math., 351:1024–1071, 2019. ISSN 0001-8708. doi:
+10.1016/j.aim.2019.05.016.
+140
+
+H. Brandt. Über eine verallgemeinerung des gruppenbegriffes. Math. Ann., 96(1):360–366, 1927.
+M. Bunge and M. Heggie. Synthetic calculus of variations. Mathematical applications of category
+theory, Proc. Spec. Sess. 89th Annu. Meet. Am. Math. Soc., Denver, CO. 1983, Contemp. Math. 30:30–
+62, 1984.
+M. Bunge, F. Gago, and A.M. San Luis Fernández. Synthetic differential topology, volume 448. Cam-
+bridge: Cambridge University Press, 2018. ISBN 978-1-108-44723-2/pbk; 978-1-108-55349-0/ebook.
+M. Burke and B. A. MacAdam. Involution algebroids: a generalisation of lie algebroids for tangent cat-
+egories, 2019. URL https://arxiv.org/abs/1904.06594v3. [math.CT](unpublished preprint).
+H. Bursztyn, A. Cabrera, and M. del Hoyo. Vector bundles over Lie groupoids and algebroids. Adv.
+Math., 290:163–207, 2016. ISSN 0001-8708.
+J. R. B. Cockett and G. S. H. Cruttwell. Differential structure, tangent structure, and SDG. Appl. Categ.
+Structures, 22(2):331–417, 2014. ISSN 0927-2852. doi: 10.1007/s10485-013-9312-0.
+J. R. B. Cockett and G. S. H. Cruttwell. The Jacobi identity for tangent categories. Cah. Topol. Géom.
+Différ. Catég., 56(4):301–316, 2015.
+J. R. B. Cockett and G. S. H. Cruttwell. Connections in tangent categories. Theory Appl. Categ., 32(26):
+835–888, 2017.
+J. R. B. Cockett and G. S. H. Cruttwell. Differential bundles and fibrations for tangent categories. Cah.
+Topol. Géom. Différ. Catég., 59(1):10–92, 2018.
+J. R. B. Cockett and J. D. Gallagher. Categorical models of the differential λ-calculus. Math. Structures
+Comput. Sci., 29(10):1513–1555, 2019. ISSN 0960-1295. doi: 10.1017/s0960129519000070.
+J. R. B. Cockett and R. A G Seely. The Faà di Bruno construction. Theory Appl. Categ., 25(15):394–425,
+2011.
+J. R. B. Cockett, G. S. H. Cruttwell, and J. D. Gallagher. Differential restriction categories. Theory Appl.
+Categ., 25(21):537–613, 2011.
+J. R. B. Cockett, G. S. H. Cruttwell, J. D. Gallagher, J.-S.P. Lemay, B. A. MacAdam, G. Plotkin, and
+D. Pronk. Reverse Derivative Categories. In Maribel Fernández and Anca Muscholl, editors, 28th
+EACSLAnnual Conference on ComputerScience Logic (CSL2020), volume 152ofLeibniz International
+Proceedings in Informatics (LIPIcs), pages 18:1–18:16, Dagstuhl, Germany, 2020. Schloss Dagstuhl–
+Leibniz-Zentrum fuer Informatik. ISBN 978-3-95977-132-0. doi: 10.4230/LIPIcs.CSL.2020.18. URL
+https://drops.dagstuhl.de/opus/volltexte/2020/11661.
+L. A. Cordero, C. T. J. Dodson, and M. de León. Differential geometry of frame bundles, volume 47 of
+Mathematics and its Applications. Kluwer Academic Publishers Group, Dordrecht, 1989. ISBN 0-
+7923-0012-2. doi: 10.1007/978-94-009-1265-6.
+T. Courant. Tangent Lie algebroids. J. Phys. A, 27(13):4527–4536, 1994. ISSN 0305-4470.
+M. Crainic and R.L. Fernandes. Integrability of Lie brackets. Ann. Math., 157(2):575–620, 2003. ISSN
+0003-486X; 1939-8980/e.
+G. S. H. Cruttwell, J. D. Gallagher, and B. A. MacAdam. Towards formalizing and extending differential
+programming using tangent categories. Proc. ACT, 2019.
+141
+
+G. S. H. Cruttwell, J. D. Gallagher, and D. Pronk.
+Categorical semantics of a simple differential
+programming language.
+Electron. Proc. Theor. Comput. Sci. EPTCS, 333:289–310, feb 2021.
+doi:
+10.4204/eptcs.333.20. URL https://doi.org/10.4204%2Feptcs.333.20.
+R. H. Cushman and Larry M. Bates. Global aspects of classical integrable systems. Birkhäuser/Springer,
+Basel, second edition, 2015.
+ISBN 978-3-0348-0917-7;
+978-3-0348-0918-4.
+doi:
+10.1007/
+978-3-0348-0918-4.
+B. Day. On closed categories of functors. In Reports of the Midwest Category Seminar IV, pages 1–38.
+Springer, 1970.
+B. Day and R. Street. Kan extensions along promonoidal functors. Theory Appl. Categ., 1(4):72–77, 1995.
+M. de León, J. C. Marrero, and E. Martínez. Lagrangian submanifolds and dynamics on Lie algebroids.
+J. Phys. A, 38(24):R241–R308, 2005. ISSN 0305-4470. doi: 10.1088/0305-4470/38/24/R01.
+M. de León, J. C. Marrero, and E. Martínez. Lagrangian submanifolds and dynamics on Lie algebroids.
+J. Phys. A Math., 38(24):R241, 2005.
+M. P. do Carmo.
+Riemannian geometry.
+Mathematics:
+Theory & Applications. Boston, MA:
+Birkhäuser Boston, Inc., 1992.
+ISBN 0-8176-3490-8.
+doi: 10.1007/978-1-4757-2201-7.
+URL
+https://doi.org/10.1007/978-1-4757-2201-7. Translated from the second Portuguese edi-
+tion by Francis Flaherty.
+E. J. Dubuc. C ∞-schemes.
+Amer. J. Math., 103(4):683–690, 1981.
+ISSN 0002-9327.
+doi: 10.2307/
+2374046.
+D. J. Eck. Product-preserving functors on smooth manifolds. J. Pure Appl. Algebra, 42(2):133–140, 1986.
+ISSN 0022-4049. doi: 10.1016/0022-4049(86)90076-9.
+C. Ehresmann. Catégories topologiques et catégories différentiables. Librairie universitaire, 1959.
+T. Ehrhard. On Köthe sequence spaces and linear logic. Math. Structures Comput. Sci., 12(5):579–623,
+2002. ISSN 0960-1295. doi: 10.1017/S0960129502003729.
+T. Ehrhard and L. Regnier. The differential lambda-calculus. Theoretical Computer Science, 309(1–3):
+1–41, 2003.
+S. Eilenberg and S. MacLane. General theory of natural equivalences. Trans. Amer. Math. Soc., 58(2):
+231–294, 1945.
+Magdalini K. F. and K. Mackenzie. Warps and duality for double vector bundles, 2019. (unpublished
+preprint).
+P. J. Freyd and G. M. Kelly. Categories of continuous functors. i. J. Pure Appl. Algebra, 2:169–191, 1972.
+ISSN 0022-4049.
+D. Fusca. A groupoid approach to geometric mechanics. PhD thesis, University of Toronto, 2018, 116pp,
+2018.
+P. Gabriel and F. Ulmer. Lokal präsentierbare kategorien, volume 221. Springer-Verlag, 2006.
+R. Garner. An embedding theorem for tangent categories. Adv. Math., 323:668–687, 2018. ISSN 0001-
+8708. doi: 10.1016/j.aim.2017.10.039.
+142
+
+J.-Y. Girard. Linear logic. Theor. Comput. Sci., 50(1):1–101, 1987.
+J. Grabowski and M. Rotkiewicz. Higher vector bundles and multi-graded symplectic manifolds. J.
+Geom. Phys., 59(9):1285–1305, 2009. ISSN 0393-0440. doi: 10.1016/j.geomphys.2009.06.009.
+P. J. Higgins and K. Mackenzie. Algebraic constructions in the category of Lie algebroids. J. Algebra, 129
+(1):194–230, 1990. ISSN 0021-8693. doi: 10.1016/0021-8693(90)90246-K.
+P. H. W. Hoffmann. A hitchhiker’s guide to automatic differentiation. Numer. Algorithms, 72(3):775–811,
+2016.
+G. Janelidze and G. M. Kelly. A note on actions of a monoidal category. Theory Appl. Categ, 9(61–91):
+02, 2001.
+B. Johnson and R. McCarthy.
+Linearization, Dold–Puppe stabilization, and MacLane’s Q-
+construction.
+Trans. Amer. Math. Soc., 350(4):1555–1593, 1998.
+ISSN 0002-9947.
+doi: 10.1090/
+S0002-9947-98-02067-4.
+P. T. Johnstone. Sketches of an elephant: a topos theory compendium. Vol. 1, volume 43 of Oxford Logic
+Guides. New York: The Clarendon Press, Oxford University Press, 2002. ISBN 0-19-853425-6.
+D. M. Kan. Functors involving c. s. s. complexes. Trans. Amer. Math. Soc., 87:330–346, 1958. ISSN
+0002-9947; 1088-6850/e.
+M. Kapranov. Free Lie algebroids and the space of paths. Sel. Math. New Ser., 13(2):277–319, 2007. ISSN
+1022-1824; 1420-9020/e.
+G. M. Kelly. On the essentially algebraic theory generated by a sketch. Bull. Aust. Math. Soc., 26(1):
+45–56, 1982.
+G. M. Kelly. Basic concepts of enriched category theory. Repr. Theory Appl. Categ., 2005(10):1–136, 2005.
+A. Kock. Synthetic differential geometry, volume 333 of London Mathematical Society Lecture Note Se-
+ries. Cambridge University Press, Cambridge, second edition, 2006. ISBN 978-0-521-68738-6; 0-521-
+68738-1. doi: 10.1017/CBO9780511550812.
+I. Kolár. Functorial prolongations of Lie groupoids. Banach Cent. Publ., 76:177, 2007.
+I. Kolár, P. W. Michor, and J. Slovák. Natural operations in differential geometry. Berlin: Springer-Verlag,
+1993. ISBN 3-540-56235-4. doi: 10.1007/978-3-662-02950-3.
+A. Kriegl and P. W. Michor. The convenient setting of global analysis, volume 53 of Mathematical Surveys
+and Monographs. American Mathematical Society, Providence, RI, 1997. ISBN 0-8218-0780-3. doi:
+10.1090/surv/053.
+S. Lack and J. Power. Gabriel–Ulmer duality and Lawvere theories enriched over a general base. J.
+Funct. Program., 19(3–4):265–286, 2009. ISSN 0956-7968. doi: 10.1017/S0956796809007254.
+R. Lavendhomme. Basic concepts of synthetic differential geometry, volume 13. Dordrecht: Kluwer
+Academic Publishers, 1996. ISBN 0-7923-3941-X.
+F. W. Lawvere. Categorical dynamics. In Topos theoretic methods in geometry, volume 30 of Various
+Publ. Ser., pages 1–28. Aarhus: Aarhus University, 1979.
+143
+
+F. W. Lawvere. Qualitative distinctions between some toposes of generalized graphs. In Categories in
+computer science and logic (Boulder, CO, 1987), volume 92 of Contemp. Math., pages 261–299. Amer.
+Math. Soc., Providence, RI, 1989. doi: 10.1090/conm/092/1003203.
+P. Leung. Classifying tangent structures using Weil algebras. Theory Appl. Categ., 32(9):286–337, 2017.
+S. Lie. Theorie der transformationsgruppen...: abschnitt. I. Die endlichen continuirlichen gruppen der
+geraden linie und der ebene. II. Die endlichen continuirlichen gruppen des gewöhnlichen raumes. III.
+Die projectiven gruppen des gewöhnlichen raumes. IV. Untersuchungen über verschiedene arten von
+gruppen des n-fach ausgedehnten raumes. V. Untersuchungen über die grundlagen der geometrie. VI.
+Allgemeine betrachtungen über endliche continuirliche gruppen. 1893, volume 3. BG Teubner, 1893.
+Fosco Loregian. (Co)end Calculus. London Mathematical Society Lecture Note Series. Cambridge Uni-
+versity Press, 2021. doi: 10.1017/9781108778657.
+R. B. B. Lucyshyn-Wright. On the geometric notion of connection and its expression in tangent cate-
+gories. Theory Appl. Categ., 33(28):832–866, 2018.
+B. A. MacAdam. Vector bundles and differential bundles in the category of smooth manifolds. Appl.
+Categ. Struct., pages 1–26, 2021.
+K. C. H. Mackenzie. Double Lie algebroids and second-order geometry. i. Adv. Math., 94(2):180–239,
+1992. ISSN 0001-8708.
+K. C. H. Mackenzie. General theory of Lie groupoids and Lie algebroids, volume 213 of London Mathe-
+matical Society Lecture Note Series. Cambridge University Press, Cambridge, 2005. ISBN 978-0-521-
+49928-3; 0-521-49928-3. doi: 10.1017/CBO9781107325883.
+K. C. H. Mackenzie. Ehresmann doubles and Drinfel’d doubles for Lie algebroids and Lie bialgebroids.
+J. Reine Angew. Math., 658:193–245, 2011. ISSN 0075-4102. doi: 10.1515/CRELLE.2011.092.
+K. C. H. Mackenzie and P. Xu.
+Integration of Lie bialgebroids.
+Topology, 39(3):445–467,
+2000.
+ISSN 0040-9383.
+doi:
+https://doi.org/10.1016/S0040-9383(98)00069-X.
+URL
+https://www.sciencedirect.com/science/article/pii/S004093839800069X.
+K.C.H. Mackenzie. Proving the Jacobi identity the hard way. In Geometric methods in physics. XXX
+workshop, Poland, June 26 – July 2, 2011. Selected papers based on the presentations at the workshop,
+pages 357–366. Basel: Birkhauser, 2013. ISBN 978-3-0348-0447-9; 978-3-0348-0448-6. doi: 10.1007/
+978-3-0348-0448-6_32.
+S. MacLane. Categories for the working mathematician. 4th corrected printing, volume 5. New York:
+Springer-Verlag, 1988. ISBN 3-540-90035-7.
+C.-M. Marle. On Henri Poincaré’s note “sur une forme nouvelle des équations de la mécanique”. J.
+Geom. Symmetry Phys., 29:1–38, 2013.
+E. Martínez. Geometric formulation of mechanics on Lie algebroids. In Proceedings of the VIII Fall
+Workshop on Geometry and Physics (Spanish) (Medina del Campo, 1999), volume 2 of Publ. R. Soc.
+Mat. Esp., pages 209–222. R. Soc. Mat. Esp., Madrid, 2001.
+E. Martínez. Classical field theory on Lie algebroids: multisymplectic formalism. J. Geom. Mech., 10
+(1):93–138, 2018. ISSN 1941-4889; 1941-4897/e.
+144
+
+E. Meinrenken and J. Pike. The Weil algebra of a double Lie algebroid. Int. Math. Res. Not., 2021(11):
+8550–8622, 2021. ISSN 1073-7928. doi: 10.1093/imrn/rnz361.
+P. W. Michor. The Jacobi flow. Rend. Semin. Mat., Torino, 54(4):365–372, 1996. ISSN 0373-1243.
+J. W. Milnor and J. D. Stasheff. Characteristic classes, volume 76. Princeton, NJ: Princeton University
+Press, 1974. doi: 10.1515/9781400881826.
+I. Moerdijk and J. Mrˇcun. On integrability of infinitesimal actions. Am. J. Appl. Math., 124(3):567–593,
+2002.
+I. Moerdijk and G.E. Reyes. Models for smooth infinitesimal analysis. New York: Springer-Verlag, 1991.
+ISBN 0-387-97489-X.
+J. Nestruev. Smooth manifolds and observables, volume 220. New York: Springer, 2003. ISBN 0-387-
+95543-7/hbk.
+H. Nishimura. Synthetic Hamiltonian mechanics. Internat. J. Theoret. Phys., 36(1):259–279, 1997. ISSN
+0020-7748. doi: 10.1007/BF02435785.
+H. Nishimura and T. Osoekawa. General Jacobi identity revisited again. Internat. J. Theoret. Phys., 46
+(11):2843–2862, 2007. ISSN 0020-7748. doi: 10.1007/s10773-007-9397-z.
+V. Nistor. Groupoids and the integration of Lie algebroids. J. Math. Soc. Jpn, 52(4):847–868, 2000.
+H. Poincaré. Sur une forme nouvelle des équations de la mécanique. CR Acad. Sci, 132:369–371, 1901.
+J. Pradines. Théorie de Lie pour les groupoïdes différentiables. Calcul différenetiel dans la catégorie
+des groupoïdes infinitésimaux. C. R. Acad. Sci. Paris Sér. A-B, 264:A245–A248, 1967. ISSN 0151-0509.
+J. Rosický. Abstract tangent functors. Diagrammes, 12:JR1–JR11, 1984. ISSN 0224-3911.
+G. J. Seal. Tensors, monads and actions. Theory Appl. Categ., 28:403–434, 2013. ISSN 1201-561X.
+G. Segal. Categories and cohomology theories. Topology, 13:293–312, 1974. ISSN 0040-9383.
+P. Selinger. A survey of graphical languages for monoidal categories. In New structures for physics, pages
+289–355. Springer, 2010.
+D Sullivan. Infinitesimal computations in topology. Publ. Mathématiques., 47:269–331, 1977. ISSN
+0073-8301; 1618-1913/e.
+R. Thom. Quelques propriétés globales des variétés différentiables. Comment. Math. Helv., 28(1):17–86,
+1954.
+L. W. Tu. An introduction to manifolds. Universitext. New York: Springer, second edition, 2011. ISBN
+978-1-4419-7399-3. doi: 10.1007/978-1-4419-7400-6.
+T. Th. Voronov. Q-manifolds and Mackenzie theory. Comm. Math. Phys., 315(2):279–310, 2012. ISSN
+0010-3616. doi: 10.1007/s00220-012-1568-y.
+M. Weber. Familial 2-functors and parametric right adjoints. Theory Appl. Categ., 18(22):665–732, 2007.
+145
+
+A. Weil. Théorie des points proches sur les variétés différentiables. In Géométrie différentielle. Collo-
+quesInternationaux du Centre National de la Recherche Scientifique, Strasbourg, 1953, pages111–117.
+Centre National de la Recherche Scientifique, Paris, 1953.
+A. Weinstein. Lagrangian mechanics and groupoids. In Mechanics days. Proceedings of a workshop,
+June 12, 1992, pages 207–231. Providence, RI: American Mathematical Society, 1996. ISBN 0-8218-
+0261-5/hbk.
+P. Wilson and F. Zanasi. Reverse derivative ascent: a categorical approach to learning Boolean circuits.
+In Proceedings of the 3rd annual international applied category theory conference 2020, ACT 2020,
+Cambridge, USA, July 6–10, 2020, pages 247–260. Waterloo: Open Publishing Association (OPA), 2021.
+URL eptcs.web.cse.unsw.edu.au/paper.cgi?ACT2020.17.
+R. Wisbauer. Regular pairings of functors and weak (co)monads. Algebra Discrete Math., 15(1):127–154,
+2013. ISSN 1726-3255.
+R. J. Wood. �V -indexed categories. In Indexed categories and their applications, volume 661 of Lecture
+Notes in Math., pages 126–140. Springer, Berlin, 1978.
+146
+
+arXiv:2301.00305v1  [math.CT]  31 Dec 2022
+UNIVERSITY OF CALGARY
+...
+by
+...
+A THESIS
+SUBMITTED TO THE FACULTY OF GRADUATE STUDIES
+IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE
+DEGREE OF ...
+...
+CALGARY, ALBERTA
+..., ...
+© ... ...
+
+Abstract
+...
+ii
+
+Table of Contents
+Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+ii
+Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+ii
+iii
+
+...
+1
+
+....
+2
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf,len=8476
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='00305v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='CT]  31 Dec 2022  UNIVERSITY OF CALGARY  The functorial semantics of Lie theory  by  Benjamin MacAdam  A THESIS  SUBMITTED TO THE FACULTY OF GRADUATE STUDIES  IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE  DEGREE OF DOCTOR OF PHILOSOPHY  GRADUATE PROGRAM IN COMPUTER SCIENCE  CALGARY,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ALBERTA  JUNE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' 2022  © Benjamin MacAdam 2022  Abstract  Ehresmann’s introduction of differentiable groupoids in the 1950s may be seen as a starting point for  two diverging lines of research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' many-object Lie theory (the study of Lie algebroids and Lie groupoids)  and sketch theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This thesis uses tangent categories to build a bridge between these two lines of  research, providing a structural account of Lie algebroids and the Lie functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  To accomplish this, we develop the theory of involution algebroids, which are a tangent-categorical  sketch of Lie algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We show that the category of Lie algebroids is precisely the category of invo-  lution algebroids in smooth manifolds, and that the category of Weil algebras is precisely the classi-  fying category of an involution algebroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This exhibits the category of Lie algebroids as a tangent-  categorical functor category, and the Lie functor via precomposition with a functor  ∂ : Weil1 → �Gpd,  bringing Lie algebroids and the Lie functor into the realm of functorial semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  ii  Preface  This thesis is the original work of the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  This thesis project arose from attempts to broadly understand differential geometry, and in partic-  ular the differential geometry of mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' First steps were taken with Jonathan Gallagher in under-  standing how the enriched perspective on category theory introduced in Garner (2018) might relate to  differential geometric structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The basic structures in this thesis, however, came out of discussions  and collaboration with Matthew Burke in the Lie theory context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Chapter 1 is an introduction to the theory of tangent categories, and contains no new results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Chap-  ter 2 began as a joint project with Matthew, who ultimately had to leave the project due to time con-  straints, although by that time we had already found the basic structure of the proof that differential  bundles are vector bundles in the category of smooth manifolds (proved in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The cur-  rent structure of the chapter, particularly with the emphasis on associative coalgebras of the weak tan-  gent comonad (T,ℓ) and the tight connection to Grabowski’s previous work on the Euler Vector Field  construction (Grabowski and Rotkiewicz (2009)), is original work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The basic results in that chapter ap-  pear in MacAdam (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' however, the results have been streamlined (there was originally a notion of  a strong differential bundle, in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4 it is observed that all differential bundles are strong), and  there are some new observations about linear connections (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Chapter3 isa rewrite ofa preprintwritten with Matthew on involution algebroids(Burke and MacAdam  (2019)), while Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4 is due to conversations with Richard Garner (we expect to release a new pa-  per on involution algebroids based on these results as a joint work).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The original idea of augmenting  an anchored bundle with an involution map is entirely due to Matthew, and the isomorphism on ob-  jects between involution and Lie algebroids follows calculations shared by Richard Garner (Proposition  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' My own contribution in this section comes from connecting this to the work of Martínez (2001),  as well as giving the bijection on morphisms for involution and Lie algebroids (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The first three chapters provide the background required for the deeper results in Chapter 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In  the work of Weinstein (1996);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Martínez (2001);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' de León et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (2005) on Lie algebroids, it was observed  that the “prolongation” of a Lie algebroid acted like a tangent bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1 makes this  intuition precise by showing the prolongation is a second tangent structure on the category of Lie al-  gebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The main theorem in this chapter (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8) shows that involution algebroids in a  tangent category � are equivalent to tangent functors (A,α) from Weil1 to � so that the functor A pre-  serves transverse limits and the natural transformation α is T -cartesian (Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' These are  entirely new observations about Lie algebroids and are original work of the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Finally, Chapter 5 puts the first four chapters into the language of enriched category theory, us-  ing Garner’s enriched perspective on tangent categories (Garner (2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The first original results in  this chapter demonstrate that differential bundles and anchored bundles are models of � -sketches  (Propositions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='10), where � is the site of enrichment for tangent categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Next, the  enriched theories framework of Bourke and Garner (2019) is used to prove Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='14, that invo-  lution algebroids are models of a nervous theory, which is the enriched version of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The  thesis concludes with the Lie Realization, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='13, which is a new characterization of Lie differ-  iii  entiation and introduces an entirely new way to construct adjunctions between categories of “smooth  groupoids” and categories of “Lie algebroids” using purely enriched-categorical methods (this is in  contrast to the geometric approach used in Crainic and Fernandes (2003) and the homotopy theoretic  approach in Sullivan (1977)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  This thesis touches on relatively advanced topics in two areas of math: differential geometry (Lie  algebroids) and enriched category theory (enriched nerve constructions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' I have striven to keep it as  self-contained as possible, introducing the category of smooth manifolds and tangent categories and  including an appendix with the basics of enriched category theory and locally presentable category the-  ory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Material on foundational category theory (which is to say, anything that can be found in MacLane  (1988)) and basic differential calculus (see any calculus textbook) is used without citation or introduc-  tion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' this includes limit, adjunction, monads and monadicity theorems, and the calculus of Kan exten-  sions and coends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Some facts about horizontal composition spans are used in Chapter 4, but nothing  that goes beyond the basic definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  iv  Acknowledgments  I would first like to thank my supervisor, Robin Cockett, for guiding this project and for his investment  of time in reading various preprints and walking through proofs with me.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' I also owe a special debt of  thankstoMatthew Burke, whowasa close collaboratoron thisprojectand spearheaded the application  of tangent categories to Lie theory, and to Jonathan Gallagher for extensive discussions about tangent  categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' I am also grateful to Rory Lucyshyn-Wright for spending a summer teaching me enriched  categorytheory, and toKristine Bauerforhelpful conversationsoverthe years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' I would alsolike tothank  all of the members of the Peripatetic Seminar and others over the years for friendly discussion and  feedback: Chad Nester, Prashant Kumar, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Lemay, Priyaa Srinivasan, Cole Comfort, Daniel Satanove,  Rachel Hardeman, and Geoff Vooys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Finally, I am grateful to my parents and family, and to my wife Niloofar for her support as I wrote  this thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  v  For my wife, Niloofar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  vi  Notation  We start with a table of symbols:  Notation  �,�,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  A (usually tangent) category, treated as a general context for mathematics, de-  noted using mathbb  � ,� ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  A small category treated as a mathematical object, denoted by mathcal  A,B,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Objects in a category and also functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' written using capital letters  f ,φ  Morphisms in a category and natural transformations: lower-case Roman and  Greek letters  f ◦ g  The composition of two maps g : A → B, f : B → C (applicative notation)  πi  The projection from the i t h component of an n-fold pullback A0q0×q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='qn−1×qnAn  or a product  �n Ai  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='G  Composition of two functors, F : � → �,G : � → � (applicative notation)  φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='G  Whiskering of a natural transformation  ⊗  Tensor product in a monoidal category  ⊠  A restricted notion of span composition, introduced in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3  T,p,0,+,ℓ,c  The data for an arbitrary tangent category, introduced in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2  D,⊙,0,!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=',δ  The data for an infinitesimal object, introduced in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7  T n  Iterated application of an endofunctor T  Tn  The n-fold pullback power of p : T → id for a tangent category, T p ×p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p ×p T  vii  We also provide a table of categories:  Notation  SMan  The category of smooth manifolds  �ℓ  The category of lifts in a tangent category �, introduced in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3  NonSing(�)  The category of non-singular lifts in a tangent category �, introduced in Defini-  tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1  PDiff(�)  The category of pre-differential bundles in a tangent category �, introduced in  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1(i)  Diff(�)  The category of differential bundles in a tangent category �, introduced in Defi-  nition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1(ii)  LieAlgd  The category of Lie algebroids, introduced in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1  InvAlgd(�)  The category of anchored bundles in a tangent category �, introduced in Defini-  tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1  Anc(A)  The category of anchored bundles in a tangent category �, introduced in Defini-  tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1  Anc� (A)  The category of involution algebroids with chosen prolongations in a tangent cat-  egory �  Weil1  The category of Weil algebras, introduced in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1  W  Notation for the Weil algebra �[x]/x 2  �  The category of transverse-limit-preserving functors Weil1 → Set, introduced in  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1  Weiln  1  The category of Weil algebras with width n, used in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8  Weil∗  1  The full subcategory of Weil1 spanned by {�,W }  �  The classifying � -category of an anchored bundles and all of its prolongations,  introduced in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='12  viii  Table of Contents  Abstract .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' 135  Bibliography .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' 140  x  Introduction  The study of Lie groupoids and Lie algebroids goes back to Charles Ehresmann and his student Jean  Pradines in the late 1950s, building upon Sophus Lie’s original research into the application of groups of  smooth symmetries to solving ordinary differential equations (Lie (1893)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Motivated by partial differ-  ential equations, Ehresmann (1959) introduced the notion of a differentiable groupoid, which models  the internal symmetries of a smooth manifold, in contrast to the external symmetries given by a Lie  group (a group object in the category of smooth manifolds) or Lie group action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Pradines (1967) ex-  tended the Lie functor (which sends Lie groups and Lie group actions to Lie algebras and Lie algebra  actions) from external to internal symmetries and introduced the notion of a Lie algebroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In doing  so, he identified some shortcomings in Ehresmann’s original definition of differentiable groupoids, in-  troducing the modern notion of a Lie groupoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Ehresmann’s investigations into differentiable groupoids initiated one of the major differential ge-  ometry research programmes of the second half of the twentieth century, the study of Lie groupoids  and Lie algebroids (which one may refer to as many-object Lie theory to distinguish it from the “single-  object” Lie groups and Lie algebras that had classically been studied).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Research into many-object Lie  theory in the 80s and 90s focused on extending Lie’s second theorem and the Cartan–Lie theorem,  which in modern terms state that Lie algebras form a coreflective subcategory of Lie groups;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' that is, the  Lie functorhasa fullyfaithful leftadjoint(Mackenzie and Xu (2000);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Moerdijk and Mrˇcun (2002);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Nistor  (2000)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The leftadjointisoften called Lie integration, and in theirfamouspaperCrainic and Fernandes  (2003) found the exactconditionsgoverningwhethera Lie algebroid integratestoa Lie groupoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Weinstein  (1996) initiated a line of research into classical mechanics on Lie algebroids and groupoids, extending  Poincarè’s development of mechanics on a space with a Lie group action and the Euler–Poincarè equa-  tions (Poincaré (1901);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' see Marle (2013) for a modern treatment);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' this has been further developed by  Eduardo Martinez and his collaborators (de León et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (2005);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Martínez (2001);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Martínez (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Fusca  (2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  However, Ehresmann’s work in differentiable groupoids signalled a change in focus for his own re-  search, as he increasingly focused within the then-new area of category theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Over the course of  the 60s and 70s Ehresmann published a string of influential papers in the nascent area of functorial  semantics, developing the formalism of sketch theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Sketch theory has proven to be highly influen-  tial in mathematical logic, and has been active for some 40 years, being extended to syntax/semantics  adjunctions Gabriel and Ulmer(2006), enriched category theory Kelly (1982) and generalized limit doc-  trines Adámek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The influence of sketch theory can still be seen on Lie theory in the work  of Kirill Mackenzie and his collaborators to develop the theory of double Lie algebroids, Lie-algebroid  groupoids, double-vector bundles, and other tensor-product theories (Mackenzie (1992, 2011)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  This thesis aims to provide a structural account of the Lie functor from Lie groupoids to Lie al-  gebroids, using tangent categories Cockett and Cruttwell (2014) to unify Ehresmann’s many-object Lie  theory and sketch theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Tangent categories provide a syntactic description of tangent structure based  on Kock and Lawvere’s synthetic differential geometry (Kock (2006), Lawvere (1979)) and the Weil func-  tor formalism (a comprehensive account may be found in Kolár et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (1993), while the explicit link to  1  abstract tangent structure is found in Leung (2017), another line of research in differential geometry  that has run parallel to modern Lie theory (albeit with some exchange, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Kolár (2007)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Recent work  has recast tangent categories as a class of enriched categories (Garner (2018)), making it possible for  modern techniques from sketch theory and functorial semantics to be applied to differential geome-  try.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In doing so, we demonstrate that the language of tangent categories sheds light on the study of  classical mechanics on Lie algebroids and groupoids, as well as Mackenzie’s investigations into “Ehres-  mann doubles” of vector bundles and Lie algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Overview  The first three chapters of this thesis build on previous work in the tangent category literature (for  example, Cockett and Cruttwell (2017, 2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Lucyshyn-Wright (2018)), providing tangent-categorical  sketches of differential-geometric structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' These structures follow Ehresmann’s original notion of  a sketch quite closely;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' they are specified as graphs (a collection of objects and arrows in the category)  with a set of diagrams that must commute and cones that must be universal, except that the data may  now include the tangent functor T and the tangent natural transformations p, 0, +, ℓ, and c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Each  sketch is accompanied by a proof that its category of models in smooth manifolds (which we shall  often write SMan) is precisely the category of geometric structures it seeks to model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The first chapter reviews the basic theory of tangent categories, paying particular attention to the  category of smooth manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The first examples of “sketches” from the tangent categories literature  are covered, namely differential objects and affine connections, which model vector spaces and con-  nections respectively (Cockett and Cruttwell (2017, 2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The chapter concludes with a study of tan-  gent submersions, which model submersions from classical differential geometry and are a useful ex-  ample of the sort of work that occurs in Chapters 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The second and third chapters develop tangent categorical sketches for vector bundles and Lie  algebroids respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Chapter 2 extends the observation due to Grabowski and Rotkiewicz (2009)  that the category of vector bundles is a full subcategory of multiplicative monoid actions by the non-  negative reals �+, and then applies the Euler vector field construction (Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The tangent  categorical sketch for a vector bundle, called a differential bundle, is then developed based on a mor-  phism λ : E → T E , and an isomorphism of categories between differential bundles in SMan and the  category of smooth vector bundles is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Chapter 3 introduces involution algebroids, which re-  place the bracket of a Lie algebroid with an involution map  σ : A̺×T πT A → A̺×T πT A  (where ̺ : A → T M is the anchor of the Lie algebroid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Using Martinez’s presentation of the structure  equations for a Lie algebroid (Martínez (2001)), we are once again able to prove an isomorphism of  categories, this time that the category of Lie algebroids is isomorphic to that of involution algebroids  in SMan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This provides the initial bridge between differential geometric structures and tangent cate-  gorical sketches, making it possible to apply more sophisticated techniques in Chapters 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The fourth chapter constructs a syntactic tangent category for Lie algebroids, and demonstrates  that Lie algebroids are precisely generalized tangent bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We call this result the Weil nerve, as it  follows the same structure as Grothendieck’s original nerve theorem Segal (1974), in this case using the  categories presentation of tangent categories due to Leung (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This result has useful implications  for the study of generalized mechanics and geometric structures on Lie algebroids, as it introduces a  novel tangent structure on the category of Lie algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This novel tangent structure corresponds to  Poincarè/Weinstein/Martinez’s characterization of classical mechanics on a Lie algebroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The fifth and final chapter introduces the enriched categories perspective on tangent categories  from Garner (2018), so that the work in Chapters 2 and 3 may be rephrased using the enriched sketches  2  of Kelly (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We construct a functor from the syntactic category of involution algebroids to the syn-  tactic category of a groupoid-in-a-tangent-category, thus giving a presentation of the Lie functor in  the spirit of Ehresmann’s sketch theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The syntactic version of the Lie functor is built by construct-  ing another novel tangent structure on the category of Lie groupoids (or more generally, groupoids in  a tangent category), which also agrees with previous investigations into classical mechanics on a Lie  groupoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' As a final result, we demonstrate that in a locally presentable tangent category we may use a  left Kan extension to construct a left adjoint to the Lie functor, which we call the the Lie realization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  3  Chapter 1  Tangent categories  Tangent categories are an example of convergent evolution in mathematics, in which two unrelated  lines of research with very different aims have arrived at a common endpoint, in this case the same  formal setting for abstract differential geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The older line of research has its roots in differential  geometry proper and, in particular, Weil’s algebraic characterization of the tangent bundle of a smooth  manifold (Weil (1953)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Weil’s work motivated Kock and Lawvere’s development of synthetic differen-  tial geometry, presented in the book of the same name Lawvere (1979);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Kock (2006) as well as the Weil  functor formalism of Kolár et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The second, more recent line of research has its foundations  in theoretical computer science following the publication of Linear Logic by Girard (1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Ehrhard  and Regnier noticed that some models of linear logic have a notion of the “Taylor series approxima-  tion” of a proof;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' this led to the development of differential linear logic by Ehrhard and Regnier (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Blute, Cockett, and Seely studied the categorical semantics of models of linear logic equipped with the  derivative operation - that is, they identified those categories whose internal language are were models  of differential linear logic - developing a categorical theory of differentiation in Blute et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (2006, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Tangent categories arise naturally in each line of research: on the first path with the distillation  of synthetic differential geometry into abstract tangent functors in Rosický (1984), and more recently  when Cockett and Cruttwell refined abstract tangent functors following their investigations into the  manifold categoriesofcartesian differential restriction categoriesin Cockett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (2011);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Cockett and Cruttwell  (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In a sense, they are categories that axiomatize Weil’s characterization of the tangent bundle as  an endofunctor in a way that captures the combinatorics of higher-order derivatives when looking at  a certain class of internal commutative monoids (Cockett and Seely (2011)), as will be made precise  in Chapter 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Tangent categories also pull Kock and Lawvere’s synthetic differential geometry into the  framework of enriched category theory, which is explored in Chapter 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Instances of tangent structure abound throughout mathematics and computer science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' For exam-  ple, many categories of geometric spaces have natural tangent structure, such as the category of conve-  nient manifolds (the categoryofmanifoldsmodelled locallybyconvenient vectorspaces Kriegl and Michor  (1997)) and the category of schemes (see point (ii) in Example 2 of Garner (2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' An example from  mathematical logic is the category of Köthe sequence spaces( Ehrhard (2002)), and categorical models  ofthe differential lambda-calculus(Cockett and Gallagher(2019)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' More recently, tangentand differen-  tial categorieshave found applicationsin differentiable programmingand machine learning(Wilson and Zanasi  (2021)), and to understanding Johnson and McCarthy’s functor calculus Bauer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  This thesis studies differential geometric structures using the language of tangent categories, fol-  lowing the tradition of synthetic differential geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' As such, this chapter will develop tangent cate-  gories with a focus on the category of smooth manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Extending the study of these formal structures  in the context of novel tangent categories is a significant endeavour and should be treated as a direc-  4  tion for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The first section introduces Cartesian differential categories as the categorical  axiomatization of multivariable calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The second section introduces the category of smooth man-  ifolds and two characterizations of its tangent bundle (kinematic versus operational), while the third  section identifies the structure of the kinematic tangent bundle that characterizes abstract tangent  structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The fourth section presents a pair of structures that allow for “local-coordinate calcula-  tions” in the tangent category of differential objects and connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The final section introduces  tangent submersions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A submersion is a differentiable map between differentiable manifolds whose  differential is everywhere surjective;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' as a preview of the work in Chapters 2 and 3, this section shows  that in the category of smooth manifolds, a tangent submersion is precisely a submersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5  first appeared in MacAdam (2021), and is the only original work in this chapter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1  Differential calculus  As with most treatments of synthetic differential geometry, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Kock (2006), it makes sense to begin  with the differential calculus - in this case, an introduction to the categorical theory of differentiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Categorical differentiation has recently gained quite a bit of attention due to its relationship with ma-  chine learning Cockett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (2020), and applications to homotopy theory Bauer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This sec-  tion will just consider the basic structures introduced in Blute et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (2009), and the canonical example  of a cartesian differential category (the category of finite-dimensional real vector spaces and smooth  maps between them).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' [Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1 Blute et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (2009)] A cartesian left additive category is a cartesian  category1 � so that:  (i) Each hom-set �(A,B) is a commutative monoid with addition +AB and zero map 0AB : A → B (the  subscript AB will be suppressed when the context is clear).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) The composition operation ◦ preserves addition on the left:  (g + h) ◦ f = g ◦ f + h ◦ f  (iii) Projection is an additive map (preserves addition):  πi ◦ (f + g ) = (πi ◦ g ◦ f ) + (πi ◦ g )  Where πi denotes the projection from the i t h component of a product or pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' j  There are various examples of cartesian left additive categories - they all fit the same pattern of a  category where each object is equipped with a non-natural, but coherent, choice of linear structure:  Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) Any category with biproducts is a cartesian left additive category where every map is additive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) The category of cartesian spacesCartSp, whose objects are finite-dimensional real vector spaces and  morphisms are smooth maps between them, is a cartesian left additive category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='" Clearly smooth  maps from A → B are closed under addition, projection is an additive map, and (g + h) ◦ f =  g ◦ f + h ◦ f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  1We use the standard notation where 1 is the terminal object, × is product, and πi is the i t h projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  5  (iii) The category of topological vector spaces and continuous morphisms is a cartesian left additive  category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  In fact, cartesian left additive categories may equivalently be described as cartesian categories  where each object has a coherent choice of commutative monoid structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3 (Blute et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (2009)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The following are equivalent:  (i) � is a cartesian left additive category  (ii) � is a cartesian category so that each object has a chosen commutative monoid structure (A,+A,0A)  where the following coherence holds:  (A × B)2  A × B  A2 × B 2  +A×B  τ  +A×+B  (where τ = ((π0 ◦ π0,π0 ◦ π1),(π1 ◦ π0,π1 ◦ π1))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iii) There is a category with biproducts �+ and a bijective-on-objects subcategory inclusion i : �+ �→ �  that creates products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Cartesian left additive categories provide an appropriate to define a differentiation operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Re-  call that the usual derivative of a map f : � → � from elementary calculus can be written  ∂ f  ∂ x : � → �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  More generally, for a map f : �n → �, one writes the Jacobian of f at x:  J [f ] : �n → (�n ⊸ �m) :=  \uf8ee  \uf8ef\uf8f0  ∂ f1  ∂ x1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  ∂ f1  ∂ xn  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  ∂ f1  ∂ x1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  ∂ f1  ∂ xn  \uf8f9  \uf8fa\uf8fb  The Jacobian, however, requires some notion of a “matrix”2 representing a linear map from �n to �m—  not every category that has a notion of differentiation supports that operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Instead, the directional  derivative:  D [f ](x,v) := lim  d→0  f (x + t · v)  t  gives an appropriately general notion of differentiation that extends to categories where the space of  linear maps A → B is not representable by an object in the category 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A cartesian differential category  axiomatizes the directional derivative as a combinator on a cartesian left-additive category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  2This would be called an internal hom in the categorical logic literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  3In the case of automatic differentiation, it is also worth noticing that computing the directional derivative of a map  �n → �m has complexity 2� (f ), while forming the Jacobian has complexity n� (f ), so the directional derivative is a more  appropriate primitive for purely practical computational reasons (see Section 5 of Hoffmann (2016) for a discussion of the  computational complexity of forward-mode automatic differentiation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  6  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' [Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1 in Blute et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (2009)] A cartesian differential category is a cartesian  left additive category equipped with a combinator (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' a function on hom-sets)  A  f−→ B  A × A −−→  D[f ] B  satisfying the following axioms:  [CD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1] Additive:  D [f + g ] = D[f ]+ D[g ]  D[0] = 0  [CD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2] Additive in the second variable:  D[f ]◦ (g ,h + k) = D[f ]◦ (g ,h) + D[f ]◦ (g ,k)  D[f ]◦ (g ,0) = 0  [CD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3] Projection is linear:  D [πi ] = πi ◦ π1  D[id ] = π1  [CD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4] Pairing:  D[(f ,g )] = (D[f ],D[g ])  [CD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5] Chain rule:  D[g ◦ f ] = D[g ]◦ (π0,D[f ])  [CD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6] Linear in the second variable:  D[D [f ]]◦ ((a,0),(0,d )) = D[f ]◦ (a,d )  [CD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7] Symmetry of partial differentiation:  D [D[f ]]◦ ((a,b ),(c ,d )) = D[D [f ]]◦ ((a,c ),(b,d ))  Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category of cartesian spaces (Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2(ii)), CartSp, is the canonical cartesian  differential category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let f : �n → �m, and consider its Jacobian at x ∈ �n, J [f ](x) ∈ �n×m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Define the  differential combinator:  D [f ]◦ (u,v) = J [f ](v) · u = lim  t →0  f (x + t · v)  t  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  In Bauer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (2018), the authors construct a cartesian differential category based on the Abelian functor  calculus of Johnson and McCarthy (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  In Wilson and Zanasi (2021), the authors consider a cartesian differential category whose objects are �2-  modules to apply gradient-based methods to learn the parameters of of Boolean circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Every cartesian differential category comes with a notion of linearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This notion of linearity is  strictly stronger than additivity - there do exist examples of non-linear additive maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6 (Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1 of Blute et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (2009)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A map f : A → B is linear whenever D[f ] ◦  (0AB,id ) = f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  We denote the category of linear maps in a cartesian differential category � as Lin(�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category  Lin(�) will have biproducts, and will be a cartesian differential subcategory of �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  7  Lemma1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7(Corollary2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3in Blute et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (2009)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let � be a cartesian differential category, and denote  its category of linear maps as Lin(�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) Linear maps preserve addition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) The category Lin(�) is a bijective-on-objects subcategory of � with biproducts, and the inclusion  Lin(�) �→ � creates products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iii) Every category with biproducts is a cartesian differential category, where  D [f ] = f ◦ π1,  and this differential structure makes the inclusion Lin(�) �→ � preserve the left additive structure  and differential combinator (that is, it is a cartesian differential functor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2  The category of smooth manifolds  Tangent categories axiomatize a more general structure than differential calculus, one in which spaces  are only “locally linear.” The category of smooth manifolds gives the historically canonical example,  and a good portion of this thesis relates to structures internal to that category, so it seems worthwhile to  set a working definition for that context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We follow Tu (2011), and allow for disconnected components  of a manifold to have different dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' [Definitions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7 in Tu (2011)] A chart on a topological space M is pair (Ui,φi :  Ui �→ �n), where Ui is an open subset Ui ⊆ M and φi : Ui → �n is a local homeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' An atlas is  a collection of charts {(Ui,φi : Ui → �n)|i ∈ I } (where n is fixed for each connected component of M ) so  that for each i, j ∈ I , the transition function ψi,j that completes the diagram  φ−1  i (Ui ∩Uj)  �n  Ui ∩Uj  Uj  φi  ⊆  φj  ψi,j  is a smooth map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A smooth manifold is a topological space equipped with a (maximal4) atlas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A mor-  phism of smooth manifolds is a topological map f : M → N that is locally smooth - for each chart pair of  charts (Ui,φi) on M and (Vj ,θj), see Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1 for an illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The map f is smooth whenever each  map fi,j that completes the diagram is smooth  Ui  Vj  M  N  f  fi,j  The category of smooth manifolds, SMan, is the category of smooth manifolds and their morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In Chapter 2, some results will implicitly use partition of unity arguments, which require  that the underlying topological space for a manifold is Hausdorff and has a countable basis (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' it is  second-countable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We will avoid any direct reference to these properties, and so we omit them from the  definition of a smooth manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  4With respect to subset inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  8  φi  φ−1  i  φ−1  j  φj  M  Ui  Uj  φi(Ui)  �m  ψi j  φj(Uj)  �m  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1:  Overlapping charts in an atlas (Credit for this tex code belongs to user Cragfelt  https://tex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='stackexchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='com/a/388493/101171.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=')  Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) Each vector space �n, for n ∈ �, has a canonical smooth manifold structure whose atlas is a single  chart (the identity map �n → �n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) Most geometric shapes that do not have any singularities or sharp edges can be equipped with an  atlas without any issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' For example, consider the circle: {(cos(x),sin(x)) : x ∈ [−π,π)}  For any appropriately small ε > 0, there are two charts from Iε = (−ε,π+ ε);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  φ0(t ) = (cos(t ),sin(t )), φ1(t ) = (cos(t − π),sin(t − π))  making the circle a smooth manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Category theory has not seen as many applications in differential geometry as it has in topology  or algebra, likely because the category of smooth manifolds (Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1) is somewhat poorly be-  haved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The two main reasons that the category of smooth manifolds is “inconvenient” are as follows:  The set of smooth maps between two manifolds M ,N fails, in general, to form a smooth mani-  fold;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' thus, the category is not cartesian closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The category of manifolds does not have quotients or arbitrary fibre products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  9  However, the category of smooth manifolds admits some limits, for example finite products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='12 Kolár et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (1993)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category SMan of smooth manifolds has finite products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Given two manifolds M ,N , take the product of their underlying topological spaces together  with the product charts  (φi × ψj ) :Ui × Vj → �n × �m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The category of smooth manifolds—using our definition where disconnected components of a  manifold may have different dimensions—does have a class of (co)limits used throughout this paper,  namely idempotent splittings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5 (Borceux and Dejean (1986)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' An idempotent is an endomorphism e : E → E so that  e ◦e = e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The splitting of an idempotent is given by a pair of maps e = s ◦r so that r ◦s = id .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The existence  of a splitting of an idempotent e is equivalent to asking that the following pair of parallel arrows has a  (co)equalizer:  E  E  e  The idempotent splitting � of a category (also known as the Cauchy completion of the category), is the  full subcategory of presheaves [�op ,Set] that are retracts of representable functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Any functor into a  category with idempotent splittings will factor through this inclusion of categories, so it is the free co-  completion of � under idempotent splittings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6 (Lawvere (1989)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category of smooth manifolds is the idempotent splitting of  the category whose objects are open subsets of Cartesian spaces and whose morphisms are smooth maps  f :U → V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  An idempotent in the idempotent splitting of a category is also an idempotent in the base category,  and thus admits a splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category of smooth manifolds is closed to idempotent splittings: for every map  e : M → M so that e = e ◦ e there exists a pair of maps r :Q → M ,s : M →Q so that e = s ◦ r,r ◦ s = id .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The main construction of interest on the category of smooth manifolds, for tangent categories at  least, is the tangent bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Given a smooth map f : M → N , restrict it to a morphism between coor-  dinate patches, so it may be regarded as a map f |U :U → V , where U ⊆ �m,V ⊆ �n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This gives a local  derivative operation (remembering that πi denotes the i t h projection from a product  �n Ai )5  D[f |U ] :U × �m → �n,  (f |U ◦ π0,D[f |U ]) :U × �m → V × �n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The tangent bundle makes this construction global;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' that is, there is a functor T : SMan → SMan giving  an assignment  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f  −→ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='N  that agrees with the local derivative on coordinate patches of M ,N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  5The wording here was originally muddled, it has since been corrected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  10  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8 (Kolár et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (1993)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Write the algebra of smooth functions on a manifold as C ∞(M ) :=  SMan(M ,�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The set SMan(�,M )/ ∼= of tangent vectors on a smooth manifold M comprises the curves  � → M subject to the equivalence relation that for a pair of curves φ,θ : � → M , φ ∼= θ if and only if  φ(0) = θ(0) and for every f ∈ C ∞(M ),  ∂ f ◦ φ  ∂ x  (0) = ∂ f ◦ θ  ∂ x  (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The set SMan(�,M )/ ∼= has a naturally determined smooth manifold structure which we call the tangent  bundle over M , T M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) For a vector space, the space of linear paths crossing through a point v ∈ V is isomorphic to V , so  T V ∼= V × V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) The tangent bundle above the circle is diffeomorphic to the cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This is follows from the classi-  cal result that a tangent vector on the circle must be perpendicular to its position vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The tangent bundle lifts the “local derivative” into a globally defined construction, so the tangent  bundle construction is functorial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The tangent bundle is a product-preserving endofunctor on the category of smooth  manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Functoriality follows by showing that a morphism of smooth manifolds preserves the equiva-  lence relation on curves that defines a tangent vector:  ∀g ∈ C ∞(M ), ∂ g ◦ φ  ∂ x  (0) = ∂ g ◦ θ  ∂ x  (0)  Note that if f : N → M ∈ C ∞(N ), so that g ◦ f ∈ C ∞(M ), the chain rule ensures that  ∀g ∈ C ∞(N ), ∂ f ◦ g ◦ φ  ∂ x  (0) = ∂ f ◦ g ◦ θ  ∂ x  (0)  To show that T is product-preserving, it suffices to show that the equivalence classes of curves are  stable under pairing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' First, note that for any M and φ ∼= θ : � → M ,  (φ,id ) ∼= (θ,id ) : � → M × �  Given a pair of curves θM ,ψM : � → M , where θM ∼= ψM and similarly θN ∼= ψN for N , this implies that  f ◦ (φM ,φN )  = f ◦ (φM × id ) ◦ (id ,φN )  = f ◦ (φM × id ) ◦ (id ,θN )  = f ◦ (id ,θN ) ◦ (φM × id )  = f ◦ (id ,θN ) ◦ (θM × id )  = f ◦ (θM × θN )  so (φM ,φN ) ∼= (θM ,θN ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  11  The scalar action by � on tangent vectors and a partially defined addition additionally give the  tangent bundle the structure of a fibered �-module (that is, an �-module in the slice categorySMan/M  whose objects are morphisms into M , f : X → M , and morphisms are commuting triangles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The tangent bundle over M is an �-module in SMan/M , as follows: let γ,ω be  tangent vectors on M , and define  p : T M → M ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p(γ) = γ(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  0 : M → T M ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0(m) = [r �→ m] (the constant map � → M sending all r ∈ � to m ∈ M )  ·p : T M × � → T M ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='γ ·p r = [x �→ γ(r · x)]  + : T M p ×p T M → T M := [γ],[ω] �→ [γ + ω] (where addition around γ(0) = ω(0) is defined using  local coordinates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The second derivative is involved in the more nuanced axioms for a cartesian differential category,  namely linearity in the vector argument and the symmetry of mixed partial derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' First, set f |U  to be the restriction of f : M → N to a map between local coordinate patches U ⊆ M ,V ⊆ N , and then  define  f0 = f |U ◦ π0 ◦ π1,  ,  and  f1 = D[f ]◦ (π0 ◦ π0,π1 ◦ π0),  f2 = D[f ]◦ (π0 ◦ π0,π0 ◦ π1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The axioms [C D C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6],[C D C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7] then give:  (U × �n) × (�n × �n)  (V × �m) × (�m × �m)  T 2M  T 2N  T M  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='N  (U × �n)  (V × �m)  T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f  ((f0,f1),(f2,D[D[f |U ]]))  (f ,D[f |U ])  ℓ  ℓ  ((π0,0),(0,π1))  ((π0,0),(0,π1))  (U × �n) × (�n × �n)  (V × �m) × (�m × �m)  T 2M  T 2N  T 2M  T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='N  (U × �n) × (�n × �n)  (V × �m) × (�m × �m)  c  ((π0◦π0,π0◦π1),(π1◦π0,π1◦π1)  T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f  T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f  c  ((π0◦π0,π0◦π1),(π1◦π0,π1◦π1)  ((f0,f1),(f2,D[D[f |U ]])  ((f0,f2),(f1,D[D[f |U ]])  The two natural transformations ℓ and c —the vertical lift and canonical flip—capture these coher-  ences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Locally, ℓ is the map inserting zeros into the second and third coordinates, while c flips the  second and third arguments, leading to the coherences established in the next proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' To capture  12  these coherences on the tangent bundle, first note that a tangent vector on T M is equivalent to an  equivalence class of surfaces on M ,  φ ∼= θ : �2 → M ⇐⇒ φ(0,0) = θ(0,0)  and ∀f ∈ C ∞(M ), ∂ f ◦ φ  ∂ xi  (0,0) = ∂ f ◦ θ  ∂ xi  (0,0),i = 0,1  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='12 (Cockett and Cruttwell (2014)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' There are two natural transformations  ℓ : T M → T 2M ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ([γ]) = [γ ◦ (π0 ·� π1)]  c : T 2M → T 2M ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c ([γ]) = [γ ◦ (π1,π0)]  satisfying the following coherences:  (i) ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ ℓ = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ◦ ℓ6  (ii) The following maps are morphisms of fibred �-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  T M  T 2M  M  T M  ℓ  p  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  0  T M  T 2M  M  T M  ℓ  p  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  0  (iii) c ◦ c = id  (iv) T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c ◦ c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c = c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c ◦ c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  (v) c ◦ ℓ = ℓ  (vi) T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c ◦ c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Observation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Equation (iv) is known as the Yang–Baxter equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' It is one of the coherences for  a symmetric monoidal category, and states that the twisting operation between two variables is coherent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  We may regard the category of endofunctors on a category as a strict monoidal category and use string  diagram notation (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Selinger (2010)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Interpreting the map c as twisting two strings, the coherence  becomes  =  The projection p : T M → M is locally trivial: for each connected component of M that is modeled  on �n, each point m lies in an open subset Um so that p −1(Um) ∼=Um ×�n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This local triviality property  leads to the following universality condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The following diagram is an equalizer:  T M  T 2M  T M  p  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  0◦p◦p  ℓ  6Recall that we are using the 2-categorical notation described in the front-matter  13  Therefore the diagram in the following corollary is a pullback:  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Write the map µ : T M p ×p T M → T 2M to be T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' + ◦(ℓ× 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The following diagram is a  pullback:  T M p ×p T M  T 2M  M  T M  0  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  µ  ⌟  The five maps (p,0,+,c ,ℓ), along with their coherences and universal properties, characterize the  kinematic tangent bundle, axiomatized as a tangent structure in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' However, there is an  equivalent characterization of the tangent bundle for a finite-dimensional smooth manifold that will  be important throughout this thesis: the operational tangent bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We first need to define the mod-  ule of vector fields on a manifold, where a vector field is essentially an ordinary differential equation  defined on a manifold rather than a cartesian space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='16 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3of Kolár et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (1993)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A vector field on a manifold M is a section X : M → T M  of the projection p : T M → M so that p ◦ X = idM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The set of vector fields on a manifold M is written  χ(M ) and carries a C ∞(M )-module structure using the fibered �-module structure on p : T M → M :  X +χ(M ) Y := +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M ◦ (X ,Y ),  0χ(M ) := 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M ,  f ·χ(M ) X (m) := f (m) ·T M X (m)  where X ,Y ∈ χ(M ), f ∈ C ∞(M ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The module χ(M ) has an important universal property as a C ∞(M )-module—it is precisely the  module of derivations of C ∞(M ):  χ(M ) = {X : C ∞(M ) → C ∞(M ) : ∀f ,g ∈ C ∞(M ),X (f ·g ) = X (f ) ·g + f · X (g )}  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='17 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4 of Kolár et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (1993)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' There is an isomorphism of C ∞(M ) modules:  Der(C ∞(M )) ∼= χ(M ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Finally, we observe that there is a C ∞(M )-Lie algebra structure on χ(M ), with two equivalent def-  initions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' First, there is the kinematic definition of the bracket, which is induced using the universality  of the vertical lift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Given vector fields X ,Y on M , note that  X = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Y ◦ X −T M c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='X ◦ Y ),  0 = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p ◦ (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Y ◦ X −T M c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='X ◦ Y )  so by Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='15, there is a unique map [X ,Y ] : M → T M so that  (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Y ◦ X −T M c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='X ◦ Y ) = µ([X ,Y ],X ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1)  Similarly, there is a bracket defined on Der(M ) using the anticommutator of derivations:  [X ,Y ](f ) = X (Y (f )) − Y (X (f )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2)  These brackets are equivalent for finite-dimensional smooth manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' [Mackenzie(2013)]Recall that a Lie algebra overa ring R isan R-module A equipped  with a bilinear map  [−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='−] : A ⊗ A → A  that is alternating and satisfies the Jacobi identity:  [X ,[Y ,Z ]]+ [Z ,[X ,Y ]]+ [Y ,[Z ,X ]] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The two brackets on χ(M ) (viewed as a Lie algebra) from Equations 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2 coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  14  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3  Tangent structures  This section develops the categorical framework to study more general categories of smooth manifolds  by axiomatizing the tangent bundle, tangent categories, which first appeared in Rosický (1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' An ar-  bitrary tangent category is significantly more general than smooth manifolds and captures examplesof  categorieswith “tangentbundle” from computerscience and logic, asdeveloped in Cockett and Cruttwell  (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' First, observe that tangent categories forget the base ring from the previous section and only  consider the fibered commutative monoid structure of the tangent bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' An additive bundle in a category � is a triple E  q−→ M ,+ : E q×qE → E ,ξ : M → E which  gives (q,+,ξ) the structure of a commutative monoid in the slice category �/M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' If (q,ξ,+),(q ′,ξ′,+′) are  both additive bundles, a bundle morphism  E  E ′  M  M ′  q  f  q  f0  is additive if f ◦+ = +′ ◦(f ◦π0, f ◦π1) and f ◦ξ = ξ′ ◦ f0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category of additive bundles in a a category  � is given by additive bundles in � and additive bundle morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  We will often write pullback powers of an additive bundle E q ×q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q ×q E as En, and use infix no-  tation to write addition so + ◦ (a,b ) becomes a + b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In the category of smooth manifolds, the tangent  bundle functor gives a well-behaved functorial vector bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A tangent category has a functorial ad-  ditive bundle and axiomatizes the coherences and universal properties of the tangent bundle from the  category of smooth manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2 (Rosický(1984);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Cockett and Cruttwell (2014)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A tangent structure consists of a functor  T : � → � equipped with natural transformations  p : T ⇒ id , 0 : id ⇒ T,+ : T p ×p T ⇒ T  ℓ : T ⇒ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  c : T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ⇒ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  satisfying the following axioms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' we call a category equipped with a tangent structure a tangent category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  [TC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1] Additive bundle axioms:  (i) Pullback powers of p exist and are preserved by T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' write these Tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) Each triple (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M : T M → M ,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M : M → T M ,+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M : T2M → T M ) is an additive bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  T  T  T2  T  id  id  id  +  p◦πi  p  p  0  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3)  [TC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2] Symmetry axioms:  (i) Involution:  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  c  c  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4)  15  (ii) Yang–Baxter: c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c ◦ c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c ◦ c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c  c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c  c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5)  (iii) Naturality equations:  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T  T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T2  T2T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T  T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ  c  c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c  ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  (c ◦T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π0,c ◦T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π1)  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='+  +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  c  c  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  c  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6)  [TC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3] Lift axioms:  (i) Naturality with addition:  T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T2  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T2  id  T  T  T  T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  (ℓ◦π0,ℓ◦π1)  +  +  ℓ  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  ℓ  p  0  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7)  (ii) Coassociativity:  T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  ℓ  ℓ  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ  ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8)  (iii) Symmetric co-multiplication:  T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  ℓ  c  ℓ  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9)  (iv) Universality: for µ := T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' + ◦(0◦ π0,ℓ◦ π1), the following diagram is a pullback for all X :  T2X  T 2X  X  T X  µ  p◦πi  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  0  16  Definition1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3(Cockett and Cruttwell (2014)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Atangent category ismonoidal whenever� isa monoidal  category, T is a monoidal functor, and +,p,c ,ℓ are monoidal natural transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A tangent cate-  gory is cartesian whenever � is cartesian and is a strict monoidal tangent category for products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Notation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Throughout this thesis, two key pieces of notation apply:  Tn denotes pullback powers of p : T ⇒ id (and more generally En for q : E → M );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' iterated powers  of T are written T n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  There will often be long strings of Tn pullbacks and functors F : A → B, so a 2-categorical notation  where functor composition is written with a period, Tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ′  m, will often be adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' While the  natural transformation c at Tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Tm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M under the image of the functor T would often be written  T (cTn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Tm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M ), in this thesis it will be written as  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Tm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M : T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Tm ⇒ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Tm  while the composition of 2-cells will be written using ◦ in applicative order, rather than the dia-  grammatic order typically used in the tangent category literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' That is, the composition  A  g−→ B  f−→ C  would be written f ◦ g rather than g f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Applying the results from section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2, the category of smooth manifolds is a tangent  category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Recall that if M is a smooth manifold, there is a coordinate patch m ∈ U �→ M around each  point m ∈ M so that U ∼=U ′ ⊆ �n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' fibre above U ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' p −1(U ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' is locally isomorphic to U ′ ×�n and similarly  (p ◦ p)−1(U ) ∼=U ′ × (�n)3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' so that:  p :U × �n → U  (m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' x) �→ m  0 :U →U × �n  m �→ (m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0)  + :U × �n × R n →U × �n  (m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' y ) �→ (m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' x + y )  ℓ :U × �n →U × �n × �n × �n  (m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' x) �→ (m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' x)  c :U × �n × �n × �n →U × �n × �n × �n  (m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='z) �→ (m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='z)  The study of tangent categories is closely related to Lawvere’ssynthetic differential geometry,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' first in-  troduced in Lawvere (1979) and laterdeveloped in Kock (2006);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Lavendhomme(1996);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Moerdijk and Reyes  (1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The setting of synthetic differential geometry is a topos � (for our purposes, we need only a com-  plete, cartesian closed category) equipped with a chosen ring object R that satisfies the Kock-Lawvere  axiom: given the object of nilpotent elements in R, D = [d : R|d 2 = 0], the following map is an isomor-  phism:  α : R × R → [D,R];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' α(a,b ) = (d �→ a + d b ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  One can find a class of objects that form a tangent category: the infinitesimally linear objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let � be a model of synthetic differential geometry where the ring object is (R,·,1,+,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  An object M in a model of synthetic differential geometry is infinitesimally linear when it satisfies the  following axioms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) The following natural morphism must be an isomorphism:  [D (2),M ] ∼= [D,M ]0∗×0∗[D,M ];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' where D(2) := [(d0,d1) ∈ D 2 : di ·d j = 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  17  (ii) M satisfies property W (credited by Kock (2006) to Gavin Wraith), namely that the following dia-  gram is a ternary equalizer:  [D,M ]  [D × D,M ]  [D,M ]  [0◦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='×D,M ]  [(0×0)◦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=',M ]  [D×0◦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=',M ]  [·,M ]  (We will often use M as shorthand for idM in diagrams).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  An infinitesimal object generalizes the object D in the category of infinitesimally linear objects in  a model of synthetic differential geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The definition adopted in this thesis is a strict generaliza-  tion of the definition given in Cockett and Cruttwell (2014): here, we work with a symmetric monoidal  closed category rather than a cartesian closed category in order to capture some examples from logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7 (Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6 Cockett and Cruttwell (2014)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' An infinitesimal object in a symmetric  monoidal category  (�,⊗,I ,α,ρ,σ)  is a tuple (⊙ : D ⊗ D → D,0 : I → D,ε : D → I ,δ : D → D(2)) (where D(n) denotes pushout powers of 0)  so that:  [IO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1] Pushout powers D(n) of 0 : I → D exist, and ε ◦ 0 = idI .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  [IO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2] ⊙ is a commutative semigroup with zero, so that the following diagrams commute:  D ⊗ D  D ⊗ D  D ⊗ D ⊗ D  D ⊗ D  D  D ⊗ D  D  D  D  D  ⊙  σ  ⊙  ⊙⊗D  ⊙  D⊗⊙  ⊙  (D,0)  ⊙  0D ◦ε  The third diagram shows that 0 is an absorbing element rather than a unit (think of 0 as being in  the commutative semigroup on the set [0,1) given by multiplication).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  [IO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3] The map δ : D → D(2) makes (0 : I → D,δ,ε) into a commutative comonoid in the coslice I /�:  D  D(2)  D  D(2)  D  D (2)  D  D (2)  D(3)  D(2)  δ  δ  D+I δ  δ+I D  δ  δ  (ι1|ι0)  (0◦ε+I D)  δ  [IO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4] The following diagram commutes (⊙ is coadditive):  D ⊗ D  D  D ⊗ D(2)  D(2)  1×δ  (ι0◦⊙|ι1◦⊙)  δ  ⊙  The notation (f |g ) : A + B → C for pushouts/coproducts is dual to pairing (f ′,g ′) : C → A × B for  pullbacks/products, just as ιi is dual to projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Therefore, (f |g ) ◦ ι0 = f just as π0 ◦ (f ,g ) = f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  18  [IO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5] The following diagram is a coequalizer:  D ⊗ I  D ⊗ D  D(2)  0◦ε⊗0  D⊗0  (⊙|I ε⊗D )◦(δ⊗id)  There are two tangent structures associated to an infinitesimal object in a symmetric monoidal  category �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The first relies on the exponentiability of the infinitesimal object, while the second tangent  structure is on the opposite category of �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The enriched perspective on tangent structure in Section  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1 will clarify the relationship between these two tangent structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let (�,⊗,I ,α,ρ,σ) be a symmetric monoidal category, and  ⊙ : D ⊗ D → D,0 : I → D,ε : D → I ,δ : D → D(2)  define an infinitesimal object in �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) There is a tangent structure on �op , where T = D ⊗ (−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) If � is also a symmetric monoidal closed category, then it is a tangent category with T = [D,−].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) The opposite category of � is a symmetric monoidal category:  (�op,⊗,I ,α−1,ρ−1,σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The tangent functor is D ⊗ (−), and the projection is  p = D ⊗ (−)  0op  −→ I ⊗ (−)  ρ−1  −→ (−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The zero map is given by  (−)  ρ−→ I ⊗ (−)  εop  −→ D ⊗ (−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Addition in �op is given by co-addition in �:  D(2) ⊗ (−)  δop ⊗(−)  −−−−→ D ⊗ (−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The lift is given by the semigroup structure (along with the monoidal coherences):  D ⊗ (−)  ⊙⊗(−)  −−−→ (D ⊗ D) ⊗ (−)  α−1  −→ D ⊗ (D ⊗ (−)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The flip is also given by monoidal coherences, combined with the symmetry on the monoidal  category:  D ⊗ (D ⊗ (−))  α−→ (D ⊗ D) ◦ (−)  σ⊗(−)  −−−→ (D ⊗ D ) ⊗ (−)  α−1  −→ D ⊗ (D ⊗ (−)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) First, we identify the following natural isomorphisms:  u : id ⇒ [I ,−],  b : [A,[B,−]] ⇒ [A ⊗ B,−].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  19  The tangent functor is [D,−],  and the triple (0  :  I  →  D,δ  :  D  →  D(2),  ε : D → I ) gives the additive bundle structure  p : [D,−]  0∗  −→ [I ,−]  u−→ id ,  0 : id  u−→ [I ,−]  [ε,−]  −−→ [D,−]  + : [D (2),−]  δ∗  −→ [D,−].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Note that by the continuity of [−,M ] : �op → �, we have [D(2),M ] ∼= [D,M ]p ×p [D,M ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The lift is given by  ℓ : [D,−]  ⊙∗  −→ [D ⊗ D,−]  b −1  −→ [D,[D,−]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The canonical flip is given by  c : [D,[D,−]]  b−→ [D ⊗ D,−]  σ∗  −→ [D ⊗ D,−]  b −1  −→ [D,[D,−]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The coherences and couniversality properties of an infinitesimal object, along with the continuity  of  [−,M ] : �op → �  then induce the coherences and universality properties for the tangent bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This completes  the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The tangent structure on �op induced by the infinitesimal object is the dual tangent structure on  �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This will be revisited in Chapters 4 and 5 when looking at tangent categories from the enriched  perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Returning to synthetic differential geometry, the co-universality conditions on an infinitesimal ob-  ject corresponds to infinitesimal linearity (Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In some sense, the category of infinitesi-  mally linear objects is the largest subcategory of � for which D is an infinitesimal object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In a model of synthetic differential geometry (�,R), the object D = [d ∈ R|d 2 = 0] is an  infinitesimal object in the category of infinitesimally linear objects in �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  This thesis makes use of the 2-category of tangent categories (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3 of Cockett and Cruttwell  (2014)), which formalizes the notion of a morphism of tangent structure and 2-cells between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  This 2-categorical framework is a departure from the classical theory of synthetic differential geometry,  where the literature only really addresses morphisms of tangent structure in the form of fully faithful  embeddings SMan �→ Microl(�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let (�,�),(�,�′) be a pair of tangent categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A pair (F : � → �,α : F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ⇒ T ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F ) is  a tangent functor if the following diagrams commute:  F  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T2  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  F  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F  T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T 2  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T 2  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T 2  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='+  α2  +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F  α  α  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0  α  ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ  α  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='α  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='α  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='α  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c  c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F  20  A tangent functor is strong whenever α is a natural isomorphism;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' for a sub-(tangent category) inclusion,  α = id .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A tangent functor (F,α) between cartesian tangent categories is a cartesian tangent functor if F  is an isomonoidal functor and α is a monoidal natural transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) The coherences on the canonical flip c guarantee that (T : � → �,c : T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ⇒ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ) is a strong tangent  endofunctor on any tangent category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) Given a pair of tangent functors (A,α) : � → �,(B,β) : � → �, the composition  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A : � → �,β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A ◦ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='α : B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T C  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T D .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A  T E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='α  β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A  )  is a tangent functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iii) A model of synthetic differential geometry (�,R) is well-adapted whenever there is a fully faithful,  strict tangent functor fromSMan to the category of microlinear spaces of �,SMan �→ Microl(�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The  original development of well-adapted models for synthetic differential geometry may be found in  Dubuc (1981), and the reader may check Bunge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (2018) for a recent account of the construction  of such models, or section 3 of Kock (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Atangentnatural transformation γ between tangent functors(F,α),(G ,β) isa natural  transformation so that the following diagram commutes:  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T (A)  G .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T (A)  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F (A)  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='G (A)  γT A  αA  βT A  T γA  If F,G are cartesian tangent functors, then γ is cartesian whenever it is an isomonoidal natural transfor-  mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We will call the 2-category of tangent categories, tangent functors, and tangent nat-  ural transformations TangCat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The 2-category of cartesian tangent categories is CartTangCat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4  Local coordinates in a tangent category  This section develops some structures to facilitate reasoning about higher tangent bundles, using “lo-  cal coordinates.” In the case of a cartesian differential category, T n(A) =  �  2n A, whereas for an open  subset U ⊆ �m the tangent bundle decomposes as T nU ∼= U × (�m)2n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This section introduces two  structures that allow for these arguments in an arbitrary tangent category: differential objects and con-  nections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1 (Cockett and Cruttwell (2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A differential object7 in a cartesian tangent category  is a commutative monoid (A,+A,0A) such that there is a section-retract pair A  λ−→ T A  ˆp−→ A which exhibits  T (A) as a biproduct in the category of commutative monoids:  A ⊕ A ∼= T A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Concretely, a differential object is a commutative monoid equipped with λ : A → T A, ˆp : T A → A, ˆp ◦λ =  id so that the following axioms hold:  7Not to be confused with 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1 from Barr (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  21  DO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1 Coherence between + and λ, ˆp:  T (A × A)  T A  A × A  A  T A × T A  T A × T A  A × A  A  T (A × A)  T A  ˆp  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='+A  (T π0,T π1)  ˆp× ˆp  +A  +A  λ×λ  (T π0,T π1)−1  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='+A  λ  DO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2 Coherence between 0A and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A:  A  1  A  1  T A  A  T A  A  0  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  0A  ˆp  λ  p  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  0A  DO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3 Coherence between +A and +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A:  T2A  A × A  A × A  T2A  T A  A  A  T A  ˆp× ˆp  +A  +  ˆp  λ×λ  +A  λ  +  DO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4 Coherence between λ and ˆp with ℓ:  T A  T 2A  A  T A  A  T A  T A  T 2A  ℓ  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ˆp  ˆp  ˆp  λ  λ  ℓ  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ  A map f : A → B between differential objects (A,λA, ˆpA,+A,0B) → (B,λB, ˆpB,+B,0B) is linear whenever  f preserves the lifts and projections  (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f ◦ λA = λB ◦ f ) and ( ˆpB ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f = f ◦ ˆpA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Following the work in Section 3 of Cockett and Cruttwell (2018), it is sufficient to check that f pre-  serves λ or ˆp (each condition implies the other).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) In the category of smooth manifolds, the real numbers are a differential object, as T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� = �[x]/x 2,  so the lift map in this case is  λ(a) = 0+ a · x,  ˆp(a + b · x) = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  More generally, finite-dimensional real vector spaces, with their canonical smooth manifold struc-  ture, are exactly differential objects in the category of smooth manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The lift map is defined  using T � ∼= �[x]/x 2, so that  V  (0,1�)  −−−→ T (V × �)  T ·V  −→ T V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  This is equivalent to the isomorphism T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V = R[x]/x 2 ⊗ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  22  (ii) Every object in a cartesian differential category has a canonical differential object structure, as  T A := A × A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Classically, the tangent space above each point of a smooth manifold is a vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We have a  similar result for differential objects from Cockett and Cruttwell (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Suppose we have the following pullback in a cartesian tangent category �, and all powers  of T preserve it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  E  T M  1  M  ι  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  p  m  There is a unique differential object structure (E ,λ, ˆp) so that ℓ◦ ι = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ι ◦ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8  There are two natural classes of morphisms between differential objects, linear and “smooth” (that  is, arbitrary morphisms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4 (Cockett and Cruttwell (2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let � be a cartesian tangent category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We define the  following categories:  (i) Diff(�) is the category of differential objects and arbitrary morphisms, so for any differential objects  A,B, we have Diff(�)(A,B) := �(A,B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) DLin(�) is the category of differential objects and linear morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The category of differential objects and smooth maps is a cartesian differential category, exhibiting  differential calculus as a specialized logic in a tangent category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5 (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5 of Cockett and Cruttwell (2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let � be a cartesian tangent category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Then:  (i) Diff(�) is a cartesian differential category, where we define the differential combinator D to be  A  f−→ B  A × A  λ×0  −−→ T (A × A)  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='+A  −−→ T (A)  T (f )  −−→ T (B)  ˆpB  −→ B  (where λA,+A are the lift and addition for the differential object structure on A, and ˆpB is the pro-  jection map on the differential object structure on B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) There is an equality of categories, Lin(Diff(�)) = DLin(�), meaning that a morphism between differ-  ential objects in the cartesian tangent category � is linear if and only if it is linear in the cartesian  differential category Diff(�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Recall that if M is an open subset of �n, the second tangent bundle of an open subsetU ⊆ �n splits  as  T 2(U ) = T (U × �n) = (U × �n) × (�n × �n) = T3U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Every smooth manifold admits such a decomposition on its second tangent bundle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' these are known  as a affine connections9 and provide a way to reason about an object as though it has local coordinates  in an arbitrary tangent category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  8Some diagrammatic notation had creeped into the original draft here, I have fixed it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  9The prefix “affine” differentiates these from more general connections on differential bundles, which are introduced in  Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  23  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6 (Cockett and Cruttwell (2017)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In a tangent category �, define the following:  (i) An affine vertical connection is a map κ : T 2M → T M so that  a) κ is a vertical descent, namely a section of the vertical lift ℓ : T M → T 2M , so κ ◦ ℓ = id ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  b) κ is compatible with both lifts on T 2M : T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='κ ◦ ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='κ ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ = ℓ◦ κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) An affine horizontal connection is a map ∇ : T2 → T 2M so that  a) ∇ is a horizontal lift, namely a section to the horizontal descent (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p) : T 2 ⇒ T2, so that  (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p) ◦ ∇ = id ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  b) ∇  is  compatible  with  the  linear  structures  on  T 2,T2:T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='∇ (ℓ  ×  0)  = ℓ◦ ∇and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='∇(0× ℓ) = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ◦ ∇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iii) An affine connection is a pair (κ,∇) comprising a vertical and horizontal connection on M satisfy-  ing the compatibility conditions:  a) T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' + ◦(+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ (ℓ◦ κ,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0◦ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ),∇(p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p)) = idT 2M ,  b) κ ◦ ∇ = 0◦ p ◦ πi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  An affine connection is torsion-free if κ ◦ c = κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The data ofa vertical connection issufficienttodefine a full connection, asobserved in Lucyshyn-Wright  (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Lemma1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7(Lucyshyn-Wright (2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Afull connection isequivalent to a vertical connection in which  the following diagram is a fiber product:  T M  T 2M  T M  M  T M  p  p  p  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  κ  Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) Every differential object in a tangent category has a canonical vertical connection given by (p ◦  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T, ˆp ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ˆp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A morphism of differential objects will preserve this vertical connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) Every smooth manifold has a non-natural choice of Riemannian metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' By the fundamental the-  orem of Riemannian geometry, there is a torsion-free connection associated with the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (See  any standard reference on Riemannian geometry, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' do Carmo (1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=')  Connections allow for arguments on higher powers of the tangent bundle to be pushed down to  pullback powers of T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Observation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Suppose M ,N each have connections (κ−,∇−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The map T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f : T 2M → T 2N can be  written using local coordinates �  T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f : T3M → T3N as  T 2M  T 2N  T3M  T3N  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='+◦(+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦(ℓ◦π2,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0◦π0),∇(π0,π1))  T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f  (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p,κN )  24  where �  T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f is given by  (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f ◦ π0,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f ◦ π1,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f ◦ π2 +N ∇[f ](π0,π1))  with ∇[f ] := κN ◦ T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f ◦ ∇M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Note that in the case that f preserves the connections, ∇[f ] = 0, as  κN ◦ T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f ◦ ∇M = T f ◦ κM ◦ ∇M = T f ◦ 0◦ p = 0◦ f ◦ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5  Submersions  The category of smooth manifolds is incomplete: there are cospans10 X  f−→ M  g←− Y for which the  pullback fails to exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Following Thom (1954), the pullback of a cospan exists and is preserved by T  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' it is a T-limit) whenever for each point f (x) = g (y ), the direct sum of the images of Tx f and Ty g is  the full vector space Tf (x)M , such cospans are called transverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Submersions, then, form a convenient  class of maps, as any cospan where one map is a submersion will be transverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' More precisely:  Definition  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  If  A  and  B  are  smooth  manifolds,  a  smooth  function  f  :  A → B is a submersion if and only if the derivative D f |a of f at every point a ∈ A is a surjective lin-  ear map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  With this definition we have the following result:  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In the category of smooth manifolds, let the class of submersions be denoted by � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) Submersions are closed under the tangent functor: f ∈ � ⇒ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f ∈ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) Submersions are closed to pullback along arbitrary maps:  X  g ∗X  X  N  M  N  M  g  f ∈�  g ∗ f ∈�  g  f  ¯g  ⌟  This will often be referred to as T -stability under reindexing, as it induces a functor between slice  categories:  g ∗ : Submersions/M → Submersions/N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The properties of the class of submersions in the category of smooth manifolds were studied in  Cockett and Cruttwell (2018) and axiomatized as a tangent display system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A tangent display system in a tangent category � is a class of maps � in � that is  stable under the tangent functor, d ∈ � ⇒ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='d ∈ �,  T -stable under reindexing (as in Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  We call any tangent display system that is closed to retracts in the arrow category a retractive display  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' If for all M , pM ∈ �, we call � a proper (retractive) display system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  10Following a general conventionincategory theory,where the prefix "co"-X means anX inthe opposite category,a cospan  in � is a span in the dual category of �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  25  This section will show that the submersions in the category of smooth manifolds give a retractive  display system, yielding a general construction of retractive display systems from display systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The definition of a submersion may be rephrased as follows: f is a submersion if and only if for all  a ∈ A and all v ∈ T (B) such that f a = pv, there exists a w ∈ T (A) such that T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f ◦w = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This is a weakly  universal cone over A  f−→ B  p←− T B: there exists at least one morphism into it for any other cone over  the diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A commuting square is a weak pullback if for any x : X → A and y : X → B so that  f x = g y , there exists a map X → W making the following diagram commute:  X  W  A  B  C  x  y  ∃  a  b  f  g  If the above diagram is a weak pullback for each T n, then it is a weak T -pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Should the pullback of A  f−→ C  g←− B exist, Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4 is equivalent to asking that the  induced map (a,b ) : W → A f ×g B be a split epimorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let r be a retract of (a,b ) : W → A f ×g B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' For any X  (x,y )  −−→ A f ×g B, the map r ◦ (x, y ) exhibits the  diagram as a weak pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' For the converse, the unique map (a,b ) : W → A f ×g B will be a section of  any map A f ×g B → W induced by weak univerality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  We now restate the submersion property for a map f using global elements (for all a ∈ A and all  v ∈ T (B) such that f a = pv, there exists a w ∈ T (A) such that T (f )w = v) using generalized elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' An arrow f : A → B in a tangent category is a tangent submersion if and only if the  naturality diagram  T A  T B  A  B  T f  p  p  f  is a weak T -pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Following Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5, in the case that the pullback exists this is equivalent to asking for a section  h : A f ×p T B → T A of the horizontal descent (p,T f ) : T A → A f ×p T B (this section is sometimes called  a horizontal lift in differential geometry literature Cordero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (1989)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In smooth manifolds, the T -  pullback along the projection p : T ⇒ id always exists, so to prove that every submersion is a tangent  submersion it suffices to show the existence of a horizontal lift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In the category of smooth manifolds, the tangent submersions are precisely the sub-  mersions (Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' There is an explicit construction of a horizontal lift for a classical smooth submersion in VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1  of Cushman and Bates (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  26  It is possible to show that the T -stability properties for submersions in the category of smooth  manifolds follow from the general theory of weak pullbacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We begin by showing that weak pullbacks  satisfy a weakened version of the pullback lemma and then show that the retract of a weak pullback is  a weak pullback (the second lemma is Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1 of Adámek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (2010)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8 (Pullback lemma).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Consider the diagram (A)  f  g (B)  (i) If f ,g are jointly monic and (A) + (B) is a weak pullback, then (A) is a weak pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) If (A),(B) are weak pullbacks then (A) + (B) is a weak pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) If (A)+(B) is a weak pullback, a map can be induced for a cone over (A) by concatenating it with (B);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  the jointly monic condition on f ,g guarantees that the map induced for (A) + (B) will commute  for (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) Given a cone for (A) + (B) induce a map for (B), which then induces a cone for (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (Weak) pullbacks are closed to retracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Suppose that S ′ is a weak pullback, and S is a retract of it in the category of commuting squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Consider the following diagram (suppressing the subscripts for s,r ):  Z  A  A′  A  B  B ′  B  C  C ′  C  D  D ′  D  x  y  s  x ′  y ′  r  x  y  w  s  w ′  r  w  z  s  z ′  r  z  s  r  Given a cone for S, there is a corresponding cone for S ′ which induces a map Z → A′ and postcompo-  sition with rA gives the desired map into A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Using these lemmas, it is straightforward to prove that the following T -stability properties hold for  tangent submersions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In any tangent category �,  (a) tangent submersions are closed to composition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (b) tangent submersions are closed to retracts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (c) any T -pullback of a tangent submersion is a tangent submersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  27  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (a) follows from Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8 while (b) follows from Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' It remains to prove (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Con-  sider a T -pullback, where u is a tangent submersion:  A  M  B  N  f  v  u  g  By naturality, the outer paths of the following two diagrams are equal:  T A  T M  M  T B  T N  N  T f  T v  p  T u  u  T g  p  =  T A  A  M  T B  B  N  p  T v  f  v  u  p  g  Note that the left diagram is a weak pullback by composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Therefore the outer perimeter of the right  diagram is a weak pullback, and the right square is a pullback, so the left square is a weak pullback by  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The class of tangent submersions is closed to retracts in the arrow category and is conditionally T -  stable under reindexing (if the T -pullback of a tangent submersion exists, it is a tangent submersion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  This stability property leads to the following result:  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let � be a tangent category that allows for reindexing of the class of tangent sub-  mersions �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Then the class of tangent submersions is a display system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Any class of maps that is closed to reindexing is a tangent display system, and the class of sub-  mersions is closed to retracts in the arrow category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The class of submersions in the category of smooth manifolds is a proper retractive  display system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  28  Chapter 2  Differential bundles  Our principal aim in this thesis is to provide an abstract tangent-categorical axiomatization for Lie  algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' To accomplish this, we must provide an axiomatization for Lie algebroids which is essen-  tially algebraic (in the sense of Freyd and Kelly (1972)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' However, in the category of smooth manifolds,  Lie algebroids are defined in terms of vector bundles and these are prima facie a highly non-algebraic  notion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  In addition to algebraic axioms which make it an �-module in the slice over its base M , a vector  bundle q : E → M satisfies a crucial topological requirement: it must be locally trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This means  that the projection q : E → M must be locally isomorphic to a projection π0 : U × �n → U for some  open subset U of M and natural number n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' It is this property that permits calculations using local  coordinates, an approach deeply enshrined in the culture of differential geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Cockett and Cruttwell (2017) introduced the algebraicnotion ofa differential bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Evidence that  differential bundles are the appropriate generalization of vector bundles was provided by showing how  classical results for vector bundles could be generalized to differential bundles in any tangent category  Cockett and Cruttwell (2017, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' However, the precise relationship in the category of smooth man-  ifolds between vector bundles and differential bundles was left open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The main result of this chap-  ter (see MacAdam (2021)) is that vector bundles and differential bundles coincide in the category of  smooth manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The axiomatization of differential bundles focuses on another important property of vector bun-  dles: given a vector bundle q : E → M and a vector v in the fibre Ex above x ∈ M , the tangent space  Tv(Ex) can be naturally identified with Ex .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This gives a lift map λ : E → T E which can be axiomatized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  While the lift map had long been noted in the differential geometry literature in the guise of the Euler  vector field (see 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='11 of Kolár et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (1993) and also Section 1 of Michor (1996) which explicitly uses the  term "lift"), it had not been adopted as the basis of an abstract axiomatization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  More recently, in the differential geometryliterature, Grabowski and Rotkiewicz (2009) and Bursztyn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (2016) realized that the multiplicative �+-action on the total space E determines the vector bundle  structure of q : E → M , and conversely such a multiplicative action determines a vector bundle pre-  cisely when its Euler vector field (Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8) satisfies an additional “non-singular” property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This  chapter extends these more recent observations on vector bundles to differential bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The chapter begins by reviewing vector bundles, describing the Euler vector field construction that  sends a vector bundle q : E → M to a "lift" map λ : E → T E from Grabowski and Rotkiewicz (2009),  whereas the rest of the chapter contains new results developed in collaboration with Matthew Burke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The second section establishes that these lifts are associative coalgebras for the weak comonad (T,ℓ),  and that there is a fully faithful functor from vector bundles into the category of lifts for smooth mani-  folds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The third section identifies the universal property satisfied by the lift (or equivalently Euler vector  29  field), while the fourth section shows that a non-singular lift corresponds precisely to a differential bun-  dle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The fifth section proves the main theorem of the chapter: vector bundles are precisely differential  bundles for smooth manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The final section contains some remarks on extending affine connec-  tions to arbitrary differential bundles, which will be useful in Chapter 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1  Vector bundles  A vector bundle over a manifold M axiomatizes the notion of a smoothly varying family of vector spaces  indexed by the points m ∈ M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The driving example is that of the tangent bundle over a smooth man-  ifold M , where the fibre above each point m ∈ M is the tangent space TmM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The manifold structure  guarantees that the projection is locally trivial: given a chart U �→ M , the next pullback splits as a  product:  U × �n  T M  U  M  p  ⌟  The local triviality of the tangent bundle is essential for various constructions and is part of the defini-  tion of a vector bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A vector bundle is a tuple  (q : E → M ,ξ : M → E ,+ : E q ×q E → E ,· : � × E → E )  of morphisms in SMan so that  (i) the tuple (q,ξ,+,·) defines an �-module in SMan/M ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) the map q : E → M is locally trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The fibred �-module structure means E is a family of vector spaces indexed by M , {Em|m ∈ M }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  It is important to note that the local triviality axiom guarantees that the projection of a vector bun-  dle is a submersion (Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' thus pullback powers of q : E → M exist and are preserved by  the tangent functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Consider the cylinder, defined as the subset of �3 spanned by C = {(x, y,z)|x 2 + y 2 =  1,z ∈ �}:  �  Above each point i ∈ S 1 = {(x, y )|x 2 + y 2 = 1} the fibre over i is �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' For each point i, we can choose a  sufficiently small ε and take the open set  Ui = {(x, y ) ∈ S 1|(ix − x)2 + (iy − y )2 ≤ ε},  which may be flattened to (−(1+ ε),1+ ε) × �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  30  The sections of a vector bundle also give rise to a C ∞(M )-module, generalizing that aspect of the  tangent bundle’s fibred �-module structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Given a vector bundle q : E → M , write the set of sections of q as Γ(q);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' as, for example,  Γ(p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M ) = χ(M ) (recall the notation from Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The set Γ(q) has a C ∞(M )-module structure  in much the same way as χ(M ):  X +Γ(q) Y := + ◦ (X ,Y ), 0Γ(q) := ξ, (f ·Γ(q) X )(m) := f (m) · X (m)  There are also a variety of general constructions that yield vector bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) The tangent bundle is a vector bundle: the construction in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2 makes it clear that the pro-  jection p : T M → M is a locally trivial, fibred �-module over the base space M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) A trivial vector bundle over M with fibres in V is the product M × V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In particular, every vector  space is a trivial vector bundle above the one-point space {∗}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iii) Each TkM will be locally trivial;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' locally it lookslike the k-fold product of the tangent space p −1  k (U ) ∼=  U × (�n)k for an n-dimensional manifold M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' More generally, one can take the fibrewise pullback  Ek = E q ×q E q ×q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q ×q E and discover a vector bundle over M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iv) The cotangent bundle of M , T ∗M , has the dual vector space of TmM above each point M : T ∗  m(M ) =  (TmM )∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This space can be appropriately topologized to be smooth, and a set of sections of Γ(T ∗M )  is isomorphic to the set of morphisms T M → � that are linear in each fibre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This construction may  be applied to any vector bundle and is called the dual vector bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (v) Consider the space Λn(E ), the alternating tensor product of E ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The set of sections of this vector  bundle is equivalent to the alternating n-linear morphism En → �;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' when restricted to the tangent  bundle, this is the space of differential n-forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  There are two constructions on vector bundles that will be necessary to prove the main theorem of  this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let (q : E → M ,ξ,+q,·q) be a vector bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) For any map f : N → M , the T -reindexing of q by f is a vector bundle:  f ∗E  E  N  M  q  f  f ∗q  ¯f  ⌟  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1)  (ii) Any retract of q in the space of arrows is a vector bundle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' that is, given  F  E  F  N  M  N  q  r ′  r  π  s  s ′  π  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2)  if there is a vector bundle structure on q, then there is a vector bundle structure on π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  31  The category of vector bundles has “locally linear” bundle morphisms as its maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A morphism of vector bundles between q : E → M and π : F → N is a commuting  square  E  F  M  N  q  f  v  π  that is fibrewise linear, so that above each fibre  f |m : Em → Fv(m)  isa linearmorphism of vectorspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Thismay equivalently be stated asa morphism of fibred �-modules,  so that the following diagrams commute:  E  F  E  F  M  N  M  N  E2  F2  � × E  � × F  E  F  E  F  q  f  v  π  ξ  ζ  f  v  +  +  f  f2  E  �×f  F  f  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) For the pullback vector bundle in Diagram 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1, the pair ( ¯f , f ) is a linear bundle morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) For the section/retract vector bundle structure from Diagram 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2, the section and retract are linear  morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Note that this is exactly the splitting of a linear idempotent on q : E → M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The lift on the tangent bundle was defined in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2 as  [γ]∼ �→ [γ ◦ ·�]∼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Instead, consider the action of � on a tangent vector:  ([γ]∼,r ) �→ [γ ◦ (r · x)]∼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Note that T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='· gives the equation  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ·◦([ω ◦ (x, y )],[(a,b ) �→ a + b · x]) = (ω ◦ (a · x,a · b · y ));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  so the lift map ℓ can be rederived as follows:  T M  T M × �  T (T M × �)  T 2M  [γ]  ([γ],1)  ([γ ◦ π0],[r �→ 1• r ])  [γ ◦ •�]  (id,1�)  0×λ  This general construction is known as the Euler vector field of a multiplicative action by �+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  32  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Consider a multiplicative monoid action a : �+ × E → E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The Euler vector field1 of  the action is the morphism λ : E → T E constructed as follows:  λ := E  (id,1�◦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=')  −−−−→ E × �  0×λ  −−→ T E × T � ∼= T (E × �)  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='a  −→ T E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Local triviality for the tangent bundle is encoded by the universality of the vertical lift condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A  similar universality condition holds for vector bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let q : E → M be a vector bundle with corresponding Euler vector field λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Then the  following diagram is a T -pullback:  E  T E  M  E × T M  (ξ,0)  q  λ  (p,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q)  Exploiting the fact that fibred �-modules have subtraction, the following result holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The following two diagrams are T-equalizers:  E2  T E  T M  T M p ×q E  T E  E  µE :=0◦π0+T q λ◦π1  νE :=T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ξ◦π0+p λ◦π1  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q◦0◦p  p  p◦T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ξ◦T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q  (recall that E2 is the pullback of a submersion along a submersion and is therefore guaranteed to exist  and be preserved by the tangent functor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Given v : X → T E so that  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q ◦ v = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q ◦ 0◦ p ◦ v  then  p ◦ (v −T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q 0◦ p ◦ v) = p ◦ v −q p ◦ v = ξ ◦ q ◦ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  So there is a unique v ′ so that  λ ◦ v ′ = (v −T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q 0◦ p ◦ v)  meaning that  v = 0◦ p ◦ v +T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q λ ◦ v ′ = µ(0◦ p ◦ v,v ′)  as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The projection q is a submersion, so the pullback E2 is preserved by the tangent functor,  as is the pullback in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9, and the same calculation may be applied for each T n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The proof  for ν follows by the same argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Recall that the class of submersions forms a retractive display system in the category of smooth  manifolds (Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3), so they are stable under reindexing and closed to retracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We may now  infer the following:  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The projection for a vector bundle is a submersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This follows from the fact that π : T E → E is a submersion, so that q ◦ π0 : E q ×p T M → M is a  submersion, so the map q : E → M is a retract of the projection p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E : T E → E in the arrow category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  1Somewhat confusingly, the Euler vector field is almost never a vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  33  Preservation of the Euler vector field is also sufficient to guarantee that a morphism f : E → F  determines a vector bundle morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let q : E → M ,π : F → N be a pair of vector bundles with Euler vector fields λE ,λF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Then a bundle morphism (f ,v) : q → π is a vector bundle morphism if and only if  λF ◦ f = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f ◦ λE  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Note that the νF map from Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='10 is monic, and if f preserves the lift, it preserves ν:  νF ◦ (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='v, f ) = + ◦ (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ζ ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='v,λF ◦ f ) = + ◦ (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='xi,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f ◦ λE ) = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f ◦ νE .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Next, observe that T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='v × f is the unique map making the following diagram commute:  M  E  T M p ×q E  T E  T N p ×q F  T F  N  F  ∃!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  ν  ν  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f  p  p  f  q◦π1  π◦π1  ζ  ξ  ⌟  ⌟  Now ν is a vector bundle morphism and monic, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f is a vector bundle morphism, so it follows  that T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='v × f is a vector bundle morphism and hence f is also a vector bundle morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The reverse  implication is immediate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2  Lifts for the tangent weak comonad  The lift ℓ : T ⇒ T 2 gives rise to a weak comonad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Weak comonads were introduced in Wisbauer (2013)  and have a natural notion of an associative algebra2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' An associative coalgebra of the weak comonad  (T,ℓ) is called a lift;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' this chapter will demonstrate that lifts provide the essential structure necessary to  formulate vector bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A weak comonad on a category � is an endofunctor S : � → � equipped with a coas-  sociative map δ : S ⇒ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='S:  S  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='S  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='S  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='S  δ  δ  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='δ  δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='S  An associative algebra of a weak comonad is an object E equipped with a map λ : E → SE so that  E  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E  λ  λ  δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ  2This is not strictly true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Wisbauer has a more nuanced hierarchy of almost-monads, and in his language (T,ℓ) would be  an endofunctor with an associative product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  34  A morphism of these algebras is a map f : (E ,λ) ⇒ (D,γ) so that  E  D  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='D  f  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f  λ  γ  Recall that for a full (co)monad, there is an adjunction between the base category and the category  of (co)algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' That result is weakened in this case:  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' For every weak comonad on a category �, there is a free coalgebra functor  F : � → CoAlg(�);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E �→ (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E ,δ : S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E → S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E )  an underlying object functor  U : CoAlg(�) → �;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='(E ,λ) �→ E  and a natural transformation  λ : id ⇒ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  E  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E  λ  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ  λ  δ  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A lift in a tangent category � is an associative coalgebra of (T,ℓ), namely, a pair (E ,λ :  E → T E ) so that the following diagram commutes:  E  T E  T E  T 2E  λ  λ  ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ  A morphism of lifts is a coalgebra morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category of lifts and lift morphisms in a tangent cate-  gory � is written Lift(�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Note that the tangent bundle is not, in general, a comonad: while the tangent projection has the  correct type for a counit, p : T ⇒ id , it does not satisfy p ◦ℓ = id .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In fact, if p were a counit, this would  force id = p ◦ ℓ = 0 ◦ p so that 0 = p −1, thus if (T,ℓ,p) is a comonad then T is naturally isomorphic to  the identity functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) For every object M in a tangent category �, the pair (T M ,ℓ : T M → T 2M ) is a lift, called the free  lift on M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) Every object M in a tangent category has a trivial lift, 0 : M → T M , where T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0◦ 0 = ℓ◦ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iii) Every differential object has a lift λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' the coherence is equivalent to axiom [D0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3] in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iv) The Euler vector field of a multiplicative �+-action h : R + ×E → E in SMan is a lift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Recall that the  Euler vector field over the scalar action sM : T M × � → � induces the vertical lift on a manifold:  ℓ = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='sM ◦ (0,λ� ◦ 1�◦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  35  In the category of smooth manifolds, T � ∼= �[x]/x 2, where λ(r ) = [x �→ r ·x] corresponds to the map  λ′(r ) = 0+r ·x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Similarly, there is an isomorphism �[x, y ]/(x 2, y 2) so that for the maps 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Tand T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T (a + r · x) ∼= a + r · x + 0y + 0x y,  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0(a + r · x) ∼= a + 0x + r · y + 0x y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Since we know that (0 + x)(0 + y ) = (0 + x y ) in �[x, y ]/(x 2, y 2) (following 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2), we can use these  isomorphisms to see that  ℓ◦ λ� ◦ 1�◦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ λ� ◦ 1�◦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=') ·T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0◦ λ� ◦ 1�◦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Consider a monoid action (�+,h) on a manifold E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The Euler vector field of this action,  λ : E  (id,1�◦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=')  −−−−→ E × �  (0,λ�)  −−−→ T E × T �  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='h  −→ T E  will define an algebra if the induced scalar action on T E commutes with the natural scalar action:  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ ◦ λ = T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='h ◦ (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0◦ λ,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ ◦ 0◦ 1�◦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=')  = T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='h ◦ (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='h ◦ (0,λ ◦ 1�◦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ),0◦ λ ◦ 1�◦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=')  = T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='h ◦ (T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='h ◦ (T 0◦ 0,T 0◦ λ ◦ 1�◦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ),0◦ λ ◦ 1�◦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=')  = T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='h ◦ (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0◦ 0,(T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0◦ λ ◦ 1�◦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=') ·T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ λ ◦ 1�◦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='))  = T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='h ◦ (ℓ◦ 0,ℓ◦ λ ◦ 1�◦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=')  = ℓ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='h ◦ (0,λ ◦ 1�◦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=')  = ℓ◦ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Observation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Recall that by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='12, morphisms preserve a monoid action if and only  if they preserve the associated Euler vector field of the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This means the Euler vector field construc-  tion gives a fully faithful functor from monoid actions to lifts in the category of smooth manifolds, and  therefore from the category of vector bundles to the category of lifts in SMan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The following proposition gives a pair of constructions on lifts—closure under the tangent functor  and finite T -limits—that will be useful in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let � be a tangent category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) The tangent functor lifts to an endofunctor on the category of lifts in �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) Given a diagram  D : � → Lift(�)  in the category of lifts of �, if the T -limit ofU .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='D exists in �, then limU .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='D has a natural lift λ′ asso-  ciated to it so that (limU .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='D,λ′) is the limit of D in Lifts(�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (That is, T -limits of lifts are computed  pointwise in the base category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=')  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) Simply check that  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='(c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ) ◦ c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c ◦ T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ ◦ c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c ◦ c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ  = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c ◦ c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  36  (ii) Concretely, a tangent terminal object will have a lift:  (1,1  0−→ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1 ∼= 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Given (E ,λ) and (F,l ), if the tangent product E × F exists there is a lift  (E × F,E × F  λ×l  −−→ T E × T F ∼= T (E × F )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Given the T -equalizer of a fork f ,g : (E ,λ) → (F,l ), the equalizer has a lift induced as follows:  C  E  F  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='C  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F  f  g  k  λ  k  λ  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='g  l  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category of lifts is a tangent category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The tangent functor sends  f : (E ,λ) → (F,l )  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f : (T E ,c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ) → (T F,c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='l ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  To see that this is still an algebra morphism, compute  c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='l ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f = c ◦ T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ = T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f ◦ c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ  The structure maps are the structure maps on the underlying object of the lift;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' the universality condi-  tions follow by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The following idempotent is key in the theory of lifts and will be used in defining non-singular  lifts (Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1), and its splitting will present the projection and zero-section of a vector bundle  (Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category of lifts in a tangent category � has a natural idempotent:  e : id ⇒ id ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e(E ,λ) : (E ,λ)  p◦λ  −−→ (E ,λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' First, we see that e = p ◦ λ is an idempotent:  p ◦ λ ◦ p ◦ λ = p ◦ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ ◦ λ = p ◦ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ ℓ◦ λ = p ◦ 0◦ p ◦ λ = p ◦ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Moreover, every f : (E ,λ) → (F,l ) preserves the idempotent:  f ◦ p ◦ λ = p ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f ◦ λ = p ◦ l ◦ f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Finally, note that the idempotent is a lift morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' :  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ ◦ λ = ℓ◦ λ = c ◦ ℓ◦ λ = c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ ◦ λ  which implies that  λ ◦ e = λ ◦ p ◦ λ = p ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ ◦ λ = p ◦ c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ ◦ λ = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ ◦ λ = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e ◦ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  37  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3  Non-singular lifts  Grabowski and Rotkiewicz (2009) introduced the notion of a non-singular lift as a means to axioma-  tize the Euler vector field of a vector bundle’s multiplicative �+-action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' While it is not immediately  clear that our definition is the same as Grabowski’s, the results of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5 will justify the use of this  language as they are necessarily the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A lift (E ,λ) in a tangent category � is non-singular whenever the following diagram is  a T -equalizer:  E  T E  T E  λ  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E  where e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E = p ◦ ℓ (the idempotent associated to the free lift on E ) and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ (the image of the  idempotent associated to (E ,λ) under the tangent functor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category of non-singular lifts is written  NonSing(�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The most prominent class of examples is given by the Euler vector field of the �-action on a vector  bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The Euler vector field of a vector bundle is a non-singular lift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let (q : E → M ,+,ξ,·) be a vector bundle with Euler vector field λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9, the  diagram  T M  E  T E  E  λ  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q  p◦T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ξ◦T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q  p  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q◦0◦p  is a T -limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The T -universality of this diagram will hold if and only if the diagram is universal after  each parallel pair of arrows is post-composed by a T -monic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A section is a T -monic, so the previous  diagram is T -universal if and only if the following diagram is T -universal:  E  T E  E  T E  T M  T E  p  ξ◦q◦p  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q  0◦q◦p  λ  0  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ξ  Now simplify this diagram using the fact that T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='(ξ ◦ q) = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e ,0◦ p = e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E :  T E  E  T E  T E  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E ◦T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E ◦T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  λ  38  Note that the the two pairs of parallel arrows have a common arrow, implying that they may be pulled  together into a single ternary equalizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' All that remains to check, then, is that for any x : X → T E ,  (e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e ◦ x = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e ◦ x = e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E ◦ x) ⇐⇒ (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e ◦ x = e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E ◦ x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The forward implication is trivial, so it remains to prove the reverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Suppose T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e ◦ x = e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E ◦ x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' then  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e ◦ x = e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e ◦ x = e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E ◦ e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E ◦ x = e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E ◦ x  giving the result, namely that the diagram  E  T E  T E  λ  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E  is a T -equalizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Every map f : E → F gives a map of free coalgebras T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f : (T E ,ℓ) → (T F,ℓ) and the idempotent e is  a coalgebra morphism by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8, so the following is immediate:  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A non-singular lift is an equalizer in the category of lifts:  (E ,λ)  (T E ,ℓ)  (T E ,ℓ)  λ  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E  Observe that the category of non-singular lifts is closed under finite limits in the category of lifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category of non-singular lifts in � is closed under T -limits:  (i) The tangent functoron liftspreservesnon-singularlifts, so that if (E ,λ) isnon-singularthen (T E ,c ◦  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ) is non-singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) The trivial lift on an object, 0 : M → T M , is non-singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iii) T -products of non-singular lifts are non-singular lifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iv) T -equalizers of non-singular lifts are non-singular lifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) This follows from the fact that the non-singularity condition is a T -limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) The zero map splits the idempotent 0◦ p, so it is the equalizer of 0◦ p,p ◦ 0 = id .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iii) This follows by stability of limits under products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iv) The following diagram commutes by naturality:  C  E  F  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='C  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='C  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F  λ  λ  f  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='g  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='g  g  λ  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='C  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  Each horizontal diagram is a T -equalizer, and the two columns on the right are T -equalizers, so  the column on the left is a T -equalizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  39  Finally, when a tangent category has certain T -equalizers, there is an idempotent monad on the  category of lifts, whose algebras are non-singular lifts:  Theorem2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let � be a tangent category with chosen T -equalizersof idempotents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Then the following  equalizer determines a left-exact idempotent monad on the category of lifts, whose algebras are non-  singular lifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (F,l )  (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E ,ℓ)  (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E ,ℓ)  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' First, take the equalizer in Lift(�);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' the functor sends (E ,λ) to the chosen limit (F,l ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='The unit of  the monad is the unique morphism from (E ,λ) to the equalizer (F,l ) induced by universality:  (F,l )  (T E ,ℓ)  (T E ,ℓ)  (E ,λ)  λ  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E  ∃!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Note that non-singular lifts are closed under finite limits, so (F,l ) is a non-singular lift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' If (E ,λ) is a  nonsingular lift then λ : (E ,λ) → (T E ,ℓ) equalizes the diagram, so there is a unique isomorphism  (F,l ) ∼= (E ,λ), making the multiplication of the monad a natural isomorphism (and thus yielding an  idempotent monad).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The functor is defined as a T -limit and therefore preserves all T -limits of lifts, so  it is left-exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  In the category of smooth manifolds, ℓ is the Euler vector field of an �+-action, this guarantees that  every λ is the Euler vector field of a multiplicative �+-action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We note the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In the category of multiplicative �+ actions in SMan, multiplication by 0 is equivalent  to the natural idempotent e in the fully faithful functor sending an �+-action to its Euler vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  By non-singularity, the following diagram is an equalizer, giving E a multiplicative action by �+ whose  Euler vector field is λ:  (E ,·E )  (T E ,·T )  (T E ,·T )  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  λ  Moreover, λ is the Euler vector field of this lift, and the following diagram commutes:  E  T E  T E  �+ × E  �+ × T E  �+ × T E  T (�+ × E )  T (�+ × T E )  T (�+ × T E )  T E  T T E  T T E  T (�+×e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E )  T (�+×T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e )  �+×λ  p  T ·p  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  T λ  T ·E  (1�,id)  (1�,id)  (1�,id)  λ�×0  λ�×0  λ�×0  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  �+×e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E  �+×T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  λ  �+×λ  40  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4  Differential bundles  This section introduces (pre-)differential bundles, which provided the rest of the data for a vector bun-  dle: namely the projection, the zero section, and the addition map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The zero section and projection  data arise by splitting the natural idempotent e : id ⇒ id , and non-singularity will induce the addi-  tion map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Every differential bundle satisfies a pair of universality diagrams, linking this presentation  of differential bundles to the original definition in Cockett and Cruttwell (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) A pre-differential bundle is a lift λ : E → T E equipped with a chosen splitting of the natural idem-  potent e = p ◦ λ from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Pre-differential bundles are formally written (q : E →  M ,ξ,λ), where q : E → M is the retract, ξ : M → E the section, and λ : E → T E the lift (the types  are only necessary for the projection as the rest may be inferred, and will generally be suppressed to  save space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) A differential bundle is a pre-differential bundle (q : E → M ,ξ,λ) with the properties that λ is  non-singular and T -pullback powers of q exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Morphisms of (pre-)differential bundles are exactly morphisms of their underlying lifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The categories  of (pre-)differential bundles are exactly (pre-)differential bundles and lift morphisms, and are written  Pre(�),DBun(�) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Recall that as e is a natural idempotent in the category of lifts, any lift morphism will preserve e  and will consequently preserve its idempotent splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Preserving the idempotent means that every  differential bundle morphism is a bundle morphism, where the base map is given by  E  F  M  N  q  f  π  m:=π◦f ◦ξ  We now look at the limits of (pre-)differential bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Observation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) The limit for a diagram of pre-differential bundles is the limit of the underlying lifts equipped with  a chosen splitting of p ◦ λ due to basic properties about idempotent splittings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) The limit for a diagram of differential bundles is the limit in the category of pre-differential bun-  dles (the lift will be universal by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4), so long as T -pullback powers of the resulting  projection exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Because there is a projection associated to a (pre-)differential bundle, the T -reindexing operation  described in Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2 can now be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This gives a pullback differential bundle in a similar  way to Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3 (Cockett and Cruttwell (2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let (q : E → M ,ξ,λ) be a (pre-) differential bundle in a  tangent category �, and consider a T -pullback in �:  u∗E  E  N  M  u  q  u∗q  ¯u  ⌟  41  Then the induced triple maps  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='u∗E  T E  N  u∗E  E  u∗E  E  N  M  N  M  T N  T M  u  q  u∗q  ¯u  ⌟  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ¯u  λ  0  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='u  0  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='u∗q  u∗λ  q  u  ⌟  ξ◦u  u∗ξ  induce a (pre-)differential bundle (u∗q : u∗E → N ,u∗ξ,u∗λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' If λ is non-singular and T -pullback pow-  ers of u∗q exist, then (u∗q,u∗ξ,u∗λ) is a differential bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Note thata pre-differential bundle (q : E → M ,ξ,λ) in � mayalsobe regarded asa pre-differential  bundle (q : (E ,λ) → (M ,0),ξ,λ) in Lift(�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Take the following pullback in Lift(�):  (u∗E ,u∗λ)  (E ,λ)  (N ,0)  (M ,0)  u  q  ⌟  It follows by construction that ι ◦ u∗λ = λ ◦ ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Thus the result holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let (q : E → M ,ξ,λ) be a non-singular pre-differential bundle in a tangent category  �, so that T -pullback powers of q exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Then there is an additive bundle structure (q,ξ,+q) so that the  differential bundle morphisms are additive:  (λ,ξ) : (q,ξ,+q) → (p,0,+)  (λ,0) : (q,ξ,+q) → (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ξ,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='+q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Furthermore, every differential bundle morphism preserves addition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Non-singularity forces the existence of an addition map:  E2  T2E  T2E  E  T E  T E  ∃!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='+q  +  λ×λ  λ  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  +  (e ×e ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E  T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  Note that this diagram commutes because T2E is a pre-differential bundle whose lift is ℓ×ℓ, and also +  is a linear morphism, so it commutes with the addition map e ◦a +e ◦b = e ◦(a ×b ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Post-composition  with λ ensuresthatξ isthe unitand thatassociativityholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Adifferential bundle morphism will induce  a morphism of the equalizer diagrams that induce each addition map to preserve addition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Recall that by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2, the Euler vector field for every vector bundle is a non-singular lift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  If T M q ×q E exists, the map  νE : T M p ×q E  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ξ×λ  −−−→ T2E  +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E  −→ T E  may be formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Similarly, using the additive bundle structure from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4, the µ map may  be formed:  µE : E q ×q E  0×λ  −−→ T (E q ×q E )  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='+q  −−→ T E  Note that any differential bundle morphism will preserve µ and ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  42  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Differential bundle maps preserve µ(x, y ) := 0◦ x +T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q λ◦ y and ν(v, y ) := T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ξ◦v +p λ◦ y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The following diagram demonstrates that lift maps preserve ν:  E q ×p T M  F q ′×p T N  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F  ν  ν  f ×T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='m  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='g  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='g  g ×T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='n  Similarly, this diagram demonstrates that lift maps preserve µ:  T E  T F  T E2  T F2  E2  F2  (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f ×T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f )  (λ×0)  (λ′×0)  (f ×f )  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='+  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='+′  T f  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Consider the full subcategory of differential bundles in � whose objects are differential  bundles (E ,λ) so that the forks  E2  T E  T M  T M p ×q E  T E  E  µE :=0◦π0+T q λ◦π1  νE :=T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ξ◦π0+p λ◦π1  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q◦0◦p  p  p◦T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ξ◦T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3)  are T -equalizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This subcategory is closed under T -equalizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Start with the T -equalizer of lifts:  C  E  F  f  g  k  Observe that k is a lift map, so by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5, the following diagram commutes:  C q c ×p T P  E q ×p T M  F q ′×p T N  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='C  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='C  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F  ν  ν  f ×T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='m  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='g  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='g  k×T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='k ′  ν  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='k  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='k  g ×T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='n  43  Next, since k is a morphism of pre-differential bundles, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5 the following diagram com-  mutes:  C2  E2  F2  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='C  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='C  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F  µ  µ  f2  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='g  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='g  g2  µ  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  The top row follows because maps satisfying Rosicky’s universality condition preserve µ, and the bot-  tom row by the naturality of e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='C and the fact that linear maps preserve the natural idempotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Thus, if  E ,F satisfy the universality diagrams in Diagram 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3, the equalizer C will as well, because T -equalizers  are closed to T -limits in the category of fork diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' For every differential bundle (q : E → M ,ξ,λ) in a tangent category, the diagram  E2  T E  T M  µE  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q◦0◦p  is a T -equalizer, and if T M p ×q E exists, then  E q ×p T M  T E  E  νE :=λ◦π0+p T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ξ◦π0  p  p◦T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ξ◦T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q  is a T -equalizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The tangent bundle satisfies both universality conditions, and by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6every differential  bundle will satisfy these conditions by the non-singularity of the lift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In a complete tangent category, the category of differential bundles is precisely the cat-  egory of algebras of the monad in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5 on pre-differential bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The universality conditions in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7 demonstrate that this definition of differ-  ential bundle agrees with that of Cockett and Cruttwell (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5  The isomorphism of categories  It is now straightforward to show that the main theorem of this chapter holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' First, observe that for  every differential bundle (q : E → M ,ξ,λ) in the category of smooth manifolds, the T -pullback E p ×q  T M exists , as p is a submersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Note that the universality of the two diagrams is equivalent:  E q ×p T M  T E  E  νE :=λ◦π0+p T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ξ◦π0  p  p◦T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ξ◦T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q  E q ×p T M  T E  M  E  νE  λ◦π0+p T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ξ◦π1  p  ξ  ⌟  Thus the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  44  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' There is an isomorphism of categories between vector bundles and differential bundles  in smooth manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Note that Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2 gives a fully faithful functor from the category of vector bundles to  differential bundles of smooth manifolds, as the lift associated to the vector bundle is non-singular and  the projection and zero section give the rest of the structure of a differential bundle: as remarked after  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1, local triviality guarantees that the projection is a submersion, so pullback powers of  the projection exist, yielding a differential bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  To see there is an isomorphism on objects, recall that every differential bundle is a fibred �-module  by Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6, and that this identification is a bijective mapping (it recovers the original �-action  from the Euler vector field of the �-action, and vice versa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Last, note that the universality condition  E q ×p T M  T E  M  E  νE  λ◦π0+p T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ξ◦π1  p  ξ  ⌟  along with Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5 ensures the local triviality of q, so that the unique fibred �-module struc-  ture associated to a differential bundle is indeed a vector bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6  Connections on a differential bundle  The connections discussed in this section generalize the notion of an affine connection to a differential  bundle, giving a “local coordinates” presentation for T E similar to the presentation of T 2M as T3M  induced by an affine connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Chapter 3 makes extensive use of connections on vector bundles, so  it is useful to set out the basic definitions before embarking on algebroid theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1 (Cockett and Cruttwell (2017)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let (q : E → M ,ξ,λ) be a differential bundle in a tan-  gent category �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  A vertical connection is a map κ : T E → E so that  (i) κ is a vertical descent, and hence a retract of the lift;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' thus, κ ◦ λ = id ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) κ is compatible with both differential bundle structures on T E , so that the maps  κ : (T E ,ℓ) → (E ,λ)  κ : (T E ,c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ) → (E ,λ)  are lift morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  A horizontal connection is a map ∇ : E q ×p T M → T E so that  (i) ∇ is a horizontal lift, and so is a retract of (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E ,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q) : T E → E q×pT M 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' thus, (p,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q)◦∇ = id ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) ∇ is compatible with each pair of lifts, so that the maps  ∇ : (E q ×p T M ,λ × ξ) → (T E ,c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ)  ∇ : (E q ×p T M ,0× ℓ) → (T E ,ℓ)  are lift morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  3That (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E ,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q) land in the pullback E q ×p T M is a consequence of naturality, as p ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q = q ◦ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  45  A full connection is a pair (κ,∇) that satisfies the following compatibility relations:  (i) κ ◦ ∇ = ξ ◦ q ◦ π0,  (ii) ∇(p,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q) +T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q µ(p,κ) = id , so that there is an isomorphism E q ×p T M p ×q E ∼= T E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  A notion that will be useful when dealing with classical differential geometry is that of a covariant  derivative, whose definition is equivalent to that of a vertical connection in the category of smooth  manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let (q : E → M ,ξ,λ,κ) be a vertical connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4 The covariant derivative associated  to (κ,∇) is the map  ∇(−)[=] : Γ(π) × Γ(p) → Γ(π);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='(A,X ) �→ (κ ◦ T A ◦ X ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Lucyshyn-Wright (2018) drastically simplified the notion of a full connection by showing that it is  exactly a vertical connection satisfying a universal property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3 (Lucyshyn-Wright (2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A connection on a differential bundle is equivalently spec-  ified by a vertical connection  κ : T E → E  so that the following diagram exhibits T E as a biproduct in the category of differential bundles over M :  E  T E  T M  M  E  p  T q  κ  q  q  p  There is also a notion of flatness for connections that extends to vertical connections on a differen-  tial bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A connection κ on a differential bundle (q : E → M ,ξ,λ) is flat whenever  κ ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='κ ◦ c = κ ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  We can see that connections are closed under similar constructions to differential bundles, in par-  ticular idempotent splittings and the reindexing construction from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let (q : E → M ,ξ,λ) be a differential bundle equipped with a vertical connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) Any linear retract of (q,ξ,λ) will have a vertical connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) The pullback differential bundle induced by pulling back q along f : N → M will have a vertical  connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  If the vertical connection is flat or is part of a full connection (that is, it satisfies the universality condition  in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3), then the induced connection will be as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The universal property induces the vertical connection in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The construction will pre-  serve flatness as it is an equational condition, and it preserves effectiveness by the commutativity of  limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  4The definition of a covariant derivative only uses a vertical connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  46  There is no reason for every differential bundle in a tangent category to have a connection (for  example, the tangent bundle in the free tangent category Weil1 in Chapter 4 is a differential bundle  that does not have a connection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' However, if the total space of a differential bundle has an affine  connection, this induces a compatible connection on the differential bundle and its base space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let (q : E → M ,ξ,λ) be a differential bundle in a tangent category in �, where E has a  (flat) vertical connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Then the total space M and the differential bundle (q,ξ,λ) each have a (flat)  vertical connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' If the connection is full (so there is a compatible horizontal connection), then the  induced connections are likewise full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' By the strong universality condition for differential bundles, the differential bundle (q ◦ π0 :  E q×pT M → M ,(ξ,0),λ×ℓ) is the pullback differential bundle of p : T E → E along ξ : M → E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This gives  (q◦π0,(ξ,0),(λ,ℓ)),a (flat, effective) vertical connection byLemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5, and soityields(q : E → M ,ξ,λ)  and (p : T M → M ,0,ℓ) as (flat, effective) vertical connections by the idempotent splitting property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Every smooth manifold has an affine connection, thus inducing a connection on any vector bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Every vector bundle has a connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  47  Chapter 3  Involution algebroids  This chapter accomplishes the first major goal of this thesis by providing a tangent-categorical axiom-  atization of Lie algebroids, namely involution algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Much like vector bundles, Lie algebroids are  a highly non-algebraic notion (in the sense of Freyd and Kelly (1972)), being vector bundles equipped  with a Lie algebra structure on the set of sections of the projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Furthermore, the bracket on sec-  tions must satisfy a product rule with respect to �-valued functions on the base space (the Leibniz  law), introducing another piece of non-algebraic structure to the definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The tangent-categorical  definition of Lie algebroids will treat the tangent bundle as the “prototypical Lie algebroid” in which  the vertical lift ℓ : T ⇒ T 2 identifies the vector bundle structure and the canonical flip c : T 2 ⇒ T 2 plays  the role of the Lie bracket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Lie algebroidsare the natural many-objectanalogue toLie algebras, in the same waythatLie groupoids  are the many-object analogue of Lie groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In the single-object case, a Lie group is classically thought  of as a space of symmetries for some smooth manifold (one often identifies a group action G ×M → M ),  and a Lie algebra may similarly be thought of as a space of derivations (often identified as a sub-Lie-  algebra of χ(M ) for a manifold M ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The extension of groups to groupoids is natural;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' in fact, Brandt’s  introduction of groupoids in Brandt (1927) predates MacLane and Eilenberg’s invention of category  theory in Eilenberg and MacLane (1945) by nearly two decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The translation of Lie algebras to the  many-object case is not as straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The first step is to replace the vector space underlying a Lie  algebra with a vector bundle (π : A → M ,ξ,λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The idea is to axiomatize this vector bundle so that each  section in Γ(π) corresponds to a derivation on C ∞(M ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The anti-commutator operation on derivations  from Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='18 suggests there should be a Lie bracket [−,−] : Γ(π) ⊗ Γ(π) → Γ(π) (similar to the  partially-defined multiplication for a groupoid), while the correspondence with derivations on C ∞(M )  suggests there be a vector bundle morphism ̺ : A → T M satisfying the Leibniz law:  [X , f · Y ] = f ·[X ,Y ]+ [X , f ]· Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  [X , f ] := ˆp ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f ◦ ̺ ◦ X 1  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1)  (the full definition of Lie algebroids may be found in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This “operational” definition of Lie alge-  broids makes it difficult to describe their morphisms, and furthermore it essentially fails to be an alge-  braic structure in the classical sense, as it axiomatizes structure on the set of sections of a map rather  than a morphism in the category itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Involution algebroids were introduced to provide a tangent-categorical presentation of Lie alge-  broids, similar to the relationship between differential bundles and vector bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Chapter 2 focused  on the Euler vector field construction on a vector bundle, showing that this induced a fully-faithful  functor from vector bundles to associative coalgebras (lifts) of the weak comonad (T,ℓ), and identified  vector bundles with a subcategory of Lift(SMan) satisfying a universal property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The corresponding  1Recall the notation from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  48  construction for Lie algebroids, then, is the canonical involution, which was identified by Eduardo  Martinez and his collaborators (a clearly written exposition may be found in Section 4 of de León et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (2005)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Given a Lie algebroid (π : A → M ,̺ : A → T M ,[−,−] : Γ(π) ⊗ Γ(π) → Γ(π)), its canonical involu-  tion is a map  σ : A̺×T πT A → A̺×T πT A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Using this σ map, there is a straightforward characterization of Lie algebroid morphisms: a Lie alge-  broid morphism is precisely a vector bundle morphism (f ,m) : A → B that preserves the anchor and  involution maps:  A  B  A̺×T πT A  B ̺B ×T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='πB T B  T M  T N  A̺×T πT A  B ̺B ×T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='πB T B  ̺A  ̺B  f  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='m  σB  f ×T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f  σA  f ×T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f  Furthermore, it is implicit in Martinez’s work (Martínez (2001)) that σ satisfies axioms corresponding  to the Lie algebroid axioms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Thus, involutivity corresponds to antisymmetry of the Lie bracket  σ ◦ σ = id ⇐⇒ [X ,Y ]+ [Y ,X ] = 0,  while the Leibniz law holds if and only if the ̺ map sends the algebroid involution to the canonical flip  on M ,  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺ ◦ π1 ◦ σ = c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π◦ ̺ ⇐⇒ ∀f ∈ C ∞(M ),[X , f · Y ] = f ·[X ,Y ]+ [X , f ]· Y  (using the same definition as before for [X , f ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The idea of an involution algebroid, then, is to axiomatize the canonical involution directly, just as  differential bundles axiomatize the Euler vector field of a differential bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' An involution algebroid  is a differential bundle equipped with a pair of structure maps  ̺ : A → T M , σ : A̺×T πT A → A̺×T πT A  satisfying a collection of axioms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Some of them are straightforward translations of the structure equa-  tions for Lie algebroids given in Martínez (2001), for instance  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺ ◦ λ = ℓ◦ ̺, σ ◦ σ = id , T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺ ◦ π1 ◦ σ = c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺ ◦ π1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  However, this requires a new coherence between the Euler vector field of the underlying vector bundle  and the involution map:  σ ◦ (ξ ◦ π,λ) = (ξ ◦ π,λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The most striking new fact about this coherence is that the Jacobi identity on the bracket [−,−] corre-  sponds to the Yang–Baxter equation on σ:  A̺×T πT AT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺×T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='πT 2A  A̺×T πT AT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺×T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='πT 2A  A̺×T πT AT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺×T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='πT 2A  A̺×T πT AT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺×T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='πT 2A  A̺×T πT AT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺×T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='πT 2A  A̺×T πT AT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺×T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='πT 2A  σ×c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A  σ×c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A  1×T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ  σ×c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A  1×T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ  This is both surprising (it is a new characterization of a central object of study in differential geom-  etry and mathematical physics) and yet in a way expected (the work in Cockett and Cruttwell (2015),  49  Mackenzie (2013) indicates that the Lie algebra structure on the set of vector fields over a manifold  follows from the Yang–Baxter equation on c ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This vector-field-free presentation of the Jacobi identity  allows for a structural approach to Lie algebroids that drives the work presented in Chapters 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  As with Chapter 2, the first section is expository, and is concerned with introducing the category of  Lie algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The second section introduces anchored bundles, together with the space of prolonga-  tions of an anchored bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The relationship between anchored bundles and involution algebroids  is equivalent to that between reflexive graphs and groupoids (the subject of Chapter 5), the space of  prolongations of an anchored bundle being equivalent to the set of composable arrows for a groupoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  This section is mostly a translation of Martinez’s prolongation construction to a general tangent cate-  gory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The rest of the chapter contains new results, developed in collaboration with Matthew Burke and  Richard Garner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Section 3 introduces involution algebroids, which are anchored bundles equipped with an involu-  tion map on their space of prolongations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Section 4 considers an anchored bundle in a tangent cate-  gory with negatives that is equipped with a connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The connection gives an involution algebroid  a “local coordinates” presentation (in the sense of Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4) that is equivalent to the local character-  ization of Lie algebroids from Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The final section of this chapter establishes the main result:  the category of Lie algebroids is isomorphic to that of involution algebroids in smooth manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1  Lie algebroids  This section reviews the basic theory of Lie algebroids: their definition and that of their morphisms,  along with some introductory examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The classical definition will not appear elsewhere in this  chapter, however, as we quickly introduce Martinez’s structure equations for a Lie algebroid (Martínez  (2001)), then translate them into tangent-categorical terms using a connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A Lie algebroid is a vector bundle π : A → M equipped with an anchor ̺ : A → T M  and a bracket [−,−] : Γ(π) ⊗ Γ(π) → Γ(π) satisfying the following axioms:  bilinear: [a X1 + b X2,Y ] = a[X1,Y ]+ b [X2,Y ] and [X ,a Y1 + b Y2] = a[X ,Y1]+ b [X ,Y2]  anti-symmetric: [X ,Y ]+ [Y ,X ] = 0  Jacobi: [X ,[Y ,Z ]] = [[X ,Y ],Z ]+ [Y ,[X ,Z ]]  Leibniz: [X , f · Y ] = f ·[X ,Y ]+ [X , f ]· Y  (where [X , f ] is defined as in Equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) The canonical example of a Lie algebroid is, of course, the tangent bundle using the operational  tangent bundle from Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) ALie algebra isa Lie algebroid overthe terminal object: fora groupG , the bundle of source-constant  tangent vectors is the usual Lie functor from Lie groups to Lie algebras, because a groupoid is a one-  object group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  50  (iii) The bundle of source-constant tangent vectors s,t : G → M of a Lie groupoid forms a Lie algebroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  This bundle is defined by the pullback  A  T G  T M  M  T M ×G  (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='s,p)  (0,e )  π  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='t  ̺  ⌟  where the projection is π and the target is given by ̺ in the diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' There is an injective �-module  morphism from sections of π, Γ(π) to vector fields on G , χ(G ), and the Lie bracket on G is closed  over the image of this lift, putting a Lie bracket on Γ(π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In particular, we can see that T M is the  bundle of source-constant tangent vectors for the pair groupoid on a manifold M :  T M  T (M × M )  M  (M × M ) × T M  (p,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π0)  (∆,0)  p  (id,0◦p)  ⌟  Given (u,v) : X → T (M × M ) above (m,m) : X → T M with u = 0 ◦ m, it follows that u = 0 ◦ p ◦ u,  so v is the unique map induced into T M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iv) Every group is a groupoid over a single object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The Lie algebroid associated with a group G , then, is  the usual Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (v) From the Hamiltonian formalism of mechanics, every Poisson manifold has an associated Lie alge-  broid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A Poisson manifold is a manifold M equipped with a Poisson algebra structure on C ∞(M ),  namely a Lie algebra that is also a derivation:  [f ·g ,h] = f ·[g ,h]+ [f ,h]·g  where · is the multiplication in the algebra C ∞(M ) as in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The cotangent bundle over  a Poisson manifold M is canonically a Lie algebroid, called a Poisson Lie algebroid Courant (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (vi) Any Lie algebra bundle—that is, a vectorbundle equipped with a Lie bracket on itsspace of sections—  is a Lie algebroid, with anchor map ξ ◦ π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Morphisms of Lie algebroids are notoriously difficult to work with, and have an involved definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let A,B be a pair of anchored bundles over M ,N , and Φ : A → B an anchored bundle  morphism over a map φ : M → N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A Φ-decomposition of X ∈ Γ(πA) is a set of Xi ∈ Γ(πB) and fi ∈ C ∞(M )  so that  Φ ◦ X =  �  i  fi · Xi ◦ φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  An anchor-preserving vector bundle morphism Φ is a Lie algebroid morphism if and only if for any X ,Y ∈  Γ(πA) and Φ-decompositions {Xi , fi},{Yj,g j} of X ,Y , the following equation holds:  Φ ◦ [X ,Y ] =  �  fi ·g j ·([Xi ,Yj]◦ φ) +  �  [X , fi ]·(Xi ◦ φ) −  �  [Y ,g j](Yj ◦ φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  51  The equation defininga Lie algebroid morphism holdsindependentlyofthe choice ofΦ-decomposition  (see Higgins and Mackenzie (1990) for a proof).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) If A,B are Lie algebroids over a base manifold M , Φ : A → B is a Lie algebroid morphism if and only  if it preserves the anchor and Lie bracket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) Given two Lie algebroid morphisms f : A → B,g : B → C , their composition g ◦ f is also a Lie  algebroid morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iii) A Poisson Sigma model (see Bojowald et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (2005)) is a morphism of Lie algebroids  φ : T Σ → T ∗M  for which Σ is a 2-dimensional manifold and T Σ denotes its tangent Lie algebroid, while M is a  Poisson manifold and T ∗M denotes the Lie algebroid structure on its cotangent bundle .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Lie algebroids are a natural generalization of Lie algebras to the “multi-object” setting, but they  are ill-suited for a functorial presentation of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A step in this direction is to consider the  coordinate-based presentation of the Lie bracket and its coherences due to Martínez (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let A  be an anchored bundle over M equipped with a bilinear bracket on its space of sections, and choose a  pair of bases for Γ(π) and χ(M ): for Γ(π) write {eα}, and for χ(M ) write { ∂  ∂ x i }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The anchor and bracket  then have a presentation in local coordinates:  ̺(eα) =  �  i  ̺i  α  ∂  ∂ x i  [eα,eβ] =  �  γ  C γ  αβ eγ  (from here on out, we use Einstein summation notation to simplify our calculations, so instead write  ̺(eα) = ̺i  α  ∂  ∂ x i  [eα,eβ] = C γ  αβ eγ  with  �  suppressed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The following characterization of the Lie algebroid axioms uses Martinez’s struc-  ture equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' [Martínez (2001)] An anchored bundle A over M equipped with a bracket is a Lie  algebroid if and only if ̺ and [−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='−] satisfy the following structure equations:  (i) Alternating:  C ν  αβ + C ν  βα = 0  (ii) Leibniz:  ̺ j  α  ∂ ̺i  β  ∂ x j = ̺i  γC γ  αβ + ̺ j  β  ∂ ̺i  α  ∂ x j  (iii) Bianchi:  0 = ̺i  α  ∂ C ν  βγ  ∂ x i + ̺i  β  ∂ C ν  γα  ∂ x i + ̺i  γ  ∂ C ν  αβ  ∂ x i + C µ  βγC ν  αµ + C µ  γαC ν  βµ + C µ  αβC ν  γµ  52  This proposition is a straightforward translation of the Lie algebroid axioms into local coordinates  using a covariant derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' First, recall that by the smooth Serre–Swan theorem (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='33 Nestruev  (2003)), the bilinearity of the bracket  [−,−] : Γ(π) × Γ(π) → Γ(π)  as a morphism of C ∞(M )-modules guarantees that it corresponds to a bilinear morphism A2 → A of  vector bundles, meaning that there exists a globally defined bilinear map A2 → A that is equal to the  Lie bracket when applied to sections of the projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We record this as a lemma:  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' For every Lie algebroid � , there is a bilinear morphism  〈−,−〉 : A2 → A  so that for any sections X ,Y ∈ Γ(π), 〈X ,Y 〉 = [X ,Y ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  There are two structure maps derived from 〈−,−〉 that encode the coherences of a Lie algebroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The first map measures the extent to which the anchor maps fail to preserve the chosen connections  on the vector bundle π : A → M and the tangent bundle p : T M → M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let A be a Lie algebroid, and for a chosen horizontal connection ∇ on A and vertical  connection κ′ on T M , set  {v, x}κ′,∇ := T M p ×q A  ∇  −→ T A  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺  −→ T 2M  κ′  −→ T M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  When the choice of connection is evident by context, we will suppress the subscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Observation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The above parentheses bracket corresponds to the symbol  {−,−} = ̺ j  β  ∂ ̺i  α  ∂ x j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The Leibniz coherence may be rewritten as follows:  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let A be an anchored bundle with a bilinear bracket (inducing an involution σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Choose  connections (κ,∇),(κ′,∇′) on A and T M respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The bracket and anchor map satisfy the Leibniz  law if and only if  ̺ ◦ 〈x, y 〉(κ,∇) + {̺x, y }(κ′,∇) = {̺y, x}(κ′,∇).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The condition is equivalent to the identity  ̺ j  α  ∂ ̺i  β  ∂ x j = ̺i  γC γ  αβ + ̺ j  β  ∂ ̺i  α  ∂ x j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The Bianchi axiom measures the failure of the Jacobi identity in local coordinates, and states that  it must be corrected for by the curvature of the brackets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We see that  C ν  αµC µ  βγ = [eα,[eβ,eγ]]  while  ∂ C ν  βγ  ∂ x i eν = κ ◦ T (〈−,−〉(κ,∇)) ◦ ∇A2 ◦ ( ∂  ∂ x i ,eβ,eγ)  53  determines a trilinear map  { ∂  ∂ x i ,eβ,eγ}(κ,∇) := κ ◦ T (〈−,−〉(κ,∇)) ◦ ∇A2 ◦ ( ∂  ∂ x i ,eβ,eγ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  This is the second derived map used in the structure equations for a Lie algebroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let (π : A → M ,ξ,λ,̺) be an anchored bundle equipped with a bilinear map  〈−,−〉 : A2 → A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The derived ternary bracket {−,−,−} : T M p ×πAπ×πA → A is defined as  {v, x, y }(κ,∇) := T M p ×πAπ×πA  ∇[A2]  −−−→ T A2  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='〈−,−〉  −−−−→ T A  κ−→ A  where ∇[A2] is the pairing (∇(π0,π1),∇(π0,π2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Observation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The ternary bracket corresponds to the following symbol:  {−,−,−}(κ,∇) := ̺i  α  ∂ C ν  βγ  ∂ x i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let A be an anchored bundle over M with a bilinear bracket and denote the induced  involution by σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Choose a pair of connections and write the derived maps 〈−,−〉,{−,−,−}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Then  (i) the bracket is antisymmetric if and only if its globalization is;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' that is, 〈eα,eβ〉 + 〈eβ,eα〉 = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) the bracket satisfies the Jacobi identity if and only if it is alternating in the last two arguments and  0 =  �  γ∈Cy(3)  〈xγ0,〈xγ1, xγ2〉〉 +  �  γ∈Cy(3)  {̺xγ0, xγ1, xγ2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) This is equivalent to C ν  αβ + C ν  βα = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) This is equivalent to  0 = C µ  βγC ν  αµ + C µ  γαC ν  βµ + C µ  αβC ν  γµ + ̺i  α  ∂ C ν  βγ  ∂ x i + ̺i  β  ∂ C ν  γα  ∂ x i + ̺i  γ  ∂ C ν  αβ  ∂ x i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A Lie algebroid is exactly an anchored vector bundle (π : A → M ,ξ,λ,̺) with a  bilinear, alternating map  〈−,−〉 : A2 → A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  〈x, y 〉 + 〈y, x〉 = 0  so that for any connection (∇,κ) on A, the maps  {v, x}(κ,∇) := κ′ ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺ ◦ ∇(v, x),  {v, x, y }(κ,∇) := κ ◦ T (〈−,−〉(κ,∇)) ◦ ∇A2 ◦ (v, x, y )  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2)  satisfy the equations  (i) ̺ ◦ 〈x, y 〉 + {̺x, y }(κ′,∇) = {̺y, x}(κ′,∇),  54  (ii)  �  γ∈Cy(3)〈xγ0,〈xγ1, xγ2〉〉 +  �  γ∈Cy(3){̺xγ0, xγ1, xγ2}(κ,∇) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  There is also a local coordinates presentation of morphisms as in Section 2 of Martínez (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' An  anchored bundle morphism A → B is a Lie algebroid morphism whenever  f β  γ Aγ  αδ + ̺i  δ  ∂ f β  α  ∂ x i = B β  θσ f θ  α f σ  δ + ̺i  α  ∂ f β  δ  ∂ x i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The A and B arguments are understood as the brackets, so this condition can be rewritten as  f β  γ Aγ  αδ = f ◦ 〈α,δ〉,  B β  θσ f θ  α f σ  δ = 〈f ◦ α, f ◦ δ〉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Set the following notation for maps between vector bundles with connection:  f : A → B  (κA,∇A,A,λA)  (κB,∇B,B,λB)  ∇[f ] : A2 → B := κB ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f ◦ ∇A  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3)  The ̺ terms are understood to be the torsion, so that  ̺i  δ  ∂ f β  α  ∂ x i = κ ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f ◦ ∇(̺eδ,eα) = ∇[f ](̺eδ,eα),  ̺i  α  ∂ f β  δ  ∂ x i = κ ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f ◦ ∇(̺α,δ) = ∇[f ](̺eα,eδ),  using the notation set up in Equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The notion of a Lie algebroid morphism, then, has the fol-  lowing presentation:  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='14 (Martínez (2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let (π : A → M ,̺A,〈−,−〉A),(q : B → N ,̺B,〈−,−〉B) be a pair of  Lie algebroids with chosen connections (κ−,∇−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' An anchor-preserving vector bundle morphism f : A →  B is a Lie algebroid morphism if and only if  f ◦ 〈eα,eδ〉 + ∇[f ]◦ (̺eδ,eα) = 〈f ◦ α, f ◦ δ〉 + ∇[f ]◦ (̺eα,eδ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2  Anchored bundles  Anchored bundlesare toLie algebroidswhatreflexive graphsare togroupoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Each theoryhas(mostly)  the same structure, but while a reflexive graph is missing a groupoid’s composition operation, an an-  chored bundle lacks a Lie algebroid’s bracket operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This section reviews the basic theory of an-  chored bundles and their prolongations (see Mackenzie (2005) for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' An anchored bundle in a tangent category isa differential bundle (A  π−→ M ,ξ,λ) equipped  with a linear morphism  A  T M  M  ̺  π  p  A morphism of anchored bundles is a linear bundle morphism (f ,v) that preserves the anchors  T A  T B  A  B  A  B  T M  T N  λ  l  f  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f  ρ  ̺  f  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='v  The category of anchored bundles and anchored bundle morphisms in a tangent category � is written  Anc(�), and a generic anchored bundle is written (A  π−→ M ,ξ,λ,̺).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  55  There are two pullbacks that are associated with every anchored bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' These play the role of the  spaces of composable arrows G2 := G t ×s G ,G3 := G t ×sG t ×sG for a reflexive graph s,t : G → M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let (A  π−→ M ,ξ,λ,̺) be an anchored bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Its first and second prolongations are  given by the limits  � (A)  T A  � 2(A)  T 2A  A  T M  T A  T 2M  A  T M  ̺  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π  ⌟  T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π  ̺  ⌟  (The notation for the fibre product is slightly non-standard, as it is not technically a pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=') Through-  out this chapter, it will always be assumed that the first and second prolongations of an anchored bundle  exist (although no choice of prolongation is explicitly made).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' It is not strictly necessary that the prolongations of an anchored bundle exist;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' this con-  dition is primarily a matter of convenience when discussing involution algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Every result in this  section, that does not explicitly mention prolongations, holds for an anchored bundle independently of  their existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) For any object M , id : T M → T M is an anchor for the tangent differential bundle, and every  f : M → N yields a morphism of anchored bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Moreover, for any anchored bundle over M the  anchor is itself a morphism of anchored bundles (A,π,ξ,λ,̺) → (T M ,p,0,ℓ,id ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This induces a  fully faithful functor:  � �→ Anc(�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  This inclusion has a left adjoint, which sends an anchored bundle (π : A → M ,ξ,λ,̺) to its base  space M (the unit is the anchor map ̺), so that � is a reflective subcategory of Anc(�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) For any differential bundle (A,π,ξ,λ), the map 0 ◦ π : A → T M is an anchor and every morphism  f : A → B of differential bundles again yields a morphism of anchored bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The naturality of  0 ensures that every differential bundle morphism will preserve this trivial anchor map, giving a  fully faithful functor  DBun(�) �→ Anc(�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  This functor has a right adjoint that replaces the anchor map with the trivial anchor map  (π : A → M ,ξ,λ,̺) �→ (π : A → M ,ξ,λ,0◦ π)  where the counit is given by the natural idempotent e = ξ ◦ π : (A,λ) → (A,λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' It is trivial to check  that the anchor map is preserved by the bundle morphism (e ,id ):  ̺ ◦ ξ ◦ π = 0◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='id ◦ π  and so the following diagram commutes:  T M  T M  A  A  0◦π  p◦λ  ̺  56  This means that differential bundles are a coreflective subcategory of anchored bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iii) Any reflexive graph (s,t : C → M ,e : M → C ) in a tangent category has an anchored bundle (when  sufficient limits exist), constructed as  C ∂  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='C  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='C  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T1  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e s  (where e s : C → C = e ◦ s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Construct a lift on C ∂ :  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='C ∂  T 2C1  T 2C1  C ∂  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='C1  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='C1  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='C1  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e s  ℓ  ℓ  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='C1  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e s  λ  This lift will be non-singular by the commutativity of T -limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The pre-differential bundle data is  given by the projection  C ∂ �→ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='C  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e s  −−→ M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The section is induced by  M  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e )  −−→ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='C  while post-composition with T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='t gives the anchor map:  C ∂ �→ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='C  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='t  −→ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The diagram is a pullback by composition, and the outer perimeter defines the pullback � (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Note  that any reflexive graph morphism will give rise to an anchored bundle morphism by naturality,  making the construction of an anchored bundle from a reflexive graph functorial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iv) For any object M in a tangent category, cM : T 2M → T 2M is an anchor on (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0,c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ), and  every map f : M → N gives a morphism of anchored bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (v) For any anchored bundle (q : E → M ,ξ,λ,̺), the differential bundle (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='q,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ξ,  c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ) has an anchor  ̺T : T E  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺  −→ T 2M  c−→ T 2M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The first prolongation of an anchored bundle � (A) behaves similarly to the second tangent bundle,  except that it does not have a canonical flip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In the definition of an affine connection, the tangent  bundle played a similar role to the “arities” of a theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' There is a lift map that makes this connection  stronger:  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let (π : A → M ,ξ,λ,̺) be an anchored bundle in a tangent category �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We define a  generalized lift:  ˆλ : A → A̺×T πT A := (ξ ◦ π,λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  This generalized lift satisfies the same coherences as the lift on the second tangent bundle:  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let (π : A → M ,ξ,λ,̺) be an anchored bundle in a tangent category �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' It follows that  (i) (Coassociativity of ˆλ) (ˆλ × ℓ) ◦ ˆλ = (id × T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ˆλ) ◦ ˆλ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  57  (ii) (Universality) the following diagram is a T -pullback:  A2  A̺×T πT A  M  A  ˆµ  π0  π◦πi  ξ  ⌟  where ˆµ := (ξ ◦ π◦ π0,µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) Compute  (ˆλ × ℓ) ◦ ˆλ = (ξ ◦ π◦ ξ ◦ π,λ ◦ ξ ◦ π,ℓ◦ λ)  = (ξ ◦ π,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (ξ ◦ π) ◦ λ,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ ◦ λ)  = (π0,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (ξ ◦ π) ◦ π1,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ ◦ π1) ◦ (ξ ◦ π,λ)  = (id × T (ˆλ)) ◦ ˆλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) Use the pullback lemma to observe that the following diagram is universal for any anchored bun-  dle:  A2  � (A)  T A  M  A  T M  ξ  ̺  0  T π  π0  π◦π1  ˆµ  π1  µ  The top triangle of the diagram commutes by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The right square and outer perimeter  are pullbacks by definition, and the bottom triangle also commutes by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The pullback  lemma ensures that the left square is a pullback, so for every anchored bundle, the general lift is  universal for � (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Now post-compose with the involution:  A2  � (A)  � (A)  M  A  A  ξ  π0  π◦π1  ˆµ  id  pπ1  σ  ˆν  It suffices to check that the top triangle commutes, so σ ◦ ˆµ = ν:  σ ◦ ˆµ ◦ (a,b ) = σ ◦ ((ξ ◦ π,0) ◦ a +π0 (ξ ◦ π,λ) ◦ b ) = (id ,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ξ ◦ ̺ ◦ a) +pπ1 (ξ ◦ π,λ) ◦ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Thus, the lift (ξπ,λ) involution algebroid is universal for � (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) For T (M ) = T (M ), the space of prolongations is T (M )id ×T p T 2M = T 2M , and the second prolon-  gation is given by T 2M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  58  (ii) In a tangent category with a tangent display system, if π ∈ � then the prolongations for (π,ξ,λ,̺)  automatically exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In particular, for every anchored bundle in the category of smooth manifolds,  all prolongations exist because the projection is a submersion (see Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iii) For any differential bundle with the anchor 0 ◦ π, it follows that � (A) ∼= Aπ×π◦πi (A2) ∼= A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The  universality of the vertical lift factors b into (a1,a2), as the following diagram is a pullback:  A3  Aπ×πA  T A  A  M  T M  π0  (π1,π2)  µ  ππi  T π  π  0  (iv) Returning to the anchored bundle constructed from a graph, the space of prolongations � (A) em-  beds into the second tangent bundle of the space of composable arrows:  A̺×T πT A �→ T 2(G t ×sG ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The category of anchored bundles is, in a sense, “tangent monadic” over the category of differential  bundles: the forgetful functor from anchored bundles to differential bundles “creates” T -limits and  the tangent structure (this is all made precise in Chapter 5 using an enriched perspective on tangent  categories).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Observation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A limit of anchored bundles is the limit of the underlying differential bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Be-  cause the anchor is preserved by every map in the diagram, this induces a natural transformation for  any D : � → Anc(C ):  �  Anc(�)  Anc  DBun(�)  D  S  U  ̺  Thus, limU .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='D computes the limit of the underlying objects, while limU .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='D computes the limit of the  underlying differential bundles in the diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The anchor/unit map, then, induces a differential bun-  dle map:  lim̺i : (limAi ,limλi ) → (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='(lim  i Mi),ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (lim  i Mi)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  This is the limit in the category of anchored bundles (so long as pullback powers of the limit projection  and the two prolongations of the limit anchor bundle exist).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The tangent structure on lifts defined in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7 lifts to a tangent structure on anchored  bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category of anchored bundles in a tangent category has a tangent structure that maps  objects as follows:  (π : A → M ,ξ,λ,̺) �→ (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π : T A → T M ,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ξ,c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ,c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺)  where the structure maps are all defined using the pointwise structure maps in �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  59  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Given an anchored bundle (q : A → M ,ξ,λ,̺), there is an anchor on the differential bundle  (T q,T ξ,c ◦ T λ) given by c ◦ T ̺;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' the diagram commutes by naturality and the coherences on c ,ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  T 2A  T 3M  T 3M  T A  T 2M  T 2M  T M  T M  T M  T q  T p  c ◦T λ  c ◦T ℓ  ℓ  c  p  T ̺  T 2̺  T c  The tangent structure maps and universality properties all follow from the forgetful property of the  functor from anchored bundles to differential bundles as a consequence of Observation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  For any anchored bundle A, there are two differential bundles associated to � (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The first is the  usual pullback differential bundle given by pulling back T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π along ̺ as in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3:  A̺×T πT A  T A  A  T M  π1  π0  T π  ̺  This gives the differential bundle structure  (A̺×T πT A  π0  −→ A,A  (id,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ξ◦̺)  −−−−−−→ A̺×T πT A,A̺×T πT A  (0,c ◦T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ)  −−−−−→ T (A̺×T πT A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Taking the fibre product in the category of anchored bundles as in Observation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8 yields the second  differential bundle structure:  (A,π,ξ,λ)  (̺,id)  −−−→ (T M ,p,0,ℓ)  (T π,π)  ←−−− (T A,p,0,ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The two lifts behave similarly to the pair (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ,ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ) on the second tangent bundle, and (ℓ,λT ) on the  tangent bundle of a pre-differential bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let � be a tangent category and Anc(�) the category of anchored bundles in �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' If T -  pullback powers of p ◦π0,π1 : A̺×T πT A → A exist, then there are two differential bundles on � (A), with  lifts induced as  (� (A),λ × ℓ)  (T A,ℓ)  (� (A),0× (c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ))  (T A,c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ)  (A,λ)  (T M ,ℓ)  (A,0)  (T M ,0)  ̺  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π  ⌟  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π  ̺  ⌟  with structure maps  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (� (A)  p◦π1  −−→ A,A  (ξπ,0)  −−−→ � (A),� (A)  λ×ℓ  −−→ T ◦ � (A)),  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (� (A)  π0  −→ A,A  (id,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ξ◦̺)  −−−−−−→ � (A),� (A)  0×c ◦T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ  −−−−→ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� (A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Furthermore, the two lifts λ� and λ̺ commute:  c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='(λ × ℓ) ◦ (0× c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ) = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='(0× c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ) ◦ (λ × ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  60  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The two differential bundles exist as a consequence of Observation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3, re-  spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The commutativity of limits follows by postcomposition, as both c ◦T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ,ℓ and λ,0 commute  by the differential bundle axioms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The above lemma determines a functor, which we denote � , that sends an anchored bundle in �  to an anchored bundle in Lift(�) (equipped with the tangent structure from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' There is a functor � from anchored bundles in � to anchored bundles in the cate-  gory of lifts Lift(�), that sends an anchored bundle (π : A → M ,ξ,λ,̺) to the tuple in Lift(�)  π� := (� (A),0× c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ)  p◦π1  −−→ (A,0),  ξ� := (A,0)  (ξ◦π,0)  −−−→ (� (A),0× c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ)  λ� := (� (A),0× c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ)  (λ×ℓ)  −−→ (T (� (A)),c ◦ T (0× c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ),  ̺� := (� (A),0× c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ)  π1  −→ (T A,c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ)  (note that this functor lands in anchored bundles of non-singular lifts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Morphisms  of anchored bundles  (f ,m) : (π : A → M ,ξ,λ,̺) → (q : E → N ,ζ,l ,δ)  are sent to  � (f ,m) := f Tm ×Tm T f : A̺×T πT A → E δ×T q T E  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' First, note that the tuple  (π� : � (A) → A,ξ� ,λ� ,̺� )  is an anchored bundle, and that each morphism is a lift morphism:  π� : (� (A),0× c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ) → (A,0) follows because  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π1 ◦ (λ × ℓ) = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p ◦ ℓ◦ π1 = 0◦ p ◦ π1  ξ� : (A,0) → (� (A),0× c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ) follows since  (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ξ ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0) ◦ 0 = (0◦ ξ ◦ π,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0◦ 0) = (0◦ ξ ◦ π,c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ ◦ 0) = (0× c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ) ◦ (ξ ◦ π,0)  λ� : (� (A),0× c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ) → (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� (A),0× c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='(c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ)) follows by the commutativity of 0× c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ)  and λ × ℓ, and  ̺� : (� (A),0× c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ) → (T A,c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ) is a lift by definition of ̺� = π1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  This gives an anchored bundle in the category of non-singular lifts in �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Next, check that the mapping is functorial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' To see that � (f ,m) preserves the lifts, note that (f ,m)  gives a morphism of diagrams for each pullback in Lift(�) defining the two lifts, so the induced map  � (f ,m) preserves each lift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' To see that � (f ,m) preserves the anchor, check that  ̺� B ◦ � (f ,m) = π1 ◦ (f × T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f ) = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f ◦ π1 = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f ◦ ̺� A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  61  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In a tangent category � where pullbacks along differential bundle projections exist,  such as a tangent category equipped with a proper retractive display system (Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3) like the  category of smooth manifolds, there is an endofunctor on the category of anchored bundles in �,  � ′ : Anc(�) → Anc(�) =U Lift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Observation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' T -Limits in Anc(Lift(�)) are computed as pointwise limits in � by Observations  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2, and � is constructed as a limit, so it follows that � preserves T -limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The prolongation endofunctor � ′ : Anc(�) → Anc(�) has natural transformations  p ′ : � ′ ⇒ id  0′ : id ⇒ � ′  +′ : � ′p ′×p ′� ′ ⇒ � ′  ℓ′ : � ′ ⇒ � ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� ′  satisfying all of the axioms of a tangent category that do not incvolve the canonical flip (Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (Sketch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Note that the full argument is given in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3, but it is not difficult to sketch it out here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  For the projection, zero map, and addition, use the differential bundle structure induced by Corollary  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' To see the lift axiom, note that up to a choice of pullback, we have:  � ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� ′(π : A → M ,ξ,λ,̺) =  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f3  (p ◦ π1,p ◦ π2) :  � 2(A) → � (A)  (ξ ◦ π◦ π0,0◦ π1,0◦ π2) :  � (A) → � 2(A)  (λ × ℓ× ℓ) :  � 2(A) → T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� 2(A)  (π1,π2) :  � 2(A) → T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� (A)  The “lift map” is then  ℓ′ : � ′ ⇒ � ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' � (A)  A×T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ˆλ  −−−→ � 2(A)  (where we recall that ˆλ = (ξ ◦ π,λ) : A → � (A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The structure described in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='11 leads to the theory of double vector bun-  dles, developed by MacKenzie and his collaborators F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' and Mackenzie (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Mackenzie (1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A dou-  ble vector bundle is a commuting square  D  B  A  M  q D  A  q D  B  q A  q B  where each projection is a vector bundle projection, and “vertical” and “horizontal” orientations of the  square are each vector bundle homomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' It was observed by Grabowski and Rotkiewicz (2009)  that this is equivalent to a pair of commuting �+-actions on the total space E ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' following the development  in Chapter 2, this is a commuting pair of non-singular lifts on E ,  λA,λB : E → T E , T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λB ◦ λA = c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λA ◦ λB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  A proper exposition of the so-called “Ehresmann doubles” (Mackenzie (2011)) for the structures in Lie  theory would substantially expand the scope of this thesis, and so it has been relegated to the margins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  62  Finally, observe that a connection (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6) on an anchored bundle’s underlying differential  bundle behaves similarly to an affine connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let (π : A → M ,ξ,λ,̺) be an anchored bundle with � (A) existing in a tangent category  �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' If (π,ξ,λ) has a connection, then define  ˆκ := κπ1,  ˆ∇ := ∇(π0,̺π1)  and note that  (i) ˆκ : � (A) → A is a retract of (ξπ,λ), and ˆκ : (� (A),l ) → (A,λ) is linear for both l = (λ×ℓ),(0×c ◦T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) ˆ∇ : A2 → � (A) is a section of (π0,p ◦ π1) and is bilinear;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iii) there is an isomorphism � (A) ∼= A3:  ∇(p ◦ π1,̺ ◦ π0) +(π,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π) ˆµ(p ◦ π1, ˆκ) = id .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Every vector bundle in the category of smooth manifolds has a connection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' it follows  that Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='16 holds for every anchored bundle in the category of smooth manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category of anchored bundles in a tangent category is almost a tangent category,  except that it lacks a symmetry map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The “differential objects” in such a category will act like a cartesian  differential category, except that the symmetry of mixed partial derivatives fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' There has been some  interest in settings for differentiable programming where the symmetry of mixed partial derivatives need  not hold (see Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4 along with the discussion at the end of Section 6 in Cruttwell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (2021));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  the category of anchored bundles in a tangent category appears to be a source of examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3  Involution algebroids  Involution algebroids are a tangent-categorical axiomatization of Lie algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Recall that a Lie al-  gebroid has almost all of the structure of the operational tangent bundle, in particular a Lie bracket  on sections that satisfies the Leibniz law so that it gives a directional derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='14,  it was demonstrated that the category of anchored bundles have almost all of the same maps as the  tangent bundle, the only structure map missing being the canonical flip  c : T 2M ⇒ T 2M  which, we may recall from the chapter introduction (and Cockett and Cruttwell (2015), Mackenzie  (2013)), is used in constructing the Lie bracket of vector fields in a tangent category with negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Thus, a natural next step is to add an involution map to an anchored bundle and require that it satis-  fies the same coherences as c from the tangent bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' An involution algebroid is an anchored bundle (A,π,ξ,λ,̺) equipped with a map  σ : � (A) → � (A) satisfying the following axioms:  (i) Involution:  � (A)  � (A)  � (A)  σ  σ  63  (ii) Double linearity:  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� (A)  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� (A)  � (A)  � (A)  σ  0×c ◦T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ  λ×ℓ  (iii) Symmetry of lift:  A  � (A)  � (A)  ˆλ=(ξ◦π,λ)  σ  ˆλ  (iv) Target:  � (A)  T A  T 2M  � (A)  T A  T 2M  σ  π1  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺  π1  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺  c  (v) Yang–Baxter:  � 2(A)  � 2(A)  � 2(A)  � 2(A)  � 2(A)  � 2(A)  σ×c  σ×c  σ×c  A×T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ  A×T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ  A×T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ  A is an almost-involution algebroid if the involution does not satisfy the Yang–Baxter equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A mor-  phism of involution algebroids is a morphism of anchored bundles, so that � (f )◦σA = σB ◦� (f ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (Note  that because σ is an isomorphism and σ = σ−1,  σ : (� (A),0× c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ) → (� (A),λ × ℓ)  is linear as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=') Write the category of involution algebroids and involution algebroid morphisms in �  is written Inv(�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Observation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' It is not immediately clear that the Yang–Baxter equation is well-typed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This follows  from the target axiom (iv) and the double linearity axiom (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Starting with (u,v,w ) : A̺×T πT AT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺×T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π  T 2A, we see that σ × c is well-typed if and only if  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺ ◦ π1 ◦ σ(u,v) = T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π◦ c ◦ w = c ◦ T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π◦ w = c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺ ◦ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Similarly, 1 × T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ is well-typed if T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺ ◦ u = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π ◦ π0 ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ(v,w );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' then use the double linearity axiom to  compute  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π◦ π0 ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ(v,w ) = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π1(v,w ) = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p ◦ w  = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p ◦ T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π◦ w = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺ ◦ v = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π◦ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  64  This perspective on Lie algebroids has already appearad in the work of Martinez and his collabora-  tors in de León et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (2005), where a “canonical involution” was derived on space of prolongations of a  Lie algebroid using the formula  σ : � (A) → � (A);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ(x, y,z) = σ(y, x,z + 〈x, y 〉).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The structure of this map has been largely unexplored;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' helpfully, involution algebroids succeed in  reverse-engineering axioms for an involution map that will induce a Lie bracket on the sections of  the projection map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The bracket from the original Lie algebroid is induced using the same formula as  for the bracket of vector fields on the tangent bundle:  λ ◦ [X ,Y ]∗ =  �  (π1 ◦ σ ◦ (id ,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='X ◦ ̺) ◦ Y −p T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Y ◦ X ◦ ̺) −T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π 0Y  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Furthermore, a morphism of anchored bundles is a Lie algebroid morphism if and only if it preserves  the derived involution map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) For any M in �, (T M ,p,0,id ,c ) is an involution algebroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Furthermore, for any involution alge-  broid anchored on M , (̺,id ) is a morphism of involution algebroids (by the target axiom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This  defines a fully faithful functor � �→ Inv(�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The same construction as the anchored bundle case in  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4(i) exhibits � as a reflective subcategory of Inv(�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) For any differential bundle, the trivial anchored bundle (π : A → M ,ξ,λ,0◦ π) has an involution  using the isomorphism � (A) ∼= A3, given by σ := (π1,π0,π2) (proving this map satisfies the involu-  tion algebroid axioms is just an exercise in combinatorics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' It follows that every differential bundle  morphism gives rise to a morphism of these trivial involution algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The same construction  from the anchored bundle case (Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4(ii)) exhibits Diff(�) as a coreflective subcategory of  involution algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iii) Consider a groupoid  s,t : G → M , e : M → G , (−)−1 : G → G , m : G2 → G .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The underlying reflexive graph has an associated anchored bundle, constructed in Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4,  and the space of prolongations of this anchored bundle includes into (T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='G2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Note that there is a  well-formed involution map:  σ(u,v) = c ◦ ((0◦ p ◦ v)−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='(T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0◦ u);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='v)  The direct proof of this involves a more conceptual construction, which is the focus of Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  An involution algebroid resembles a generalized tangent bundle, and so the lift (ξ◦π,λ) : A → � (A)  satisfies the same universality conditions as ℓ : T ⇒ T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The double linearity condition is equivalent to  the naturality condition for c ,ℓ in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2:  � (A)  � (A)  T 2  T 2  � 2(A)  � 2(A)  � 2(A)  T 3  T 3  T 3  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ  c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c  c  ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  id×T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ˆλ  σ  σ×c  id×T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ  ˆλ×ℓ  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4)  65  which, using string diagrams for monoidal categories (Selinger (2010)), is the equation  =  where the circle denotes the lift and c is the crossing of two lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let (π : A → M ,ξ,λ,̺) be an anchored bundle, and suppose that  σ : � (A) → � (A)  satisfies the involution axiom (i) and the symmetry of lift axiom (iii),and furthermore that the involution  “exchanges” the idempotents associated to the two lifts (λ×ℓ) and (0×c ◦T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ) on � (A) (Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8)  σ ◦ ((p ◦ λ) × (p ◦ ℓ)) = (p ◦ 0) × (p ◦ c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ) = id × (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ)  Then the double linearity axiom is equivalent to the left-hand commuting diagram in Diagram 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4:  (ˆλ × ℓ) ◦ σ = (id × T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ) ◦ (σ × c ) ◦ (id × T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ˆλ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Starting with the left-hand side of the equation,  (id × T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ) ◦ (σ × c ) ◦ (id × T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ˆλ) ◦ (u,v)  = (id × T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ) ◦ (σ × c ) ◦ (u,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ξ ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π◦ v,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λv)  = (id × T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ) ◦ (σ ◦ (u,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ξ ◦ ̺ ◦ u),T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λv)  = (id × T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ) ◦ (ξ ◦ π◦ u,0◦ ̺ ◦ u,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λv)  = (id × T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ) ◦ (ξ ◦ π◦ u,0◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π◦ v,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λv)  = (ξ ◦ π◦ u,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ ◦ ˆλ◦ v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  For the right-hand side:  (ˆλ,ℓ) ◦ σ ◦ (u,v)  = (ξ ◦ π◦ π0 ◦ σ ◦ (u,v),(λ × ℓ) ◦ σ ◦ (u,v))  = (ξ ◦ π◦ p ◦ v,(λ × ℓ) ◦ σ ◦ (u,v))  = (ξ ◦ p ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π◦ v,(λ × ℓ) ◦ σ ◦ (u,v))  = (ξ ◦ p ◦ ̺ ◦ u,(λ × ℓ) ◦ σ ◦ (u,v))  = (ξ ◦ π◦ u,(λ × ℓ) ◦ σ ◦ (u,v))  = (ξ ◦ π◦ u,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ ◦ (0× c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ) ◦ (u,v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  So the naturality equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4 is equivalent to  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ ◦ (0× c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ) = (λ × ℓ) ◦ σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The category of involution algebroids is “tangent monadic” over the category of anchored bundles,  in the same sense that anchored bundles are tangent monadic over the category of differential bundles,  or internal categories over reflexive graphs in a category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The “tangent monadicity” leads to a similar  observation about T -limits of involution algebroids as in Observation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  66  Observation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The forgetful functor from involution algebroids to anchored bundles creates limits;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  that is to say, the T -limits of the underlying anchored bundles give the limits of involution algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Recall that by Observation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='13, the limit � (limAi ) = lim� (Ai ) in the category of anchored bundles,  and this induces a map between objects in �  limσi : lim� (Ai ) → lim� (Ai ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The axioms for an involution algebroid are induced by universality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Note that a point-wise tangent structure may be defined following the anchored bundles example  from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9:  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' For a tangent category �, the category of involution algebroids has a “point-wise”  tangent structure that maps objects as follows:  (π : A → M ,ξ,λ,̺,σ) �→ (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ξ,c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺,σT )  where σT is defined as  σT := (1×c c ) ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ ◦ (1×c c ) : � (AT ) → � (AT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The tangent structure on involution algebroids is inherited from the functor Inv(�) → Anc(�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Thus, it suffices to give the involution map for the tangent involution algebroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Note that given (π : A → M ,ξ,λ,̺,σ), the space of prolongations on (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ξ,  c ◦T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ,c ◦T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺) is T Ac ◦T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺×T 2πT 2A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We can construct an isomorphism between the objects � (T A) and  T (� (A)) in � using the cospan isomorphism  T A  T 2M  T 2A  T A  T 2M  T 2A  c ◦T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺  T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺  T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π  id  c  c  thus inducing a map  T Ac ◦T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺×T 2πT 2A  id×c  −−→ T (A̺×T πT A)  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ  −→ T (A̺×T πT A)  id×c  −−→ T Ac ◦T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺×T 2πT 2A  which we call σT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The linearity and involution axioms follow by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Now check the rest of the axioms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' For the unit:  (1×c c ) ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ ◦ (1×c c ) ◦ (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='(ξ ◦ π),c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ)  = (1×c c ) ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ ◦ (T (ξπ),T λ) = (1×c c ) ◦ (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='(ξ ◦ π),T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ) = (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='(ξ ◦ π),c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  For the anchor:  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺T ◦ π1 ◦ σT = T (c ◦ T ̺) ◦ π1 ◦ (1×c c ) ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ ◦ (1×c c )  = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c ◦ c ◦ T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺ ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π1 ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ ◦ (1×c c ) = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c ◦ c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c ◦ T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺ ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π1 ◦ (1×c c )  = c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c ◦ c ◦ T 2̺ ◦ c ◦ π1 = c ◦ T c ◦ T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺ ◦ π1 = c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺T ◦ π1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The Yang–Baxter equation is straightforward to check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The tangent bundle is a canonical involution algebroid on every object in a tangent category, and  the anchor induces a morphism from an involution algebroid to the tangent involution algebroid on  its base space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The anchor acts as a reflector from involution algebroids in � to � itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  67  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Any tangent category � is a reflective subcategory of the category of involution alge-  broids in �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' First, observe that the inclusion of � into Inv(�) (Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1) is fully faithful because the  anchor on the tangent involution algebroid is id : T M → T M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This means that the only involution  algebroid morphisms T A → T B are pairs (T f , f ), f : A → B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Now consider the functor Inv(�) → � that  sends (π : A → M ,ξ,λ,̺,σ) to M : this gives an endofunctor S : Inv(�) → Inv(�) along with a natural  transformation ̺ : id ⇒ S, so that S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺ = ̺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='S = id , given by the anchor map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Thus, the category � is  the category of algebras for an idempotent monad on Inv(�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let  �A = (π : A → M ,ξ,λ,̺,σ)  be an involution algebroid in a tangent category �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) The morphism  (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π,π) : T A → T M  is an involution algebroid morphism from the tangent involution algebroid on A to the tangent  involution algebroid on M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) The morphism  (̺,id ) : �A → T M  is an involution algebroid morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  If pullback powers of p ◦ π1 : � (A) → A exist, then  (p ◦ π1 : � (A) → A,(ξ ◦ π,0),(λ × ℓ))  is a differential bundle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' note that π0 acts as an anchor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' If the prolongations exist, then  (p ◦ π1 : � (A) → A,(ξ ◦ π,0),(λ × ℓ),π0,σ′)  is an involution algebroid;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' this follows from computing the pullback in the category of involution alge-  broids (σ′ is induced as in Observation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The above corollary puts an involution algebroid structure on the differential bundle (� (A),λ × ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Note that the map σ gives an isomorphism of differential bundles  (� (A),λ × ℓ) → (� (A),0× c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Martinez observed that the canonical involution σ puts a unique Lie algebroid structure on (� (A),0×  c ◦T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ), and σ isa uniquelydetermined isomorphism ofinvolution algebroids(see de León et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (2005)):  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' For an involution algebroid (π : A → M ,ξ,λ,̺,σ), the isomorphism of differential  bundles  σ : (� (A),0× c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ) → (� (A),λ × ℓ)  induces a second involution algebroid structure on � (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  68  Recall that Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='14 sketched out a proof that the category of anchored bundles in � has  an endofunctor � ′ and natural transformations p ′,0′,+′,ℓ′ satisfying the axioms of a tangent structure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  this endofunctor and the natural transformations all lift to Inv(�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The involution map σ is the missing  piece that gives a tangent structure on Inv(�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The construction may be spelled out here at a big-picture  level, but the actual proof brings up tricky coherence issues that make up the bulk of Chapter 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category of involution algebroids in a tangent category � has a second tangent  structure, where the tangent functor is given by  � ′ : Inv(�) → Inv(�)  and the tangent natural transformations are given as in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='14, with the canonical flip  σ′ : � ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� ′ → � ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� ′ := � 2(A)  1×T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ  −−−→ � 2(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Starting with an involution algebroid, � ′(A) and � ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� ′(A) are given by  � ′(A)  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  π′ :  � (A)  p◦π1  −−→ A  ξ′ :  A  (ξ◦π,0)  −−−→ � (A)  λ′ :  � (A)  λ×ℓ  −−→ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� (A)  ̺′ :  � (A)  π1  −→ T A  σ′ :  � 2(A)  σ×c  −−→ � 2(A)  � ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� ′(A)  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  π′′  � 2(A)  (p◦π1,p◦π2):  −−−−−−−→ � (A)  ξ′′ :  � (A)  (ξ◦π◦π0,0◦π1,0◦π2)  −−−−−−−−−−−→ � 2(A)  λ′′ :  � 2(A)  (λ×ℓ×ℓ)  −−−−→ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� 2(A)  ̺′′ :  � 2(A)  (π1,π2)  −−−→ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� (A)  σ′′ :  � 3(A)  (σ×c ×c )  −−−−−→ � 3(A)  (where � 3(A) = A̺×T πT AT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺×T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='πT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (A̺×T πT A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The tangent natural transformations are given by  p : � ′ ⇒ id ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' � (A)  π0  −→ A  0 : id ⇒ � ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A  (id,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ξ◦T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π)  −−−−−−−→ � ′(A)  + : � ′  2 ⇒ � ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A̺×T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π◦πi T2A  id×+  −−−→ � (A)  ℓ : � ′(A) ⇒ � ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' � (A)  1×T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (ξ◦π,λ)  −−−−−−→ � 2(A)  ̺′ : � ′ ⇒ T  c : � ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� ′(A) ⇒ � ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' � 2(A)  1×T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ  −−−→ � 2(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Deferred to Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4  Connections on an involution algebroid  In this section we take an involution algebroid with a chosen linear connection on its underlying an-  chored bundle (Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1), and rederive Martinez’s structure equations for a Lie algebroid  (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  In a tangent category � with negatives, there is a natural transformation  n : T ⇒ T  69  making each fibred commutative monoid (p : T M ⇒ M ,0,+,n) a fibred abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Because the  additive bundle structure on differential bundles is induced via universality (Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4), in a  tangent category with negatives the additive bundle structure for differential bundles will likewise have  negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We adopt the following notation for the “fibred linear algebra” used in this section, as there  are a significant number of computations done on bundles with multiple choices of additive bundle  structure (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' the second tangent bundle of a differential bundle has three additive bundles structures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Notation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let E be an object with multiple differential bundle structures (πi : E → M i ,ξi,λi)  in a tangent category with negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Addition over a specific differential bundle is written using infix  notation, where a subscript is added to the symbol denoting the projection of the differential bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Letting x, y : X → E denote a pair of generalized elements for which the πi -addition operation is well-  defined;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' we set  x +π[i] y = X  (x,y )  −−→ E π[i]×π[i]E  +i  −→ E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Similar notation is used for subtraction:  x −π[i] y = X  (x,y )  −−→ E π[i]×π[i]E  id×n[i]  −−−−→ E π[i]×π[i]E  +i  −→ E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Throughout this section, we work in a tangent category � with negatives and a chosen anchored  bundle (π : A → M ,ξ,λ,̺), equipped with a connection (κ,∇), whose base object has a torsion-free  connection (κ′,∇′) and a morphism  σ : � (A) → � (A)  that exchanges the two projection maps p ◦ π1,π0, meaning that the following diagram commutes:  � (A)  A  A  � (A)  π0  p◦π1  σ  p◦π1  π0  The following notation will be useful when working with local coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Notation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' First, recall the ∇-notation for morphisms of differential bundle where each morphism  has a choice of connection from Equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3:  f : A → B  (κA,∇A,A,λA)  (κB,∇B,B,λB)  ∇[f ] := κB ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f ◦ ∇A : A2 → B  Let  (π : A → M ,ξ,λ,̺),(q : B → N ,ζ,l ,ρ)  be a pair of anchored bundles equipped with connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' “Hatting” a map f : � (A) → � (B) refers to  the map:  �f : A3  ˆν(π0,π2)+ ˆ∇(π0,π1)  −−−−−−−−−−→ � (A)  f−→ � (B)  (π0,p◦π1,κ◦π1)  −−−−−−−−→ B3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Similarly, for f : T A → T B,  �f : T M p ×πAπ×πA  ν(π0,π2)+∇(π0,π1)  −−−−−−−−−−→ T A  f−→ T B  (π0,p◦π1,κ◦π1)  −−−−−−−−→ T M p ×q B π×q B  70  Clearly, �  f ◦ g = �f ◦ �g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Similarly, a map A3 → B3 may be “barred” to form a map � (A) → � (B):  g : � (A)  (π0,p◦π1,κ◦π1)  −−−−−−−−→ A3  g−→ B3  ˆν(π0,π2)+ ˆ∇(π0,π1)  −−−−−−−−−−→ � (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  It is straightforward to see that �f = f , �g = g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' σ : � (A) → � (A) induces a bracket on Γ(π):  λ ◦ [X ,Y ] = (σ ◦ (id ,T X ◦ ̺) ◦ Y −T π (id ,T Y ◦ ̺) ◦ X ) − 0◦ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let X ,Y ∈ Γ(π) and compute:  p ◦ π1 ◦ (σ ◦ (id ,T X ◦ ̺) ◦ Y −π0 (id ,T Y ◦ ̺) ◦ X )  = p ◦ (π1 ◦ σ ◦ (id ,T X ◦ ̺) ◦ Y −T π T Y ◦ ̺ ◦ X )  = p ◦ π1 ◦ σ ◦ (id ,T X ◦ ̺) ◦ Y −π p ◦ T Y ◦ ̺ ◦ X  = π0 ◦ (id ,T X ◦ ̺) ◦ Y − Y ◦ p ◦ ̺ ◦ X  = Y − Y = ξ,  π0 ◦ (σ ◦ (id ,T X ◦ ̺) ◦ Y −π0 (id ,T Y ◦ ̺) ◦ X )  = π0 ◦ σ ◦ (id ,T X ◦ ̺) ◦ Y −π π0 ◦ (id ,T Y ◦ ̺) ◦ X  = p ◦ π1 ◦ (id ,T X ◦ ̺) ◦ Y −π X  = p ◦ T X ◦ ̺ ◦ Y −π X = X −π X = ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The universality of the lift induces a new section [X ,Y ] so that  λ ◦ [X ,Y ] = (σ ◦ (id ,T X ◦ ̺) ◦ Y −T π (id ,T Y ◦ ̺) ◦ X ) − 0Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The map σ : � (A) → � (A) is linear whenever the two bundle morphisms  σ : (� (A),λ × ℓ) → (� (A),0× c ◦ T λ) σ : (� (A),0× c ◦ T λ) → (� (A),λ × ℓ)  are linear, and cosymmetric if  σ ◦ �λ = �λ = (ξπ,λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Note that whenever σ is linear and cosymmetric, σ ◦ ˆµ(u,v) = ˆν(u,v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Linearity and unit axioms, along with the connection on the differential bundle, force the existence  of a bilinear bracket 〈−,−〉 as in the definition of a Lie algebroid in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' For an anchored bundle (π : A → M ,ξ,λ,̺) with connection (κ,∇), a cosymmetric  and bilinear σ is equivalent to a bilinear 〈−,−〉 : A2 → A, with the correspondence given by  〈−,−〉 : κ ◦ σ ◦ ∇−π κ ◦ ∇  σ : � (A) −−−−−−−−−−→  (π1,π0,π2+〈π0,π1〉)  � (A)  71  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Derive  �σ ◦ (x, y,z) = (y, x,κ ◦ σ ◦ (∇(̺x, y ) +p◦π1 µ(y,z)))  = (y, x,κ ◦ σ ◦ ∇(̺x, y ) +π κ ◦ σ ◦ µ(y,z))  = (y, x,κ ◦ σ ◦ ∇(̺x, y ) +π κ ◦ ν ◦ (y,z))  = (y, x,κ ◦ σ ◦ ∇(̺x, y ) +π z)  =: (y, x,〈x, y 〉 +π z)  where 〈−,−〉 is certainly bilinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The converse is immediate: take  �σ(u,v,w ) := (v,u,w + 〈u,v〉).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  It is easy to see that σ is linear and cosymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The linear bracket must be involutive for the bilinear bracket to be alternating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' If σ is cosymmetric and linear, then the bilinear bracket 〈−,−〉 is alternating if and  only if σ ◦ σ = id .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' First, note that σ ◦ σ = id if and only if �σ ◦ �σ = id .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Then check that  �σ ◦ �σ(u,v,w ) = �σ(v,u,w + 〈u,v〉) = (u,v,w + 〈u,v〉 + 〈v,u〉).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  By the cancellativity of addition on A,  w = w + 〈u,v〉 + 〈v,u〉 ⇐⇒ 0 = 〈u,v〉 + 〈v,u〉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Observation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A bilinear 〈−,−〉 on an anchored bundle with a connection induces the maps  {v, x}(κ,∇) := κ′ ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺ ◦ ∇(v, x),  {v, x, y }(κ,∇) := κ ◦ T (〈−,−〉(κ,∇)) ◦ ∇A2 ◦ (v, x, y )  from Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2) in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let σ be cosymmetric and bilinear, with the associated bracket 〈−,−〉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Then  T ̺ ◦ π1 ◦ σ = c ◦ T ̺ ◦ π1  if and only if the Leibniz equation is satisfied:  ̺ ◦ 〈u,v〉 + {̺v,u} = {̺u,v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='}  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Following the given notation and using the hypothesis that the connection is torsion-free on  M ,  �  T ̺ ◦ π1(u,v,w ) = (̺u,̺v,w + {u,v})  �c (x, y,z) = (y, x,z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Computing each side,  �  T ̺ ◦ �  π1 ◦ �σ ◦ (u,v,w ) = �  T ̺ ◦ �  π1 ◦ (v,u,w + 〈u,v〉)  = �  T ̺ ◦ (̺v,u,w + 〈u,v〉)  = (̺v,̺u,̺w + ̺〈u,v〉 + D[̺](̺v,u))  = (̺v,̺u,̺w + ̺〈u,v〉 + {v,u}),  �c ◦ �  T ̺ ◦ �  π1(u,v,w ) = �c (̺u,̺v,̺w + {u,v})  = (̺v,̺u,̺w + {u,v}),  so it follows that the two terms are equal if and only if the desired equality holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  72  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let σ be linear and cosymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Then  (i) T (〈−,−〉) : T A2 → T A satisfies  �  T (〈−,−〉)(ax,uy ,uz,ux y ,ux z)  = (ax ,〈uy ,uz〉,{ax,uy , yz} + 〈uy ,ux z〉 + 〈ux y ,uz〉);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ satisfies  �  (id × T (�σ))(ux ,uy ,ux y ,uz,ux z,uy z,ux y z)  = (ux ,uz,ux z,uy ,ux y ,uy z + 〈uy ,uz〉,  ux y z + {ax,uy , yz} + 〈uy ,ux z〉 + 〈ux y ,uz〉).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) By the universality of the vertical lift and bilinearity of 〈−,−〉, the outer squares below are pull-  backs:  A4  A2  T (A)  T (A)  T M  T M  M  M  µA2  (〈π0,π1〉,〈π0,π3〉+〈π1,π2〉)  µA  T 〈,〉  T π◦T πi  T π  0  0  so that  T (〈,〉)(0,uy ,uz,ux y ,ux z) = µA(〈uy ,uz〉,〈uy ,ux z〉 + 〈ux y ,uz〉).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Now we compute  T (〈,〉)(µA2((uy , yz),(ux y ,ux z)) +p ∇A2(ax,(uy ,uz)))  = T (〈,〉)µA2((uy ,uz),(ux y ,ux z)) +p T (〈,〉) ◦ ∇(ax,(uy ,uz))  = µA(〈uy ,uz〉,〈uy ,ux z〉 + 〈ux y ,uz〉) +p T (〈,〉) ◦ ∇(ax,(uy ,uz))  and then postcompose this with (T π,p,κ) to obtain  (0,〈uy ,uz〉, 〈uy ,ux z〉 + 〈ux y ,uz〉) +p (ax,〈uy ,uz〉,{ax,uy , yz})  = (ax,〈uy ,uz〉,{ax,uy , yz} + 〈uy ,ux z〉 + 〈ux y ,uz〉).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) Consider the following diagram:  T M p ×πA3π×πA3  T M p ×πA3π×πA3  T (A3)  T (A3)  T ◦ � (A)  T ◦ � (A)  �  T �σ  ∇A3+p µA3  T �σ  T (π1,π0,π2+〈π0,π1〉)  T (∇′+π0µ′))  (T (π3),p,κA3)  T σ  T (π0,p◦π1,κ◦π1)  73  We want to find �  T �σ =  �  T (π1,π0,π2 + 〈π0,π1〉).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Note that  T (π1,π0,π2 +π 〈π0,π1〉) = T (π1,π0,π2) +T π T (ξππi ,π0,〈π0,π1〉).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The left term is straightforward:  �  T (π0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π2)(ax ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='(uy ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ux y ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='(uz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ux z),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='(uy z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ux y z))  = (ax,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='(uz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ux z),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='(uy ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ux y ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='(uy z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ux y z))  and for the right term,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' use part (i) of this lemma:  �  T 〈−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='−〉◦ �  T (π0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π1))(ax,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='(uy ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ux y ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='(uz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ux z),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='(uy z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ux y z))  = �  T 〈−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='−〉(ax,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='uy ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ux y ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='uz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ux y z)  = (ax ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='〈uy ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='uz〉,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='{ax,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='uy ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' yz} + 〈uy ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ux z〉 + 〈ux y ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='uz〉)  Then compute  �  T �σ(ax ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='uy ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ux y ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='uz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ux z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='uy z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ux y z)  = (ax ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='uz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ux z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='uy ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ux y ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='uy z + 〈uy ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='uz〉,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ux y z + q)  where q = {ax ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='uy ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' yz} + 〈uy ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ux z〉 + 〈ux y ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='uz〉,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' giving the desired equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let σ be cosymmetric, doubly linear, and involutive, and satisfy the target axiom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Then σ satisfies the Yang–Baxter equation if and only if 〈−,−〉 and {−,−,−} satisfy the Bianchi identity:  0 =  �  γ∈Cy(3)  〈xγ0,〈xγ1, xγ2〉〉 +  �  γ∈Cy(3)  {̺xγ0, xγ1, xγ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='}  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We expand �  id × T σ and �  σ × c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Start with T (id × σ), which was derived in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9:  σ1(u) =  �  (id × T (�σ))(ux,uy ,ux y ,uz,ux z,uy z,ux y z)  = (ux ,uz,ux z,uy ,ux y ,uy z + 〈uy ,uz〉,  ux y z + 〈uy ,ux z〉 + 〈ux y ,uz〉 + {̺ ◦ ux,uy ,uz}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Using the fact that κ′ is torsion free, so ˆc (ux z,uy z,ux z y ) = (uy z,ux z,ux y z), it follows that  σ2(ux,uy ,ux y ,uz,ux z,uy z,ux y z) = (uy ,ux,ux y + 〈ux,uy 〉,uz,uy z,ux z,ux y z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Then compute  σ2σ1σ2(u)  =  �  uz,uy ,uy z + 〈uy ,uz〉,ux,ux z + 〈ux,uz〉,ux y + 〈ux ,uy 〉,z1  �  ,  σ1σ2σ1(u)  =  �  uz,uy ,uy z + 〈uy ,uz〉,ux,ux z + 〈ux,uz〉,ux y + 〈ux ,uy 〉,z2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  74  Note that the first five terms are equal, so it suffices to check z1 = z2 for z1 = π6σ1σ2σ1(u),z2 =  π6σ2σ1σ2(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  z1 = ux y z + 〈uy ,ux z〉 + 〈ux y ,uz〉 + {̺ ◦ ux,uy ,uz} + {̺ ◦ uz,ux,uy }  + 〈ux,uy z + 〈uy ,uz〉〉 + 〈ux z + 〈ux ,uz〉,uy 〉  = ux y z + 〈uy ,ux z〉 + 〈ux y ,uz〉 + {̺ ◦ ux,uy ,uz} + {̺ ◦ uz,ux,uy }  + 〈ux,uy z〉 + 〈ux,〈uy ,uz〉〉 + 〈ux z,uy 〉 + 〈〈ux ,uz〉,uy 〉  = ux y z + 〈ux y ,uz〉 + {̺ ◦ ux,uy ,uz} + {̺ ◦ uz,ux,uy }  + 〈ux,uy z〉 + 〈ux,〈uy ,uz〉〉 + 〈〈ux,uz〉,uy 〉,  z2 = ux y z + 〈ux,uy z〉 + {̺ ◦ uy ,ux,uz} + 〈ux y + 〈ux ,uy 〉,uz〉  = ux y z + 〈ux,uy z〉 + {̺ ◦ uy ,ux,uz} + 〈ux y ,uz〉 + 〈〈ux,uy 〉,uz〉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  So z1 = z2 is equivalent to requiring  ux y z + 〈ux y ,uz〉 + {̺ ◦ ux,uy ,uz} + {̺ ◦ uz,ux,uy }  + 〈ux,uy z〉 + 〈ux,〈uy ,uz〉〉 + 〈〈ux,uz〉,uy 〉  = ux y z + 〈ux,uy z〉 + {̺ ◦ uy ,ux,uz} + 〈ux y ,uz〉 + 〈〈ux,uy 〉,uz〉;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  cancelling alike terms, this is equivalent to  〈ux,〈uy ,uz〉〉 + 〈〈ux,uz〉,uy 〉 + {̺ ◦ ux,uy ,uz} + {̺ ◦ uz,ux,uy }  = {̺ ◦ uy ,ux,uz} + 〈〈ux,uy 〉,uz〉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Using the fact that 〈−,−〉 and {−,−,−} are alternating in the last two arguments, this is equivalent to  0 = 〈ux,〈uy ,uz〉〉 + 〈〈ux,uz〉,uy 〉 + 〈uz,〈ux,uy 〉〉  + {̺ ◦ ux,uy ,uz} + {̺ ◦ uz,ux,uy } + {̺ ◦ uy ,uz,ux}  = 〈ux,〈uy ,uz〉〉 + 〈uy ,〈uz,ux〉〉 + 〈uz,〈ux,uy 〉〉  + {̺ ◦ ux,uy ,uz} + {̺ ◦ uz,ux,uy } + {̺ ◦ uy ,uz,ux}  giving the desired identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let (π : A → M ,ξ,λ,̺) be an anchored bundle in a tangent category with negatives,  with anchored connection (∇,κ) on A and torsion-free affine connection (∇′,κ′) on M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' An involution  algebroid structure on A is equivalent to a bilinear map  〈−,−〉 : A2 → A  with derived maps  {−,−} : Aπ×p T M → T M := κ′ ◦ T ̺ ◦ (π0,π1),  {−,−,−} : T M p ×π◦πi A2 → A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='{a,u1,u2} := κ ◦ T (〈−,−〉) ◦ (∇(a,u1),∇(a,u2))  satisfying  (i) 〈−,−〉 is linear and cosymmetric,  (ii) 〈−,−〉 is alternating,  75  (iii) 〈−,−〉 and {−,−} satisfy the Leibniz equation, Equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iv) 〈−,−〉,{−,−}, and {−,−,−} satisfy the Bianchi identity, Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Morphisms of involution algebroids may also be characterized by preservation of the tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let A,B be a pair of involution algebroids with chosen connections in a tangent  category with negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Then an anchored bundle morphism f : A → B is an involution algebroid  morphism if and only if (recalling the notation from Equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3)  ∇[f ](x, y ) + 〈f ◦ x, f ◦ y 〉 = ∇[f ](y, x) + f ◦ 〈x, y 〉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Note that  (σ ◦ � (f ) = � (f ) ◦ σ)  ⇐⇒ (π1,π0,π2 + 〈π0,π1〉) ◦ (f ◦ x, f ◦ y, f ◦ z + ∇[f ](x, y ))  = (f , f , f ◦ π2 + ∇[f ](π0,π1)) ◦ (y, x,z + 〈x, y 〉)  while the second condition reduces to  ∇[f ](x, y ) + 〈f ◦ x, f ◦ y 〉 = ∇[f ](y, x) + f ◦ 〈x, y 〉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5  The isomorphism of Lie and involution algebroid categories  Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4 have made the relationship between involution algebroids and Lie algebroids clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  It is important to note that while the proofs used connections as a tool to identify the local coherences  satisfied by involution and Lie algebroids, the construction of a Lie algebroid from an involution alge-  broid (or vice versa) is independent of the choice of connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' There is an isomorphism of categories between Lie algebroids and involution algebroids  in smooth manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' For the equivalence of categories, note that by Propositions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='13 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='14, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='11,  and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='12 there is an isomorphism of categories between involution algebroids with a  choice of connection and Lie algebroids with a choice of connection (morphisms are not restricted to  connection preserving morphisms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This allows us to chain together isomorphisms  Inv(SMan) ∼= Inv(SMan)ChosenConn ∼= LieAlgdChosenConn ∼= LieAlgd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  To complete the proof, we must show that the assignment that sends an involution algebroid to a  Lie algebroid whose bracket is given by  λ ◦ [X ,Y ]∗ =  �  (π1 ◦ σ ◦ (id ,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='X ◦ ̺) ◦ Y −p T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Y ◦ ̺◦) −T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π 0◦ Y  �  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8)  is a bijection on objects, which brings up some subtleties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' First, while an involution map  σ : A̺×T πT A → A̺×T πT A  76  is defined with respect to a particular choice of pullback A̺ ×T πT A, the category of involution alge-  broids does not distinguish between different choices of this pullback (and therefore different repre-  sentations of the map σ), and it is not part of the data of an involution algebroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' It is immediate by  universality that the left-hand-side of Equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8 is independent of the choice of pullback A̺×T πT A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Now, recall thatthe canonical involution ofa Lie algebroid isuniquelybyTheorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7ofde León et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (2005) (this was also mentioned in Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Once we make a choice of prolongation A̺×T πT A,  we have made a choice of pullback � (A) in LieAlgd, which uniquely determines the canonical involu-  tion  σ : A̺×T πT A → A̺×T πT A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  While the exact map σ depends on the choice of pullback A̺ ×T π T A, they all determine the same  involution algebroid, thus proving the bijection correspondence of objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  77  Chapter 4  The Weil nerve of an algebroid  The first three chapters of this thesis demonstrated that tangent categories allow for an essentially alge-  braic description of Lie algebroids by axiomatizing the behaviour of the tangent bundle, and showing  that a Lie algebroid over a manifold M is a “generalized tangent bundle”, namely an involution al-  gebroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This chapter will make precise the sense in which an involution algebroid is a generalized  tangent bundle, by showing that the category of involution algebroids in a tangent category � is equiv-  alent to a certain tangent-functor category from the free tangent category over a single object to �, or  more generally that there is a fully faithful functor  Inv(�) �→ TangLax(FreeTangCat(∗),�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  This functor, the Weil nerve of an involution algebroid, builds a functor from the free tangent category  over a single object to � using a span construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This chapter primarily builds on two pieces of work:  Leung’s construction of the free tangent category Weil1 (Leung (2017)) , and Grothendieck’s original  nerve construction (first published in Segal (1974)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  To understand Leung’s construction of the free tangent category, and more generally his actegory-  theoretic presentation of tangent categories (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2), we first look at Weil’s original insight relating  the kinematic and operational descriptions of the tangent bundle in SMan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The definition of a tangent  vector on a manifold M as an equivalence class of curves (Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8) puts a bijective correspon-  dence between tangent vectors and �-algebra homomorphisms from the ring of smooth functions  C ∞(M ) to the ring of dual numbers:  C ∞(M ) → �[x]/x 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The hom-set �Alg(C ∞(M ),�[x]/x 2) is precisely the set of derivations on C ∞(M ), which defines the  operational tangent bundle discussed in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='16: there is a natural smooth manifold structure  on this set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The Weil functor formalism, most notably developed in Kolár et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (1993), extends this ob-  servation to a general class of endofunctors on SMan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' For example, the fibre product T2M corresponds  to �-algebra morphisms,  C ∞(M ) → R[x, y ]/(x 2, y 2, x y ),  while the second tangent bundle corresponds to �-algebra morphisms into the tensor product,  C ∞(M ) → R[x]/(x 2) ⊗ R[y ]/(y 2) = R[x, y ]/(x 2, y 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  By applying Milnor’s exercise (Problem 1-C Milnor and Stasheff (1974)), which states that the C ∞ func-  tor  SMan → �Algop;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' M �→ C ∞(M ) = SMan(M ,�)  78  is fully faithful, the structure maps occur as natural transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' For example, the tangent projec-  tion is induced by the morphism  p : �[x]/x 2 a+b x�→a  −−−−−→ �,  so that  T M  p−→ M = [C ∞(M ),�[x]/x 2]  (p)∗  −→ [C ∞(M ),�].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The zero map and addition are similarly induced by  0 : �  a�→a+0x  −−−−−→ R[x]/x 2 and + : R[x, y ]/(x 2, y 2, x y )  a+b x+c y  −−−−−−−→  �→a+(b+c )x R[x]/x 2,  respectively, while the lift and flip are induced by the morphisms  ℓ : �[x]/x 2  a+b x  −−−−−→  �→a+b x y �[x, y ]/(x 2, y 2)  and  c : �[x, y ]/(x 2, y 2)  a+b x+c y +d x y  −−−−−−−−−−−→  �→a+c x+b y +d x y �[x, y ]/(x 2, y 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  More generally, there is a monoidal category of Weil algebras (Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1) which has a monoidal  action on the category of smooth manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The Weil functor formalism, then, studies differential ge-  ometric structures from the perspective of the endofunctors and natural transformations induced by  this action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Leung’s insight is that there is an analogous category of commutative rigs1 built by replac-  ing �[x]/x 2 with �[x]/x 2, called Weil1 (Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' a tangent structure is precisely a monoidal  action by Weil1 satisfying some universal properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In particular, this category Weil1 is precisely the  free tangent category over a point, FreeTang(∗), so that every object A in a tangent category � deter-  mines a strict tangent functor  T (−)A : Weil1 → �;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V �→ T V A  and morphisms f : A → B are in bijective correspondence with tangent-natural transformationsT (−)A ⇒  T (−)(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The axioms of an involution algebroid in a tangent category � correspond bijectively with those  of the tangent bundle - this suggests that an involution algebroid should determine a tangent functor  from Weil1 to �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A first guess would lead one to think that p : �[x]/x 2 → � is sent to π : A → M , 0 to  ξ, and + to +A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' As the space of prolongations � (A) = A̺ ×T πT A plays the role of the second tangent  bundle, we can see that  ℓ : N [x]/x 2 → N [x, y ]/(x 2, y 2) �→ (ξ ◦ π,λ) : A → � (A),  and  c : N [x, y ]/(x 2, y 2) → N [x, y ]/(x 2, y 2) �→ σ� (A) → � (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  This pattern may be neatly summed up using span composition - we will construct a functor that sends  �[x]/x 2 to the span  A  M  T M  ̺  π  1A rig is a ring without negatives, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' a commutative monoid equipped with a bilinear multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  79  and the tensor product �[x]/x 2 ⊗ �[x]/x 2 to the span composition (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' the pullback)  � (A)  A  T A  M  T M  T 2M  ̺  π  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺  ⌟  which is the space of prolongations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This leads to the first major result of this chapter, the Weil Nerve  (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9), which states there is a fully faithful embedding  NWeil : Inv(�) �→ [Weil1,�].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  This bears a strong similarity to Grothendieck’s original nerve theorem, which takes an internal cate-  gory s,t : C → M and constructs a functor ∆op → � (where ∆op is the monoidal theory of an internal  monoid) by sending tensor to span composition, and the composition and unit maps given span mor-  phisms  C2  M  M  M  M  M  C  C  s◦π0  t ◦π1  s  t  m  s  t  e  while the unit and associativity axioms for a category are exactly the unit and associativity laws for  a monoid in this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The Segal conditions identify exactly the functors C : ∆op → � that lie in  the image of the nerve functor N as those whose [n]t h object is sent to the wide pullback C ([n]) =  C [2]t ×s C [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' t ×s C [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The corresponding result for involution algebroids is found in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8,  which states that a tangent functor (A,α) : Weil1 → � is the nerve of an involution algebroid if and  only if A preserves tangent limits and α is a T -cartesian natural transformation (this forces A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (W ⊗n) =  A̺×T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='πT A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T n−1̺×T nπT nA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The similarity between the Weil nerve and Grothendieck/Segal’s nerve  runs deep, and in Chapter 5 we demonstrate that the enriched perspective on tangent categories puts  these both into the same formal framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Sections4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2give a detailed introduction tothe Weil functorformalism (Kolár et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (1993),Bertram and Souvay  (2014)) and Leung’s unification of Weil functors with tangent categories Leung (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The rest of the  chapter contains contains new results developed by the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3 proves the embedding  part of the Weil nerve theorem, that the category of involution algebroids embeds into the category  of tangent functors and tangent natural transformations [Weil1,�].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4 identifies exactly those  tangent functors (A,α) : Weil1 → � that are the nerve of an involution algebroid, completing the proof  of the Weil nerve theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5 uses the Weil nerve to develop a novel tangent structure on the  category of involution algebroids in a tangent category (in particular, the category of Lie algebroids will  have this novel tangent structure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1  Weil algebras and tangent structure  This section gives a more thorough introduction to the Weil functor formalism of Kolár et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (1993),  and in particular how the structure maps of a tangent category may be teased out of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We begin by  80  introducing Weil algebras, and the prolongation of a smooth manifold by a Weil algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (The relation-  ship with prolongations of involution algebroids from Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1 will be made clear in Section  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=')  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' An �-Weil algebra2 is a finite-dimensional �-algebra V so that V = � ⊕ ˙V as �-  modules and ˙V is a nilpotent ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category �Weil is the full subcategory of �Alg spanned by the  �-Weil algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The prolongation of a manifold by a Weil algebra V is given by the manifold  T V M := �Alg(C ∞(M ,�),V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Eck (1986) showed that every product-preserving endofunctor on the category of smooth mani-  folds is constructed as the Weil prolongation by some Weil algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Consequently, �-algebra homo-  morphisms induce natural transformations between these product-preserving endofunctors on the  category of smooth manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Consider the following �-Weil algebras and their associated prolongation functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) Prolongation by � induces the identity functor, and the tangent bundle is given by �[x]/x 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The  tangent projection, then, is equivalent to the �-algebra morphism  p : �[x]/x 2 → �;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' p(a + b x) = a  while the 0-map induces the zero vector field:  0 : � → �[x]/x 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' 0(a) = a + 0x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) The algebra �[xi ]1≤i≤n/(xi x j )1≤i≤j ≤n = (�[x]/x 2)n is the wide pullback TnM = T M p×pT M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p×p  T M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In particular, prolongation by �[x, y ]/(x 2, y 2, x y ) gives the bundle T2M = T M p×pT M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The  �-algebra morphism  + : �[x, y ]/(x 2, y 2, x y ) → R[x]/x 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' +(a0 + a1x + a2y ) = a0 + (a1 + a2)x  corresponds to the addition of tangent vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iii) The algebra �[x, y ]/(x 2, y 2) = (�[x]/x 2) ⊗ (�[x]/x 2) is the second tangent bundle T 2M = T T M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The vertical lift T → T 2 is induced by the morphism  ℓ : �[x]/x 2 → �[x, y ]/(x 2, y 2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ℓ(a + b x) = a + b x y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iv) The monoidal symmetry map induces c : T 2 ⇒ T 2, as follows:  c : (�[x]/x 2) ⊗ (�[y ]/y 2) → (�[y ]/y 2) ⊗ (�[x]/x 2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  c (a + b1x + b2y + b3x y ) = a + b2x + b1y + b3x y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (v) For n ≥ 2, the algebra �[x]/x n gives the n-jet bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Note that this is the equalizer of ⊗n�[x]/x 2  by the symmetry actions of Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  2Not to be confused with the normal usage of “Weil algebra” in Lie theory, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Meinrenken and Pike (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  81  Further examples may be found in the monograph Kolár et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Tangent categories bridge  the gap between the Weil functor approach to studying the differential geometry of smooth manifolds  and the synthetic differential geometry approach of axiomatizing a tangent bundle using nilpotent in-  finitesimals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The main structure axiomatized here is that of monoidal action of a symmetric monoidal  category on a category M × � → �, or equivalently, a lift from a category to the category of complexes  � → [M ,�], which involves translating a bit of classical category theory to the 2-categorical setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2 leads to the classical theorem that the category of smooth manifolds has an action by  the category of �-Weil algebras that preserves all connected limits that exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' These “natural” universal  properties (in the sense of Kolár et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (1993)) is foundational to synthetic differential geometry;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' see,  for example, Chapter Two of Lavendhomme (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Unfortunately, Weil algebras are not an ideal syn-  tactic presentation: they are not a finitely presentable category, and it is not immediately clear when  a diagram is a connected limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3 Moving from �-algebras to commutative rigs and restricting to an  appropriate subcategory solves this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3 (Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1 Leung (2017)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category Weil1 is defined to be the full subcategory  of commutative rigs, CRig, generated by the rig of dual numbers W := �[x]/x 2, constructed as follows:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Start with finite product powers of W in CRig, and make a strict choice of presentation:  W0 = �, Wn := �[xi ]/(xi x j )i≤j ,0 ≤ i < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Then take the closure of Wn under coproduct of commutative rigs, written ⊗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Again, make a strict  choice of presentation:  Wn(0) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='⊗ Wn(m−1) := �[xi,j ]/(xi j xik)j ≤k,0 < i < m,0 < j < n(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Note that we will often suppress the tensor product ⊗ and simply write  U V :=U ⊗ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4 (Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3 Leung (2017) ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) The category Weil1 is a symmetric strict monoidal category with unit � and coproduct ⊗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) � is a terminal object in Weil1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Note that there is a forgetful functor  Weil1 → (CMon/�) → CMon  that reflects connected limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This gives the following class of limits, identified in Leung (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We say the following pullback diagrams in Weil1 are transverse:  W  W  W 2  W  W 2  W W  W  W  W  �  �  W  id  id  id  id  ⌟  π0  π1  p  p  ⌟  p◦πi  µ  0  pW  where µ(a +a1x +a2y ) = a +a1x +a2x y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The ⊗-closure of these three pullbacks is the set of transverse  squares, and they are also pullback squares by Leung (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  3It should be noted that Nishimura and Osoekawa (2007) made progress applying techniques from computer algebra to  latter problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  82  To see that each transverse square in the ⊗-closure is a pullback diagram, take the two non-identity  squares and rewrite them in CMon:  � × (� × �)  � × �  � × (� × �)  � × (� × � × �)  � × �  �  �  � × �  �×π0  �×π1  π0  π0  ⌟  π0  �×(π0,0◦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=',π1)  (id,0◦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=')  �×π1  ⌟  The coproduct of Weil algebras is the tensor product of the underlying commutative monoids, which  are finite-dimensional and free, so these limits are closed under ⊗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6 (Leung (2017) Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category Weil1 is a tangent category, where the  tangent functor is  T := W ⊗ (_) : Weil1 → Weil1  and the natural transformations are given by  p : W ⊗ (_)  p⊗(_)  −−→ (_), 0 : (_)  0⊗(_)  −−→ W ⊗ (_), + : W2 ⊗ (_)  +⊗(_)  −−→ W ⊗ (_),  ℓ : W ⊗ (_)  ℓ⊗(_)  −−→ W ⊗ W ⊗ (_),  c : W ⊗ W ⊗ (_)  p⊗(_)  −−→ W ⊗ W ⊗ (_)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The category Weil1 is, in some sense, a finitely presented theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' It is precisely the free tangent  category on a single object:  Proposition4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7(Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5, Leung(2017)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The categoryWeil1 isgenerated by the maps{p,0,+,ℓ,c }  from Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2, closed under composition, tensor, and maps induced by transverse limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category Weil1 is the free tangent category over a single object: every object C in a  tangent category � determines a strict tangent functor T−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='C : Weil1 → �, mapping  V = W n1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='⊗ W n(k) �→ Tn(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Tn(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='C = T V C  so that there isan isomorphism of categoriesbetween � and the category of strict tangent functorsWeil1 →  � with tangent natural transformations as morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Notation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Throughout this section, the notation T V C will refer to the action of the Weil algebra  V on an object C in a tangent category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In particular, we will make use of the isomorphism T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T V C =  T U V C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2  Tangent structures as monoidal actions  The presentation of Weil1 as the free tangent category situates the formal theory of tangent categories  as an instance of more general categorical machinery, namely monoidal actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Recall that in a sym-  metric monoidal category (�,⊗,I ), an internal monoid (C ,•,i) determines a monad:  (C ⊗ _ : � → �,µ : � ⊗ (� ⊗ _)  ⊗_  −→ � ⊗ _,η : _  ρ−→ I ⊗ _  e ⊗_  −−→ C ⊗ _).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  83  The category of algebras for this monad is exactly the category of C -modules, objects with an associa-  tive and unital action by C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A morphism will be a map on the base object that preserves the action:  C  M ⊗ C  M ⊗ M ⊗ C  M ⊗ C  C  M ⊗ C  C  M ⊗ C  M ⊗ D  C  D  M ⊗∝  ⊗M  ∝  ∝  (u,C )  ∝  f  ∝C  ∝D  M ⊗f  Strict actegories are the 2-categorical generalization of modules over a monoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The coherences for a 2-  monad follow from the coherences from a strict monoidal category in the 2-category of categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The  following proposition relies on a few facts from enriched category theory (treating the cartesian closed  category Cat as a Cat-enriched category, per Kelly (2005)) but a more general treatment of non-strict  actegories may be found in Janelidze and Kelly (2001):  A 2-functor and 2-natural transformations are exactly a functor and natural transformations  that satisfy extra coherences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' These coherences follow for free by constructing the monad and  comonad � × _,[� ,_].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  An algebra of the underlying 1-monad is exactly an algebra of the 2-monad (the same result holds  for comonads).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  When working with algebras of a 2-monad, four different notions of morphisms can come into play  (Lack and Power (2009)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' These arise through using the 2-categorical data to weaken the notion of a  morphism:  (i) Strict: this is exactly a morphism of the underlying algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Write the 2-category of strict � -  actegories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) Strong: the morphisms preserve the action to an isomorphism:  � × �  � × �  �  �  � ×F  ∝C  ∝D  F  α  (iii) Lax: the 2-cell is no longer an isomorphism:  � × �  � × �  �  �  � ×F  ∝C  ∝D  F  α  (iv) Oplax: the 2-cell travels in the opposite direction (these will not figure into this account).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  2-cells between actegory morphisms must satisfy a coherence between the natural transformation  parts of the actegory morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  84  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In the case of strict, strong, and lax tangent functors, the same notion of a 2-cell applies:  a natural transformation γ : F ⇒ G satisfying the following coherences with the natural transformations  α and β:  � × �  � × �  � × �  � × �  �  �  �  �  � ×F  ∝�  ∝�  F  α  G  � ×G  � ×F  ∝�  ∝�  G  β  γ  � ×γ  =  We call these actegory natural transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Note that for strict actegory morphisms, this condition holds for any natural transformation γ : F ⇒  G .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Now consider the following three 2-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let (� ,⊗,I ) be a strict monoidal category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Define the following three 2-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' �Actstrict: the 2-category of strict � -actegories, strict actegory morphisms, and natural transfor-  mations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' �Actstrong: the 2-category of strict � -actegories, strong actegory morphisms, and actegory nat-  ural transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' �Actlax: the 2-category of strict � -actegories, lax actegory morphisms, and actegory natural  transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Note that the inclusions of these 2-categories are locally fully faithful, so only the 1-cells differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The case where the action preserves certain limits in the monoidal category is of particular interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  A small category equipped with a class of chosen limits is known as a sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The previous correspon-  dence restricts to the class of limit-preserving actions in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A sketch is a small category with a class of chosen limits, and a sketch morphism is a  functor sending chosen limits to the chosen limits in the domain (up to isomorphism).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category of  models of a sketch � in a category �, Mod(� ,�), is the full subcategory [� ,�] whose functors preserve  the chosen limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A monoidal sketch, then, is a sketch (� ,� ) equipped with a symmetric monoidal  category structure on � so that _ ⊗ _ preserves limits in each argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Now use the fact that the category Weil1 is a monoidal sketch, since it is a small, strict monoidal  category equipped with a class of limits stable under the tensor product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4 (Theorem 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1, Leung (2017)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let � be a category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The following are equivalent:  (i) A tangent structure on �,  (ii) A sketch action ∝: Weil1 × � → �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' There is a coalgebraic perspective on tangent categories, coming from the equiva-  lence between algebrasof the 2-monad (Weil1×(_),⊗,I : 1 → Weil1) and the 2-comonad ([Weil1,_],[⊗,_],[I ,_]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  For any category �, there is a free tangent category given by  Weil1 × �  85  and this agrees with the free Weil1-actegory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' However, for the cofree tangent category, take  Mod(Weil1,�),  the category of transverse-limit-preserving functors Weil1 → �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  We can use Leung’s theorem to induce a monoidal functor Weil1 → � when a tangent structure is  induced by a single object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let (�,⊗,I ) be a strict monoidal category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' If an additive bundle (p : A → I ,+ : A2 →  A,0 : I → A) equipped with morphisms  A ⊗ A  c−→ A ⊗ A  A  ℓ−→ A ⊗ A  determines a tangent structure on � using the endofunctor A⊗(−), then A determines a strict, transverse-  limit-preserving, monoidal functor  A(−) : Weil1 → �;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Wn[1] ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='⊗ Wn[k] �→ An[1] ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='⊗ An[k] = Tn[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Tn[k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='I  Note that this allows for a more conceptual description of representable tangent structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In a symmetric monoidal closed category, an infinitesimal object is exactly a strict  symmetric monoidal functor D : Weil1 → �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  This presentation of an infinitesimal object makes it tautological that �op has a tangent structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Given a strict symmetric monoidal functor  D : Weil1 → �op  there is a strict action of Weil1 on �op given by  Weil1 × �op D op ⊗�  −−−−→ �op × �op  ⊗−→ �op .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  There is a clear correspondence between the notions of a (strict, strong, lax) tangent functor and  a (strict, strong, lax) actegory morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This proposition extends to the following equivalence of 2-  categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9 (Theorem 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1 Leung (2017)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The following pairs of 2-categories are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) The 2-category of tangent categoriesand strict tangent functorsisthe full sub-2-category ofWeil1ActStrict  spanned by sketch actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) The 2-category of tangent categoriesand strong tangent functorsisthe full sub-2-category ofWeil1Actstrong  spanned by sketch actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iii) The 2-category of tangent categoriesand lax tangent functorsisthe full sub-2-category ofWeil1ActLax  spanned by sketch actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  86  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3  The Weil nerve of an involution algebroid  The construction in this section is analagous to the nerve of an internal category—hence the “Weil  nerve” construction—and deals with similar technical issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In particular, the construction in this  section will mimic the nerve construction for internal categories by replacing the tensor product of  Weil1 with span composition in the domain category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Recall that every anchored bundle or internal  category has a canonical span associated with it:  A  C  M  T M  M  M  π  ̺  s  t  In any category �, there is a category of spans in � as well as span composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A span from A to B in a category � is a diagram of the form  X  A  B  l  r  There is a notion of span composition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' so given a span X : A → B and Y : B → C ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' then the composition  of X and Y is the pullback (if it exists):  Z  X  Y  A  B  C  l  r  l ′  r ′  ⌟  l ′′  r ′′  A morphism of spans is a commuting diagram of the form  A  X  B  C  Y  D  l  r  l ′  fl  fr  r ′  fc  Note that if f and g are span morphisms with fr = gl ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' then the horizontal composition may be formed  if each respective span composition exists: X  W  A  B  E  C  D  F  Y  Z l  r  l ′  fl  fr =gl  r ′  fc  gr  gc  ⌟  ⌟  87  When discussing span composition in a tangent category,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' it is assumed that the pullback is a T -pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Note that the category of spans in � is a functor category, so that limits are computed  pointwise in �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This also means that the horizontal composition operation, when it exists, preserves  limits in either argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  These span constructions can be helpful in constructing functors from a monoidal category into  a non-monoidal category �, by forming a monoidal category from � using spans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In the case of an  internal category over M , one takes the slice category �/(M × M ) where the tensor product is span  composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' An internal category s,t : C → M is a monoid in this category of spans over M , so that it  determines a monoidal functor  C : ∆op → �/(M × M ),  remembering that ∆op is the monoidal theory for monoid (every monoid in a monoidal category �  determines a monoidal functor ∆ → �).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The construction of the corresponding monoidal category for  spans is more nuanced, as the category Weil1 is not �-indexed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Observe that the prolongation of an  anchored bundle is constructed as a span composition:  � (A)  A  T A  M  T M  T 2M  π  ̺  T π  T ̺  π0  π1  ⌟  The third prolongation is given by span composition as well:  � 2(A)  A  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� (A)  M  T M  T 2M  π  ̺  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π◦π0  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺◦π1  π0  π1  ⌟  This horizontal composition will play the same role as the tensor product in �/(M × M ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In a tangent category �, consider a pair of spans  X : M → T U M , Y : M → T V M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Define X ⊠ Y to be the horizontal composition (when it exists):  Z  X  T U Y  M  T U M  T U V M  lX  rY  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='lY  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='r  ⌟  88  (recall that we will often suppress the ⊗ in Weil1 to save space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' So the span composition is  M  X−→ T U M  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Y  −−−→ T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T V M  M  X  T U M  M  Y  T U M  M  A  T V  M  B  T V ′  lX  lY  θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M  rX  rY  f  rA  rB  ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M  g  lB  lA  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1)  The horizontal composition f ⊠ g is defined as f ×θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='g :  Z  X  Y  M  T U M  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T V M  M  T U ′M  T U ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T V ′M  A  T U ′B  C  lX  lA  θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M  rX  rA  f  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='rY  T U ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='rB  θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M  θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='g  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='lY  T U ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='lB  ⌟  ⌟  In any tangent category with a tangent display system (Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3), the category of spans on  M whose maps are of the form given by Equation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1 with l ∈ � is a monoidal category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Any anchored  bundle in a tangent category gives rise to a monoidal category after a strict choice of T -pullbacks (as-  suming those T -pullbacks exist).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let (π : A → M ,ξ,λ,̺) be an anchored bundle in a tangent category �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Write the span  �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Wn := (M  π◦πi  ←−− An  ̺n  −→ TnM ),  �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� := (M = M = M ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  A choice of prolongations for (π,ξ,λ,̺) is a strict choice of horizontal composition for each V ∈ Weil1:  �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V = �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (Wn[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Wn[k]) := �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Wn[1] ⊠ ··· ⊠ �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Wn[k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  We will write the span as follows:  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V  M  T V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M  πV  ̺V  (notice that the apex is not hatted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Given a choice of prolongations for an anchored bundle (π,ξ,λ,̺),  the category Span(π,ξ,λ,̺) is defined as follows:  89  Objects are �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V for V ∈ Weil1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Morphisms are given by pairs  (f ,φ) : �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V → �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U  where f : A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V → A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U and φ : V →U determine a span morphism of the form  M  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V  T V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M  M  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M  πV  ̺V  φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M  πV  ̺U  f  as discussed in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Tensor structure: The tensor product is defined using the horizontal composition ⊠ as defined in  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The idea is to show that an involution algebroid structure on an anchored bundle induces a tangent  structure on the monoidal categoryofprolongations, and then toapplyLeung’stheorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The following  two lemmas will simplify this proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let (π : A → M ,ξ,λ,̺) be an anchored bundle with chosen prolongations in a tangent  category �, and identify the monoidal category Span(π,ξ,λ,̺).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) There is a functor  U ̺ : Span(π,ξ,λ,̺) → �  constructed by sending a span morphism to the morphism between the objects at its apex:  M  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V  T V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V  M  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U  πV  ̺V  φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M  πV  ̺U  f  f  (ii) Suppose we have a square  �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U  �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Y  �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='X  �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Z  (f ,φ)  (g ,ψ)  (l ,α)  (r,β)  whose image under U ̺ is a T -pullback in �, and so that the square in Weil1 is a transverse T -  pullback:  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Y  U  Y  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='X  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Z  X  Z  f  g  l  r  φ  ψ  β  α  ⌟  ⌟  Then U ̺ reflects the limit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' that is, the original square in Span(π,ξ,λ,̺) is a T -pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iii) T -pullbacks of the form described in (ii) are closed under ⊠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  90  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The functor in in (i) is straightforward to construct, as it simply forgets the left and right legs  of the spans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' For (ii), note that because the Weil1 part of the diagram is a transverse T -pullback, then  given a pair of maps  �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V  �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U  �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Y  �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='X  �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Z  (f ,φ)  (g ,ψ)  (l ,α)  (r,β)  (x,ω)  (y,γ)  a unique span morphism �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V → �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U may be induced using the apex map from �, and the unique  map induced in Weil1 by the universality of transverse squares (this square is also universal in �), so  the span morphism diagram will commute by universality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  For (iii), T -pullback squares of the form in (ii) are closed under ⊠ as transverse squares in Weil1  are closed under ⊗, so the result follows by the commutativity of limits and by applying part (ii) of this  lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' It will be useful to have a “flat” presentation of the prolongation A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Wn(1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Wn(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The higher prolongations of an anchored bundle may be concretely described as the T -pullback of the  zig-zag below:  ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Wn(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Wn(k)  ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Wn(1)  Tn[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Wn(2)  Tn(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='n(k−1) ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Wn(k)  Tn(1)M  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Tn(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺n(2)  Tn(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='n(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (π◦π0)  ̺n(1)  Tn[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (π◦π0)  so that the prolongation A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='W n[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='W n[k] may be written concretely as  (u1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=',uk) : An[1]̺′×T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π′Tn[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='An[2]T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺×T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺′×T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π′Tn[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='n[k−1]An[k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Furthermore, the choice of prolongation identifies the following limits:  ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U V  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V  ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U  T U M  ̺U  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='πV  ⌟  ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U V  T U ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V  T U M  T U M  ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U  T U M  ⌟  ̺U  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='πV  so that  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U V = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U ⊠ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U ⊠ idM ⊠ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V  where idM is the span M = M = M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Note that a ̺ sends the involution algebroid structure map to its corresponding tangent structure  map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Each of the structure maps, then, gives a span morphism where ⊠ is well defined:  91  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let (π : A → M ,ξ,+q,λ,̺,σ) be an involution algebroid in � with chosen prolonga-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Then define the following maps in Span(π,ξ,λ,̺):  The projection p : ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='W → ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='�,  M  A  T M  M  M  M  ̺  π  p  π  The zero map 0 : ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� → ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='W ,  M  M  M  M  A  T M  0  π  ̺  ξ  The addition map + : ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='W2 → ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='W ,  M  A2  T2M  M  A  T M  ̺2  π◦πi  +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M  π  ̺  +  The lift map ℓ : ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='W → ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='W W ,  M  A  T M  M  A̺×T πT A  T 2M  π  ̺  ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M  π◦π0  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺◦π1  (ξπ,λ)  The flip map c : ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='W W → ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='W W ,  M  A̺×T πT A  T 2M  M  A̺×T πT A  T 2M  c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M  π◦π0  ̺◦π1  π◦π0  ̺◦π1  σ  The idea is to show that the monoidal category of chosen prolongations Span(π,ξ,λ,̺) for an in-  volution algebroid has a tangent structure generated by the structure maps in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7 and the  endofunctor �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='W ⊠(−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Using the flat presentation, we can then show thatU ̺ will determine a tangent  functor in �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The following lemma about ̺ will be useful in constructing the natural transformation  part of a tangent functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let (π : A → M ,ξ,λ,̺) be an anchored bundle with chosen prolongations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Recall that  by Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4, the right leg of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U is written ̺, so it induces a span map:  M  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U  T U M  M  T U M  T U M  ̺U  ̺U  πU  pU  92  This map has a flat presentation as  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U V  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U ̺U ×T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='πV T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V  T U M id ×T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='πV T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V  ̺U ⊠A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V  πU ×T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V  We write the map  ̺U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V := ̺U ⊠ ( �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V )  which corresponds to the following span morphism:  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U V  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V  M  T U M  T U V M  T U M  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V  ̺U  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺V  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='πV  ̺U  πU  ⌟  pU .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='πV  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9 (The Weil Nerve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' There is a fully faithful functor  NWeil : Inv(�) → [Weil1,�]  that sends an involution algebroid to the transverse-limit-preserving tangent functor:  ( �A,α) : Weil1 → �  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' For the first step of this proof, we show that an involution algebroid structure on an anchored  bundle (π : A → M ,ξ,λ,̺) determines a tangent category structure on the monoidal category Span(π :  A → M ,ξ,λ,̺).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  We check that the endofunctor ˆA ⊠ (−) determines a tangent structure, with the structure maps  given by Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7:  [TC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1] Additive bundle axioms:  (i) Use Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5 to see that  ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='W2 = ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='W p ×p ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='W Aπ×πA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  this is preserved by ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V ⊠ (−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) The triple ( ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='+, ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p, ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0) = (+q ,π,ξ) is an additive bundle induced by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4,  and ⊠ preserves pullbacks (and therefore additive bundles), so the additive bundle axioms  hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  93  [TC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2] Symmetry axioms:  (i) ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c ◦ ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c = id follows from the involution axiom σ ◦ σ = id .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) For Yang–Baxter, note that  ( ˆA ⊠ c ) ◦ (c ⊠ ˆA) ◦ ( ˆA ⊠ c ) = (c ⊠ ˆA) ◦ ( ˆA ⊠ c ) ◦ (c ⊠ ˆA)  follows from the Yang–Baxter equation on an involution algebroid  (σ × c ) ◦ (id × T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ) ◦ (σ × c ) = (id × T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ) ◦ (σ × c ) ◦ (id × T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ),  since  σ × c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A = ( ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c ) ⊠ ( ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='W ) and id × T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ = ( ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='W ) ⊠ c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iii) For the naturality conditions:  (a) The interchanges of +,0,p all follow from the fact that  σ : (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='W W,λ × ℓ) → (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='W W,id × c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ)  is linear, and so is an additive bundle morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (b) The axiom  ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ c = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c ◦ c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ  is equivalent to the equation  (σ × c ) ◦ (1× T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ) ◦ (ˆλ× ℓ) = (1× T ˆλ) ◦ σ  which is equivalent to the double linearity axiom on σ by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  [TC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3] The lift axioms:  (i) The additive bundle equations are a consequence of λ being a lift and + being the addition  induced by the non-singularity of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) The coassociativity axiom  ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ ℓ = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ◦ ℓ  is equivalent to  (ˆλ× ℓ) ◦ ˆλ = (id × T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ˆλ) ◦ ˆλ  proved in (i) of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iii) The symmetry of comultiplication, c ◦ℓ = ℓ, is given by the unique equation for an involu-  tion algebroid, so that σ ◦ (ξ ◦ π,λ) = (ξ ◦ π,λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iv) The universality of the lift follows from part (ii) of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5 ensures  that for any V ∈ Weil1, �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V ⊠ µ and µ ⊠ �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V are universal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  This lemma puts a tangent structure on Span(π,ξ,λ,̺).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Now consider the functor sending spans to  the apex map,  U ̺ : Span(π,ξ,λ,̺) → �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The family of maps  {̺U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V : ̺U ⊠ �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V |U ,V ∈ Weil1}  94  gives a family of natural transformations  ̺U : A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T U ⇒ T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A,  so that the following pair constitute a tangent functor  (U ̺,̺) : Span(π,ξ,λ,̺) → �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Because the universality conditions on Span(π,ξ,λ,̺) followed by reflecting limits in � using Lemma  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5, it follows that (U ̺,̺) will preserve the tangent-natural limits in Span(π,ξ,λ,̺) corresponding to  transverse limits in Weil1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  By Leung’s Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7 (by way of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6), the tangent structure on Span(π,ξ,λ,̺) in-  duces a strict, monoidal, transverse-limit-preserving functor  ¯A : Weil1 → Span(π,ξ,λ,̺)  that sends the tensor product ⊗ to the span composition ⊠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' By composing the strict tangent functor  ( ¯A,id ) and (U ̺,̺), we have a lax, transverse-limit-preserving, tangent functor:  (A,̺) : Weil1 → �;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V �→ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V  Now, check the bijection on morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Starting with an involution algebroid morphism (f ,m) :  A → B, note that this gives a span morphism ˆf :  M  A  T M  N  B  T N  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='m  m  f  This gives a natural definition of ˆf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V using the horizontal composition of span morphism, so that  ˆf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (U V ) = ˆf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U ⊠ ˆf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V and ˆf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� = m,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2)  giving a family of maps { ˆfU : U ∈ objects(Weil1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Because f will commute with the structure maps  {π,ξ,+,(ξ ◦ π,λ),σ}, it follow immediately that ˆf is a natural transformation, because the following  calculation holds for each θ : X → Y ∈ {p,0,+,ℓ,c }:  ˆf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U Y V ◦ ( ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U ⊠ θ ⊠ ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V )  = ( ˆf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U ⊠ ˆf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Y ⊠ ˆf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V ) ◦ ( ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U ⊠ θ ⊠ ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V )  = ˆf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U ⊠ ( ˆf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Y ◦ θ) ⊠ ˆf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V  = ˆf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U ⊠ (θ ◦ ˆf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='X ) ⊠ ˆf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V  = ( ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U ⊠ θ ⊠ ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V ) ◦ ˆf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U X V  Tangent naturality will follow by the preservation of the anchor map by f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The equality, for any Weil  algebra U , of the diagrams  M  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U  T U M  M  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U  T U M  N  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U  T U N  M  T U M  T U M  N  T U N  T U N  N  T V N  T V N  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='m  m  πU  ̺U  ̺U  ˆf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U  ̺U  pU  πU  pU  ̺U  pU  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='m  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='m  m  ̺U  =  95  is precisely the tangent-naturality condition from Definitions 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='12, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  For the inverse of this mapping, consider a tangent natural transformation (Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='12)  γ : (A,α) → (B,β),  A ◦ T (A)  B ◦ T (A)  T ◦ A(A)  T ◦ B(A)  γT A  αA  βT A  T γA  where (A,α) and (B,β) are tangent functors Weil1 → � built out of involution algebroids with chosen  prolongations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' For any U ,V , the map γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U V decomposes as γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U ⊠ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V :  �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  �B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' �B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' �A  �B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' �B  �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T =π0  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T =π1  �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  �B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' �B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  �B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T =π0  β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T =π1  γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='γ  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T T  Applying this relationship inductively, it is clear that the base maps γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='W and γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� determine the entire  morphism γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V :  M  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U  T U M  N  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U  T U N  M  An[1]  Tn[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Tn[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Tn[k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M  N  Bn[1]  Tn[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Tn[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Tn[k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='N  Tn[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Tn[k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='�  γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='�  ̺n[1]  π◦πi  π◦πi  ̺n[1]  γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Wn[1]  Tn[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='�  Tn[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (π◦πi)  Tn[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (π◦πi)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='�  πU  πU  ̺U  ̺U  γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='�  γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U  =  Thus, every tangent-natural transformation is constructed out of a pair  (γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� : M → N ,γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='W : A → B)  using the ⊠ construction from Equation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' All that remains to show is that this pair is an involution  algebroid morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Tangent naturality gives the following two coherences:  ̺B ◦ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='W = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� ◦ ̺A and σB ◦ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='W W = γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='W W ◦ σA  since ̺B = β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='W,̺A = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='W,σB = B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c , and σA = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The following diagram proves  that γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='W preserves the lifts, so that (γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='W,γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='�) is an involution algebroid morphism:  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  (ξπ,λA)  π1  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='γ  γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  m  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ  β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  (ξπ,λB )  β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  λA  λB  96  Thus, a tangent natural transformation ( �A,α) → ( �B,β) is exactly a morphism of involution algebroids  A → B, proving the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Now, the projection for a Lie algebroid is a submersion, as we may make a choice of prolongations  for each U ∈ Weil1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' These prolongations lead to a new observation about Lie algebroids: they embed  into a category of functors into smooth manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Using the Weil nerve construction, the category of Lie algebroids embeds into the  tangent-functor category:  LieAlgd �→ [Weil1,SMan].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4  Identifying involution algebroids  This section identifies those tangent functors  (A,α) : Weil1 → �  that are involution algebroids as precisely those where A preserves transverse limits and α is a T -  cartesian natural transformation (Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' These conditions will force each A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V to be the  V -prolongation of the underlying anchored pre-differential bundle:  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p : A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T → A, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0 : A → A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ  −→ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  −→ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T, α : A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T → T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A)  (these conditions also ensure that this tuple is an anchored differential bundle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Initially, it is only clear that α is T -cartesian for the projection p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Indeed, recall that the prolonga-  tion A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U V is defined to be the T -pullback of the cospan:  �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U  αU  −→ T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='�  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p V  ←−−−−− T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V  Then consider the following diagram:  �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U W V  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T V  �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V  �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T U  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='�  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V  �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T V  �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p V  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p V  ⌟  ⌟  This means that every naturality square of α for p is a T -pullback;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' natural transformations satisfying  this property for every map in the domain category are called T -cartesian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A natural transformation γ : F ⇒ G is cartesian whenever each naturality square  F C  G C  F D  G D  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f  γ  γ  G .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f  ⌟  is a pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A natural transformation between functors into a tangent category is T -cartesian when-  ever each component square is a T -pullback (we will generally suppress the T when the context is clear).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  97  Now, recall that the Weil complex determined by an involution algebroid has A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V determined  by the following T -pullback squares:  �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V  �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T U  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='�  �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p V  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p V  ⌟  Then it is not difficult to show that the T -cartesian condition on a Weil complex forces it to be an  involution algebroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We first need:  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A T -cartesian Weil complex in � is a tangent functor  (A,α) : Weil1 → �  for which A sends transverse limits to T -limits and α is a T -cartesian natural transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The first condition to check is that a T -cartesian Weil complex gives a natural anchored bundle ˆA  whose Weil prolongations coincide with the functor assignments on objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let (A,α) be a T -cartesian Weil complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Then we have an anchored bundle  (M := A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='�,  ˆA := A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='W, π := A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π, ξ := A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ξ, λ := α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Furthermore,  � ( ˆA) = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='W W,  � 2( ˆA) = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='W W W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Suppose we have a tangent functor (F,α) : � → � and a differential bundle (π,ξ,λ) in �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' If F pre-  serves T -pullbacks of π, it preserves the additive bundle structure on (π,ξ,+), so to show (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π,F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ξ,α ◦  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ) is universal it suffices to show that the following diagram is a T -pullback in �:  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A2  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π2  µα◦F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π  Expand this to  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A2  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π2  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='µλ  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M  In this case, it restricts to the diagram  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T2  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T 2  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  A  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='µ  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  α  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0  ⌟  98  Each square is a T -pullback by hypothesis, so the universality of the lift follows by the T -pullback  lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Because the complex is T -cartesian, the assignment A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V gives a coherent choice of prolonga-  tions by the T -pullback  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p V  αU  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p V  αU  There is, of course, a natural candidate for the involution map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let (π : A → M ,ξ,λ,̺) be the anchored bundle induced by a T -cartesian Weil complex  in a tangent category �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Then we have an involution map  σ : � (A)  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c  −→ � (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The equations for an involution algebroid should follow immediately by functoriality;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' one need  only ensure that the maps take the correct form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let A be a T -cartesian Weil complex in a tangent category �, with (π : A → M ,ξ,λ,σ) its  underlying anchored bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Then we have:  (i) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T = σ × c ,  (ii) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c = 1× T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ,  (iii) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T = ˆλ × ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A,  (iv) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ = id × T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ˆλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) Consider the diagram  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  αT T  αT T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='A  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  ⌟  ⌟  Observe that this forces A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c × c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T = σ × c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  99  (ii) Likewise, the diagram  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='T T  αT  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='p  αT  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='T T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c  ⌟  ⌟  forces A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c = id × T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c = id × T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iii) The diagram  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  αT T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A  α  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  αT T  ⌟  ⌟  ⌟  ⌟  forces A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ× ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A = ˆλ × ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iv) As α is T -cartesian, the following diagram is a T -pullback:  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ  ⌟  Using previous results, this means that A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T is the unique map making the following diagram  commute:  A̺×T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='πT A  A̺×T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='πT AT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺×T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='πT 2A  T A  T AT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺×T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='πT 2A  (π1,π2)  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ˆλ  π1  which we can see is id × ˆλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Pulling together this lemma and the previous proposition, the following is now clear:  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A T -cartesian Weil complex determines an involution algebroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  However, we have not yet exhibited an isomorphism of categories between the image of the Weil  nerve functor and T -cartesian Weil complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' At first glance, the Weil nerve construction only gives  a Weil complex that is T -cartesian for the tangent projection p ∈ Weil1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Being T -cartesian for p is,  100  however, sufficient: a Weil complex that is T -cartesian for tangent projections will be T -cartesian for  every map in Weil1 (a similar result appears in the context of differentiable programming languages;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  see Cruttwell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (2019)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A lax tranverse-limit-preserving tangent functor (F,α) : Weil1 → � for which F pre-  serves pullback powers of each T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p is T -cartesian if and only if each  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  αA  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  αB  is a T -pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We only check the converse since the forward implication is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We make use of the T -  pullback lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) c is an isomorphism, so its naturality square is a T -pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) For projections T2 → T , the retract of a T -pullback diagram is a T -pullback, so the following  diagram is universal:  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T2  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T2  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T2  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T2  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T2  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T2  α  α  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='πi  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='πi  α  α  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='πi  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='πi  (iii) For 0, observe that the following two diagrams are equal:  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='0  α  α  α  α  ⌟  ⌟  The right diagram is a T -pullback, and the right square of the left diagram is a T -pullback by  hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' By the T -pullback lemma, the left square of the left diagram is a T -pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  ⌟  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  α  ⌟  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  ⌟  The outer perimeter of the right diagram is a T -pullback (left square by hypothesis, right square  by (ii)), as is the right square of the left diagram (by hypothesis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' By the T -pullback lemma, the  left square of the left diagram is a T -pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  101  (v) For +, observe that  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T2  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T2  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T2  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T2  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='+  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='+  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  ⌟  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='πi  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='πi  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T2  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  ⌟  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  α  ⌟  The outer diagram on the right is a T -pullback by composition, and the right square on the left  diagram is a T -pullback by hypothesis, so the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  To check that the naturality square is a T -pullback for every map in Weil1, we once again use Leung’s  characterization of maps in Weil1 from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Inductively, the set of maps generated by  {p,0,+,ℓ,c } closed under ⊗ and ◦ follows as T -pullback squares are closed to composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' For maps  induced by a tranverse limit in Weil1, F preserves transverse limits so this follows by the commutativity  of limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' For any tangent category �, the replete image of the Weil nerve functor  Inv(�) �→ [Weil1,�]  is precisely the category of T -cartesian Weil complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' That α : A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ⇒ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A is T -cartesian is equivalent to requiring that the tangent functor  (A,α) : Weil1 → �  restricts to an anchored bundle  (π : A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p  −→ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='�, ξ : A  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0  −→ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T, λ : A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ  −→ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T T  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  −→ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T, ̺ : A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  α−→ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A)  and each A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T V is the V -prolongation of this anchor bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The condition in Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9 is analogous to the Segal conditions identifying those  simplicial complexes  ∆ → �  that are internal categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Note that every simplicial object has an underlying reflexive graph  tr1(X ) := (s,t : X ([1]) → X ([0]),i : X ([0]) → X ([1]))  where X ([n]) is isomorphic to the object of n-composable arrows for the underlying reflexive graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Notably, being T -cartesian for p is enough to force that a natural transformation is  T -cartesian for the other tangent-structural natural transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This has consequences when one  uses partial maps to combine topological notions with tangent categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In this context, a partial map  N → X with domain M �→ N is a span  M  N  X  f  m  whose right leg is monic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The intuition is that the map f is defined on the subobject M of N , which  introduces a new problem: what is the proper notion of a subobject in a tangent category?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Such a notion  102  should give rise to a stable class of monics: one that is closed under horizontal span composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' One  answer is the notion of etale monics: a morphism is etale whenever the naturality square for p is a T -  pullbacks:  T M  T N  M  T N  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='f  p  p  ⌟  Geometrically, this means that the morphism is a local diffeomorphism;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' for example, an etale subobject  of �n in the Dubuc topos is precisely an open subset in the usual sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' An endofunctor lifts to the partial  map category whenever it preserves the class of monics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A natural transformation lifts to endofunctors  on the partial map category whenever it is T -cartesian for the class of monics, and the same proof will  show that this property holds for etale monics Cruttwell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5  The prolongation tangent structure  One of the most important consequences of the Weil Nerve Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9 is that the category of in-  volution algebroids (with chosen prolongations) may be equipped with two tangent structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The  first tangent structure is the pointwise tangent structure described in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The tangent  functor sends  (A,α) �→ (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A : Weil1 → �,c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='α : T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ⇒ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A)  (recall the composition of tangent functors given in Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='11 (ii)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The structure morphisms will  be given by whiskering, so in this case θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A,θ ∈ {p,0,+,ℓ,c }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The restriction to tangent functors that  preserve transverse limits along with the fact that the natural part α is T -cartesian, however, ensures  that precomposition with the tangent functor  (A,α) �→ (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T : Weil1 → �,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='α ◦ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c : A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ⇒ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T )  returns an involution algebroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The structure maps are once again given by whiskering, with the pre-  composition tangent structure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='θ,θ ∈ {p,0,+,ℓ,c }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Preservation of transverse limits guarantees that  this tangent structure will satisfy the necessary universality conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1 (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category of involution algebroids with chosen prolongations  in a tangent category � has a second tangent structure, where the action by Weil1 is given by  (A,α) �→ (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T : Weil1 → �,α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c : ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ⇒ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The proposition statement means that the structure morphisms for this new involution alge-  broid are given by  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T,α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c ) ∼=  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c = ̺′ :  � (A)  π1  −→ T A  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p = π′ :  � (A)  p◦π1  −−→ A  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0 = ξ′ :  A  (ξ◦π,0)  −−−→ � (A)  ̺′ ◦ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ = λ′ :  � (A)  λ×ℓ  −−→ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� (A)  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c = σ′ :  � 2(A)  σ×c  −−→ � 2(A)  103  Similarly, we can see that  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T,α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c )  =  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c = ̺′′ :  � 2(A)  (π1,π2)  −−−→ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� (A)  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p = π′′ :  � 2(A)  (p◦π1,p◦π2):  −−−−−−−→ � (A)  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0 = ξ′′ :  � (A)  (ξ◦π◦π0,0◦π1,0◦π2)  −−−−−−−−−−−→ � 2(A)  ̺′′ ◦ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='ℓ = λ′′ :  � 2(A)  (λ×ℓ×ℓ)  −−−−→ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� 2(A)  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='c = σ′′ :  � 3(A)  (σ×c ×c )  −−−−−→ � 3(A)  These coincide with the involution algebroids � ′(A),� ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� ′(A) in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='10: the second tan-  gent structure follows from the fact that the natural transformations for the tangent structure there are  given by  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='φ : A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='U ⇒ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V,φ :U → V ∈ {p,0,+,ℓ,c }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The result follows as a corollary of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The Jacobi identity for involution algebroids  Classically, the theory of Lie algebroids uses the algebra of sections Γ(π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' One key observation is that  when using the Lie tangent structure (Inv(�),� ), sections of π are in bijective correspondence with  χ� (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This observation allows for different statements about Lie algebroids to be translated into for-  mal statements about the tangent bundle in (Inv(�),� ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let A be an involution algebroid in �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' There is a bijection between the sections of π  in � and the vector fields on A in Inv(�):  X ∈ Γ(π) �→ ((id ,T X ◦ ̺),X ) : A → TL(A);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  ˆX ∈ χ� (A) �→ ˆXR : A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='R → A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Recall the coherence for tangent natural transformations γ : (H ,φ) ⇒ (G ,ψ):  H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='H  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='K  γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  φ  ψ  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='γ  We specify this to a morphism ˆX : (A,α) ⇒ TL(A,α) = (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T,α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ) at �,W :  A  A̺×T πT A  T M  T A  X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T  ̺  π1  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='X  and so infer that, if we set X := XR , we have π1 ◦ X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T = T (X ) ◦ ̺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Furthermore, the condition that  pL ◦ X = id forces id = π0 ◦ X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' thus, we can see that every section X of pL is given by a morphism of  the form ((id ,T X ◦ ̺),X )) on the underlying involution algebroids, where π◦ X = id .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  We now show that every X ∈ Γ(π) gives rise to a section of πL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Observe that the following is a mor-  phism of involution algebroids:  A  A̺×T πT A  M  A  π  (id,T X ◦̺)  p◦π1  X  104  Note that it is well typed, as T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='π◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='X ◦ ̺ = ̺ ◦ id .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Check that it is a bundle morphism:  p ◦ π1 ◦ (id ,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='X ◦ ̺) = p ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='X ◦ ̺ = X ◦ p ◦ ̺ = X ◦ π  and that it is linear:  (λ ◦ π0,ℓ◦ π1) ◦ (id ,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='X ◦ ̺) = (λ,ℓ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='X ◦ ̺)  = (λ,T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='X ◦ ℓ◦ ̺) = (λ,T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='X ◦ λ) = T (id ,T X ) ◦ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Then check that it preserves the anchor:  π1 ◦ (id ,T X ◦ ̺) = T X ◦ ̺  and the involution:  (π0,π1,T 2X ◦ T ̺π1) ◦ σ = (σ,T 2X ◦ T ̺ ◦ π1σ)  = (σ,T 2X ◦ c T ̺ ◦ π1) = (σ(π0,π1),c π3) ◦ (id ,T 2X ◦ T ̺ ◦ π1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Thus we have that (id ,T X ◦̺π1) is a morphism of involution algebroids, inducing a morphism of  T -cartesian Weil complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Lastly, we check that it is a section of p L  A , but this is clear, since  π0 ◦ (id ,T X ◦ ̺) = id ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  thus we have the desired bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Recall that given an involution on an anchored bundle, there is a bracket on its set of sections (see  the explicit construction in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Given an X ,Y ∈ Γ(π), there is a bracket defined as follows:  ˆλ ◦ [X ,Y ]A +1 (ξπ,0) ◦ Y = ((σ ◦ (id ,T Y ◦ ̺) ◦ X −2 (id ,T X ◦ ̺) ◦ Y )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  A direct proof of the Jacobi identity is a detailed calculation (see the original preprint on involution al-  gebroids Burke and MacAdam (2019)) and still relies on Cockett and Cruttwell’s result for an arbitrary  tangent category with negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' As a result of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2, we can instead use Cockett and Crut-  twell’s result directly:  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let A be a complete involution algebroid in a tangent category � with negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' There  is a Lie bracket defined on Γ(π), [−,−] induced by  ˆλ ◦ [X ,Y ]A +1 (ξπ,0) ◦ Y = ((σ ◦ (id ,T Y ◦ ̺) ◦ X −2 (id ,T X ◦ ̺) ◦ Y )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The bracket is induced by Rosicky’s universality diagram, as  0 = p ◦ ((σ ◦ (id ,T Y ◦ ̺) ◦ X − (id ,T X ◦ ̺) ◦ Y )) − 0Y  = T p ◦ ((σ ◦ (id ,T Y ◦ ̺) ◦ X − (id ,T X ◦ ̺) ◦ Y )) − 0Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  We look at the Lie tangent structure for Inv∗(A);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' this is precisely the vector field induced by  e vR([ ˆX , ˆY ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  We complete the proof by using the result that for any A in a tangent category with negatives, the  bracket on χ(A) that is defined by  ℓ◦ [X ,Y ] = (c ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='X ◦ Y − T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Y ◦ X ) − 0X  satisfies the Jacobi identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  105  Identifying categories of involution algebroids  Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4 identified whenever a functor Weil1 → � is an involution algebroid, whereas this section  identifies tangent categories � that embed into the category of involution algebroids in some tangent  category �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We call this structure an abstract category of involution algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This notion involves  some 2-category theory, using a modified notion of codescent (see Bourke (2010) for a development of  codescent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Recall that for any tangent category �, the category of involution algebroids has � as a reflective  subcategory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Furthermore, because limits of involution algebroids are computed pointwise, this reflec-  tor is left-exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This left-exact reflection is the main structure we axiomatize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' An abstract category of involution algebroids is a tangent category � with a left-exact  T -cartesian tangent localization (Z ,̺) : � → �, where L satisfies a codescent condition:  TangCatStrict(Weil1,�) �→ TangCatLax(Weil1,�)  L∗  −→ TangCatLax(Weil1,�)  (where L∗ denotes post-composition by L) is fully faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category of involution algebroids in any tangent category � is an abstract category  of involution algebroids using (Inv(C ),� ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The reflector is the functor sending an involution algebroid  to its base space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' the T -cartesian natural transformation is the anchor map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Any tangent subcategory of  Inv(�) that contains � as a full subcategory will give rise to an abstract category of involution algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let � �→ � be an abstract category of involution algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Then there is an embed-  ding � �→ Inv(�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The proof follows by treating objects in � as strict tangent functors Weil1 → � and morphisms  as tangent natural transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Weil1  �  �  Z  � (−,A)  � (−,B)  � (−,f )  The natural part of Z is T -cartesian, and the functor part preserves limits, so Z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='� (−,A) =: Z [A] de-  termines an involution algebroid in �, and f a morphism of involution algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The embedding is  guaranteed by the codescent condition so that the post-composition functor is fully faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' An abstract category of involution algebroids � �→ � is exactly a full subcategory � �→  � �→ Inv(�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  106  Chapter 5  The infinitesimal nerve and its realization  The main thrust of Chapters 2, 3, and 4 has been that the tangent categories framework allows for Lie  algebroids to be regarded as tangent functors  Weil1 → SMan  which satisfy certain universality conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This chapter, which is more experimental than the pre-  vious four chapters and represents work still in progress, puts Lie algebroids into the framework of  enriched functorial semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This new perspective on algebroids uses Garner’s enriched perspective  on tangent categories (Garner (2018)) and the enriched theories paradigm from Bourke and Garner  (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The functorial-semantics presentation of the Lie functor will generalize the Cartan–Lie the-  orem (that the category of Lie algebras is a coreflective subcategory of Lie groups) into a statement  within the general theory of functorial semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The goal is to show that the infinitesimal approximation of a groupoid, as discussed in Example  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2, may be constructed as a nerve, just like Kan’s original simplical approximation of a topological  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The nerve of a functor K : � → � approximates objects and morphisms in � by � -presheaves,  so it sends an object in � to the � -presheaf  NK : � → �;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='C �→ �(K −,C ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Thus there will be an infinitesimal object,  ∂ : Weilop  1  → Gpd(� )  where Gpd(� ) denotes groupoids in the category � of Weil spaces (formally defined in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The nerve of ∂ has a left adjoint, the Lie realization, given by the left Kan extension (just as Kan’s geo-  metric approximation of a simplicial set is, in Kan (1958)):  Weilop  1  Gpd(� )  [Weil1,� ]  �  ∂  Lan� ∂  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1)  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='13 (The Lie Realization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' There is a tangent adjunction between the category of involution  algebroids and groupoids in � , where each functor preserves products and the base spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Gpd(� )  Inv(� )  N∂  |−|∂  ⊣  107  Note, however, that the left Kan extension in Equation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1 does not immediately give the desired  adjunction of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='13, as there is no guarantee that the nerve functor N∂ lands in the category  of algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' To prove this we must revisit the work in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3 presenting involution algebroids as  those functors A : Weil1 → � for which each A(V ) is the prolongation of its underlying anchored bundle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  this leads naturally to the formalism for enriched theories developed in Bourke and Garner (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The presentation of algebroids as models of an enriched theory requires situating the categories  of differential bundles, anchored bundles, and involution algebroids in � as full subcategories of � -  presheaf categories on small � -categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1 reviews the work in Garner (2018) characteriz-  ing tangent categories as categories enriched in � = Mod(Weil1,Set) (the cofree tangent category on  Set, by Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2 reconfigures the content of Chapter 2 using the enriched perspective, so that lifts are � -  functors from the � -monoid D + 1, while differential bundles are a reflective subcategory of functors  from its idempotent splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Similarly, the category of anchored bundles are a reflective subcategory  of the category [Weil1  1,�], where Weil1  1 is the category of 1-truncated Weil algebras (Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3 reviews the basic idea of an enriched nerve/approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A particularly important ex-  ample is the linear approximation of a reflexive graph, the functor introduced in Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7, which  is the nerve of a functor  ∂ : (Weil1  1)op → Gph(� ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4 appliesthe enriched theoriesframework introduced in Bourke and Garner(2019), where  a dense subcategory of a locally presentable � �→ � forms the “arities”, and a bijective-on-objects func-  tor � → � is the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category of models is the pullback of (enriched) categories:  ��  [� ,� ]  �  [� op,� ]  The first step is to freely complete the � -category Weil1  1 of truncated Weil algebras (Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8)  so that its base anchored bundle has all prolongations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' call this � -category � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Then, in every tangent  category�, the categoryofanchored bundleswith chosen prolongations(Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4) in � embeds  fully and faithfully into the functor category:  Anc� (�) �→ [� ,�]  (here, Anc� (�) denotes the category of anchored bundles with chosen prolongations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In particular,  the tangent bundle on � in Weil1 determines a bijective-on-objects functor  � → Weil1  so that the category of involution algebroids in any tangent category � is the pullback in � Cat:  Inv∗(�)  [Weil1,�]  Anc∗(�)  [� op,�]  ⌟  This means that the category of involution algebroids in � is monadic over the category of anchored  bundles in � using the monad-theory correspondence from Bourke and Garner (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  108  The final section looks at the category of � -groupoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Essentially, the free groupoid over the lin-  ear approximation of a graph will now give an infinitesimal object Weil1 → Gpd(� ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The nerve of  ∂ : Weilop  1  → Gpd(� )—the infinitesimal approximation—has a right adjoint via the realization of the  nerve functor from Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This is used to prove the culminating Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The individual pieces of categorical machinery used in this chapter are not new (the enriched per-  spective on tangent categories, enriched nerve constructions, enriched theories).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' However, all of the  results dealing with the application of enriched nerve constructions and enriched theories to tangent  categories is original work of the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1  Tangent categories via enrichment  This section gives a quick introduction to Garner’s enriched perspective on tangent categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The  enriched approach to tangent categories first appeared in Garner (2018) and builds on the category  perspective on tangent categories introduced in Leung (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Garner was able to exhibit some of the  major results from synthetic differential geometry as pieces of enriched category theory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' for example,  the Yoneda lemma implies the existence of a well-adapted model of synthetic differential geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The category of Weil spaces is the site of enrichment for tangent categories and is closely related to  Dubuc’s Weil topos from his original work on models of synthetic differential geometry Dubuc (1981);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  a deeper study of this topos may be found in Bertram (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Recall that the category Weil1 is the free  tangent category over a single object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category of Weil spaces is the cofree tangent category over  Set, which is the category of transverse-limit-preserving functors Weil1 → Set by Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Call this the category of Weil spaces, and write it � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Just as a simplicial set S : ∆ → Set is a gadget  recording homotopical data, a Weil space records infinitesimal data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A Weil space is a functor Weil1 → Set that preserves transverse limits (Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5):  that is, the ⊗-closure of the set of limits  \uf8f1  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f3  Tn+m  Tm  Tn  �  ⌟  ,  T2  T 2  �  T  0  µ  ⌟  ,  Tn  Tn  Tn  Tn  \uf8fc  \uf8f4  \uf8f4  \uf8fd  \uf8f4  \uf8f4  \uf8fe  A morphism of Weil spaces is a natural transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Write the category of Weil spaces as � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) Every commutative monoid may be regarded as a Weil space canonically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Observe that for every V ∈  Weil1 and commutative monoid M , one hasthe free V -module structure on M given by |V |⊗CMonM  (here |V | is the underlying commutative monoid of V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The commutative monoid |V | is exactly  �dimV, so that  |V |⊗CMon M ∼= ⊕dimV M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  This agrees with the usual tangent structure on a category with biproducts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) Following (i), any tangent category � that is concrete—that is, admitting a faithful functorU : � →  Set—will have a natural functor into Weil spaces (copresheaves on Weil1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Every object A will have  an underlying Weil space V �→ USet(T V (A)), and whenever U preserves connected limits (such as  the forgetful functor from commutative monoids to sets), each of the underlying copresheaves will  be a Weil space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  109  (iii) Consider a symmetric monoidal category with an infinitesimal object, which by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7 is  a transverse-colimit-preserving symmetric monoidal functor D : Weil1 → �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Then for every object  X , the nerve (Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1) ND (X ) : �(D −,X ) : Weil1 → Set is a Weil space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iv) For any pair of objects A,B in a tangent category, �(A,T (−)B) : Weil1 → Set is a Weil space by the  continuity of �(B,−) : � → Set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Unlike the category of simplicial sets, the category of Weil spaces is not a topos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category of  Weil spaces does, however, inherit some nice properties from the topos of copresheaves on Weil1 by  applying results from Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3, as it is locally presentable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The basics of locally presentable categories  can be found in the Appendix 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Roughly speaking, a cocomplete category � has a subcategory of  finitely presentable objects �f p, those C so that  �(C ,−) : � → Set  preserves filtered colimits (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' those colimit diagrams that commute with all finite limits in Set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' � is  locally finitely presentable whenever every object is given by the coend  C ∼=  � X ∈�f p  �(X ,C ) · X  where S · X is the (possibly infinite) product X |S| with |S| the cardinality of the set S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This means that  � = Lex(�op  f p,Set), where Lex means the category of finite-limit-preserving functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The third pointofCorollary5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4 below, that� islocallyfinitelypresentable asa cartesian monoidal  category, means that the category �f p is closed under products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This implies that � is locally pre-  sentable as a � -category (this ends up being an important technical condition whereby locally pre-  sentable � -categories make sense).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Whenever we discuss an arbitrary � -category, we will assume  that � is locally presentable as a monoidal category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3 (Garner (2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category of Weil spaces is a cartesian-monoidal reflective sub-  category of [Weil1,Set].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category of Weil spaces is  (i) a cartesian closed category;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) a representable tangent category, where the infinitesimal object is given by the restricted Yoneda  embedding � : Weilop  1  → � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iii) Locally finitely presentable as a cartesian monoidal category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The cofree tangent structure on � is given by precomposition, that is:  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (V ) = M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (U ⊗ V ) = M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  This tangent coincides with the representable tangent structure induced by the Yoneda embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The proof is an application of the Yoneda lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Observe that  [D,M ](V ) ∼= [Weil1,Set](D × � (V ),M ) ∼= [Weil1,Set](� (W V ),M ) ∼= M (W V )  where D ×� (V ) = � (W V ) follows because the tensor product in Weil1 is cocartesian and the reflector  is cartesian monoidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  110  Notation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The tangent category � is a representable tangent category, where D = � W (Definition  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We will write the Yoneda functor � : Weilop  1  → � as D(−), so it is closer to the usual notation  used in representable tangent categories or synthetic differential geometry (a single D may be used as  shorthand for D(W ), D (n) for D(Wn), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Note that  D(V ) = D (⊗kWn(i)) =  K  �  D(ni)  and that D(ℓ) = ⊗,D(c ) = (π1,π0), D(+) = δ, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  At this point, we are ready to move to the enriched perspective on tangent categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The basics  of enriched category theory may be found in the Appendix 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2, but one definition in particular is im-  portant to include here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let � be a � -category for � a closed symmetric monoidal category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' For J ∈ � , the  power by J of an object C ∈ � is an object C J so that the following is an isomorphism:  ∀D : � (J ,�(D,C )) ∼= �(D,C J )  whereas the copower is given by  ∀D : � (J ,�(C ,D)) ∼= �(J • C ,D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Let � ←� � be a full monoidal subcategory of � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A � -category � has coherently chosen powers by � if  there is a choice of � -powers (−)J so that  (C J )K = C J ⊗K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Likewise, coherently chosen copowers are a choice of � -copowers so that  K • (J • C ) = (K ⊗ J ) • C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The sub-2-categories of � -categories equipped with coherently chosen powers and copowers are � Cat�  and � Cat� , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Wood (1978) proved that the 2-category of actegories over a monoidal � is equivalent to the 2-  category of [�,Set]-enriched categories with powers by representable functors (Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6), us-  ing the monoidal structure on [�,Set] induced by Day convolution1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Moreover, Garner (2018) showed  that a monoidal reflective subcategory � �→  ˆ  � exhibits the 2-category of � -categories as a reflective  sub-2-category of � -categories;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' this proves that tangent categories are equivalent to a particular class  of enriched category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7 (Garner (2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A tangent category is exactly a � -category with powers by repre-  sentables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' For every A,B ∈ �0, the Weil space is defined as  �(A,B) :=U �→ �(A,T U B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  1The Day convolution tensor product of presheaves X ,Y : �op → Set is given by X �⊗Y := Lan⊗(X ⊠Y ), where (X ×Y )(V ) =  X (V ) × Y (V ) (Day (1970)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  111  The functor �(A,−) is continuous and T −B is an infinitesimally linear functor, so this is a Weil space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The following diagram gives the composition map :  �(B,C ) × �(A,B)  �(A,C )  �(B,T U C ) × �(A,T V B)  �(A,T V U C )  �(T V B,T V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T U C ) × �(A,T V B)  T V ×id  m  Note that it is natural in U and V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  By the Yoneda lemma (as the internal hom in � is the internal hom of copresheaves on Weil1),  �(A,T V B) = (U �→ �(A,B)(V ⊗U )) = [D (V ),�(A,B)]  so this category has coherently chosen powers by representable functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Now the original notions of (lax, strong, strict) tangent functors can be shown to correspond to  power-preservation properties of � -functors between � -categories with coherently chosen powers:  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8 (Garner (2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We have the following equivalences of 2-categories:  (i) the 2-category of � -categories with coherently chosen powers and TangCatLax;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) the 2-category of � -categories with coherently chosen powers and power-preserving � functors  and TangCatStrong;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iii) The 2-category of � -categories with coherently chosen powers and chosen-power-preserving �  functors and TangCatStrict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Note that for a lax tangent functor (F,α) : � → �, the map  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='X : F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='X → T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='X  can be seen as the unique morphism induced by universality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Conversely, for a strong tangent functor,  the natural isomorphism α is the isomorphism from a power F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='X to the coherently chosen power  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In contrast, a strict tangent functor preserves the coherent choice of powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The ability to work with � -categories that do not have powers by representables allows for signif-  icant flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' It is useful to observe that there are � -categories which are not tangent categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) Every monoid (M ,m,e ) in � gives rise to a one-object � -category whose hom-object is M , com-  position is m, and unit is e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) Given a tangent category �, it is possible to take the full � -category over some set of objects �,  even though D may not be closed under iterated applications of the tangent functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iii) The dual of a � -category is a � category, where  �op (A,B) = �(B,A) : Weil1 → Set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Dually, in the case that � is a tangent category, �op will have coherent copowers by representables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  112  (iv) If a cartesian category � hasan infinitesimal object D (Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7), it hasa natural � -category  structure (with coherent copowers by representables)  �(A,B) : Weil1 → Set := �(A × D (−),B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Using the dual tangent structure on �op from Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8, the enrichment on � is exactly the  enrichment found by regarding � as the dual � -category of the tangent category �op .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  As a final remark, note that the Yoneda lemma applies to � -categories, so there is an embedding  � �→ [�op ,Set].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The powers and copowers by representables are computed pointwise in a presheaf category, so they  inherit the coherent choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Thus the following holds:  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='10 (Garner (2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Every tangent category embeds into a � -cocomplete representable  tangent category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (In fact, showing that the Yoneda embedding applies to tangent categories is the main theorem of  Garner (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=')  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2  Differential and anchored bundles as enriched structures  This section gives an enriched-categorical reinterpretation of the work in Chapter 2 regarding differen-  tial bundles and Chapters 3 and 4 regarding anchored bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Differential bundles as enriched structures  As a first case study using the enriched perspective for tangent categories, consider differential bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Most of the work in Chapter 2 uses the intuition that differential bundles are some sort of tangent-  categorical algebraic theory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' this section will make that intuition concrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Recall that a lift (Definition  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3) is a map λ : E → T E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Using the enriched perspective and treating T E as a power (recall Defini-  tion 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6), this gives the following correspondence:  λ : 1 → �(E ,E D )  ˆλ : D → �(E ,E )  The commutativity condition for ˆλ, then, is translated as follows:  E  T E  D × D  �(E ,E ) × �(E ,E )  T E  T 2E  D  �(E ,E )  ⊙  mE ,E ,E  ˆλ×ˆλ  ˆλ  λ  λ  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='λ  ℓ  That is, ˆλ is a semigroup morphism D → �(E ,E ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In any cartesian closed category with coproducts,  a semigroup may be freely lifted to a monoid using the "exception monad" (−) + 1 from functional  programming (see, for example, Seal (2013)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Regard the following monoid as the one-object � -category Λ:  D × D + D + D + 1  (ιL ◦m|ιL|ιL |ιR )  −−−−−−−−→ D + 1  m : (D + 1) × (D + 1) → D + 1  113  Thus, a lift λ is exactly a functor ˆλ : Λ → �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Now check that morphisms are tangent natural transfor-  mations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Note that the semigroup D is commutative, so Λ = Λop;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' the choice of using Λop in the next  lemma is to be consistent with conventions used in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category of lifts in a tangent category � is isomorphic to the category of � -functors  and � -natural transformations Λop → �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Check that a � -natural transformation is exactly a morphism of lifts f : λ → λ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Start with the  � -naturality square:  D + 1  �(A,A) × �(A,B)  �(A,B) × �(B,B)  �(A,B)  (λ,f ◦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=')  (f ◦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=',l )  mAB B  mAAB  Now, rewriting D + 1 → �(A,B) as a semigroup map D → �(A,B), we have:  D  (λ,f ◦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=')  −−−→ �(A,A) × �(A,B)  mAAB  −−−→ �(A,B)  1  (λ′,f )  −−→ �(A,T A) × �(A,B)  1×T  −−→ �(A,T A) × �(T A,T B)  mA,T A,T B  −−−−−→ �(A,T B)  1  T f ◦λ′  −−−→ �(A,T B)  Similarly, the other path is exactly λ′ ◦ f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Thus, a � -natural transformation is exactly a morphism of  lifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  It is a classical result in synthetic differential geometry that the object D has only one point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In the  case of � , this follows from the Yoneda lemma (regarding � as a Set-category):  � (1,D) = Weil1(�[x]/x 2,�) = {!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' : �[x]/x 2 → �}  Note that the natural idempotent e : id ⇒ id from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8, then, must be the point 0 : 1 → D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Note that this idempotent is an absorbing element of the monoid D + 1, so for any f : X → D + 1, it  follows that m(f ,0◦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=') = m(0◦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=', f ) = 0◦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='. A pre-differential bundle is exactly a lift with a chosen splitting  of the natural idempotent p ◦ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' For every lift ¯λ : Λop → �, the natural idempotent e = p ◦ λ is exactly  1  0−→ D  ιL−→ D + 1 → �(E ,E ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Now that the natural idempotent is understood as a map in Λ, that idempotent splits to give the  theory of a pre-differential object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The � -category Λ+ is given by the set of objects {0,1} with hom-Weil-spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Specifi-  cally,  The hom-spaces are Λ+(1,1) = D + 1, otherwise Λ+(i, j) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Composition (writing the original composition from Λ as m) is given by  m111 : (D + 1) × (D + 1)  m  −→ (D + 1)  m101 : 1× 1  ιR−→ (D + 1)  otherwise: mi j k =!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  114  Idempotent splittings are absolute (co)limits, and are preserved by all functors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' as this is a limit  completion, we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Lemma5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category of pre-differential bundles is exactly the category of �-functors Λ+ → � (that  is, �-valued presheaves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  It is straightforward to exhibit the category of differential bundles as a reflective subcategory of  pre-differential bundles in [Λ+,�] (so long as � has equalizers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category of differential bundles in a tangent category � with T -equalizers and  T -pullbacks is a reflective subcategory of [Λ+,�].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category of pre-differential bundles in � is isomorphic to [Λ+,�].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' By Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8, the  category of differential bundles is the category of algebras for an idempotent monad on the category  of pre-differential bundles in �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The reflector sends a pre-differential bundle to the T -equalizer:  A  T E  T E  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e  This equalizer will always exist if � has equalizers, and pullbacks of the projection will exist if � has  pullbacks, so the pullback is a differential bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This reflection gives a left-exact idempotent monad  on [Λ+,�] whose algebras are differential bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Now, in the case that � is a locally presentable tangent category (such as � ), [Λ+,�] is locally pre-  sentable and so is the reflective subcategory of differential bundles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' thus the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' If � is locally presentable, then DBun(�) is a locally presentable category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Anchored bundles  There are two ways to think about anchored bundles:  (i) an anchored bundle is a differential bundle with an anchor A → T M , or  (ii) an anchored bundle is an involution algebroid without an involution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  These two perspectives can be unified by regarding A as a cylinder for the weighted limit T M , so that  the anchor is induced by the unique map A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0 → T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0:  Λ+  �  A  T M  ̺  That is, the syntactic category for anchored bundles is constructed as a full � -category of Weil1 that  doesn’t include the map c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This may be found by taking the full subcategory of Weil1 whose objects  are constructed out of Wn,n ∈ �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A Weil algebra has width k ∈ � if it can be written  V = ⊗0≤i<kWn(i), n(i) ∈ �  The category of k-truncated Weil algebras, written Weilk  1 , is the full � -subcategory of Weil1 of Weil  algebras of width k or less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The full subcategory whose objects are {�,W } will be given the special notation Weil∗  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  115  Note that for each V in Weilk  1 , the enrichment is given by  Weilk  1 (U ,V ) := (X �→ Weil1(U ,X ⊗ V )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The maps in Weil1  1—that is, the full subcategory of Weil1 whose objects are  {Wn|n ∈ �}  —have the useful property that they may be written without the flip c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This makes Weil1  1 a natural  candidate for the syntactic category of anchored bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Every morphism  Wn → V ∈ Weil1  may be written without c ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' that is, it is generated by the set of maps {p,+,0,ℓ} closed under tensor, com-  position, and maps induced by transverse limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Every map Wn → V in Weil1 is the finite sum  v �→  �  x  (Ax v) • x  where x ∈ var(V ), and Ax ∈ �n so that An v is the ordinary dot product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Each term can be written  without c , and the whole term is then constructed by adding each component using the appropriate  U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V -symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' 2  It is possible, then, to show that the category of anchored bundles in � is a full sub-tangent-category  of functors Weil1  1 → �;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' note that W acts as a cone for (−)D , so this induces a map  ̺ : F (W ) → T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F (�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Every anchored differential bundle in � determines a functor Weil1  1 → �;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' an an-  chored bundle morphism is exactly an enriched natural transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Start with an anchored bundle (q : E → M ,ξ,λ,̺);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' then for hom-objects with domain �,  Weil1  1(�,�) = 1 �→ idM ,  Weil1  1(�,T ) = 1 �→ ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Forthe hom-objectswith domain T , the problem isslightlymore difficult, asWeil1  1(T,�)(T V ) = Weil1(T,T V )  and Weil1  1(T,T )(T ) = Weil1(T,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This part of the proof amounts to constructing maps  Weil1(T,T V ) → �(E ,T V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M ),  Weil1(T,T V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ) → �(E ,T V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E )/  The first mapping is straightforward: send θ to θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M ◦̺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' For the second map, observe that the following  diagrams commute:  T A  T 2M  A  T M  A  T M  A2  T2M  A  T M  M  M  M  M  A  T M  ̺  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='̺  λ  ℓ  ξ  0  ̺  ̺  π  p  +q  ̺2  +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M  ̺  2This may also be regarded as a consequence of the graphical notation for maps in Weil1 in Table 1 on page 308 of Leung  (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  116  The idea is to take an anchored bundle morphism f and rewrite it as a string of compositions that does  not include c , switching out every occurrence of  T V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M ,θ ∈ {p,0,+,ℓ}  and replacing it with T V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (θ ′), where θ ′ is the corresponding map in {q,ξ,+q,λ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This induces a map  Weil1  1(W,W ) → �(E ,E ) ∈ �  and the ̺ is exactly the unique α : F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='W → T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F induced by universality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  For morphisms, the inclusion (Λ+)op → Weil1 ensures that any tangent natural transformation will  be a linear morphism on the underlying differential bundle, and the tangent natural transformations  coherences ensure that a tangent natural transformation will preserve the anchor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Conversely, an an-  chored bundle morphism will preserve each of the constructed morphisms E → T V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E ,E → T V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='M  (as it preserves each of {q,+q,ξ,λ}), giving a natural tangent transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Thus there is a faithful  embedding Anc(�) �→ [Weil1  1,�].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The converse, identifyingthose functorsWeil1  1 → � thatdetermine anchored bundles, isimmediate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category of anchored bundles comprises precisely the � -functors and � -natural  transformations A : Weil∗  1 → � (Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8) so that the precomposition Λ+ → Weil∗  1 → � determines  a differential bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' That is to say, the category of anchored bundles in � is the following pullback in  � Cat:  Anc(�)  [Weil1  1,�]  DBun(�)  [Λ+,�]  ⌟  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3  Enriched nerve constructions  Nerve constructionspresenta powerful generalization of the Yoneda functor that sends an object C ∈ �  to the representable presheaf �(−,C ) ∈ [�op ,� ] (for a general reference on nerve constructions and  their realizations, see Chapter 3 of Loregian (2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The Yoneda lemma states that this functor is an  embedding (that is, it is fully faithful), so no information is lost when embedding a category into its cat-  egory of presheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Nerve constructions, then, move this towards an approximation of the original  category � by some subcategory � �→ �, or more generally by some functor K : � → �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5,  the “infinitesimal” approximation of a Lie groupoid as a Lie algebroid will be exhibited as approxima-  tion by a � -functor ∂ : Weilop  1  → Gpd(� ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  We work with a � that is locally presentable as a monoidal category, such as � ,Set, or the category  of commutative monoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The nerve of a � -functor K : � → � is the functor  NK : � → [� op,� ];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  C �→ �(K −,C )  Any presheaf A : � → � that is in the image of NK is a K -nerve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  117  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The “K -nerve” terminology seems to go back to Grothendieck/Segals’s original intuition  for the nerve construction of a category3 (Segal (1974)) and appears in, for example, Berger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (2012)  and Bourke and Garner (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' However, the “approximation” of a category by a functor K : � → �  originally used in topology seems to be a more intuitive description of the functor NK .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) The nerve of the identity functor � = � is the usual Yoneda embedding � �→ [�op ,� ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) The first example of a nerve construction in the mathematical literature is the simplicial approxi-  mation of a topological space by Kan (1958).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Recall the original construction of the simplicial nerve  of a topological space X , where Xn = Top(∆n,X ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This is exactly the nerve of the functor ∆ → Top  that sends n to the n-simplex  �  x ∈ �n|  �  xi = 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iii) The following example figures into Segal’s original nerve construction for a category, and is revisited  in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Define a reflexive graph to be a presheaf over the full subcategory ofCatwhose objects  are the two preorders  [0] = 0, [1] = 0 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  This is equivalent to the free category with two parallel arrows and a common retract,  [1]  [0]  t  s  e  so that a graph in a category has an object-of-vertices V and an object-of-edges E ,along with source  and target maps s,t : E → V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' morphisms of graphs are maps (fE , fV ) that commute with the source  and target maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  By Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3, any full subcategory containing the representables will be dense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Define the  category of paths, Pth, to be full subcategory of graphs generated by  [0]  [0]  [1]  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  [1]  [n]  [t ]  [s]  [s]  [t ]  ⌟  ,n ∈ �  so that Gph([n],G ) picks out the set of paths of length n in a graph G .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We call this subcategory  P : Pth → Gph, and see that NP sends a graph to its Pth-presheaf of composable paths:  En  E  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  E  V  V  t  s  t  s  ⌟  3Segal published the result, but seems to have credited the theorem to Grothendieck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  118  The pushout [n] is precisely the graph  0 −→ 1 −→ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' −→ (n − 1) −→ n  where the reflexive map e at each vertex is suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Recall the functor sending reflexive graphs to anchored bundles from Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7(iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This may  be restated as a nerve construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' There is a functor  ∂ : Weil∗  1 → Gph(� )  so that N∂ : Gpd(� ) → [Weil∗  1,� ] is the linear approximation of a reflexive graph as described in Exam-  ple 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7 (iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let s,t : C → M ,e : M → C be a reflexive graph, and recall that the anchored bundle C ∂ → C is  induced by the equalizer  C ∂  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='C  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='C  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T1  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e s  The category Gph is a representable tangent category, where D is the discrete graph D = D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' By the  Yoneda lemma, we have that  [Gph,� ](� 1,G ) = G1,[Gph,� ](� 0,G ) = G0,[Gph,� ](D × � (i),G ) = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='Gi,  so the limit diagram defining the infinitesimal approximation of a graph becomes  C ∂  Gph(D × I ,C )  Gph(D × I ,C ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (e ×I )∗  (D×e −)∗  Now observe thatGph(� )op with the dual tangentstructure from Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8 hasa reflexive graph  object  s,t : 1 → I ,!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' : I → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Construct its linear approximation as in Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7(iv) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' take the coequalizer of � -graphs):  D × I  D × I  ∂ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  e ×I  D×e s  By Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='10, this determines a functor  ∂ : Weil1  1 → Gph(� )op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  By the continuity of the hom-functor, the nerve N∂ : Gph(� ) → [Weil∗  1,� ] lands in the category of  anchored bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The continuity of the hom-functor ensures that this is indeed the linear approxi-  mation from Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7 (iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The simplicial localization in Kan (1958) has a left adjoint, the geometric realization, that constructs  a topological space using the data of a simplicial set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This realization may be constructed using a left  Kan extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  119  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let K : � → � be a � -functor for a cocomplete �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The realization of K is the left Kan  extension  A  �A  �  K  �  |−|:=LanK �  |C |K :=  � A  �(K A,C ) • K A  A nerve/realization context is a � -functor K : � → � from a small A to a cocomplete �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The adjunction between simplicial sets and topological spaces follows from general categorical  machinery as a nerve/realization context:  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' For every nerve/realization context K : � → �, the realization of K is left adjoint to the  nerve of K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4  Nervous monads and algebroids  Transposing the characterization of algebroids from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9 to the enriched perspective intro-  duces the challenge of finding an appropriate framework to describe algebroids as enriched structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Kelly’s theory of enriched sketches (see Chapter 6 of Kelly (2005)), small � -categories equipped with  a chosen class of limits cones, seems to be a natural candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' However, when regarding a tangent  functor as a � -functor, the natural part  α : F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ⇒ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F  is the unique morphism induced by the universality of T as a weighted limit, so it becomes unclear  how to translate the condition that α is a cartesian natural transformation (Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  A clue for how to proceed may be found in Kapranov (2007), which proved that Lie algebroids  are monadic over anchored bundles (when allowing for infinite-dimensional vector bundles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Recent  work in Bourke and Garner (2019) and Berger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (2012) has developed the appropriate notion of (en-  riched) theories that correspond to monads over general locally presentable categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A critical in-  sight is that for a filtered-colimit-preserving monad � on Set, the opposite category of the Lawvere  theory � is precisely the full subcategory of the Kleisli category whose objects are [n] =  �  n 1, and the  nerve of the inclusion  � op �→ Set�  is fully faithful;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' that is, when the functor K is dense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Formally, a functor is dense whenever its nerve  behaves like the Yoneda embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The theory of dense functors is developed in Chapter 5 of Kelly  (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We continue to assume that an arbitrary site of enrichment � is locally presentable as a � -  category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A functor K : � → � is dense whenever the nerve of K is fully faithful, and a subcate-  gory inclusion that is dense will be called a dense subcategory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Dense functors are poorly behaved under composition, but there is a useful cancellativity result  from Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2 of Kelly (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Consider a diagram of � -categories  X  Z  Y  K  F  J  α  120  where α is a natural isomorphism and K is dense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' If this diagram exhibits (α, J ) as the left Kan extension  of K along F , then J is also dense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' If K is dense, F is fully faithful, and J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F = K , then F is a dense subcategory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Generally speaking, we will often refer to dense subcategories rather than dense functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This is  achieved by factoring the dense functor  K = �  K ′  −→ im(K ) �→ �  where im(K ) is the � -category whose objects are those of � and whose hom-objects are given by  im(K )(A,B) = �(K A,K B), so that the inclusion of im(K ) into � is fully faithful and therefore dense  by Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) The category of finite cardinals Σ is the full subcategory of Set whose objects are given by finite  coproducts of the terminal object, [n] =  �  n 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This is a skeleton of the category of finite sets, and a  dense subcategory of Set (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Bourke and Garner (2019)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) Recall that the category of � -presheaves on � , [� op,� ] is the free colimit completion of � (see  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Kelly (2005)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' If � is cocomplete, and K : � → � is dense, then the realization |− |K exhibits �  as a reflective subcategory of [� op,� ], and is this a locally presentable category4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Conversely, if �  is a reflective subcategory of [� op ,� ] that contains the representable functors, then � is a dense  subcategory of �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iii) By Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7, the category of differential bundles in � is a reflective subcategory of [Λ+op,� ],  so � : Λ+ �→ [Λ+op,� ] is a dense subcategory of differential bundles in � following the argument  in the above example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Nervous theories (Bourke and Garner (2019)) are generalizations of Lawvere theories that extend to  arbitrary locally finitely presentable � -categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Recall that a classical Lawvere theory is a bijective-  on-objects, product-preserving functor  t : Σ → �  where � is a cartesian category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Nervous theories replace Σ with a dense subcategory of some locally  presentable � -category, and the product preservation condition with conditions on the nerve of the  theory map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let K : � → � be a dense � -subcategory of a locally finitely presentable �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We call  the replete image of NK in [� op,� ] the category of K -nerves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' An � -theory is a bijective-on-objects  � -functor J : � → � , where each  � (J −,a) : � → �  is a K -nerve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category of models for an � -theory is the pullback in � CAT:  ��  �  �  �  �  �  J ∗  NK  ⌟  (These are called concrete models in Bourke and Garner (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=')  4In fact, an equivalent definition of a locally presentable category is as a cocomplete category with a dense subcategory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  121  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category of models for a theory is monadic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The core of the argument is due to Weber’s  nerve theorem, found in Weber (2007), but an exposition on that result is beyond the scope of this thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) A functor t : Σ �→ � is a Σ-theory if and only if � is a Lawvere theory, where we use the fact that  Σ (Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4) a skeletal subcategory of finite sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The nerve conditions in this case identify the  models of the Lawvere theory, as the nerve of Σ �→ Set sends a set to the strict product-preserving  functor [n] �→ An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) As shown in Berger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (2012) and Bourke and Garner (2019), the original nerve construction  from Segal (1974) may be restated as saying that small categories arise as models of a Pth-theory  (Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The set Gph([n],G ) is the set of paths of through the graph G that have n non-  identity elements, as  G t ×s G ∼= G t ×id M id ×s G ∼= G t ×s◦e ◦sG t ◦e ◦t ×sG .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Next, recall that the category ∆ may be regarded as follows:  Objects: A strict order [n] = 0 < 1 < ··· < n, for n ≥ 0, regarded as a category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Maps: Functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  There is a bijective-on-objects functor from Pth → ∆ that sends the graph [n] to the pre-order [n],  as every graph homomorphism between paths will be order-preserving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Now observe that a model  is precisely a simplicial set, where  X ([0]) = M ,X ([1]) = C ,X ([2]) = C t ×s C ,X ([3]) = C t ×s C t ×s C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Furthermore, the map  0  0  1  1  2  in ∆ becomes a composition map:  X ([2]) → X ([1])  C t ×s C → C  Associativity and unitality (for the section e : M → C ) follow by functoriality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Thus, a model of  Pth → ∆ is a small category, and a morphism is exactly a functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='11, then, gives a monadicity result for � -anchored bundles over � -differential bun-  dles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' � -anchored bundles are models of the theory (Λ+)op → (Weil∗  1)op (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  122  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Note that the inclusion Λ+ → Weil1  1 is a differential bundle, so it is a nerve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Then the diagram:  Anc(� )  [Weil∗  1,� ]  DBun(� )  [Λ+,� ]  ⌟  exhibits � -anchored bundles as models of a (Λ)op -theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  As a corollary, we see that the category of anchored bundles is locally presentable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category of anchored bundles is locally presentable, and (Weil∗  1)op �→ Anc(� ) is  dense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  In Bourke and Garner (2019), the authors identify exactly those monads on a locally presentable  � -category that correspond to the models of a theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Recall the notation that the category of algebras  for a monad is �T and the category of free coalgebras is �T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' For a dense subcategory � �→ �, use �T  for the category of free algebras over objects in � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Recall that the Lawvere theory  K : FinSet → �  for a filtered-colimit-preserving monad � on Set may be re-derived as the full subcategory of free alge-  bras over finite sets, KT : FinSet� �→ Set� �→ Set�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Nervous monads abstract this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let � be a locally presentable � -category, with K : � → � a dense sub-� -category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  A � -monad � over � is K -nervous if  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' the inclusion KT : �T �→ �T is dense;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' the following diagram is a pullback in � CAT:  �T  [� op  T  ,� ]  �  [� op,� ]  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='11 (Bourke and Garner (2019)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let K : � �→ � be a dense sub-� -category of a cocomplete  � -category �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' There is an equivalence of categories between � -theories and � -nervous monads on �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The first step in showing that algebroids are monadic over anchored bundles is to construct a dense  subcategory of Anc(� ) that plays the role of the V -prolongations from Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The enriched  framework makes this straightforward: prolongations are weighted limits, with which we may freely  complete Weil∗  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Consider the category Anc(� ) as a representable tangent category, and take the dual  tangent structure on Anc(� )op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category � is defined as the full subcategory of Anc(� )op whose  objects are generated by the prolongations of the Yoneda embedding � : Weil∗  1 → Anc(� ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  For any small � -category �, the free completion of � is the opposite � -category of the category of  copresheaves on �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5 Thus, � is the free completion of Weil1  1 with prolongations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Therefore, a choice  5This is the dual statement of the classical theorem that the category of presheaves is the free cocompletion of a small  category �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  123  of prolongations on an anchored bundle A determines a unique functor � → � given by right Kan  extension (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Kelly (1982)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The continuity of  Anc(� )(−,A) : Anc(� )op → �  ensures that Anc(� )(LV ,A) is AV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9, the category of involution algebroids in � is a full sub-� -category of [Weil1,�]  that has a forgetful functor down to the category of anchored bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A consequence of Theorem  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8 is that a functor Weil1 → � is an involution algebroid if and only if the precomposition  Weil∗  1 �→ Weil1 → �  restricts to an anchored bundle, with each V sent to the V -prolongation of this anchored bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In  other words, algebroids are models of the following enriched theorem (as in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Define the � op-theory of algebroids as the functor  a : � → Weil1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Thisfunctorisbijective-on-objectsby definition, and satisfiesthe nerve condition because the V -prolongation  of the tangent bundle is T V , so each presheaf  Weil1(a−,V ) : � → �  is an � -nerve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The Weil nerve, then, translates to the following characterization of the category of involution alge-  broids in a tangent category �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category of involution algebroids with chosen prolongations in a tangent category  � is precisely the pullback in � CAT:  I nv ∗(�)  [Weil1,�]  Anc ∗(�)  [� ,�]  ⌟  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2)  Consequently, in � involution algebroids are monadic over anchored bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Recall the correspondence  ¯A : Weil1 → � ∈ � Cat  ( ˆA,α) : Weil1 → � ∈ TangCatl a x  If  Weil∗  1 → � → Weil1 → �  determines an anchored bundle (π : A → M ,ξ,λ,̺) whose V -coprolongation is ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='V , so that ˆA is the  nerve of an involution algebroid by Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Thus, we may characterize � -involution algebroids as algebras for a � -nervous monad on the  category of � -anchored bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  124  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category of involution algebroids in � is equivalent to the category of algebras  for the � -nervous monad on Anc(� ) generated by the theory  a : � → Weil1  from Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='13, using Theorem 19 from Bourke and Garner (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The construction of involution algebroids as the category of models for a � -theory is  remarkably similar to the original nerve construction for categories in Segal (1974), with the symmetric  nerve construction for groupoids replacing ∆ with the category of finite sets Σ (see Example 44 (iv) of  Bourke and Garner (2019)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Each construction truncates the original category to two objects and builds  a new category with a bijective set of objects by freely adding limits to the two-object truncation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5  The infinitesimal approximation of a groupoid  This section contains the main theorem of the chapter, namely that there is an adjunction  Gpd(� )  Inv(� )  ⊣  This adjunction follows by inducing a nerve/realization context on the category of groupoids in � :  ∂ : Weilop  1  → Gpd(� ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Asimplified version ofthisfunctorappeared in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4, the linearapproximation ofa reflexive  graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In fact, ∂ will be exactly the free groupoid over the graph ∂ from Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4 (if we had  worked in the category of reflexive graphs equipped with an involution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Recall that groupoids and  Weil spaces are both models of a sketch (recall Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The symmetry of the theories gives two  equivalent presentations of groupoids in � :  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The tangent category of � -groupoids is equivalently  (i) the tangent category of internal groupoids in � , where the tangent structure is computed point-  wise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) the tangent category of transverse-limit-preserving functors Weil1 → Gpd(Set) with the cofree tan-  gent structure from Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Itisimportanttonote thatthe tangentstructure on � -groupoidsisrepresentable, asitisa cartesian  closed category with an infinitesimal object given by Weil1 �→ � �→ Gpd(� ) (a proof that Gpd(�) is  cartesian closed for a cartesian closed category � with sufficient limits a may be found in Section B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3  of Johnstone (2002)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  There are two classes of “trivial” � -groupoids that will be useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) Every small groupoid G may be regarded as a groupoid in � as the constant functor Weil1 →  Gpd(Set) sending V to the groupoid G preserves transverse limits, we may then apply the symmetry  of theories to find a groupoid in the category of Weil spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  125  (ii) Every object M in a category has a cofree groupoid given by s,t : M = M , this is the discrete  groupoid whose only morphisms are the identity maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Thus, every Weil space M ∈ � has a corre-  sponding discrete groupoid (this will often be written M = M to denote the discrete groupoid over  M ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Using the cartesian closure of � and the full subcategories of trivial and discrete groupoids, we  have the following:  Observation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The category of � -groupoids is both  (i) a � -category, with powers by � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The power of a � -groupoid � = s,t : G → M by a Weil space E  is given by the � -groupoid  [E ,�] = {[E ,s],[E ,t ] : [E ,G ] → [E ,M ], [E ,e ] : [E ,M ] → [E ,G ]};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) aGpd-enriched category with powersby small groupoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The powerof a � -groupoid � : Gpd → �  by a groupoid � is the � -groupoid  [� ,�](V ) = [� ,�(V )]Gpd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The following two � -groupoids form the basic building block for the main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) Set D := � W in � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The discrete groupoid D = D represents the tangent functor internally, and the  copresheaf Gpd(D,−) : Gpd(� ) → � sends a groupoid s,t : G → M to T M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) The arrow groupoid is the free groupoid generated by the graph:  → •  Write the trivial � -groupoid on this groupoid as I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Note that the power by I will send a groupoid  to its “arrow groupoid” � →, the groupoid whose objects are arrows in �, with a map u → v being  a commuting square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' It follows that Gpd(I ,�) is the space of arrows of the groupoid, Gpd(I × I ,�)  the space of commuting squares, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The next proposition gives the necessary properties of I to construct the Lie derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) For every groupoid � = s,t : G → M , there is an isomorphism  G t ×t G s ×t G ∼= Gpd(� )(I × I ,�);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  this corresponds to the unique filler for the diagram u  v  w  ∃!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='w −1◦v◦u  126  (ii) The arrow groupoid has a semigroup with a zero structure (like an infinitesimal object D), and the  multiplication is the coequalizer  1× I  I × I  I × I  I  I × 1  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='×I  s×I  I ×!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  I ×s  m  The dual result of part (ii) is somewhat more obvious to see, as the following fork is always an  equalizer in � for a groupoid G :  G  G □  G □  G e [s]×I  G I ×e [s]  G m  (where we write G as the object of arrows and Gpd(� )(I × I ,G ) as G □).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Note that G e [s]×I and G I ×e [s]  correspond to the idempotents on G □: q  u  v  w  u  u  q  q  G I ×e [s]  G e [s]×I  Also note that a commuting square is equalized by these two maps if and only if u = q = id , which  forces w = v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' these commuting squares are exactly the image of G m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The semigroup structure on I represents the map sending  Y  X  Y  X  X  X  u  u  u  Now look at the ∂ defined in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4, and take the same colimit in Gpd(� ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Groupoids are  monadic over reflexive graphs with an involution, so this is essentially the free groupoid over that graph  (as the free functor is a left adjoint and therefore preserves colimits).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Define the � -groupoid ∂ to be the � -coequalizer  D × I  D × I  ∂  e ×I  D×e [s]  where e = 0◦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=',e [s] = i ◦ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Note that just as in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4, ∂ is an anchored bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The properties of the arrow groupoid make it possible to prove the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The object ∂ determines a cartesian � -functor  ∂ (−) : Weilop  1  → Gpd(� )  that is both an infinitesimal object in � -groupoids and an involution algebroid in Gpd(� )op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  127  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Write the pushout powers of ξ : 1 → ∂ as ∂ (n), and for any Weil algebra V = Wn(1) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ⊗ Wn(k),  write ∂ (V ) = ∂ (n1)×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='×∂ (nk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Composition for internal categories will be written in the diagramattic  order, writing composition as an infix semicolon “;”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' to keep it distinct from composition of morphisms  in � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The proof has two main steps:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' For every Weil algebra V , there is an isomorphism  ∂V ∼= ∂ (V )  where ∂V is the V -prolongation of the anchored bundle in Gpd(� )op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The universal lift for the anchored bundle, given as a coequalizer  ∂ × ∂  ∂ × ∂  ∂  e ∂ ×∂  e ∂ ×∂  ⊡  is a semigroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  From (1) and (2), we may infer that the object ∂ is an infinitesimal object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The semigroup map ⊡ is  commutative and has a zero by (2), and satisfies all of the couniversality axioms by (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The proof of this step follows by induction on the width (Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8) of the Weil algebra V ,  where the cases 0,1 both hold by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In the case n = 2, we need only check this holds for  ∂ × ∂ ∼= ∂W W :  D × ∂  D  L2  ∂ × ∂  ∂  (id,0)  δ  ⌟  (δ,id)  (id,0)  For any G , a map X : ∂ × ∂ → G corresponds to a commuting square in T 2G of the form T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e ◦u  v  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦v  u  so that T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e ◦ v = id and e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ u = id .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' It follows that u = (e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ v)−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='(T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e ◦ u);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='v (Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5 (i)),  and that precomposition with the uniquely induced map determines ( ¯u,v) : X → G L(2), where  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0◦ ¯u = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Observe that any pair ( ¯u,v) : X → G L(2) determines a square T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0◦ ¯u  v  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦v  128  where T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e ◦ v = id and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e ◦ ¯u = id .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Note that T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0 ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='p ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0 ◦ ¯u = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0 ◦ ¯u, and that the bottom  horn is u = (e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ v)−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='(T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0◦ ¯u);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' now check  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e ◦ u = (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e ◦ e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ v)−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='(T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0◦ ¯u);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='(T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e ◦ v)  = id ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0◦ ¯u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='e ◦ v = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0◦ ¯u  and  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ u = e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ (e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ v)−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='(T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0◦ ¯u);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='v  = (e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ v)−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='(e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0◦ ¯u);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='(e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ v)  = (e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ v)−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='id ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='(e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T ◦ v) = id  thus determining a map X → G ∂ ×∂ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The two maps are inverse to each other, giving an isomor-  phism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  For the inductive case, look at the prolongations of anchored bundles and recall that  AU V ⊠M AZ = AU V ⊠� (U ,A) AU Z  Where AU V and AU Z are treated as spans over AU , so that  AU  id⊠πV  ←−−− AU V  an c U ⊠AV  −−−−−−→ T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='AV  AU  id⊠πZ  ←−−− AU Z  an c U ⊠AZ  −−−−−−→ T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='AZ  and observe that their span composition is  AU V Z  AU V  T V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='AU Z  AU  T V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='AU  T Z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='T V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='AU  ̺U ⊠AV  T U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (̺U ⊠AZ )  ⌟  So it now suffices to prove that the diagram  ∂U × DV  ∂U × (DV × ∂Z )  ∂U × ∂V  ∂U × (∂V × ∂Z )  ⌟  is a pushout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' But by the inductive hypothesis,  DV  DV × ∂Z  ∂V  ∂V × ∂Z  ⌟  so the result follows by the cocontinuity of ∂U × (−), and Gpd(� ) is a cartesian closed category  (so X × (−) is cocontinuous).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  129  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Recall that the fork  I × I  I × I  I  m  e [s]×I  I ×e [s]  is a coequalizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Also note that e ∂ is the coequalizer of e ,e [s].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The commutativity of colimits  then ensures that there is a multiplication map induced by the following diagram:  (D × I )2  (D × I )2  ∂ 2  (D × I )2  (D × I )2  ∂ 2  D × I  D × I  ∂  Thus the multplication is associative, is commutative, and has a zero given by 1  0×s  −→ D × I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Aan immediate corollary of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7 is that the ∂ determines a nerve/ realization where the  nerve factorsthrough the categoryofinvolution algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Thisputsthe Lie functorintoa nerve/realization  context (recall Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The Lie realization is the left Kan extension  |− |∂ = Lan∂ : Inv(� ) → Gpd(� ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  This functor is well behaved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' First, ote that it preserved products:  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The realization functor preserves products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Product preservation is a consequence of |−|∂ being the left Kan extension of a cartesian functor  along a cartesian functor (Day and Street (1995)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Note that this implies  � v ∂ (v) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Next, we see that the realization of an involution algebroid has the same base space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The base space of the groupoid ∂ is D v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' When constructing the colimit of groupoids, the colimit’s base space is the ordinary colimit for  the diagram of the base spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The reflector from simplicial objects to groupoids preserves products,  so it suffices to check that the base space of ∂ (n) is D(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The base space of I is 1+1, and the map e s  0 is given by δ−◦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='. Since D is a discrete cubical object, its  base space is D × 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Thus the coequalizer defining ∂ is  D + D  D + D  ∂0  0◦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='+0◦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  id+0◦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  A map γ : D + D → M is a pair of maps γ0,γ1 : D → M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We can see that γ0 and γ1 agree at 0 (they are  both γ0(0)), and γ(0) is a constant tangent vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' It follows that γ1 ◦ (id |id ) = γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  130  Now recall the co-Yoneda lemma: for any � -presheaf A : �op → � ,  F (C ) =  � C ′∈�  �(C ,C ′) ⊗ F (C ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  In particular, for an involution algebroid in � (or any � -presheaf on Weilop  1 ),  A(U ) =  � V ∈Weil1  Weilop  1 (U ,V ) × A(V ) =  � V ∈Weil1  Weil1(V,U ) × A(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' For any involution algebroid A,  A(R) =  � V ∈Weil1  A(V ) × D V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Use the tangent structure on Weil1 to regard it as a � -enriched category:  D V = � (V ) = Weil1(V,−) = (U �→ Weil1(V,U ))  = (U �→ Weil1(V,U ⊗ R)) = Weil1(V,R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The following computation gives the result:  A(R) =  � V  Weil1(V,R) × A(V ) =  � V  D V × A(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  We can now see that the base space of an involution algebroid is isomorphic to the base space of  its realization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' That is to say, the realization sends an involution algebroid over a Weil space M to a  groupoid over the Weil space M (up to isomorphism).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let A be an involution algebroid in � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Then |A|([0]) = A(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Use the Yoneda lemma, and the fact that � [0] = 1 is a small projective so that Gpd(1,−) is � -  cocontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Now apply Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='10 and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='11:  |A|([0]) = Gpd(1,|A|)  = Gpd  �  1,  � v∈Weil1  A(v) • ∂ v  �  =  � v  A(v) ×Gpd(1,∂ v)  =  � v  A(v) × D v = A(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Thus, as a final result, we have achieved an adjunction between the category of involution alge-  broids and groupoids in � that is product-preserving and stable over the base spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  131  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='13 (The Lie Realization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' There is a tangent adjunction between the category of involution  algebroids and groupoids in � , where each functor preserves products and the base spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Gpd(� )  Inv(� )  N∂  |−|∂  ⊣  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Just as the introduction to this thesis begins with the work of Charles Ehresmann, we  should take a moment to see how the Lie realization relates to his original research into sketch theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Rather than sketches, we use their closely related cousins, essentially algebraic theories, which in this  case are small, finitely complete � -categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We may regard Gpd(� ) and Inv(� ) as � -functor cate-  gories  Lex(�Gpd,� ),  Inv(�Gpd,� ),  respectively (where Lex means finite-limit-preserving � -functors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The functor ∂ : Weilop  1  → Gpd(� )  induces a left-exact � -functor  ˆ∂ : �Inv → �Gpd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  This means the functor from Lie groupoids to Lie algebroids is induced by a morphisms of essentially  algebraic theories, and may thus be presented as a morphism of sketches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  132  Chapter 6  Conclusions and future work  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1  Conclusions  We now take stock of what this thesis has accomplished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Enrichedessentiallyalgebraicpresentationsofgeometricstructures  Thisthesishasbeen concerned  with categories of vector bundles and Lie algebroids in smooth manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' While these are not models  of a limit sketch in the category of smooth manifolds, classical results such as the Serre–Swann theo-  rem or Vaintrob’s presentation of Lie algebroids indicate that these categories do have some algebraic  description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' By combining the results in Chapters 2 and 3, we can see that vector bundles and Lie  algebroids are characterized by tangent categorical gadgets, differential bundles (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1) and  involution algebroids (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' By combining these observations with the enriched perspec-  tive on tangent categories, the results in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2 exhibit vector bundles and Lie algebroids as models  of enriched sketches (Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6 and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='14), as can also be found in Chapter 6 of Kelly  (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Note that from this perspective, the construction of the Weil nerve of an involution algebroid  is mostly important as an intermediate step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  New tangent structures for Lie algebroids and groupoids  It is well known that the categories of  Lie algebroids and groupoids have tangent structures induced by post-composition with the tangent  functor, yielding the tangent algebroid and tangent groupoid, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The key result in Chapter  4 exhibiting the category of Lie algebroids as transverse-limit-preserving cartesian-tangent functors  Weil1 → SMan, then, gives the category of Lie algebroids a second tangent structure that corresponds  to Martinez’s prolongation Lie algebroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Similarly, the construction of the infinitesimal nerve of a lo-  cal groupoid restricts to a new tangent structure on the category of Lie groupoids;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' in this case, the  base space is the Lie algebroid of the groupoid G , and the total space is the Lie algebroid of the arrow  groupoid G 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  While this may seem like a piece of formal category theory, it is closely related to symmetry re-  duction in classical mechanics using Lie theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This goes back to Poincaré’s celebrated note Poincaré  (1901), which introduced the Euler-Poincare formulation ofLagrangian mechanicson a manifold equipped  with a Lie group action, where the tangent bundle is replaced with a Lie algebra action (see Marle  (2013) for a modern exposition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Poincaré’s formalism has been extended to Lie groupoids and Lie  algebroids in Weinstein (1996) and Martínez (2001), and the thesis Fusca (2018) investigates fluid me-  chanics through this lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This work may be interpreted as using the novel tangent structures on Lie  groupoids and Lie algebroids given in Chapters 4 and 5, and warrants further investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  133  Functorial semantics of Lie theory  The work in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5 puts the Lie derivative on a new formal  grounding:  (i) In the enriching category � , the Lie derivative functor now arises as a nerve/realization context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  This guarantees the existence of a left adjoint—the realization—that constructs a Lie groupoid  from an algebroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The realization preserves products and is stable over the base space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) It is only a small extension of the work in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5 to show that the Lie derivative of a groupoid  in a tangent category may generally be regarded as a weighted limit, where AV = {∂ (V ),G } for  each Weil prolongation of the involution algebroid of the groupoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  These appear to be new results, although in a nerve/realization approach they had previously appeared  in Lie theory in the form of Sullivan’s construction (Sullivan (1977)), which takes the differential graded  algebra of a Lie algebroid and constructs a simplicial set:  DGAop(Ω(∆n),A)  where Ω(∆n) is the de Rham cohomology of the n-simplex in cartesian space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2  Future work  This section outlines various lines of research that were either cut from the thesis while writing due to  time and/or space constraints, or new lines of research that the thesis-writing process has motivated  but which have not yet been pursued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Enriched sketches and Mackenzie theory  Following Voronov (2012), Mackenzie theory refers to the  body of research developed by Kirill Mackenzie and his collaborators into Lie theory, particularly struc-  tures like double Lie algebroids (Mackenzie (1992)), VB-Lie algebroids (Bursztyn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (2016)), double  Lie groupoids and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Intuitively, a double Lie algebroid is a Lie algebroid in the category of Lie alge-  broids, while a VB-Lie algebroid is a vector bundle in the category of Lie algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' These structures  have an intuitive relationship with tensor products of sketches;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' that is, for limit sketches A,B there is  a chain of isomorphisms of categories:  Mod(A,Mod(B,Set)) ∼= Mod(A ⊗ B,Set) ∼= Mod(B,Mod(A,Set)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  We have already demonstrated that the Lie algebroids and vector bundles are � -sketchable, so the  natural next step is to revisit Mackenzie theory via this lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Certain results such as the symmetry of  partial Lie derivatives from Mackenzie (1992) (given a double Lie groupoid, the order in which the Lie  functor is applied doesn’t matter) should be immediate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  A tangent categorical formalization of mechanics  Several papers in synthetic differential geometry  have translated aspectsofLagrangian and Hamiltonian mechanicsintothe syntheticsetting(Bunge and Heggie  (1984);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Nishimura (1997)), and to a degree this has made it difficult to build enthusiasm for a tangent  categorical presentation of mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' These approaches generally emphasize the ability to construct  function spaces, or the use of a topos-theoretic internal language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The novel tangent structures for  Lie algebroids and groupoids presented here, however, provide a new line of inquiry: to develop a uni-  fied framework for Lagrangian or Hamiltonian mechanics in a tangent category that agrees with the  algebroid and groupoid approaches to mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  134  Appendix: Background on locally  presentable � -categories  Synthetic differential geometry uses a topos-theoretic framework, such as can be found in the final  chapters of Lavendhomme (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The internal language of a topos is a key tool, and models of syn-  thetic differential geometry are constructed using sheaf constructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Tangent categories have an  analagous toolbox, drawing from enriched locally presentable categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' We begin with locally pre-  sentable categories;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' basic material on these may be found in Adámek and Rosický (1994), and on en-  riched categories in Kelly (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The following class of colimits is foundational in the theory of locally  presentable categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A category is filtered whenever any finite diagram in � has a cocone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A colimit whose  diagram category is filtered is called a filtered colimit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Note that a category � is filtered if and only if  every colimit diagram � → Set commutes with every finite limit in Set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Example .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) Any finite category � with a terminal object is filtered: for any diagram D : � → � , the natural  transformation !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' : id ⇒ K1 is a cocone via whiskering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  �  �  �  D  K1  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) Any category with finite colimits is filtered, because each diagram has a colimit (and therefore a  cocone).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' An object in a cocomplete � is finitely presentable whenever �(C ,−) preserves filtered  colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A cocomplete category � is locally finitely presentable whenever it has a small subcategory  �f p of finitely presentable objects so that every object in � is given by the coend  C ∼=  � X ∈�f p  �(X ,C ) · X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Example .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In the category Set, the subcategory of finitely presentable objects is precisely the category  of finite sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Every set is a filtered colimit of finite sets, so that Set is locally finitely presentable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Intuitively, this means that every object in � is generated by gluing together finitely presentable  objects in a canonical way;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' hence the name locally finitely presentable categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The subcategory of finitely presentable objects in a cocomplete category is itself finitely cocom-  plete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This follows from the fact that �(−,X ) : �op → Set is a continuous functor (it sends colimits in �  135  to limits in Set) and also that filtered colimits commute with finite limits in Set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Thus, for any filtered  colimit of representables,  �(colD,limF ) ∼= limX limY �(D(X ),F (Y )) ∼= limY limX �(D(X ),F (Y )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  There are several ways to characterize locally finitely presentable categories;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' these may be found in  Adámek and Rosický (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' For a category � the following are equivalent:  (i) � is a locally finitely presentable category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) For a small cocomplete category � , � = Lex(� ,Set) (where Lex means finite-limit-preserving) and  � = �op  f p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iii) � is the category of models for some limit sketch (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' a small category � equipped with a class of  cones � , where a model is a functor into Set sending cones to limit diagrams).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iv) � is a reflective subcategory of a presheaf topos for some small category � (this holds tautologically  for presheaf topoi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (v) � is a full subcategory of a locally presentable category that is closed under limits (this relies on a  large cardinal axiom known as Vopenka’s principle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  One of the remarkable aspects of the theory of locally finitely presentable categories is that its re-  sults translate over to enrichment in a monoidal category � without any significant changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let � be a monoidal category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A � -graph � is given by a (large) set of objects �0, and a  map  � : �0 × �0 → � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Given a collection of maps  mABC : �(B,C ) × �(A,B) → �(A,C ),  jA : I → �(A,A)  we say � is a � -category whenever the following associativity and unitality diagrams commute:  �(B,D) ⊗ �(A,B)  (�(C ,D) ⊗ �(B,C )) ⊗ �(A,B)  �(A,D )  �(C ,D) ⊗ (�(B,C ) ⊗ �(A,B))  �(C ,D ) ⊗ �(A,C )  m  m⊗id  m  α  id⊗m  I ⊗ �(B,C )  �(B,B) ⊗ �(B,C )  �(B,C )  �(B,C )  �(B,C ) ⊗ I  �(B,C ) ⊗ �(C ,C )  λ  j ⊗id  m  ρ  id⊗j  m  136  Examples of enrichment abound throughout category theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Example .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) Any category with biproducts is enriched in the category of commutative monoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) A symmetric monoidal closed category is self-enriched using the internal hom [−,−].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iii) A 2-category is a category enriched in Cat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iv) The unit � category is the 1-object � category whose hom-object is I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (v) When � is symmetric monoidal, we can construct the opposite � -category of � , where �op (A,B) =  �(B,A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The new composition is defined by  mop : �op (B,C ) ⊗ �op(A,B)  σ  −→ �(B,A) ⊗ �(C ,B)  m  −→ �(C ,A) = �op (A,C )  There is a 2-category of � -categories for any � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A � -functor F : � → � is an assignment on objects F0 : �0 → �0 and a collection of  maps FA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='B : �(A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='B) → �(F0A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F0B) so that the following diagrams are satisfied:  �(B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='C ) ⊗ �(A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='B)  �(F0B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F0C ) ⊗ �(F0A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F0B)  I  �(A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='A)  �(A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='C )  �(F0A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F0C )  �(F0A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F0A)  m C  F ⊗F  m D  F  F  j C  A  j D  F0A  A � -natural transformation F ⇒ G is a collection of maps γA : I → �(F0A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='G0A) so that the following  diagram commutes:  �(F0B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F0C )  �(F0B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F0C ) ⊗ I  �(F0B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F0C ) ⊗ �(G0B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F0B)  �(B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='C )  �(G0B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F0C )  �(G0B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='G0C )  I ⊗ �(G0B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='G0C )  �(G0B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='F0B) ⊗ �(G0B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='G0C )  G  F  ρ  id⊗γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='B  m  λ  γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='C ⊗id  m  � Cat is the 2-category of � -categories, � -functors and � -natural transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Enriched categories have a richer notion of (co)limit than ordinary categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A limit for a functor  is defined as a universal cone;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' that is, a right Kan extension along !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' :  1  �  �  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  KA  F  β  so that the universal property  ∀C : �(C ,limF ) ∼= [�,Set](KC ,F −)  137  is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Dually, a colimit is a universal cocone, and so a left Kan extension:  �  �  1  F  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  KA  β  where the couniversal property  ∀C : [�op ,Set](F −,KC ) ∼= �(colimF,C )  is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This does not translate over to � -categories, for a few reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' An obvious one is that �  need not be cartesian monoidal, so there is no reason a � -category should have a functor � → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The  notion of a weighted colimit is more appropriate:  Definition .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Let � be a � -category, and call W : � → � the weight and D : � → � the diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The  limit of D weighted by W , or the weighted limit {W,�}, is an object satisfying the following universal  property:  ∀C : � (W,[�,� ](F −,KC )) ∼= �(F,W ,C ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Dually, given a diagram P : �op → �, the colimit of P weighted by W , W ∗P, is an object in � satisfying  the universal property  ∀C : � (W,[�,� ](KC ,F −)) ∼= �(C ,F ∗ W ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Example .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) A conical (co)limit is weighted by the constant functor into the unit I , KI : � → w .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' This coincides  with the definition of ordinary (co)limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) The power of C by an object V ∈ � is an object C V satisfying the universal property that  �(B,C V ) ∼= � (V,�(B,C )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Dually, the copower by V is given by  �(B • V,C ) ∼= � (V,�(B,C )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iii) In ordinary categories, the power of an object C by V is given by the (possibly infinitary) V -indexed  coproduct:  C V ∼=  �  v∈V  C  whereas the copower is the possibly infinitary V -index product of C :  V • C ∼=  �  v∈V  C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iv) In a symmetric monoidal closed category � , the power by � is given by [V,−] and the copower by  V ⊗ _.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Following Kelly’s work, we observe that every finite weighted limit is composed of powers and con-  ical limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Every weighted (co)limit is the composition of conical (co)limits and (co)powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  138  Corollary .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Weighted limits are exactly conical/ordinary limits in ordinary categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  The previous definitions relating to locally presentable categories transfer over mostly unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Definition .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (i) A monoidal closed � is presentable as a closed category if �f p is a monoidal subcategory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (ii) Given � as in (i), a weight W : � → � is filtered whenever W ⋆ (−) : �  � → � preserves finite limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iii) An object in a cocomplete � category is finitelypresentable when �(C ,−) preservesfiltered weighted  colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  (iv) A cocomplete � -category � is locally finitely presentable whenever every object C is the coend  C ∼=  � X ∈�f p  �(X ,C ) · X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Proposition .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='14 (Freyd and Kelly (1972);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Kelly (2005)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' All of the previous results about locally pre-  sentable categories lift up to locally presentable � -categories, provided that � is presentable as a closed  category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  139  Bibliography  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Adámek and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Rosický.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Locally presentable and accessible categories, volume 189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Cambridge: Cam-  bridge University Press, 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ISBN 0-521-42261-2/pbk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Adámek, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Borceux, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Lack, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Rosický.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A classification of accessible categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Pure Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Algebra, 175(1):7–30, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ISSN 0022-4049.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' doi: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1016/S0022-4049(02)00126-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  URL https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='sciencedirect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='com/science/article/pii/S0022404902001263.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Spe-  cial Volume celebrating the 70th birthday of Professor Max Kelly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Adámek, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Gumm, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Trnková.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Presentation of set functors: a coalgebraic perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Logic  Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=', 20(5):991–1015, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ISSN 0955-792X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1093/logcom/exn090.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Barr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Acyclic models, volume 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' American Mathematical Society (AMS), Providence, RI, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ISBN  0-8218-2877-0/hbk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Bauer, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Johnson, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Osborne, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Riehl, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Tebbe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Directional derivatives and higher-order chain  rules for abelian functor calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Topology Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=', 235, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Berger, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Melliès, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Weber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Monads with arities and their associated theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Pure Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Algebra, 216(8-9):2029–2048, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ISSN 0022-4049.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='jpaa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='039.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Bertram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Weil spaces and weil-lie groups, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' URL https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='org/abs/1402.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2619v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='GR](unpublished preprint).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Bertram and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Souvay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A general construction of Weil functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Cahiers Topologie Géom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Différen-  tielle Catég.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=', 55(4):267–313, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ISSN 1245-530X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Blute, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Cockett, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Seely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Differential categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Cordero, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Dodson, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' Differential geometry of frame bundles, volume 47 of  Mathematics and its Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Kluwer Academic Publishers Group, Dordrecht, 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' Courant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Tangent Lie algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' A, 27(13):4527–4536, 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' Crainic and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Fernandes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Integrability of Lie brackets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=', 157(2):575–620, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ISSN  0003-486X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Day and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=', 1(4):72–77, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' A, 38(24):R241–R308, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=', 38(24):R241, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' do Carmo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Riemannian geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Mathematics:  Theory & Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Boston, MA:  Birkhäuser Boston, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=', 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Dubuc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' C ∞-schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=', 103(4):683–690, 1981.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  ISSN 0002-9327.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='2307/  2374046.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Eck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Product-preserving functors on smooth manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Pure Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Algebra, 42(2):133–140, 1986.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  ISSN 0022-4049.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' Catégories topologiques et catégories différentiables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Librairie universitaire, 1959.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' On Köthe sequence spaces and linear logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' Structures Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=', 12(5):579–623,  2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Ehrhard and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Regnier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The differential lambda-calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Theoretical Computer Science, 309(1–3):  1–41, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Eilenberg and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=', 58(2):  231–294, 1945.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Magdalini K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Mackenzie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Warps and duality for double vector bundles, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (unpublished  preprint).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Freyd and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Kelly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Categories of continuous functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Pure Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Algebra, 2:169–191, 1972.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  ISSN 0022-4049.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' A groupoid approach to geometric mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' PhD thesis, University of Toronto, 2018, 116pp,  2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Gabriel and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Ulmer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Lokal präsentierbare kategorien, volume 221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Springer-Verlag, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Garner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' An embedding theorem for tangent categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=', 323:668–687, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ISSN 0001-  8708.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='aim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='039.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  142  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Girard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Linear logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=', 50(1):1–101, 1987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Grabowski and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Rotkiewicz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Higher vector bundles and multi-graded symplectic manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=', 59(9):1285–1305, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ISSN 0393-0440.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='geomphys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Higgins and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Mackenzie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Algebraic constructions in the category of Lie algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Algebra, 129  (1):194–230, 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ISSN 0021-8693.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1016/0021-8693(90)90246-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Hoffmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A hitchhiker’s guide to automatic differentiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Numer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Algorithms, 72(3):775–811,  2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Janelidze and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Kelly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A note on actions of a monoidal category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Theory Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Categ, 9(61–91):  02, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Johnson and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' McCarthy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Linearization, Dold–Puppe stabilization, and MacLane’s Q-  construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=', 350(4):1555–1593, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  ISSN 0002-9947.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1090/  S0002-9947-98-02067-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Johnstone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Sketches of an elephant: a topos theory compendium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' 1, volume 43 of Oxford Logic  Guides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' New York: The Clarendon Press, Oxford University Press, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ISBN 0-19-853425-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Kan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Functors involving c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=', 87:330–346, 1958.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ISSN  0002-9947;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' 1088-6850/e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Kapranov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Free Lie algebroids and the space of paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Sel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' New Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=', 13(2):277–319, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ISSN  1022-1824;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' 1420-9020/e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Kelly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' On the essentially algebraic theory generated by a sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Aust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=', 26(1):  45–56, 1982.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Kelly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Basic concepts of enriched category theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Repr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Theory Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Categ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=', 2005(10):1–136, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Kock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Synthetic differential geometry, volume 333 of London Mathematical Society Lecture Note Se-  ries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Cambridge University Press, Cambridge, second edition, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ISBN 978-0-521-68738-6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' 0-521-  68738-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1017/CBO9780511550812.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Kolár.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Functorial prolongations of Lie groupoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Banach Cent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=', 76:177, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Kolár, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Michor, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Slovák.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Natural operations in differential geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Berlin: Springer-Verlag,  1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ISBN 3-540-56235-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1007/978-3-662-02950-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Kriegl and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Michor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The convenient setting of global analysis, volume 53 of Mathematical Surveys  and Monographs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' American Mathematical Society, Providence, RI, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ISBN 0-8218-0780-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' doi:  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1090/surv/053.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Lack and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Gabriel–Ulmer duality and Lawvere theories enriched over a general base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=', 19(3–4):265–286, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ISSN 0956-7968.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1017/S0956796809007254.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Lavendhomme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Basic concepts of synthetic differential geometry, volume 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Dordrecht: Kluwer  Academic Publishers, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ISBN 0-7923-3941-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Lawvere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Categorical dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In Topos theoretic methods in geometry, volume 30 of Various  Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=', pages 1–28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Aarhus: Aarhus University, 1979.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  143  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Lawvere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Qualitative distinctions between some toposes of generalized graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In Categories in  computer science and logic (Boulder, CO, 1987), volume 92 of Contemp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=', pages 261–299.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=', Providence, RI, 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1090/conm/092/1003203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Leung.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Classifying tangent structures using Weil algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Theory Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Categ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=', 32(9):286–337, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Lie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Theorie der transformationsgruppen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=': abschnitt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Die endlichen continuirlichen gruppen der  geraden linie und der ebene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Die endlichen continuirlichen gruppen des gewöhnlichen raumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Die projectiven gruppen des gewöhnlichen raumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Untersuchungen über verschiedene arten von  gruppen des n-fach ausgedehnten raumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Untersuchungen über die grundlagen der geometrie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Allgemeine betrachtungen über endliche continuirliche gruppen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' 1893, volume 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' BG Teubner, 1893.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Fosco Loregian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' (Co)end Calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' London Mathematical Society Lecture Note Series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Cambridge Uni-  versity Press, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1017/9781108778657.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Lucyshyn-Wright.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' On the geometric notion of connection and its expression in tangent cate-  gories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Theory Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Categ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=', 33(28):832–866, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' MacAdam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Vector bundles and differential bundles in the category of smooth manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=', 94(2):180–239,  1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ISSN 0001-8708.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' ISBN 978-0-521-  49928-3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' Ehresmann doubles and Drinfel’d doubles for Lie algebroids and Lie bialgebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=', 658:193–245, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Integration of Lie bialgebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Topology, 39(3):445–467,  2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  ISSN 0040-9383.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Mackenzie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Proving the Jacobi identity the hard way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In Geometric methods in physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' XXX  workshop, Poland, June 26 – July 2, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Selected papers based on the presentations at the workshop,  pages 357–366.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Basel: Birkhauser, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ISBN 978-3-0348-0447-9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' 978-3-0348-0448-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1007/  978-3-0348-0448-6_32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' MacLane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Categories for the working mathematician.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' 4th corrected printing, volume 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' New York:  Springer-Verlag, 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ISBN 3-540-90035-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Marle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' On Henri Poincaré’s note “sur une forme nouvelle des équations de la mécanique”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Symmetry Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=', 29:1–38, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Martínez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Geometric formulation of mechanics on Lie algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In Proceedings of the VIII Fall  Workshop on Geometry and Physics (Spanish) (Medina del Campo, 1999), volume 2 of Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Esp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=', pages 209–222.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Esp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=', Madrid, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' Classical field theory on Lie algebroids: multisymplectic formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=', 10  (1):93–138, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ISSN 1941-4889;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='  144  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Meinrenken and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Pike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The Weil algebra of a double Lie algebroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=', 2021(11):  8550–8622, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ISSN 1073-7928.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1093/imrn/rnz361.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Michor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' The Jacobi flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Rend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Semin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=', Torino, 54(4):365–372, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ISSN 0373-1243.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Milnor and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Stasheff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Characteristic classes, volume 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Princeton, NJ: Princeton University  Press, 1974.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1515/9781400881826.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Moerdijk and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Mrˇcun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' On integrability of infinitesimal actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=', 124(3):567–593,  2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Moerdijk and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Reyes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Models for smooth infinitesimal analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' New York: Springer-Verlag, 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  ISBN 0-387-97489-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Nestruev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Smooth manifolds and observables, volume 220.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' New York: Springer, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ISBN 0-387-  95543-7/hbk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Nishimura.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Synthetic Hamiltonian mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Internat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Theoret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=', 36(1):259–279, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ISSN  0020-7748.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1007/BF02435785.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Nishimura and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Osoekawa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' General Jacobi identity revisited again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Internat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Theoret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=', 46  (11):2843–2862, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ISSN 0020-7748.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='1007/s10773-007-9397-z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Nistor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Groupoids and the integration of Lie algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Jpn, 52(4):847–868, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Poincaré.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Sur une forme nouvelle des équations de la mécanique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' CR Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Sci, 132:369–371, 1901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Pradines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Théorie de Lie pour les groupoïdes différentiables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Calcul différenetiel dans la catégorie  des groupoïdes infinitésimaux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Paris Sér.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A-B, 264:A245–A248, 1967.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ISSN 0151-0509.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Rosický.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Abstract tangent functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Diagrammes, 12:JR1–JR11, 1984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ISSN 0224-3911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Seal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Tensors, monads and actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Theory Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Categ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=', 28:403–434, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ISSN 1201-561X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Segal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Categories and cohomology theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Topology, 13:293–312, 1974.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ISSN 0040-9383.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Selinger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' A survey of graphical languages for monoidal categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' In New structures for physics, pages  289–355.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Springer, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  D Sullivan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Infinitesimal computations in topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' Mathématiques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=', 47:269–331, 1977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' ISSN  0073-8301;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content=' 1618-1913/e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
+page_content='  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='  A THESIS  SUBMITTED TO THE FACULTY OF GRADUATE STUDIES  IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE  DEGREE OF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+page_content='  Abstract  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfdffn/content/2301.00305v1.pdf'}
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+XXX-X-XXXX-XXXX-X/XX/$XX.00 ©20XX IEEE 
+FIPS Compliant Quantum Secure Communication 
+using Quantum Permutation Pad  
+Abstract—Quantum computing has entered fast development 
+track since Shor’s algorithm was proposed in 1994. Multi-cloud 
+services of quantum computing farms are currently available. One 
+of which, IBM quantum computing, presented a road map 
+showing their Kookaburra system with over 4158 qubits will be 
+available in 2025. For the standardization of Post-Quantum 
+Cryptography or PQC, the National Institute of Standards and 
+Technology or NIST recently announced the first candidates for 
+standardization with one algorithm for key encapsulation 
+mechanism (KEM), Kyber, and three algorithms for digital 
+signatures. NIST has also issued a new call for quantum-safe 
+digital signature algorithms due June 1, 2023. This timeline shows 
+that FIPS-certified quantum-safe TLS protocol would take a 
+predictably long time. However, ‘steal now, crack later’ tactic 
+requires protecting data against future quantum threat actors 
+today. NIST recommended the use of a hybrid mode of TLS 1.3 
+with its extensions to support PQC. The hybrid mode works for 
+certain cases but FIPS certification for the hybridized 
+cryptomodule might still be required. This paper proposes to take 
+a nested mode to enable TLS 1.3 protocol with quantum-safe data, 
+which can be made available today and is FIPS compliant. We 
+discussed the performance impacts of the handshaking phase of 
+the nested TLS 1.3 with PQC and the symmetric encryption phase. 
+The major impact on performance using the nested mode is in the 
+data symmetric encryption with AES. To overcome this 
+performance reduction, we suggest using quantum encryption 
+with a quantum permutation pad for the data encryption with a 
+minor performance reduction of less than 10%. 
+Keywords—quantum communication, quantum encryption, 
+quantum decryption, quantum security, secure communication, 
+QPP, FIPS, TLS 1.3.  
+I. INTRODUCTION  
+Peter Shor proposed his celebrated quantum algorithm in 
+1994 [1], which solves the NP-hard problem of prime integer 
+factorization in polynomial time.  At its beginning, quantum 
+computing, 
+especially 
+universal 
+gate-based 
+quantum 
+computing, experienced a slow development phase for about 
+two decades. In 2019, Arute et al. from Google claimed 
+Quantum Supremacy with their 53-qubits Sycamore processor 
+[2]. This marked the start of global quantum computing race. 
+Since IBM released their 5-qubit quantum computer for public 
+access with Qiskit tool in 2017, IBM recently announced their 
+433-qubit quantum computer and plan to double its qubits every 
+year for 2023 to reach over 4,000-qubits by 2025, outlined in 
+their development roadmap [3].  
+The fundamental shift from classical computing to quantum 
+computing is the shift in computing algebra from the classical 
+Boolean algebra to linear algebra, used in quantum computing. 
+That is, from classical logic gates, implemented in CPU, to 
+quantum logic gates to be implemented in QPU. That means, 
+quantum computing indicates a quadratic speedup of classical 
+computers or �2��2, where � denotes the number of qubits of a 
+QPU. This exponential computing power would break today’s 
+RSA-2048 in 10 seconds if a fault tolerant quantum computer 
+reaches 4099 qubits, while a classical computer would take 300 
+trillion years. Mosca predicted that there is a “1/2 chance of 
+breaking RSA-2048 by 2031” [4].  The year from quantum 
+computing to break classical public key RSA has been called 
+the Year to Quantum threat or Y2Q.  
+Symmetric cryptography such as the well-known Advanced 
+Encryption Standard or AES also suffered quadratic speedup of 
+the best quantum attack on the key space using Grover’s 
+algorithm proposed by Grover in 1996 [5]. That requires the 
+key length to be doubled in comparison with the equivalent 
+classical security level. For example, the classical AES-128 
+would be replaced by AES-256 for quantum security.  
+It has been well-understood that the upcoming quantum 
+computing systems will destroy the foundation of classical 
+public key infrastructure or PKI for both key establishment 
+such as RSA, Diffie-Hellman or elliptical curve Diffie-Hellman 
+and digital signature such as Digital Signature Algorithm or 
+DSA. National Institute of Standards and Technology or NIST 
+has announced its standardization process of Post-Quantum 
+Cryptography or PQC in November of 2017. In July 2022 [6], 
+NIST announced the lattice-based Kyber [7] to be its first 
+standardized key encapsulation mechanism or KEM. And 
+lattice-based Dillithium [8] Falcon [9], as well as hash-based 
+SPHINCS+ [10] to be first standardized digital signature 
+schemes. Other KEM candidates BIKE [11], Classic McEliece 
+[12], HQC [13], and SIKE [14] moved to the 4th round to be 
+considered further. NIST has also announced its reopening 
+Alex He  
+Quantropi Inc. 
+Ottawa, Canada 
+alex.he@quantropi.com 
+Dafu Lou 
+Quantropi Inc. 
+Ottawa, Canada 
+dafu.lou@quantropi.com 
+Eric She 
+DLS Technology Corporation  
+Ottawa, Canada 
+eshe@dlstech.com 
+Shangjie Guo 
+FinQ Tech Inc.  
+College Park, Maryland 
+sguo@finq.tech 
+Hareesh Watson   
+DLS Technology Corporation  
+Ottawa, Canada 
+hwatson@dlstech.com 
+Sibyl Weng   
+DLS Technology Corporation  
+Ottawa, Canada 
+sweng@dlstech.com 
+Maria Perepechaenko 
+Quantropi Inc. 
+Ottawa, Canada 
+maria.perepechaenko@quantropi.com 
+ 
+Randy Kuang 
+Quantropi Inc. 
+Ottawa, Canada 
+randy.kuang@quantropi.com 
+ORCID: 000-0002-5567-2192-
+0002-5567-2192 
+OORCID: RCID iD: 0000-0002-
+
+submissions for digital signature standardization due Jaune 
+2023 [15].   
+In 2022 some PQC algorithms such as Rainbow (a digital 
+signature scheme based on Multivariate Public Key 
+Cryptography or MPKC) [16] and Supersingular Isogeny 
+Diffie-Hellman protocol or SIDH [17, 18] proved to be 
+vulnerable to classical attacks. Another very interesting 
+cryptoanalysis was reported in the late 2022 by Wenger et al. 
+[19]. The team used transformers (a deep learning model) to 
+develop an attack on certain lattice-based schemes. They 
+noticed that the basic equation system for Learning With Error 
+or LWE can be expressed as linear regression used in machine 
+learning. If trained transformers, especially combined with 
+quantum computational advantage, can solve the Short Vector 
+Problem, then PQC algorithms face an enormous challenge.  
+Recall that most of the PQC standard schemes are lattice-based. 
+This issue is especially concerning since estimation on Y2Q has 
+not taken the power of quantum machine learning into account. 
+Some recent novel PQC algorithms were proposed by 
+Kuang, Perepechaenko, and Barbeau for KEM [20, 21] and 
+digital signature [22, 23, 24, 25], based on NP-complete 
+Modular Diophantine Equation Problem or MDEP. These 
+novel schemes share a foundation which we call Multivariate 
+Polynomial Public Key or MPPK. MPPK is built on two vector 
+spaces: a linear multivariate vector space { ��, … , �� } 
+containing noise variables used for obscurity and polynomial 
+vector space { 1, ��, … , ���� } containing secret message 
+variable.  MPPK offers small parameter sizes at the level of 
+hundreds of bytes for key, ciphertext, and signature and 
+outperforms NIST finalists in key generation, encryption, 
+decryption, as well as signing and signature verification. They 
+could be considered as generic PQC algorithms for a wide range 
+of devices and systems.  
+NIST recommended using a hybrid scheme for quantum 
+resistant TLS 1.3 [26]. By leveraging the keyShare extension of 
+TLS 1.3, one can combine a NIST-approved classical key 
+establishment algorithm such as ECCDH in TLS 1.3 with one 
+or more PQC KEM algorithms and a NIST-approved digital 
+signature algorithm such as DSA with a list of PQC digital 
+signature algorithms for a chain signatures.  This hybrid scheme 
+offers crypto agility as the ongoing process of PQC 
+standardization and cryptanalysis. However, this hybrid 
+scheme brings two potential limitations: 1) the hybrid TLS 1.3 
+may still require FIPS certification although its core 
+cryptomodule is certified in classical TLS 1.3. In general, the 
+FIPS certification comes at a cost for both time and money, 
+although it would be possible to receive the certificate; 2) The 
+certified hybrid TLS 1.3 must be integrated with an application 
+for a quantum resistant service. But in some cases, this 
+integration may be difficult for some applications running 
+inside web browsers.   
+This paper proposes a new hybrid scheme by nesting PQC 
+inside classical TLS 1.3, creating nested TLS 1.3, to overcome 
+the limitations in the above hybrid scheme. The nested TLS 1.3 
+does not require a new FIPS certification because it does not 
+change the existing certified TLS. Moreover, it supports TLS 
+1.3 as well as any certified TLS such as TLS 1.2 or even earlier. 
+One drawback of the nested scheme is that it may reduce the 
+performance of the encryption and decryption operations 
+because the transmitted data would be encrypted twice if AES 
+is used for both encryptions. To minimize the performance 
+impact, we propose to use quantum encryption with a Quantum 
+Permutation Pad algorithm or QPP [27, 28]. The QPP will be 
+used to encrypt the raw data first, producing a quantum-
+encrypted message used as a TLS 1.3 message to be encrypted 
+with AES.    
+II. FIPS COMPLIANT QUANTUM ENCRYPTION WITH QPP 
+In this section, we first introduce the concept of a nested TLS 
+1.3 protocol, that uses quantum-safe cryptosystems. The nested 
+TLS 1.3 allows for a smooth transition from classical era to 
+quantum era while maintaining FIPS compliance. We then 
+discuss the nested TLS 1.3 handshake process. Next, we 
+consider the symmetric data encryption with QPP for 
+considerations of both performance and security.  
+ 
+A. Nested TLS 1.3 with PQC in TLS Handshaking Proccess 
+The proposed nested TLS 1.3 is illustrated in Fig. 1. The 
+figure illustrates an Open Systems’ Interconnection model 
+(OSI-model) consisting of 7 layers: physical, data link, internet, 
+transport, 
+session, 
+presentation, 
+and 
+application. 
+The 
+functionality of each layer is illustrated in [29]. On the right-
+hand side, we mark the corresponding layers in TCP/IP model.  
+The existing TLS cryptomodule is in the transport layer, while 
+the nested TLS is in the application layer. White arrows denote 
+a FIPS certified TLS 1.3 for clientHello request from the client 
+and serverHello response from the server to establish a shared 
+session for session data encryption and decryption. The NIST-
+Approved key agreement protocol, ECCDH, is used for forward 
+secrecy in the existing TLS. The RSA algorithm is excluded 
+from the key agreement protocol. RSA, DSA, and ECDSA are 
+paired with hash functions for digital signature in the existing 
+TLS.  The current conventional FIPS certified TLS 
+cryptomodule will be vulnerable to quantum computing attacks 
+once the fault tolerable quantum computers are available. 
+However, the "steal now crack later" tactics are already in use, 
+meaning all encrypted data today is at risk. Immediate action is 
+imperative to protect data against future quantum threat actors. 
+If sensitive information requires to remain secret for over 10 
+years, then it would not be wise to wait for the FIPS certified 
+TLS 1.3 with quantum resistance. 
+The proposed nested TLS 1.3 with PQC cryptographic 
+modules is independent from the FIPS certified TLS 
+cryptomodule in a sense that packets from the nested TLS 1.3 
+become data packets for the FIPS certified TLS. Since the outer 
+classical TLS cryptomodule is not altered, the nested TLS 1.3 
+does not violate the FIPS certification. This solution can be 
+considered as a promising FIPS compliant TLS 1.3 for quantum 
+security. The nested TLS 1.3 can be used to turn “steal now, 
+crack later” into “steal now, safe forever”. 
+Nested TLS 1.3 is based on the Open Quantum Safe or OQS 
+OpenSSL to support PQC KEM algorithms such as the NIST 
+finalists Kyber and Saber, as well as MPPK [20] and 
+Homomorphic Polynomial Public Key further evolved from 
+MPPK [30]. For the digital signatures, nested TLS 1.3 supports 
+PQC digital signature algorithms such as NIST finalists Falcon, 
+
+Dilithium, Rainbow, as well as MPPK/DS to be submitted to 
+NIST for standardization in 2023 [22].  
+ 
+B. Performance of a Nested TLS 1.3 Handshake 
+We tested the performance of a nested TLS1.3 handshake 
+in a local machine on a 16-core Intel®Core™ i7-10700 CPU at 
+2.90 GHz system for all the measured primitives. Fig. 2 
+illustrates the performance of TLS 1.3 handShake for each pair 
+of KEM and digital signature schemes in terms of NIST 
+security levels. In general, Rainbow digital signature 
+demonstrates the worst performances for TLS handshake with 
+about 500 connections/second at all NIST security level I, 65 
+connections/second at all NIST security level III, and 30 
+connections/second at all NIST security level V.  
+ 
+By pairing MPPK/DS with Saber, Kyber, MPPK KEM, and 
+HPPK, the MPPK/DS scheme outperforms digital signature 
+schemes Falcon and Dilithium over 30% at security level I, over 
+35% at security level III, and 40% at security level V, 
+respectively. On the other hand, pairing MPPK KEM and 
+HPPK with NIST digital signature finalists Falcon and 
+Dilithium, as well as MPPK/DS, the pairs of MPPK KEM with 
+MPPK/DS and HPPK with MPPK/DS outperform NIST 
+finalists Falcon and Dilithium. HPPK pairing with MPPK/DS 
+demonstrates a slightly better performance than MPPK KEM 
+pairing with MPPK/DS, about 10% for all three security levels.   
+Fig. 2 also demonstrates that the average TLS 1.3 
+handShake can be completed at sub-million second. For 
+example, MPPK/DS paired with MPPK KEM and HPPK would 
+establish about 5000 TLS 1.3 connections per second which 
+gives 0.2 ms/connection. In an actual cloud environment, the 
+performance would be reduced due to the network latency. A 
+typical network latency is at 10 ms level, so a sub-million 
+second processing time contribute no impact on a practical TLS 
+1.3 handShake. That means, a nested TLS 1.3 inside the 
+existing TLS cryptomodule would not impact the overall 
+performance if we consider the fact that there is only one 
+handshaking per session.      
+ 
+C. Nested TLS 1.3 with PQC in Symmetric Encryption 
+After the handshake process of a nested TLS is complete, 
+communication peers establish a shared session key for 
+symmetric encryption during the session. Undoubtedly, the 
+NIST-Approved AES-256 can be used for data encryption in 
+the nested TLS, and the produced ciphertext would be 
+Quantum Encryption 
+Encryption 
+1. ClientHello 
+
+
+ 
+
+
+
+ 
+
+
+
+
+
+
+
+
+
+
+ 
+2. ServerHello   
+
+
+ 
+Nested TLS 1.3 
+1. ClientHello 
+
+
+ 
+2. ServerHello   
+
+
+ 
+Existing TLS  
+Figure 21. Illustration of nested TLS with PQC and quantum 
+encryption in OSI and TCP/IP models. The colorful OSI model is 
+taken from literature [29]. The TCP/IP model is indicated on the right-
+hand side. The white arrows refer to the existing certified TLS 1.x and 
+green arrows denote the nested TLS 1.3 with quantum resistant 
+cryptographic modules. 
+Figure 12. TLS 1.3 handshake performances (connections/second) are 
+illustrated in terms of PQC KEM algorithms paired with PQC digital 
+signature algorithms. The x-axis is marked with KEM schemes and y-
+axis represents handshaking connections per second. Digital signature 
+schemes are listed in the legends. NIST security level I, III, and V are 
+associated with the top, middle, and bottom graphs respectively. 
+
+Connectior
+2000
+1000
+66
+65
+66
+65
+0
+Saber
+Kyber768
+MPPKL 416
+HPPKL 214
+Key ExchangeMechanism
+TLs 1.3 handshake performance of different asymmetric algorithms - security level 5
+Signature algorithm
+second
+4188
+4000
+Falcon-1024
+3584
+Dilithium5
+3025
+per
+3000
+2893
+Rainbow-V-Classic
+2506
+Connections
+2432
+2239
+2225
+MPPKDS 623
+2089
+2000
+1000
+30
+30 
+31
+30
+0
+FireSaber
+Kyber1024
+MPPKL 417
+HPPKL_215
+KeyExchangeMechanismTLS1.3handshakeperformanceofdifferentasymmetricalgorithms-securitylevel1
+5219
+Signaturealgorithm
+5000
+4762
+4709
+Falcon-512
+Dilithium2
+persecond
+4000
+3852
+3983
+3943
+3849
+Rainbow-l-Classic
+3782
+MPPKDS_223
+3000
+3047
+Connections
+2000
+1000
+529
+477
+480
+508
+0
+LightSaber
+Kyber512
+MPPKL415
+HPPKL_213
+KeyExchangeMechanism
+TLs 1.3 handshake performance of different asymmetric algorithms - security level 3
+5000
+5037
+persecond
+Signature algorithm
+4591
+4277
+Dilithium3
+4000
+Rainbow-lll-Classic
+3319
+3140
+MPPKDS 423
+3000
+2778
+2776
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+P
+Et
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+da
+ho
+on
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+dr
+ne
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+e
+.HnSession7.Application
+tion4.Transport6:Data-
+Google
+Networkprocessto application
+7.Application
+DNS,WWW/HTTP,P2P,EMAIL/POP,SMTP,Telnet,FTP
+Presentation
+Datarepresentationand encryption
+Recognizingdata:HTML,DOC,JPEG,MP3,AvI,Sockets
+Session
+Interhostcommunication
+SessionestablishmentinTCP,SiP,RTP,RPC-Namedpipes
+End-to-endconnectionsandreliability
+4.Transport
+TCP,UDP,SCTP,SSL,TLS
+Pathdeterminationandlogicaladdressing
+3.Network
+IP,ARP,IPseCICMP.IGMP.OSPF
+Router
+MAC
+MAC
+Physicaladdressing
+2.Data Link
+Ethernet,8o2.11,MAC/LLC,VLAN,ATM,HDP,FibreChannel
+FrameRelay,HDLC,PPP,Q.921,TokenRing
+Media,signal,and binarytransmission
+1.Physical
+RS-232,RI45.V.34.100BASE-TX,SDH.DSL,802.11encrypted again by the outer FIPS certified AES-256 with its 
+own session key. In this case, the session keys may be 
+potentially obtained by attackers using “Steal now, crack later” 
+strategy, and later decrypted using quantum attacking 
+mechanisms. However, if the nested TLS uses quantum-safe 
+algorithms for encryption, then vulnerability of the outer 
+session does not influence the security of transmitted data to a 
+great extent. That is, if the nested TLS 1.3 with PQC establishes 
+a secure session key for session data encryption, then the 
+attacker with quantum resources will not be able to decrypt the 
+data encrypted in the nested TLS layer even if they were able 
+to decrypt the data encrypted in the outer classical layer. This 
+nested TLS 1.3 with PQC stops the “steal now, crack later” and 
+offers the “steal now, safe forever” services. 
+If AES-256 is used for encryption in conventional outer and 
+nested layer, that would cause the overall performance to drop 
+by 50%. However, there is no requirement to use AES-256 for 
+the nested layer encryption. Under the consideration of the FIPS 
+compliance, the inner data encryption does not need to be the 
+NIST-Approved and FIPS certified, we can choose quantum 
+encryption with Quantum Permutation Pad or QPP [28], 
+implemented classically with permutation matrices. Unlike 
+AES-256 encryption with 14 rounds, QPP encryption follows 
+the same way as quantum gate operations or matrix vector 
+multiplications. QPP encryption is bijective transformation so 
+typical pre-randomization and dispatch techniques would be 
+applied before the gate operations to avoid statistic patterns 
+appearing in the ciphertext. Quantum encryption with QPP 
+demonstrates excellent performance in both encryption and 
+decryption, being over 10x faster than AES-256 [31, 32, 33, 34] 
+Using QPP algorithm for encryption in the inner layer does not 
+impact the performance as greatly as AES-256. The drop in 
+performance with QPP is less than 10%. At this cost we can 
+offer “steal now, safe forever” services. In addition, quantum 
+encryption with QPP has been implemented into IBM quantum 
+computers and compiled into 2-qubit and 3-qubit quantum 
+circuits [35, 36, 37]. 
+For the detailed discission of quantum encryption with QPP, 
+please refer to [27-34]. Here we briefly summarize classical or 
+quantum implementation of QPP as follows shown in Fig. 3: 
+1. Choose the number of n-bits or n-qubits used to 
+generate the permutation gates, for example, � �
+4, 8.  
+2. Choose the number of gates to be used, M. M=64 
+for � �  8  in the digital QKD implementation 
+[38]. It can be reduced to 8 permutation gates with 
+� � 4.  
+3. Session key is first expanded for M permutation 
+gates and then mapped to a set of permutation 
+gates through the Init module where the Fisher-
+Yates shuffling algorithm. 
+4. The session key is also used to seed a 
+cryptographic pseudo random number generator or 
+PRNG 
+5. The pseudo random number generated by PRNG 
+is used to pre-randomize the plaintext m with XOR 
+and then dispatch the randomized data to the 
+indexed permutation gate for encryption and then 
+the ciphertext c is output accordingly. 
+6.  At the receiving side, the process is symmetric to 
+the encryption side, but the permutation gate must 
+be reversed or transposed. The pseudo random 
+number is first used to dispatch the ciphertext to 
+the indexed permutation for decryption and after 
+then derandomize with the pseudo random 
+number, the original plaintext m is obtained.  
+From Fig. 3, we can see that quantum encryption with QPP 
+only consists of three steps to complete its encryption: 
+randomization to remove any statistic bias, dispatching to the 
+indexed permutation gate/matrix, and then gate operation 
+applied to the randomized plaintext. Overall, the entire 
+encryption process may take at most the process time of a single 
+round in AES encryption. That is why QPP would be 10x faster 
+than AES.  Therefore, the nested TLS 1.3 with PQC only has a 
+minor impact on the communication performance, less than 
+10%. 
+If the attackers apply the “steal now, crack later” strategy 
+and wait for the quantum computers to break the public key for 
+the session key, then they can decrypt the outer AES encryption 
+and obtain the quantum encrypted ciphertext. The quantum 
+encrypted ciphertext is then requires to be cracked. However, 
+QPP as well as PQC algorithms and any other quantum-safe 
+algorithms are designed to withstand classical and quantum 
+attacks.  
+We have performed randomness analysis on ciphertext 
+produced by QPP. Table 1 illustrates the randomness analysis 
+of very biased English character files and ciphertext encrypted 
+with QPP using ENT test tool. ENT randomness test tool is very 
+sensitive to bit and byte level bias, especially detected in the 
+Chi Square values.  ENT outputs six reports on their entropy 
+per 8 bits, Chi Square value, p-value, arithmetic mean, Monte 
+Carlo �, and serial correlation. Table 1 shows that the total 
+biased English plaintext is encrypted with QPP into ciphertexts 
+which demonstrate excellent randomness, especially the Chi 
+Square value 233.2 with a p-value 0.83. The acceptable p-value 
+a good randomness is from 0.01 to 9.99. The ciphertext also 
+demonstrated excellent value for arithmetic mean 127.49, 
+Monte Carlo � = 3.14198164, and finally the serial correlation 
+at 9.3�10��.  
+Init 
+PRNG 
+�
+⬚
+⋯
+⬚
+⋮
+⋱
+⋮
+⬚
+⋯
+⬚
+� 
+�
+⬚
+⋯
+⬚
+⋮
+⋱
+⋮
+⬚
+⋯
+⬚
+� 
+…….. 
+�
+⬚
+⋯
+⬚
+⋮
+⋱
+⋮
+⬚
+⋯
+⬚
+� 
+�
+⬚
+⋯
+⬚
+⋮
+⋱
+⋮
+⬚
+⋯
+⬚
+� 
+Dispatcher 
+QPP 
+Key   
+m 
+c 
+Figure 3. Quantum encryption with QPP is illustrated with the session 
+key, plaintext data, and ciphertext together with related modules. 
+
+ Indeed, the nested TLS 1.3 with PQC offers FIPS 
+compliant solution with a quantum encryption component. We 
+this this solution as a good strategy for transition from classical 
+security to quantum security without waiting for NIST 
+standardization and FIPS certification to complete necessary 
+processes. Crypto agility is essential nowadays, since PQC and 
+other algorithms are novel and might have undiscovered 
+attacks, especially as quantum computing matures. So, 
+whenever a specific algorithm is found to be vulnerable, the 
+algorithm can be easily removed from the nested cryptomodule 
+and replaced with a new one in a convenient and quick manner. 
+When all PQC KEMs and digital signature algorithm are 
+standardized and the FIPS certification is required, then the 
+whole TLS 1.3 cryptomodule would be used for FIPS 
+certification. Once it is FIPS certified, the nested TLS 1.3 
+automatically becomes certified quantum resistant TLS 1.3 
+cryptomodule to replace the classical TLS 1.3.  With this, we 
+feel confident to turn the “steal now, crack later” into “Steal 
+now, safe forever.”  
+ Table 1. ENT testing is tabulated for statistically biased plaintext 
+inputs and ciphertext encrypted with QPP, together with their ideal 
+values. 
+ 
+III. CONCLUSION 
+In this work, we propose a FIPS compliant TLS 1.3 solution 
+with a nested quantum-secure TLS component. This solution 
+makes seamless transition and mitigation from classical 
+security to quantum security, that does not require any wait time 
+for standardization and certification. Given that the 
+standardization and FIPS certification process might take 10 
+years, adversaries can use this time to take advantage of the 
+“steal now, crack later” strategy. The proposal of the nested 
+TLS 1.3 with quantum-safe component could turn the “steal 
+now, crack later” into “steal now, safe forever”, while 
+preserving FIPS certified outer TLS layer. Therefore, this 
+proposed work is critical for protecting sensitive data today 
+with long shelf life against future quantum threats, which may 
+impact sectors including public health, insurance, genetics, 
+retirement. Any symmetric algorithm can be used in the nested 
+TLS layer. However, to overcome performance impact in 
+symmetric encryption, we suggest using quantum encryption 
+with QPP to further enhance the data security even with the 
+successful crack the outer public key with quantum computer, 
+the inner TLS 1.3 is still secure. In the future, we plan to build 
+the nested TLS 1.3 with PQC and test its real performance in a 
+cloud environment in comparison with normal TLS 1.3 with 
+PQC.  
+ACKNOWLEDGMENT 
+We acknowledge that Ryan Toth provided the OQS 
+OpenSSL diagram shown in Fig. 2. The overall performance of 
+TLS 1.3 hand shaking with MPPK/DS and MPPK KEM, as 
+well as HPPK would be published separately. 
+ 
+We acknowledge that the image in Fig.1 has been originally 
+taken from [29]. The image is licensed under the Creative 
+Commons Attribution 4.0 International [39]. We have made 
+modifications to the original image available in its original form 
+in [29]. 
+REFERENCES 
+[1] 
+P. W. Shor, "Algorithms for quantum computation: discrete 
+logarithms and factoring," Proceedings 35th Annual Symposium on 
+Foundations of Computer Science, 1994, pp. 124-134, doi: 
+10.1109/SFCS.1994.365700 
+[2] 
+Arute, F., Arya, K., Babbush, R. et al. Quantum supremacy using a 
+programmable superconducting processor. Nature 574, 505–510 
+(2019). https://doi.org/10.1038/s41586-019-1666-5 
+[3] 
+IBM 
+Newsroom: 
+https://newsroom.ibm.com/2022-11-09-IBM-
+Unveils-400-Qubit-Plus-Quantum-Processor-and-Next-Generation-
+IBM-Quantum-System-Two 
+[4] 
+M. Mosca, “Cybersecurity in an era with quantum computers: will we 
+be ready?”, Cryptology ePrint Archive, https://eprint.iacr.org/ 
+2015/1075. 
+[5] 
+L. K. Grover, A Fast Quantum Mechanical Algorithm for Database 
+Search, Proceedings of the Twenty-Eighth Annual ACM Symposium 
+on Theory of Computing, pp. 212-219, 1996. 
+[6] 
+NIST. Status report on the third round of the nist post-quantum cryptography 
+standardization 
+process. 
+https://csrc.nist.gov/publications/detail/ nistir/ 8413 /final (2022). 
+[7] 
+Bos J, Ducas L, Kiltz E, Lepoint T, Lyubashevsky V, Schanck JM, 
+Schwabe P, Seiler G, Stehle D (2018) CRYSTALS - Kyber: A CCA-
+secure module-lattice-based KEM. 2018 IEEE European Symposium 
+on Security and Privacy (EuroS P), pp 353–367. 
+[8] 
+Lyubashevsky V (2009) Fiat-Shamir with aborts: Applications to 
+lattice and factoring-based signatures. Advances in Cryptology – 
+ASIACRYPT 2009, ed Matsui M (Springer Berlin Heidelberg, Berlin, 
+Heidelberg), pp 598–616. 
+[9] 
+Gentry C, Peikert C, Vaikuntanathan V (2008) Trapdoors for hard 
+lattices and new cryptographic constructions. Proceedings of the 
+Fortieth Annual ACM Symposium on Theory of Computing STOC 
+’08 (Association for Computing Machinery, New York, NY, USA), pp 
+197–206. https://doi.org/10.1145/1374376.1374407. 
+[10] 
+Bernstein DJ, Hulsing ¨ A (2019) Decisional second-preimage 
+resistance: When does SPR imply PRE? Advances in Cryptology – 
+ASIACRYPT 2019, eds Galbraith SD, Moriai S (Springer 
+International Publishing, Cham), pp 33–62. 
+[11] 
+Misoczki R, Tillich JP, Sendrier N, Barreto PSLM (2013) MDPC-
+McEliece: New McEliece variants from moderate density parity-check 
+codes. 2013 IEEE International Symposium on Information Theory, 
+pp 2069–2073. https://doi.org/10.1109/IS IT.2013.6620590. 
+[12] 
+McEliece RJ (1978) A Public-Key Cryptosystem Based On Algebraic 
+Coding Theory. Deep Space Network Progress Report 44:114–116. 
+[13] 
+Aguilar-Melchor C, Blazy O, Deneuville JC, Gaborit P, Zemor ´ G 
+(2018) Efficient encryption from random quasi-cyclic codes. IEEE 
+Transactions 
+on 
+Information 
+Theory 
+64(5):3927–3943. 
+https://doi.org/10.1109/TIT.2018.2804444. 
+[14] 
+Jao D, De Feo L (2011) Towards quantum-resistant cryptosystems 
+from 
+supersingular 
+elliptic 
+curve 
+isogenies. 
+Post-Quantum 
+Cryptography, ed Yang BY (Springer Berlin Heidelberg, Berlin, 
+Heidelberg), pp 19–34. 
+[15] 
+NIST. 
+https://csrc.nist.gov/csrc/media/Projects/pqc-dig-
+sig/documents/call-for-proposals-dig-sig- sept-2022.pdf. 
+[16] 
+Beullens, W. Breaking rainbow takes a weekend on a laptop. 
+Cryptology 
+ePrint 
+Archive, 
+Paper 
+2022/214 
+(2022). 
+https://eprint.iacr.org/2022/214. 
+[17] 
+Robert, D. Breaking sidh in polynomial time. Cryptology ePrint 
+Archive, 
+Paper 
+2022/1038 
+(2022). 
+https://eprint.iacr.org/ 
+2022/1038. 
+ENT 
+Plaintexts 
+Ciphertext  
+Ideal Value 
+Entropy (bits) 
+4.224280 
+7.999998 
+8.000000 
+Chi Square 
+1821992676 
+233.20 
+256 
+p-Value  
+0.0001 
+0.83 
+0.5 
+Arithmetic Mean 
+97.9686 
+127.4953 
+127 
+Monte Carlo   
+4.000000000 
+3.14198164 
+3.14159265 
+Serial Correlation 
+-0.138722 
+- 0.000093 
+0 
+
+[18] 
+ Castryck, W. & Decru, T. An efficient key recovery attack on sidh 
+(preliminary 
+version). 
+Cryptology 
+ePrint 
+Archive, 
+Paper 
+2022/975 (2022). https://eprint.iacr.org/2022/975. 
+[19] 
+Wenger, E., Chen, M., Charton, F. & Lauter, K. Salsa: Attacking 
+lattice cryptography with transformers. Cryptology ePrint 
+Archive, Paper 2022/935 (2022). https://eprint.iacr.org/2022/935. 
+14/15 
+[20] 
+Kuang, R., Perepechaenko, M. & Barbeau, M. A new post-quantum 
+multivariate polynomial public key encapsulation algorithm. Quantum 
+Inf Process 21, 360 (2022). https://doi.org/10.1007/s11128-022-
+03712-5 
+[21] 
+R. Kuang, "A Deterministic Polynomial Public Key Algorithm over a 
+Prime Galois Field GF(p)," 2021 2nd Asia Conference on Computers 
+and 
+Communications 
+(ACCC), 
+2021, 
+pp. 
+79-88, 
+doi: 
+10.1109/ACCC54619.2021.00020. 
+[22] 
+Kuang, R., Perepechaenko, M. & Barbeau, M. A new quantum-safe 
+multivariate polynomial public key digital signature algorithm. Sci 
+Rep 12, 13168 (2022). https://doi.org/10.1038/s41598-022-15843-x 
+[23] 
+R. Kuang and M. Perepechaenko, "Digital Signature Performance of a 
+New 
+Quantum 
+Safe 
+Multivariate 
+Polynomial 
+Public 
+Key 
+Algorithm," 2022 7th International Conference on Computer and 
+Communication Systems (ICCCS), 2022, pp. 419-424, doi: 
+10.1109/ICCCS55155.2022.9846785. 
+[24] 
+R. Kuang and M. Perepechaenko, "Digital Signature Performance of a 
+New 
+Quantum 
+Safe 
+Multivariate 
+Polynomial 
+Public 
+Key 
+Algorithm," 2022 7th International Conference on Computer and 
+Communication Systems (ICCCS), 2022, pp. 419-424, doi: 
+10.1109/ICCCS55155.2022.9846785. 
+[25] 
+R. Kuang, M. Perepechaenko, R. Toth and M. Barbeau, "Benchmark 
+Performance of a New Quantum-Safe Multivariate Polynomial Digital 
+Signature Algorithm," 2022 IEEE International Conference on 
+Quantum Computing and Engineering (QCE), 2022, pp. 454-464, doi: 
+10.1109/QCE53715.2022.00067. 
+[26] 
+NIST. https://csrc.nist.gov/Projects/post-quantum-cryptography/faqs 
+[27] 
+R. Kuang and N. Bettenburg, “Shannon Perfect Secrecy in a Discrete 
+Hilbert Space”, 2020 IEEE International Conference on Quantum 
+Computing and Engineering (QCE), pp. 249-255, 2020.  
+[28] 
+Kuang, R., Barbeau, M. Quantum permutation pad for universal 
+quantum-safe cryptography. Quantum Inf Process 21, 211 (2022). 
+https://doi.org/10.1007/s11128-022-03557-y  
+[29] 
+The Cyber Security Hub™: Information Technology and Services, 
+London, UK, 2021.  
+[30] 
+Kuang, R. & Perepechaenko. A new symmetric homomorphic 
+functional encryption over a hidden ring for polynomial public key 
+encapsulation, under review at Scientific Reports, 2022 
+[31] 
+D. Lou, R. Kuang and A. He, "Entropy Transformation and Expansion 
+with Quantum Permutation Pad for 5G Secure Networks," 2021 IEEE 
+21st International Conference on Communication Technology (ICCT), 
+2021, pp. 840-845, doi: 10.1109/ICCT52962.2021.9657891.  
+[32] 
+ R. Kuang, D. Lou, A. He, C. McKenzie and M. Redding, "Pseudo 
+Quantum Random Number Generator with Quantum Permutation 
+Pad," 2021 IEEE International Conference on Quantum Computing 
+and 
+Engineering 
+(QCE), 
+2021, 
+pp. 
+359-364, 
+doi: 
+10.1109/QCE52317.2021.00053. 
+[33] 
+R. Kuang, D. Lou, A. He and A. Conlon, "Quantum Safe Lightweight 
+Cryptography with Quantum Permutation Pad," 2021 IEEE 6th 
+International Conference on Computer and Communication Systems 
+(ICCCS), 
+2021, 
+pp. 
+790-795, 
+doi: 
+10.1109/ICCCS52626.2021.9449247. 
+[34] 
+R. Kuang, D. Lou, A. He, A. Conlon "Quantum Secure Lightweight 
+Cryptography with Quantum Permutation Pad", Advances in Science, 
+Technology and Engineering Systems Journal, vol. 6, no. 4, pp. 401-
+405 (2021). 
+[35] 
+M. 
+Perepechaenko 
+and 
+R. 
+Kuang, 
+"Quantum 
+Encrypted 
+Communication between Two IBMQ Systems Using Quantum 
+Permutation 
+Pad," 2022 
+11th 
+International 
+Conference 
+on 
+Communications, Circuits and Systems (ICCCAS), 2022, pp. 146-152, 
+doi: 10.1109/ICCCAS55266.2022.9824836. 
+[36] 
+Kuang, R., Perepechaenko, M. Quantum encryption with quantum 
+permutation pad in IBMQ systems. EPJ Quantum Technol. 9, 26 
+(2022). https://doi.org/10.1140/epjqt/s40507-022-00145-y 
+[37] 
+M. 
+Perepechaenko 
+and 
+R. 
+Kuang, 
+"Quantum 
+Encrypted 
+Communication in IBMQ Systems Using Quantum Permutation 
+Pad," J. Comm.,Vol. 17, No. 12, December 2022. https://doi.org/ 
+10.12720/jcm.17.12.972-978. 
+[38] 
+D. Lou et al., "Benchmark Performance of Digital QKD Platform 
+Using Quantum Permutation Pad," in IEEE Access, vol. 10, pp. 
+107066-107076, 2022, doi: 10.1109/ACCESS.2022.3212738. 
+[39] 
+Creative Commons, Attribution 4.0 International (CC BY 4.0), source: 
+https://creativecommons.org/licenses/by/4.0/. 
+Last 
+accessed 
+on 
+12/13/22.  
+
diff --git a/-dAyT4oBgHgl3EQfRPaF/content/tmp_files/load_file.txt b/-dAyT4oBgHgl3EQfRPaF/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf,len=537
+page_content='XXX-X-XXXX-XXXX-X/XX/$XX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='00 ©20XX IEEE  FIPS Compliant Quantum Secure Communication  using Quantum Permutation Pad  Abstract—Quantum computing has entered fast development  track since Shor’s algorithm was proposed in 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Multi-cloud  services of quantum computing farms are currently available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' One  of which, IBM quantum computing, presented a road map  showing their Kookaburra system with over 4158 qubits will be  available in 2025.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' For the standardization of Post-Quantum  Cryptography or PQC, the National Institute of Standards and  Technology or NIST recently announced the first candidates for  standardization with one algorithm for key encapsulation  mechanism (KEM), Kyber, and three algorithms for digital  signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' NIST has also issued a new call for quantum-safe  digital signature algorithms due June 1, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' This timeline shows  that FIPS-certified quantum-safe TLS protocol would take a  predictably long time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' However, ‘steal now, crack later’ tactic  requires protecting data against future quantum threat actors  today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' NIST recommended the use of a hybrid mode of TLS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='3  with its extensions to support PQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' The hybrid mode works for  certain cases but FIPS certification for the hybridized  cryptomodule might still be required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' This paper proposes to take  a nested mode to enable TLS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='3 protocol with quantum-safe data,  which can be made available today and is FIPS compliant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' We  discussed the performance impacts of the handshaking phase of  the nested TLS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='3 with PQC and the symmetric encryption phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  The major impact on performance using the nested mode is in the  data symmetric encryption with AES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' To overcome this  performance reduction, we suggest using quantum encryption  with a quantum permutation pad for the data encryption with a  minor performance reduction of less than 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  Keywords—quantum communication, quantum encryption,  quantum decryption, quantum security, secure communication,  QPP, FIPS, TLS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' INTRODUCTION  Peter Shor proposed his celebrated quantum algorithm in  1994 [1], which solves the NP-hard problem of prime integer  factorization in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' At its beginning, quantum  computing,  especially  universal  gate-based  quantum  computing, experienced a slow development phase for about  two decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' In 2019, Arute et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' from Google claimed  Quantum Supremacy with their 53-qubits Sycamore processor  [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' This marked the start of global quantum computing race.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  Since IBM released their 5-qubit quantum computer for public  access with Qiskit tool in 2017, IBM recently announced their  433-qubit quantum computer and plan to double its qubits every  year for 2023 to reach over 4,000-qubits by 2025, outlined in  their development roadmap [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  The fundamental shift from classical computing to quantum  computing is the shift in computing algebra from the classical  Boolean algebra to linear algebra, used in quantum computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  That is, from classical logic gates, implemented in CPU, to  quantum logic gates to be implemented in QPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' That means,  quantum computing indicates a quadratic speedup of classical  computers or �2��2, where � denotes the number of qubits of a  QPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' This exponential computing power would break today’s  RSA-2048 in 10 seconds if a fault tolerant quantum computer  reaches 4099 qubits, while a classical computer would take 300  trillion years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Mosca predicted that there is a “1/2 chance of  breaking RSA-2048 by 2031” [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' The year from quantum  computing to break classical public key RSA has been called  the Year to Quantum threat or Y2Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  Symmetric cryptography such as the well-known Advanced  Encryption Standard or AES also suffered quadratic speedup of  the best quantum attack on the key space using Grover’s  algorithm proposed by Grover in 1996 [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' That requires the  key length to be doubled in comparison with the equivalent  classical security level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' For example, the classical AES-128  would be replaced by AES-256 for quantum security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  It has been well-understood that the upcoming quantum  computing systems will destroy the foundation of classical  public key infrastructure or PKI for both key establishment  such as RSA, Diffie-Hellman or elliptical curve Diffie-Hellman  and digital signature such as Digital Signature Algorithm or  DSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' National Institute of Standards and Technology or NIST  has announced its standardization process of Post-Quantum  Cryptography or PQC in November of 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' In July 2022 [6],  NIST announced the lattice-based Kyber [7] to be its first  standardized key encapsulation mechanism or KEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' And  lattice-based Dillithium [8] Falcon [9], as well as hash-based  SPHINCS+ [10] to be first standardized digital signature  schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Other KEM candidates BIKE [11], Classic McEliece  [12], HQC [13], and SIKE [14] moved to the 4th round to be  considered further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' NIST has also announced its reopening  Alex He  Quantropi Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  Ottawa, Canada  alex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='he@quantropi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='com  Dafu Lou  Quantropi Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  Ottawa, Canada  dafu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='lou@quantropi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='com  Eric She  DLS Technology Corporation  Ottawa, Canada  eshe@dlstech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='com  Shangjie Guo  FinQ Tech Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  College Park, Maryland  sguo@finq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='tech  Hareesh Watson  DLS Technology Corporation  Ottawa, Canada  hwatson@dlstech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='com  Sibyl Weng  DLS Technology Corporation  Ottawa, Canada  sweng@dlstech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='com  Maria Perepechaenko  Quantropi Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  Ottawa, Canada  maria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='perepechaenko@quantropi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='com  Randy Kuang  Quantropi Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  Ottawa, Canada  randy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='kuang@quantropi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='com  ORCID: 000-0002-5567-2192-  0002-5567-2192  OORCID: RCID iD: 0000-0002-  submissions for digital signature standardization due Jaune  2023 [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  In 2022 some PQC algorithms such as Rainbow (a digital  signature scheme based on Multivariate Public Key  Cryptography or MPKC) [16] and Supersingular Isogeny  Diffie-Hellman protocol or SIDH [17, 18] proved to be  vulnerable to classical attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Another very interesting  cryptoanalysis was reported in the late 2022 by Wenger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' The team used transformers (a deep learning model) to  develop an attack on certain lattice-based schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' They  noticed that the basic equation system for Learning With Error  or LWE can be expressed as linear regression used in machine  learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' If trained transformers, especially combined with  quantum computational advantage, can solve the Short Vector  Problem, then PQC algorithms face an enormous challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  Recall that most of the PQC standard schemes are lattice-based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  This issue is especially concerning since estimation on Y2Q has  not taken the power of quantum machine learning into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  Some recent novel PQC algorithms were proposed by  Kuang, Perepechaenko, and Barbeau for KEM [20, 21] and  digital signature [22, 23, 24, 25], based on NP-complete  Modular Diophantine Equation Problem or MDEP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' These  novel schemes share a foundation which we call Multivariate  Polynomial Public Key or MPPK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' MPPK is built on two vector  spaces: a linear multivariate vector space { ��, … , �� }  containing noise variables used for obscurity and polynomial  vector space { 1, ��, … , ���� } containing secret message  variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' MPPK offers small parameter sizes at the level of  hundreds of bytes for key, ciphertext, and signature and  outperforms NIST finalists in key generation, encryption,  decryption, as well as signing and signature verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' They  could be considered as generic PQC algorithms for a wide range  of devices and systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  NIST recommended using a hybrid scheme for quantum  resistant TLS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='3 [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' By leveraging the keyShare extension of  TLS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='3, one can combine a NIST-approved classical key  establishment algorithm such as ECCDH in TLS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='3 with one  or more PQC KEM algorithms and a NIST-approved digital  signature algorithm such as DSA with a list of PQC digital  signature algorithms for a chain signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' This hybrid scheme  offers crypto agility as the ongoing process of PQC  standardization and cryptanalysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' However, this hybrid  scheme brings two potential limitations: 1) the hybrid TLS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='3  may still require FIPS certification although its core  cryptomodule is certified in classical TLS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' In general, the  FIPS certification comes at a cost for both time and money,  although it would be possible to receive the certificate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' 2) The  certified hybrid TLS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='3 must be integrated with an application  for a quantum resistant service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' But in some cases, this  integration may be difficult for some applications running  inside web browsers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  This paper proposes a new hybrid scheme by nesting PQC  inside classical TLS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='3, creating nested TLS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='3, to overcome  the limitations in the above hybrid scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' The nested TLS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='3  does not require a new FIPS certification because it does not  change the existing certified TLS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Moreover, it supports TLS  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='3 as well as any certified TLS such as TLS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='2 or even earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  One drawback of the nested scheme is that it may reduce the  performance of the encryption and decryption operations  because the transmitted data would be encrypted twice if AES  is used for both encryptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' To minimize the performance  impact, we propose to use quantum encryption with a Quantum  Permutation Pad algorithm or QPP [27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' The QPP will be  used to encrypt the raw data first, producing a quantum-  encrypted message used as a TLS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='3 message to be encrypted  with AES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' FIPS COMPLIANT QUANTUM ENCRYPTION WITH QPP  In this section, we first introduce the concept of a nested TLS  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='3 protocol, that uses quantum-safe cryptosystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' The nested  TLS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='3 allows for a smooth transition from classical era to  quantum era while maintaining FIPS compliance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' We then  discuss the nested TLS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='3 handshake process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Next, we  consider the symmetric data encryption with QPP for  considerations of both performance and security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Nested TLS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='3 with PQC in TLS Handshaking Proccess  The proposed nested TLS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='3 is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' The  figure illustrates an Open Systems’ Interconnection model  (OSI-model) consisting of 7 layers: physical, data link, internet,  transport,  session,  presentation,  and  application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  The  functionality of each layer is illustrated in [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' On the right-  hand side, we mark the corresponding layers in TCP/IP model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  The existing TLS cryptomodule is in the transport layer, while  the nested TLS is in the application layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' White arrows denote  a FIPS certified TLS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='3 for clientHello request from the client  and serverHello response from the server to establish a shared  session for session data encryption and decryption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' The NIST-  Approved key agreement protocol, ECCDH, is used for forward  secrecy in the existing TLS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' The RSA algorithm is excluded  from the key agreement protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' RSA, DSA, and ECDSA are  paired with hash functions for digital signature in the existing  TLS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' The current conventional FIPS certified TLS  cryptomodule will be vulnerable to quantum computing attacks  once the fault tolerable quantum computers are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  However, the "steal now crack later" tactics are already in use,  meaning all encrypted data today is at risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Immediate action is  imperative to protect data against future quantum threat actors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  If sensitive information requires to remain secret for over 10  years, then it would not be wise to wait for the FIPS certified  TLS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='3 with quantum resistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  The proposed nested TLS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='3 with PQC cryptographic  modules is independent from the FIPS certified TLS  cryptomodule in a sense that packets from the nested TLS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='3  become data packets for the FIPS certified TLS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Since the outer  classical TLS cryptomodule is not altered, the nested TLS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='3  does not violate the FIPS certification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' This solution can be  considered as a promising FIPS compliant TLS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='3 for quantum  security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' The nested TLS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='3 can be used to turn “steal now,  crack later” into “steal now, safe forever”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  Nested TLS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='3 is based on the Open Quantum Safe or OQS  OpenSSL to support PQC KEM algorithms such as the NIST  finalists Kyber and Saber, as well as MPPK [20] and  Homomorphic Polynomial Public Key further evolved from  MPPK [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' For the digital signatures, nested TLS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='3 supports  PQC digital signature algorithms such as NIST finalists Falcon,  Dilithium, Rainbow, as well as MPPK/DS to be submitted to  NIST for standardization in 2023 [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Performance of a Nested TLS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='3 Handshake  We tested the performance of a nested TLS1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='3 handshake  in a local machine on a 16-core Intel®Core™ i7-10700 CPU at  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='90 GHz system for all the measured primitives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' 2  illustrates the performance of TLS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='3 handShake for each pair  of KEM and digital signature schemes in terms of NIST  security levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' In general, Rainbow digital signature  demonstrates the worst performances for TLS handshake with  about 500 connections/second at all NIST security level I, 65  connections/second at all NIST security level III, and 30  connections/second at all NIST security level V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  By pairing MPPK/DS with Saber, Kyber, MPPK KEM, and  HPPK, the MPPK/DS scheme outperforms digital signature  schemes Falcon and Dilithium over 30% at security level I, over  35% at security level III, and 40% at security level V,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' On the other hand, pairing MPPK KEM and  HPPK with NIST digital signature finalists Falcon and  Dilithium, as well as MPPK/DS, the pairs of MPPK KEM with  MPPK/DS and HPPK with MPPK/DS outperform NIST  finalists Falcon and Dilithium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' HPPK pairing with MPPK/DS  demonstrates a slightly better performance than MPPK KEM  pairing with MPPK/DS, about 10% for all three security levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' 2 also demonstrates that the average TLS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='3  handShake can be completed at sub-million second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' For  example, MPPK/DS paired with MPPK KEM and HPPK would  establish about 5000 TLS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='3 connections per second which  gives 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='2 ms/connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' In an actual cloud environment, the  performance would be reduced due to the network latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' A  typical network latency is at 10 ms level, so a sub-million  second processing time contribute no impact on a practical TLS  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='3 handShake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' That means, a nested TLS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='3 inside the  existing TLS cryptomodule would not impact the overall  performance if we consider the fact that there is only one  handshaking per session.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Nested TLS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='3 with PQC in Symmetric Encryption  After the handshake process of a nested TLS is complete,  communication peers establish a shared session key for  symmetric encryption during the session.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Undoubtedly, the  NIST-Approved AES-256 can be used for data encryption in  the nested TLS, and the produced ciphertext would be  Quantum Encryption  Encryption  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' ClientHello \uf0e0  \uf0e0  \uf0e0  \uf0e0  \uf0df  \uf0df  \uf0df  \uf0df  \uf046  \uf046  \uf046  \uf046\uf049  \uf049  \uf049  \uf049\uf050  \uf050  \uf050  \uf050  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' ServerHello   \uf0e0  \uf0e0  \uf0e0  \uf0e0  Nested TLS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' ClientHello \uf0e0  \uf0e0  \uf0e0  \uf0e0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' ServerHello   \uf0e0  \uf0e0  \uf0e0  \uf0e0  Existing TLS  Figure 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Illustration of nested TLS with PQC and quantum  encryption in OSI and TCP/IP models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' The colorful OSI model is  taken from literature [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' The TCP/IP model is indicated on the right-  hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' The white arrows refer to the existing certified TLS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='x and  green arrows denote the nested TLS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='3 with quantum resistant  cryptographic modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' TLS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='3 handshake performances (connections/second) are  illustrated in terms of PQC KEM algorithms paired with PQC digital  signature algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' The x-axis is marked with KEM schemes and y-  axis represents handshaking connections per second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Digital signature  schemes are listed in the legends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' NIST security level I, III, and V are  associated with the top, middle, and bottom graphs respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  Connectior  2000  1000  66  65  66  65  0  Saber  Kyber768  MPPKL 416  HPPKL 214  Key ExchangeMechanism  TLs 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='3 handshake performance of different asymmetric algorithms - security level 5  Signature algorithm  second  4188  4000  Falcon-1024  3584  Dilithium5  3025  per  3000  2893  Rainbow-V-Classic  2506  Connections  2432  2239  2225  MPPKDS 623  2089  2000  1000  30  30  31  30  0  FireSaber  Kyber1024  MPPKL 417  HPPKL_215  KeyExchangeMechanismTLS1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='3handshakeperformanceofdifferentasymmetricalgorithms-securitylevel1  5219  Signaturealgorithm  5000  4762  4709  Falcon-512  Dilithium2  persecond  4000  3852  3983  3943  3849  Rainbow-l-Classic  3782  MPPKDS_223  3000  3047  Connections  2000  1000  529  477  480  508  0  LightSaber  Kyber512  MPPKL415  HPPKL_213  KeyExchangeMechanism  TLs 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='3 handshake performance of different asymmetric algorithms - security level 3  5000  5037  persecond  Signature algorithm  4591  4277  Dilithium3  4000  Rainbow-lll-Classic  3319  3140  MPPKDS 423  3000  2778  2776  SUP  P  Et  Frgn  da  ho  on  on  abl  CO  UD  CT  ep  min  RP,  ca  dr  ne  2  e  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='HnSession7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='Application  tion4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='Transport6:Data-  Google  Networkprocessto application  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='Application  DNS,WWW/HTTP,P2P,EMAIL/POP,SMTP,Telnet,FTP  Presentation  Datarepresentationand encryption  Recognizingdata:HTML,DOC,JPEG,MP3,AvI,Sockets  Session  Interhostcommunication  SessionestablishmentinTCP,SiP,RTP,RPC-Namedpipes  End-to-endconnectionsandreliability  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='Transport  TCP,UDP,SCTP,SSL,TLS  Pathdeterminationandlogicaladdressing  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='Network  IP,ARP,IPseCICMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='IGMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='OSPF  Router  MAC  MAC  Physicaladdressing  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='Data Link  Ethernet,8o2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='11,MAC/LLC,VLAN,ATM,HDP,FibreChannel  FrameRelay,HDLC,PPP,Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='921,TokenRing  Media,signal,and binarytransmission  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='Physical  RS-232,RI45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='100BASE-TX,SDH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='DSL,802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='11encrypted again by the outer FIPS certified AES-256 with its  own session key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' In this case, the session keys may be  potentially obtained by attackers using “Steal now, crack later”  strategy, and later decrypted using quantum attacking  mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' However, if the nested TLS uses quantum-safe  algorithms for encryption, then vulnerability of the outer  session does not influence the security of transmitted data to a  great extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' That is, if the nested TLS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='3 with PQC establishes  a secure session key for session data encryption, then the  attacker with quantum resources will not be able to decrypt the  data encrypted in the nested TLS layer even if they were able  to decrypt the data encrypted in the outer classical layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' This  nested TLS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='3 with PQC stops the “steal now, crack later” and  offers the “steal now, safe forever” services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  If AES-256 is used for encryption in conventional outer and  nested layer, that would cause the overall performance to drop  by 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' However, there is no requirement to use AES-256 for  the nested layer encryption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Under the consideration of the FIPS  compliance, the inner data encryption does not need to be the  NIST-Approved and FIPS certified, we can choose quantum  encryption with Quantum Permutation Pad or QPP [28],  implemented classically with permutation matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Unlike  AES-256 encryption with 14 rounds, QPP encryption follows  the same way as quantum gate operations or matrix vector  multiplications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' QPP encryption is bijective transformation so  typical pre-randomization and dispatch techniques would be  applied before the gate operations to avoid statistic patterns  appearing in the ciphertext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Quantum encryption with QPP  demonstrates excellent performance in both encryption and  decryption, being over 10x faster than AES-256 [31, 32, 33, 34]  Using QPP algorithm for encryption in the inner layer does not  impact the performance as greatly as AES-256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' The drop in  performance with QPP is less than 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' At this cost we can  offer “steal now, safe forever” services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' In addition, quantum  encryption with QPP has been implemented into IBM quantum  computers and compiled into 2-qubit and 3-qubit quantum  circuits [35, 36, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  For the detailed discission of quantum encryption with QPP,  please refer to [27-34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Here we briefly summarize classical or  quantum implementation of QPP as follows shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' 3:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Choose the number of n-bits or n-qubits used to  generate the permutation gates, for example, � �  4, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Choose the number of gates to be used, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' M=64  for � �  8  in the digital QKD implementation  [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' It can be reduced to 8 permutation gates with  � � 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Session key is first expanded for M permutation  gates and then mapped to a set of permutation  gates through the Init module where the Fisher-  Yates shuffling algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' The session key is also used to seed a  cryptographic pseudo random number generator or  PRNG  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' The pseudo random number generated by PRNG  is used to pre-randomize the plaintext m with XOR  and then dispatch the randomized data to the  indexed permutation gate for encryption and then  the ciphertext c is output accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' At the receiving side, the process is symmetric to  the encryption side, but the permutation gate must  be reversed or transposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' The pseudo random  number is first used to dispatch the ciphertext to  the indexed permutation for decryption and after  then derandomize with the pseudo random  number, the original plaintext m is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' 3, we can see that quantum encryption with QPP  only consists of three steps to complete its encryption:  randomization to remove any statistic bias, dispatching to the  indexed permutation gate/matrix, and then gate operation  applied to the randomized plaintext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Overall, the entire  encryption process may take at most the process time of a single  round in AES encryption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' That is why QPP would be 10x faster  than AES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Therefore, the nested TLS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='3 with PQC only has a  minor impact on the communication performance, less than  10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  If the attackers apply the “steal now, crack later” strategy  and wait for the quantum computers to break the public key for  the session key, then they can decrypt the outer AES encryption  and obtain the quantum encrypted ciphertext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' The quantum  encrypted ciphertext is then requires to be cracked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' However,  QPP as well as PQC algorithms and any other quantum-safe  algorithms are designed to withstand classical and quantum  attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  We have performed randomness analysis on ciphertext  produced by QPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Table 1 illustrates the randomness analysis  of very biased English character files and ciphertext encrypted  with QPP using ENT test tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' ENT randomness test tool is very  sensitive to bit and byte level bias, especially detected in the  Chi Square values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' ENT outputs six reports on their entropy  per 8 bits, Chi Square value, p-value, arithmetic mean, Monte  Carlo �, and serial correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Table 1 shows that the total  biased English plaintext is encrypted with QPP into ciphertexts  which demonstrate excellent randomness, especially the Chi  Square value 233.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='2 with a p-value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' The acceptable p-value  a good randomness is from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='01 to 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' The ciphertext also  demonstrated excellent value for arithmetic mean 127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='49,  Monte Carlo � = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='14198164, and finally the serial correlation  at 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='3�10��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  Init  PRNG  �  ⬚  ⋯  ⬚  ⋮  ⋱  ⋮  ⬚  ⋯  ⬚  �  �  ⬚  ⋯  ⬚  ⋮  ⋱  ⋮  ⬚  ⋯  ⬚  �  ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='.  �  ⬚  ⋯  ⬚  ⋮  ⋱  ⋮  ⬚  ⋯  ⬚  �  �  ⬚  ⋯  ⬚  ⋮  ⋱  ⋮  ⬚  ⋯  ⬚  �  Dispatcher  QPP  Key  m  c  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Quantum encryption with QPP is illustrated with the session  key, plaintext data, and ciphertext together with related modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  Indeed, the nested TLS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='3 with PQC offers FIPS  compliant solution with a quantum encryption component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' We  this this solution as a good strategy for transition from classical  security to quantum security without waiting for NIST  standardization and FIPS certification to complete necessary  processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Crypto agility is essential nowadays, since PQC and  other algorithms are novel and might have undiscovered  attacks, especially as quantum computing matures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' So,  whenever a specific algorithm is found to be vulnerable, the  algorithm can be easily removed from the nested cryptomodule  and replaced with a new one in a convenient and quick manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  When all PQC KEMs and digital signature algorithm are  standardized and the FIPS certification is required, then the  whole TLS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='3 cryptomodule would be used for FIPS  certification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Once it is FIPS certified, the nested TLS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='3  automatically becomes certified quantum resistant TLS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='3  cryptomodule to replace the classical TLS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' With this, we  feel confident to turn the “steal now, crack later” into “Steal  now, safe forever.”  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' ENT testing is tabulated for statistically biased plaintext  inputs and ciphertext encrypted with QPP, together with their ideal  values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' CONCLUSION  In this work, we propose a FIPS compliant TLS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='3 solution  with a nested quantum-secure TLS component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' This solution  makes seamless transition and mitigation from classical  security to quantum security, that does not require any wait time  for standardization and certification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Given that the  standardization and FIPS certification process might take 10  years, adversaries can use this time to take advantage of the  “steal now, crack later” strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' The proposal of the nested  TLS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='3 with quantum-safe component could turn the “steal  now, crack later” into “steal now, safe forever”, while  preserving FIPS certified outer TLS layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Therefore, this  proposed work is critical for protecting sensitive data today  with long shelf life against future quantum threats, which may  impact sectors including public health, insurance, genetics,  retirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Any symmetric algorithm can be used in the nested  TLS layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' However, to overcome performance impact in  symmetric encryption, we suggest using quantum encryption  with QPP to further enhance the data security even with the  successful crack the outer public key with quantum computer,  the inner TLS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='3 is still secure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' In the future, we plan to build  the nested TLS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='3 with PQC and test its real performance in a  cloud environment in comparison with normal TLS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='3 with  PQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  ACKNOWLEDGMENT  We acknowledge that Ryan Toth provided the OQS  OpenSSL diagram shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' The overall performance of  TLS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='3 hand shaking with MPPK/DS and MPPK KEM, as  well as HPPK would be published separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  We acknowledge that the image in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='1 has been originally  taken from [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' The image is licensed under the Creative  Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='0 International [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' We have made  modifications to the original image available in its original form  in [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  REFERENCES  [1]  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Shor, "Algorithms for quantum computation: discrete  logarithms and factoring," Proceedings 35th Annual Symposium on  Foundations of Computer Science, 1994, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' 124-134, doi:  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='1109/SFCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='365700  [2]  Arute, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=', Arya, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=', Babbush, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Quantum supremacy using a  programmable superconducting processor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Nature 574, 505–510  (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='1038/s41586-019-1666-5  [3]  IBM  Newsroom:  https://newsroom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='ibm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='com/2022-11-09-IBM-  Unveils-400-Qubit-Plus-Quantum-Processor-and-Next-Generation-  IBM-Quantum-System-Two  [4]  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Mosca, “Cybersecurity in an era with quantum computers: will we  be ready?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=', Cryptology ePrint Archive, https://eprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='iacr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='org/  2015/1075.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  [5]  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Grover, A Fast Quantum Mechanical Algorithm for Database  Search, Proceedings of the Twenty-Eighth Annual ACM Symposium  on Theory of Computing, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' 212-219, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  [6]  NIST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Status report on the third round of the nist post-quantum cryptography  standardization  process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  https://csrc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='nist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='gov/publications/detail/ nistir/ 8413 /final (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  [7]  Bos J, Ducas L, Kiltz E, Lepoint T, Lyubashevsky V, Schanck JM,  Schwabe P, Seiler G, Stehle D (2018) CRYSTALS - Kyber: A CCA-  secure module-lattice-based KEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' 2018 IEEE European Symposium  on Security and Privacy (EuroS P), pp 353–367.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  [8]  Lyubashevsky V (2009) Fiat-Shamir with aborts: Applications to  lattice and factoring-based signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Advances in Cryptology –  ASIACRYPT 2009, ed Matsui M (Springer Berlin Heidelberg, Berlin,  Heidelberg), pp 598–616.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  [9]  Gentry C, Peikert C, Vaikuntanathan V (2008) Trapdoors for hard  lattices and new cryptographic constructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Proceedings of the  Fortieth Annual ACM Symposium on Theory of Computing STOC  ’08 (Association for Computing Machinery, New York, NY, USA), pp  197–206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='1145/1374376.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='1374407.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  [10]  Bernstein DJ, Hulsing ¨ A (2019) Decisional second-preimage  resistance: When does SPR imply PRE?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Advances in Cryptology –  ASIACRYPT 2019, eds Galbraith SD, Moriai S (Springer  International Publishing, Cham), pp 33–62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  [11]  Misoczki R, Tillich JP, Sendrier N, Barreto PSLM (2013) MDPC-  McEliece: New McEliece variants from moderate density parity-check  codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' 2013 IEEE International Symposium on Information Theory,  pp 2069–2073.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='1109/IS IT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='6620590.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  [12]  McEliece RJ (1978) A Public-Key Cryptosystem Based On Algebraic  Coding Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Deep Space Network Progress Report 44:114–116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  [13]  Aguilar-Melchor C, Blazy O, Deneuville JC, Gaborit P, Zemor ´ G  (2018) Efficient encryption from random quasi-cyclic codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' IEEE  Transactions  on  Information  Theory  64(5):3927–3943.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='1109/TIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='2804444.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  [14]  Jao D, De Feo L (2011) Towards quantum-resistant cryptosystems  from  supersingular  elliptic  curve  isogenies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  Post-Quantum  Cryptography, ed Yang BY (Springer Berlin Heidelberg, Berlin,  Heidelberg), pp 19–34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  [15]  NIST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  https://csrc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='nist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='gov/csrc/media/Projects/pqc-dig-  sig/documents/call-for-proposals-dig-sig- sept-2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  [16]  Beullens, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Breaking rainbow takes a weekend on a laptop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  Cryptology  ePrint  Archive,  Paper  2022/214  (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  https://eprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='iacr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='org/2022/214.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  [17]  Robert, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Breaking sidh in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Cryptology ePrint  Archive,  Paper  2022/1038  (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  https://eprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='iacr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='org/  2022/1038.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  ENT  Plaintexts  Ciphertext  Ideal Value  Entropy (bits)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='224280  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='999998  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='000000  Chi Square  1821992676  233.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='20  256  p-Value  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='0001  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
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+page_content='5  Arithmetic Mean  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='9686  127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='4953  127  Monte Carlo  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='000000000  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='14198164  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='14159265  Serial Correlation  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='138722  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='000093  0  [18]  Castryck, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' & Decru, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' An efficient key recovery attack on sidh  (preliminary  version).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  Cryptology  ePrint  Archive,  Paper  2022/975 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' https://eprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='iacr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='org/2022/975.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  [19]  Wenger, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=', Chen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=', Charton, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' & Lauter, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Salsa: Attacking  lattice cryptography with transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Cryptology ePrint  Archive, Paper 2022/935 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' https://eprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='iacr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='org/2022/935.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  14/15  [20]  Kuang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=', Perepechaenko, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' & Barbeau, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' A new post-quantum  multivariate polynomial public key encapsulation algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Quantum  Inf Process 21, 360 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='1007/s11128-022-  03712-5  [21]  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Kuang, "A Deterministic Polynomial Public Key Algorithm over a  Prime Galois Field GF(p)," 2021 2nd Asia Conference on Computers  and  Communications  (ACCC),  2021,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  79-88,  doi:  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='1109/ACCC54619.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='00020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  [22]  Kuang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=', Perepechaenko, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' & Barbeau, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' A new quantum-safe  multivariate polynomial public key digital signature algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Sci  Rep 12, 13168 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='1038/s41598-022-15843-x  [23]  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Kuang and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Perepechaenko, "Digital Signature Performance of a  New  Quantum  Safe  Multivariate  Polynomial  Public  Key  Algorithm," 2022 7th International Conference on Computer and  Communication Systems (ICCCS), 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' 419-424, doi:  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='1109/ICCCS55155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='9846785.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  [24]  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Kuang and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Perepechaenko, "Digital Signature Performance of a  New  Quantum  Safe  Multivariate  Polynomial  Public  Key  Algorithm," 2022 7th International Conference on Computer and  Communication Systems (ICCCS), 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' 419-424, doi:  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='1109/ICCCS55155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='9846785.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  [25]  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Kuang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Perepechaenko, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Toth and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Barbeau, "Benchmark  Performance of a New Quantum-Safe Multivariate Polynomial Digital  Signature Algorithm," 2022 IEEE International Conference on  Quantum Computing and Engineering (QCE), 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' 454-464, doi:  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='1109/QCE53715.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='00067.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  [26]  NIST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' https://csrc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='nist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='gov/Projects/post-quantum-cryptography/faqs  [27]  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Kuang and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Bettenburg, “Shannon Perfect Secrecy in a Discrete  Hilbert Space”, 2020 IEEE International Conference on Quantum  Computing and Engineering (QCE), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' 249-255, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  [28]  Kuang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=', Barbeau, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Quantum permutation pad for universal  quantum-safe cryptography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Quantum Inf Process 21, 211 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='1007/s11128-022-03557-y  [29]  The Cyber Security Hub™: Information Technology and Services,  London, UK, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  [30]  Kuang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' & Perepechaenko.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' A new symmetric homomorphic  functional encryption over a hidden ring for polynomial public key  encapsulation, under review at Scientific Reports, 2022  [31]  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Lou, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Kuang and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' He, "Entropy Transformation and Expansion  with Quantum Permutation Pad for 5G Secure Networks," 2021 IEEE  21st International Conference on Communication Technology (ICCT),  2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' 840-845, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='1109/ICCT52962.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='9657891.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  [32]  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Kuang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Lou, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' He, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' McKenzie and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Redding, "Pseudo  Quantum Random Number Generator with Quantum Permutation  Pad," 2021 IEEE International Conference on Quantum Computing  and  Engineering  (QCE),  2021,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  359-364,  doi:  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='1109/QCE52317.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='00053.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  [33]  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Kuang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Lou, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' He and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Conlon, "Quantum Safe Lightweight  Cryptography with Quantum Permutation Pad," 2021 IEEE 6th  International Conference on Computer and Communication Systems  (ICCCS),  2021,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  790-795,  doi:  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='1109/ICCCS52626.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='9449247.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  [34]  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Kuang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Lou, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' He, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Conlon "Quantum Secure Lightweight  Cryptography with Quantum Permutation Pad", Advances in Science,  Technology and Engineering Systems Journal, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' 6, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' 401-  405 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  [35]  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  Perepechaenko  and  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  Kuang,  "Quantum  Encrypted  Communication between Two IBMQ Systems Using Quantum  Permutation  Pad," 2022  11th  International  Conference  on  Communications, Circuits and Systems (ICCCAS), 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' 146-152,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='1109/ICCCAS55266.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='9824836.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  [36]  Kuang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=', Perepechaenko, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Quantum encryption with quantum  permutation pad in IBMQ systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' EPJ Quantum Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' 9, 26  (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='1140/epjqt/s40507-022-00145-y  [37]  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  Perepechaenko  and  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  Kuang,  "Quantum  Encrypted  Communication in IBMQ Systems Using Quantum Permutation  Pad," J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=',Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' 17, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' 12, December 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='org/  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='12720/jcm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='972-978.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  [38]  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' Lou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=', "Benchmark Performance of Digital QKD Platform  Using Quantum Permutation Pad," in IEEE Access, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content=' 10, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='  107066-107076, 2022, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
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+page_content='  [39]  Creative Commons, Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
+page_content='0 International (CC BY 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
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+page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf'}
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+1
+Sample-Efficient Unsupervised Domain Adaptation
+of Speech Recognition Systems:
+A case study for Modern Greek
+Georgios Paraskevopoulos Student Member, IEEE, Theodoros Kouzelis, Georgios Rouvalis, Athanasios
+Katsamanis Member, IEEE, Vassilis Katsouros Member, IEEE, Alexandros Potamianos Fellow, IEEE
+Abstract—Modern speech recognition systems exhibits rapid
+performance degradation under domain shift. This issue is
+especially prevalent in data-scarce settings, such as low-resource
+languages, where diversity of training data is limited. In this work
+we propose M2DS2, a simple and sample-efficient finetuning
+strategy for large pretrained speech models, based on mixed
+source and target domain self-supervision. We find that including
+source domain self-supervision stabilizes training and avoids
+mode collapse of the latent representations. For evaluation, we
+collect HParl, a 120 hour speech corpus for Greek, consisting
+of plenary sessions in the Greek Parliament. We merge HParl
+with two popular Greek corpora to create GREC-MD, a test-
+bed for multi-domain evaluation of Greek ASR systems. In our
+experiments we find that, while other Unsupervised Domain
+Adaptation baselines fail in this resource-constrained environ-
+ment, M2DS2 yields significant improvements for cross-domain
+adaptation, even when a only a few hours of in-domain audio
+are available. When we relax the problem in a weakly supervised
+setting, we find that independent adaptation for audio using
+M2DS2 and language using simple LM augmentation techniques
+is particularly effective, yielding word error rates comparable to
+the fully supervised baselines.
+Index Terms—Unsupervised Domain Adaptation, Automatic
+Speech Recognition, Multi-Domain Evaluation, Greek Speech
+I. INTRODUCTION
+Automatic Speech recognition (ASR) models have matured
+to the point where they can enable commercial, real-world
+applications, e.g., voice assistants, dictation systems, etc., thus
+being one of machine learning’s success stories. However,
+the performance of ASR systems rapidly deteriorates when
+the test data domain differs significantly from the training
+data. Domain mismatches can be caused by differences in
+the recording conditions, such as environmental noise, room
+reverberation, speaker and accent variability, or shifts in the
+target vocabulary. These issues are extenuated in the case of
+low-resource languages, where diversity in the training data
+is limited due to poor availability of high-quality transcribed
+audio. Therefore, specialized domain adaptation approaches
+need to be employed when operating under domain-shift.
+Unsupervised Domain Adaptation (UDA) methods are of
+special interest, as they do not rely on expensive annotation
+G. Paraskevopoulos is with the Graduate School of ECE, National Technical
+University of Athens, Athens, Greece
+G. Paraskevopoulos, T. Kouzelis, G. Rouvalis, A. Katsamanis, V. Katsouros
+are with the Institute for Speech and Language Processing, Athena Research
+Center, Athens, Greece
+A. Potamianos is with the Faculty of ECE, National Technical University
+of Athens, Athens, Greece
+of domain-specific data for supervised in-domain training.
+In contrast to supervised approaches, where the existence of
+labeled data would allow to train domain-specific models,
+UDA methods aim to leverage data in the absense of labels
+to improve system performance in the domain of interest [1],
+[2]. In the context of speech recognition the importance of
+UDA is extenuated, as the transcription and alignment pro-
+cess is especially expensive and time-consuming. Adaptation
+methods have been explored since the early days of ASR,
+at different levels of the system and different deployment
+settings [3]. UDA has been used to improve the robustness
+of ASR on a variety of recording conditions including far-
+field speech, environmental noise and reverberation [4], [5],
+[6]. Furthermore, UDA has been used for speaker adaptation,
+and to improve performance under speaker, gender and accent
+variability [7], [8]. UDA has also been employed for multilin-
+gual and cross-lingual ASR, in order to improve ASR models
+for low-resource languages [9], adapt to different dialects [10],
+and even train speech recognition systems for endangered
+languages [11].
+Classical speech adaptation techniques involve feature-
+based techniques, e.g., speaker normalization [12], feature-
+based approaches [13]–[15], or multi-condition training [16].
+Generally, traditional approaches require some knowledge
+about the target domain, and the domain mismatch, e.g.,
+regarding the noise and reverberation variability [17], and
+require specific engineering for each adaptation scenario.
+Modern ASR pipelines, increasingly rely on end-to-end
+neural networks, e.g., [18], [19], or large pretrained models
+with self-supervised objectives [20], [21]. The key approaches
+employed for UDA of end-to-end ASR models can be grouped
+in three categories, namely, teacher-student learning [10],
+domain adversarial training [22], and target domain self-
+supervision [23]. The benefit of these techniques is that they
+do not require any special knowledge about the source or
+the target domain. This makes end-to-end UDA approaches
+versatile and able to be utilized in a larger array of adaptation
+scenarios. In particular, adaptation through self-supervision
+has been shown to be a robust, simple and efficient technique
+for adaptation of state-of-the-art speech models [24].
+Here, we leverage in-domain self-supervision to propose
+the Mixed Multi-Domain Self-Supervision (M2DS2) finetun-
+ing strategy, enabling sample-efficient domain adaptation of
+wav2vec2 [20] based speech recognition models, even when
+available in-domain data are scarce. Our key contributions are
+arXiv:2301.00304v1  [cs.CL]  31 Dec 2022
+
+2
+TABLE I
+SUMMARY OF RELATED WORKS ON UNSUPERVISED DOMAIN ADAPTATION FOR ASR.
+Work
+Method
+Model
+Adaptation Setting
+Language
+[23], [25], [26]
+Teacher-Student
+Hard and soft labels
+Conformer RNN-T [27]
+Transformer CTC
+RNN-T [19]
+News speech, Voice search, Far-field,
+Telephony, YouTube
+English
+[4], [5]
+Teacher-Student
+Soft labels
+TDNN-LSTM [28]
+Noise, Far-field
+English
+[29]
+Teacher-Student
+Hard and soft labels
+NiN-CNN [30]
+Dialects
+Children speech
+Japanese
+[31]
+Teacher-Student
+Soft labels
+Streaming RNN-T [32]
+Multilingual
+English,
+Brazilian Portuguese,
+Russian,
+, Turkish,
+Nordic/Germanic
+[6], [33], [34]
+Domain Adversarial Training
+TDNN Kaldi [35], [36]
+DNN-HMM
+DNN-HMM
+Noise, Channel
+English
+[37]
+Domain Adversarial Training
+RNN-CTC [38]
+Far-field
+English
+[8], [39]
+Domain Adversarial Training
+TDNN Kaldi
+RNN-T
+Accent
+Mandarin
+[7], [40]
+Domain Adversarial Training
+DNN-HMM
+CNN-DNN
+Speaker, Gender,
+Accent
+English
+[9]
+Domain Adversarial Training
+DSN [41]
+Multilingual
+Hindi, Sanskri
+[24], [42]
+Continual Pre-Training
+wav2vec2 [20]
+Audiobooks, Accents,
+Ted Talks, Telephony,
+Crowd-sourced, Parlamentary speech
+English
+[43]
+Continual Pre-Training
+wav2vec2
+Cross-lingual
+Korean
+[11], [44]
+Continual Pre-Training
+XLSR-53 [21]
+wav2vec2
+Low resource languages
+Ainu
+Georgian, Somali,
+Tagalog, Farsi
+organized as follows:
+1) Inspired by recent advances on UDA for Natural Lan-
+guage Processing systems [45], we propose a finetuning
+strategy for speech models, where the self-supervised
+objective is based on a contrastive loss in Section III.
+Contrary to prior works, who leverage only in-domain
+self-supervision, we find that in this contrastive setting
+this leads to mode-collapse of the latent representations,
+and mixed source and target domain self-supervision
+is essential. We demonstrate this empirically in Sec-
+tion VII-B.
+2) We collect and curate HParl, the largest publicly avail-
+able1 speech corpus for Greek, collected from plenary
+sessions in the Greek Parliament between 2018 and
+2022. We establish a data collection, pre-processing
+and alignment pipeline that can be used for continuous
+data integration, as the parliamentary proceedings get
+regularly uploaded. We provide a detailed description of
+our data collection process and the dataset statistics in
+Section IV-A. HParl is merged in Section IV with two
+popular Greek corpora (Logotypografia and Common-
+Voice) to create GREC-MD, a testbed for multi-domain
+evaluation of ASR systems in Greek.
+3) We demonstrate that, while other baselines fail at UDA
+in our resource-constrained setting, M2DS2 can improve
+model performance in the target domain in multiple
+adaptation scenarios in Section VII. Specifical emphasis
+is given in the sample efficiency of our approach in Sec-
+1We plan to release this version of HParl under the CC BY-NC 4.0 license
+upon publication. The other corpora used in this work are available through
+their respective distributors.
+tion VII-A, where we demonstrate successful adaptation
+even when we reduce the available in-domain data.
+4) When we relax the problem to a weakly supervised
+adaptation setting, where some in-domain text is avail-
+able but the pairing between audio and text is unknown,
+we find that M2DS2 can be effectively combined with
+simple N-gram adaptation techniques to get compara-
+ble performance with the fully supervised baseline in
+Section VIII. Furthermore we find that a simple text
+augmentation approach, based on perplexity filtering of a
+large corpus can produce strong adaptation results, even
+for small amounts of in-domain text.
+Additionally, we provide a formulation of the UDA problem
+for ASR in Section II-A and link prior works to this formu-
+lation in Sections II-B, II-C and II-D. We provide detailed
+experimental settings for reproducibility in Section V, and
+an upper-bound estimation for UDA performance with fully
+supervised finetuning in Section VI.
+II. BACKGROUND
+We start by formally defining the Unsupervised Domain
+Adaptation (UDA) problem. Initially, we formulate the prob-
+lem in a classification setting and then we extend it for
+speech recognition. We then provide an overview of different
+adaptation approaches in the literature, and link each approach
+to the UDA problem formulation. Table I presents a summary
+of the key adaptation settings and applications that are ex-
+plored in the literature. We see, that a relatively small amount
+of methods, and their variants, is used to address multiple
+real-world ASR problems, for example, cross-lingual, accent,
+speaker and noise adaptation. Furthermore, while the majority
+
+3
+of the works focus on the English language, there is an effort
+to explore other popular languages, e.g., Mandarin, and under-
+resourced languages, e.g., Ainu, Somali etc.
+A. Problem Definition
+Formally, the problem of UDA can be defined as follows.
+Let X ⊆ Rn be a real-valued space that consists of n-
+dimentional feature vectors x ∈ X, and Y a finite set of
+labels y ∈ Y , i.e., Y = {1, 2, . . . , L}. Furthermore, assume
+two different distributions, i.e., the source domain distribution
+S(x, y) and the target domain distribution T (x, y), defined on
+the cartesian product X × Y .
+The goal is to train a model that learns a mapping between
+feature vectors xT to their respective labels yT for samples
+drawn from the target distribution (xT , yT ) ∼ T .
+At training time we have access to samples from the source
+distribution S(x, y) and the marginalized target distribution
+T (x), i.e., no target labels are provided. We define the training
+dataset D as the concatenation of the source and target training
+sets, D = (DS, DT ). DS and DT are defined as sequences of
+tuples, i.e.,
+DS = {(xi, yi) | (xi, yi) ∼ S(x, y), 1 ≤ i ≤ N}
+DT = {(xi, ∅) | xi ∼ T (x), 1 ≤ i ≤ M},
+(1)
+where we draw N samples from S(x, y) and M samples
+from T (x). Finally, we augment tuples in D with a domain
+indicator function:
+D = {(xi, y′
+i, 1i) | 1 ≤ i ≤ N + M}
+1i =
+�
+0
+if xi ∼ S(x),
+1
+if xi ∼ T (x).
+y′
+i =
+�
+yi
+if xi ∼ S(x),
+∅
+if xi ∼ T (x).
+(2)
+1) Unsupervised (Acoustic) Adaptation for ASR: The above
+definition can be directly extended in the case of speech
+recognition, with some modifications. In detail, we modify
+the feature space X, to be the set of (finite) sequences of
+real-valued feature vectors (xk)k∈N\{∞} ∈ X ⊆ (Rn)∗.
+Furthermore, the label space Y is modified to be the set
+of sequences (ym)m∈N\{∞}, where Y
+= ({1, 2, . . . , L})∗
+contains finite-length sequences over a finite lexicon. For
+CTC training we make the assumption that k > m for any
+sample (xk, ym), i.e., feature sequences are longer than their
+respective label sequences [46]. The rest of the definitions need
+no modifications.
+2) Unsupervised (Language) Adaptation for ASR: Adapta-
+tion for ASR systems can also be performed at the language
+level, i.e., the label space. In this setting, we assume that
+the target domain samples are drawn from the marginalized
+target distribution T (y). The target dataset DT now consists of
+tuples in the form (∅, yi), where yi is the label word sequence
+(ym)m∈N\{∞} for the i-th sample.
+3) Weakly supervised Adaptation for ASR: The last setting
+we explore is the case were both audio and language in-
+domain samples are available, but the mapping between them
+is unknown. This situation can be encountered in real-world
+settings, e.g., in the case in-domain audio and text are collected
+independently. For example consider the case where audio
+clips from news casts are collected, along with contemporary
+newspaper articles. Another example is the case where long
+audio clips alongside with transcriptions are available, but no
+fine-grained time alignments2. In this case the target domain
+samples are drawn independently from the marginalized dis-
+tributions T (x) and T (y), and the target dataset DT consists
+of tuples in the form (xi, ∅) and (∅, yi).
+B. Teacher-Student Models
+Teacher-Student learning or self-training, is one of the
+earliest methods in semi-supervised learning [47]–[49]. The
+key idea is to reduce the problem of unsupervised learning
+of the task at hand in the target domain to a supervised one.
+The general methodology is to train a teacher model gS using
+the labeled data in the source domain DS, and then use this
+for inference on the target domain to produce pseudolabels
+ˆyi = gS(xi), xi ∼ T (x). The target domain dataset DT is
+augmented with these silver labels, to contain tuples (xi, ˆyi).
+Finally, a student model gT is trained in a supervised fashion,
+using the augmented DT or a combination of DS and DT .
+This process is usually repeated, with the student model
+serving as the teacher model for the next iteration, until no
+further improvement is observed. More recently, soft target
+Teacher-Student learning has been explored for ASR [26],
+[31], [50], where the KL divergence between the teacher and
+student output label distributions is used as the loss function.
+Being trained only on the source domain data the teacher
+model is susceptible to error propagation. Filtering is a com-
+monly used technique to achieve the right balance between
+the size of the target domain used for training the student
+model and the noise in the pseudolabels. Confidence scoring
+based on the likelihood is usually applied, discarding those
+utterances for which the hypothesized labels are untrustworthy
+[51]. In [25] dropout is used to measure the model uncertainty.
+The agreement between model predictions with and without
+dropout are used for confidence scoring. In [23] a multi-task
+training objective with a confidence loss is applied to minimise
+the binary cross entropy between the estimated confidence and
+the binary target sequence. In order to learn more robust and
+generalizable features from the teacher model, Noisy Student
+Training (NST) has been proposed in [52]. The teacher models
+generates pseudolabels for DT while the student models are
+trained on a heavily augmented version of DT [52]. In [52],
+[53] the augmentation of the input target data is performed
+with SpecAugment [54], while in [29] a spectrum frequency
+augmentation is performed.
+In [4] Teacher-Student learning with soft labels is introduced
+for ASR to tackle noisy, far-field, and children speech. In
+2While a fully supervised in-domain dataset can be constructed in this
+case using long / forced alignment methods, this is not a focal point for the
+experimental part of this work.
+
+4
+[5], this approach is extended for LF-MMI based models and
+used for noisy, far-field and bandwidth adaptation. In [29] a
+weighted sum of hard and soft target cross entropy losses
+is used for Japanese dialects and children speech adaptation.
+Ramabhadran et al. [31] propose a self-adaptive distillation,
+and a method for distilling from multiple teachers that is
+applied across several multilingual ASR systems for different
+language groups. A comparison between soft and hard targets
+for RNN-T models [19] showed that soft targets perform better
+when both the teacher and student models have the same
+architecture. Otherwise, hard targets are superior [50].
+C. Domain Adversarial Training
+Domain Adversarial Training (DAT) was initially introduced
+for image classification [55]. The key idea is to train a
+model that learns deep features that solve the task at hand
+in the source domain, while being invariant with respect
+to the domain shift. Concretely, the model is trained end-
+to-end using a combination of the supervised task loss Lt,
+learned on DS, and the domain discrimination loss La, i.e.,
+L = Lt − αLa. The loss La is binary cross-entropy, trained
+for domain discrimination using the tuples (xi, 1i). Notice
+the − sign in the loss indicates adversarial learning, i.e., the
+model should learn features that cannot discriminate between
+domains, while solving the task.
+In [6] DAT is employed for noise adaptation on a noise
+corrupted version of WSJ [56] as the target dataset. Using the
+Aurora-4 [57] dataset which has labels associated to the noise
+type, Serdyuk et al. [33] train an adversarial noise classifier. In
+[8] and [39] DAT is utilized for accent adaptation for Mandarin
+and English respectively. Anoop C.S. et al. [9] propose DAT,
+to address the scarcity of data in low-resource languages which
+share a common acoustic space with a high-resource language,
+namely Sanskrit and Hindi. They empirically demonstrate the
+effectiveness of adversarial training, presenting experiments
+with and without the reversal of the domain classification loss.
+D. Leveraging In-domain Self-supervision
+These lines of work have roots in Natural Language Pro-
+cessing tasks [45], [58], and explore domain adaptation by
+leveraging the in-domain data DT for self-supervised learning.
+The core focus is domain adaptation of large pre-trained
+models, e.g., [59], and self-supervision is achieved by use
+of the pre-training self-supervised loss Ls. This process can
+either take part in stages, via continual pre-training [58], or by
+constructing a multitask objective L = Lt + αLs, as in [45].
+Continual Pre-Training (CPT) has been explored for adap-
+tation of ASR models. Robust wav2vec2 [24] explores the
+effectiveness of CPT for domain adaptation, indicating the
+importance of utilizing unlabeled in-domain data. In CASTLE
+[42], CPT is combined with an online pseudolabeling strategy
+for domain adaptation of wav2vec2. Cross-dataset evaluation
+for popular English speech corpora indicates that CPT helps
+to reduce the error rate in the target domain. In [43] and [11]
+CPT is utilized for cross-lingual adaptation of wav2vec2 for
+Korean and Ainu respectively. Notably for Ainu, which is an
+endagered language, CPT has resulted in significant system
+Fig. 1. Target-domain adaptation through self-supervision. In the left we see
+the general pre-training stage of XLSR-53 using the self-supervised loss Ls.
+General pre-training is performed on 56, 000 hours of audio in 53 languages.
+In the right, we see the proposed domain-adaptive finetuning stage, where the
+speech recognition task is learned using transcribed source domain data, while
+adaptation to the target domain is performed by including the self-supervised
+loss over (audio-only) source and target domain data
+improvement. DeHaven and Jayadev [44] compare CPT and
+pseudolabeling for adapting XLSR-53 to four under-resourced
+languages, i.e., Georgian, Somali, Tagalog and Farsi. They find
+that both approaches yield similar improvements, with CPT
+being the more computationally efficient approach.
+While CPT yields significant improvements in a variety of
+tasks, one common theme in these works is the assumption
+of hundreds or thousands of hours of available in-domain
+data, mostly from online resources, e.g., YouTube. This can be
+infeasible when we consider more niche adaptation settings,
+or possible privacy concerns, e.g., how would one collect
+1000 hours of psychotherapy sessions in Greek? In this work,
+we explore domain adaptation methods in a more resource-
+constrained environment.
+III. DOMAIN ADAPTATION THROUGH MULTI-DOMAIN
+SELF-SUPERVISION
+The proposed approach is based on end-to-end adaptation of
+a large pre-trained speech model during the finetuning phase,
+by including in-domain self-supervision. We extend UDALM
+[45], that has shown promise for NLP tasks, for adaptation of
+wav2vec2 based acoustic models, and specifically XLSR. We
+focus on the problem of UDA in the context of a low-resource
+language, i.e., Greek. The key finding of our exploration is
+that straight-forward extension of UDALM, i.e., by using only
+target domain self-supervision, underperforms in this setting,
+and use of both source and target domain data is essential for
+successful adaptation. In this section, first, we will present
+a quick overview of the XLSR-53 training procedure, and
+then we are going to outline the proposed domain adaptation
+approach, which is shown in Fig. 1.
+A. XLSR-53
+XLSR-53 [21] is a massively pre-trained speech model,
+trained on 56, 000 hours of multilingual speech, covering 53
+languages. The model is based on wav2vec2 [20], which is
+composed of a multi-layer convolutional feature encoder, that
+
+General Pretraining
+Finetuning
+Ls
+LCTC
+Ls
+Masked Transformer
+Masked Transformer
+XLSR
+XLSR
+MLS, CommonVoice and BABEL
+Source Domain
+Target Domain
+56.0000 of speech data from 53 languages5
+TABLE II
+THE GREC-MD CORPUS. WE CAN SEE THE DURATION OF EACH SPLIT IN H O U R S:M I N U T E S:S E C O N D S FORMAT, AS WELL AS THE NUMBER OF
+SPEAKERS FOR EACH OF THE SUB-CORPORA.
+Dataset
+Domain
+Speakers
+Train
+Dev
+Test
+Total Duration
+HParl
+Public (political) speech
+387
+99:31:41
+9:03:33
+11:12:28
+119:47:42
+CV
+Crowd-sourced speech
+325
+12:16:17
+1:57:44
+1:59:19
+16:13:20
+Logotypografia
+News casts
+125
+51:58:45
+9:08:35
+8:59:22
+70:06:42
+Total
+-
+713
+163:46:43
+20:09:52
+22:11:44
+206:08:19
+extracts audio features zt from the raw audio, and a trans-
+former context encoder that maps the latent audio features to
+the output hidden states ct. Each latent feature zt corresponds
+to 25 ms of audio with stride 20 ms. A contrastive objective Lc
+is used for pre-training. For this, product quantization [60] is
+applied to the features zt, and then a discrete approximation of
+zt is obtained by sampling from a Gumbel-softmax distribution
+[61], to obtain discrete code vectors qt, organized into G = 2
+codebooks with V
+= 320 vocabulary entries each. The
+contrastive loss aims to identify the correct code vector for
+a given time step, among a set of distractors Qt, obtained
+through negative sampling from other timesteps. To avoid
+mode collapse, a diversity loss Ld is included by maximizing
+the entropy over the averaged softmax distribution over the
+code vector entries ¯pg. The total loss is:
+Ls = −log
+es(zt,qt)
+�
+˜q∼Qt es(zt,˜q)
+�
+��
+�
+Contrastive Loss
+Diversity Loss
+�
+��
+�
+− 1
+GV
+G
+�
+g=1
+V
+�
+v=1
+¯pg,vlog(¯pg,v)
+(3)
+B. Domain Adaptive finetuning for Contrastive Learning of
+Speech Representations
+Fig. 1 shows the proposed finetuning process. The key
+intuition is that we want the model to synergistically learn
+the task at hand (in our case ASR), while being adapted to
+the target domain by in-domain self-supervision. In the left
+we see the general pre-training stage of XLSR-53, which is
+pre-trained on 56K hours of multilingual audio corpora using
+the contrastive pre-training objective. In the right we see the
+proposed finetuning stage, which is inspired by [45].
+During finetuning we form a mixed objective function:
+L = LCT C(xs, ys) + αLs(xs) + βLs(xt),
+(4)
+where (xs, ys) ∼ S(x, y), xt ∼ T (x), LCT C is the CTC
+objective function, optimized using transcribed source domain
+data, and Ls is the contrastive loss from Eq. (3). We scale the
+contribution of each term using hyper-parameters α and β.
+Note that contrary to [45], who use only in-domain self-
+supervision, we leverage both source and target domain sam-
+ples for the mixed self-supervision. We find that this is essen-
+tial in our case to avoid mode collapse, i.e., the model using
+only a few of the available discrete code vectors. Simultaneous
+self-supervision on both the source and target data alleviates
+mode collapse by anchoring the target code vector space to
+have a similar structure as the source code vectors.
+Hence we refer to this approach as Mixed Multi-Domain
+Self-Supervision (M2DS2).
+IV. THE GREC-MD CORPUS
+For our experiments we compose a speech corpus for the
+Greek language, that is suitable for multi- and cross-domain
+evaluation. The GREC-MD corpus contains 206 hours of
+Greek speech. Audio is segmented into individual utterances
+and each utterance is paired with its corresponding tran-
+scription. Table II summarizes the included sub-corpora, as
+well as the train, development and test splits. The dataset is
+constructed with three core principles in mind:
+1) Data Volume: We collect the largest publicly available
+speech recognition corpus for the Greek language, able
+to scale to hundreds of hours of transcribed audio.
+2) Temporal Relevance: Language changes over time. We
+aim at an up-to-date corpus that encompasses the latest
+terms and topics that appear in daily speech.
+3) Multi-Domain Evaluation: Single domain evaluation
+can lead to misleading estimations of the expected
+performance for ASR models. For example, state-of-
+the-art ASR models [27] achieve under 5% Word Error
+Rate (WER) on Librispeech [62] test sets, but this is
+an over-estimation of system performance in the field.
+This is extenuated when considering different acoustic
+conditions or terminology. We consider multi-domain
+evaluation essential when developing and deploying
+real-world ASR models.
+To satisfy the first two points, we collect data from a public,
+continuously updated resource, i.e., the Hellenic Parliament
+Proceedings, where recordings of the parliamentary sessions
+are regularly uploaded. The benefit of using this resource is the
+straight-forward collection of a continuously growing, multi-
+speaker corpus of transcribed audio that is always up-to-date,
+as the parliamentary discussions revolve around current affairs.
+We refer to this corpus as HParl. For the multi-domain evalua-
+tion, we merge HParl with two publicly available corpora, that
+have different acoustic and language characteristics. We refer
+to the merged, multi-domain corpus as GREC-MD. In this
+Section, we will describe the collection and curation process
+of HParl, and present the relevant statistics for the experiments.
+TABLE III
+PLENARY SESSIONS INCLUDED IN HPARL. THE HOURS COLUMN REFERS
+TO THE RAW (UNSEGMENTED) HOURS OF COLLECTED AUDIO.
+Start date
+End date
+#Sessions
+Hours
+15-02-2022
+01-03-2022
+10
+55
+18-01-2019
+01-02-2019
+10
+52
+28-03-2019
+10-05-2019
+20
+108
+10-12-2018
+21-12-2018
+10
+88
+
+6
+Fig. 2. Overview of the Hellenic Parliament Chamber. The chamber has an
+amphitheatrical shape and can accomodate approximately 400 − 450 people.
+The positions of the key speakers, i.e., current speaker and the parliament
+president are annotated in the image.
+A. Collection and Curation of HParl
+Modern technological advances allow for more direct gov-
+ernment transparency, through the commodification of storage
+and internet speeds. In this spirit, the records of plenary ses-
+sions of the Hellenic Parliament are made publicly available,
+for direct access through a webpage3. The available video
+recordings date back to 2015. For each plenary session, a
+video recording is uploaded, along with a full transcription
+that is recorded verbatim, and in real time by the parlia-
+ment secretaries. For the creation of HParl, we build a web-
+crawler that can traverse and download the video recordings,
+along with the transcriptions from the official website. The
+collection process is parallelized over multiple threads, and
+parameterized by a range of dates and, optionally, a target
+corpus size in GB or in hours. For this version of HParl, we
+collect the plenary sessions in four date ranges, as described in
+Table III. The majority of the collected sessions are from 2019,
+but we also include sessions from 2018 and 2022 to include
+coverage of different topics. The individual components of the
+HParl curation pipeline are: Audio Pre-processing, Text Pre-
+processing, Alignment, Post-processing, and dataset Splitting.
+1) Audio Pre-processing: Fig. 2 shows the layout of the
+Hellenic Parliament Chamber. Plenary sessions mainly take
+place in this room, or in the secondary House Chamber that
+has similar setup but is smaller in size. Because of the room
+and microphone characteristics, the captured audio in the
+video streams contains reverberation, due to sound reflections.
+We employ a light preprocessing pipeline, by passing the
+input video streams through FFmpeg, and converting them to
+monophonic, lossless audio format at 16000 Hz sampling rate.
+The resulting audio is not passed through any de-reverberation
+or speech enhancement software. The resulting audio files have
+a minimum, average and maximum duration of 6 minutes, 6
+hours and 16 hours respectively.
+2) Text Pre-processing: The text files contain full, word-
+by-word transcription of the speeches and questions asked by
+members of the audience, as well as extra annotations made
+by the parliament secretaries. Some annotations are relevant,
+3https://www.hellenicparliament.gr/en/
+i.e., the speaker name, while others are plain text descriptions
+of events happening during the session and need to be filtered
+out (e.g., “The session is interrupted for a 15 minute break”).
+We use a rule-based system, based on regular expressions,
+that filters the unnecessary information, keeping only the
+transcriptions and the speaker names. The speaker labels are
+created by transliterating their names and roles from Greek
+to Greeklish using the “All Greek to Me!” tool [63]. Text is
+lower-cased and normalized to remove multiple whitespaces.
+The result is a text file containing the raw transcriptions, and
+a mapping from speaker labels to their respective text parts.
+3) Aligment and Segmentation: The primary challenge of
+exploiting the plenary sessions for ASR purposes is the length
+of the plenary recordings, as their durations vary from 6
+minutes to 16 hours in length. However, data samples used to
+train ASR are generally less than 30 seconds long. Computa-
+tional challenges have limited the length of training utterances
+for HMM-GMM models [64], and continue to do so in the
+contemporary neural network models. Therefore, we need to
+segment the sessions into smaller pieces more suitable for ASR
+training. A second challenge is posed by mismatches between
+audio and transcripts. Parliamentary proceedings do not fully
+capture everything that is said during the parliamentary ses-
+sions, and do not account for speech disfluencies.
+In order to obtain smaller, clean segments, that are suit-
+able for ASR training we follow the segmentation procedure
+proposed by [65]. Initially the raw recordings are segmented
+into
+30 second segments and the transcriptions are split
+into smaller segments of approximately 1000 words called
+documents. Each segment is decoded using a seed acoustic
+model trained on the Logotypografia corpus [66] and a 4-
+gram biased LM trained on the corresponding transcription
+of each recording. The best path transcript of each segment
+is obtained and paired with the best matching document via
+TF-IDF similarity. Finally each hypothesis is aligned with the
+transcription using Smith-Waterman alignment [67] to select
+the best matching sub-sequence of words. The above method
+yields a list of text utterances, with their corresponding start
+and end times in the source audio files. The procedure yields
+120 hours of useable segmented utterances out of the original
+303 hours of raw audio, or a ratio of 39.6%.
+4) Post-processing: After the segments are extracted, we
+filter out extremely short segments (less than 2 words).
+Moreover, the iterative alignment algorithm may replace some
+intermediate words with a <spoken-noise> tag. When this
+tag is inserted, we match the surrounding text with the raw
+transcriptions and re-insert the missing words. Furthermore,
+we match each segment to its corresponding speaker label.
+Segments without a speaker label are discarded. Lastly, speak-
+ers are associated to their gender based on name suffixes, using
+a simple, Greek language-specific, rule: Speaker names which
+end in a(α), h(η), w(ω) or is(ις) are classified as female, while
+the rest as male. We format the segments, speaker and gender
+mappings in the standard folder structure used by the Kaldi
+speech recognition toolkit [36].
+5) Data Splitting: We provide an official train - devel-
+opment - test split. The development set contains 3 plenary
+sessions, one from 2018, one from 2019 and one from 2022,
+
+Current Speaker
+Parliament President7
+resulting to 9 hours of segmented speech. Similarly, the test
+set contains one session from each year, resulting to 11 hours
+of segmented speech. The rest 99 hours of segmented speech
+are assigned to the training set.
+B. Including corpora from different domains
+We merge HParl with two publicly available corpora to
+create GREC-MD for multi-domain evaluation.
+1) Common Voice: Common Voice (CV) [68] is a crowd-
+sourced, multi-lingual corpus of dictated speech, created by
+Mozilla. The data collection is performed by use of a web
+app or an iPhone app. Contributors are presented with a
+prompt and are asked to read it. The prompts are taken from
+public domain sources, i.e., books, wikipedia, user submitted
+prompts and other public corpora. The maximum prompt
+length is 15 words. A rating system is built into the plat-
+form, where contributors can upvote or downvote submitted
+<audio,transcript> pairs. A pair is considered valid, if
+it receives two upvotes. Speaker independent train, develop-
+ment and test splits are provided. The dataset is open to the
+research community, released under a permisFsive Creative
+Commons license (CC0). In this work, we use version 9.0
+of CV, accessed on April 27, 2022. We keep only the valid
+utterances, i.e., 16 hours of speech from 325 contributors
+(19 − 49 years old, 67% male / 23% female).
+2) Logotypografia: Logotypografia [66] is one of the first
+corpora for Large Vocabulary Continuous Speech Recognition
+in Greek. The dataset contains 33, 136 newscast utterances, or
+72 hours of speech. The utterances were collected from 125
+speakers (55 male, 70 female), who were staff of the popular
+“Eleftherotypia” newspaper in Greece, under varied acoustic
+conditions. Approximately one third of the utterances were
+collected in a sound proof room, one third in a quiet room and
+the last third in an office room. The average utterance duration
+is 7.8 seconds. The transcriptions contain several speech and
+non-speech events (e.g., <cough>), lower-cased Greek words
+and stress marks. Numbers are expanded to full words. We
+use the whole dataset, and perform light preprocessing in
+the transcriptions, by discarding the annotated events and
+punctuation.
+We hence refer to each dataset by the abbreviations: HParl:
+HP, CommonVoice: CV, Logotypografia: LG.
+V. EXPERIMENTAL SETTINGS
+For our experiments we use the following hyper-parameter
+settings, unless explicitly stated otherwise. For model training,
+we use AdamW optimizer [69] with learning rate 0.0003. We
+apply warmup for the first 10% of the maximum training
+steps, and a linear learning rate decay after that. Models
+are finetuned for a maximum of 10000 steps. For speech
+recognition training, we make use of the Connectionist Tem-
+poral Classification (CTC) loss [70], optimized using the
+available transcribed data in each scenario. Validation runs
+every 500 steps on the development set, and early stopping
+is employed on the development CTC loss with patience 5.
+Batch size is set to 8 during finetuning for all scenarios,
+except for M2DS2. In the case of M2DS2 we create mixed
+batches of size 12, containing 4 transcribed source domain
+samples and 8 unlabeled target domain samples and train
+for 10, 000 CTC updates. For memory reasons we split the
+mixed batches in mini-batches of 4 and interleave them during
+model training. Gradients are accumulated over 3 interleaved
+batches. For the self-supervised objective, we create masks
+of maximum timestep length 10, with masking probability
+0.4. We weigh the contributions of the source and target
+domain contrastive objectives, and bring them to the same
+order of magnitude as the CTC loss, by setting α = 0.01 and
+β = 0.02. The convolutional feature encoder is kept frozen
+for all experiments. Our code is based on the huggingface 4
+implementation of XLSR. For all experiments we resample
+the audio files to 16 kHz and downsample to single channel
+audio. We exclude utterances in the training set that are longer
+than 12 seconds. All experiments are run on a single NVIDIA
+RTX 3090 GPU, with mixed precision training.
+For the Language model training, we create a large corpus
+for the Greek language using a subset of the Greek part of CC-
+Net [71] (approximately 11 billion tokens) and combine it with
+1.5 billion tokens from the Greek version of Wikipedia and the
+Hellenic National Corpus (HNC) [72]. During preprocessing,
+we remove all punctuation and accents, deduplicate lines and
+convert all letters to lowercase. We will refer to this corpus as
+the Generic Greek Corpus (GGC). We train a 4-gram language
+model on GGC using KenLM [73] and prune bigrams, trigrams
+and four-grams with counts less than 3, 5 and 7 respectively.
+We incorporate the n-gram LMs at inference time using the
+pyctcdecode framework5. We use language model rescoring
+over a beam search decoder with 13 beams.
+The evaluation metric is the Word Error Rate (WER) over
+the target test set. For assessing the adaptation effectiveness we
+also report the relative WER improvement over the unadapted
+baseline in appropriate scenarios, which is defined in Eq. (5).
+We refer to this metric as Relative Adaptation Improvement
+(RAI) for the rest of this paper:
+RAI = −WERadapted − WERunadapted
+WERunadapted
+× 100%
+(5)
+The minus sign is included, so that RAI takes negative
+values when the adaptation fails, i.e., when WERunadapted <
+WERadapted.
+TABLE IV
+ASR PERFORMANCE OF XLSR-53 OVER THE THREE CORPORA FOR FULLY
+SUPERVISED IN-DOMAIN FINETUING (WER)
+Dataset
+LM
+No LM
+4g GGC
+HP
+26.21
+15.64
+CV
+29.33
+9.52
+LG
+31.94
+26.45
+VI. SUPERVISED IN-DOMAIN TRAINING
+In the first set of experiments, we explore the performance
+of supervised finetuning of XLSR-53 for each domain. This
+4https://huggingface.co/docs/transformers/
+5https://github.com/kensho-technologies/pyctcdecode
+
+8
+TABLE V
+M2DS2 PERFORMANCE USING GREEDY DECODING FOR UDA BETWEEN HP, CV, AND LG. A → B INDICATES THAT A IS THE SOURCE DOMAIN AND B IS
+THE TARGET DOMAIN. (G) INDICATES GREEDY DECODING. (LM) INDICATES BEAM SEARCH WITH LM RESCORING. WE REPORT THE WER ON THE
+TARGET TEST SET, AS WELL AS THE RAI (%) OVER THE SO (UNADAPTED) BASELINE. WER: LOWER IS BETTER. RAI: HIGHER IS BETTER.
+Method
+SO (G)
+CPT (G)
+PSL (G)
+M2DS2 (G)
+SO (LM)
+CPT (LM)
+PSL (LM)
+M2DS2 (LM)
+Setting
+WER
+WER
+RAI
+WER
+RAI
+WER
+RAI
+WER
+WER
+RAI
+WER
+RAI
+WER
+RAI
+HP → CV
+55.9
+59.68
+−6.8
+55.3
+1.2
+52.95
+5.3
+25.26
+26.44
+−4.7
+24.24
+4.0
+18.35
+27.4
+HP → LG
+48.65
+52.63
+−8.2
+57.68
+−18.6
+58.99
+−21.3
+30.34
+32.27
+−6.4
+39.32
+−29.6
+32.58
+−7.4
+LG → CV
+59.57
+66.43
+−13.4
+81.90
+−39.8
+51.31
+12.4
+25.96
+31.51
+−21.4
+52.05
+−100.5
+17.30
+33.4
+LG → HP
+62.13
+67.51
+−8.7
+71.46
+−15.0
+60.09
+3.3
+31.48
+31.58
+−0.3
+45.36
+−44.1
+31.36
+0.4
+CV → LG
+69.55
+71.12
+−2.3
+71.34
+−2.6
+63.40
+8.8
+50.80
+52.40
+−3.2
+48.68
+4.2
+36.93
+27.3
+CV → HP
+70.72
+73.83
+−4.4
+78.05
+−10.4
+68.70
+2.9
+52.09
+52.18
+−0.2
+54.82
+−5.2
+41.88
+19.6
+will give an upper bound estimation for UDA performance.
+We finetune XLSR-53 on CV, HP and LG (separately) and
+perform in-domain evaluation on the respective test sets.
+Results are summarized in Table IV. The first row indicates the
+performance of greedy decoding, while in the second row we
+report the performance of the beam search decoder, rescored
+using the scores of the 4-gram GGC language model. We
+observe that the greedy decoding performance is under 30
+WER for both HP and CV, while for LG we achieve ∼ 32
+WER. This makes sense, as LG is the most diverse dataset,
+with respect to the included acoustic conditions. Furthermore,
+we observe that the incorporation of a language model results
+in an impressive WER reduction on CV, followed by HP and
+then LG. While CV includes relatively simple phrases with
+common vocabulary, HP and LG contain more specialized
+terminology.
+VII. UNSUPERVISED DOMAIN ADAPTATION USING
+IN-DOMAIN AUDIO
+Here, we evaluate the effectiveness of M2DS2 for UDA.
+We compare with three baselines:
+1) Source Only Training (SO): We perform supervised
+finetuning of XLSR-53 (CTC) using only the source-
+domain data, and run decoding on the target domain
+test set. No in-domain data are used for adaptation.
+2) Continual Pre-Training (CPT): We perform a pre-
+training phase using the loss in Eq. (3) on the target
+domain train set, to create adapted versions of XLSR.
+Pre-training is run for 20000 steps with batch size
+4. Only the audio is used, without transcriptions. The
+adapted checkpoints are then finetuned by use of CTC
+loss on the source domain transcribed data. Evaluation
+is performed on the target test set.
+3) Pseudolabeling (PSL): We finetune XLSR-53 using the
+source domain data with CTC loss. Then we run infer-
+ence on the source model, to extract silver transcriptions
+for the target domain training set. We use the silver
+transcriptions for supervised finetuning on the target
+domain.
+In Table V we compare M2DS2 with the SO, CPT and
+PSL baselines for six adaptation scenarios, i.e., cross dataset
+evaluation between the three datasets in GREC-MD. The left
+half corresponds to greedy decoding, while for the right half
+we use the 4-gram LM trained on GGC. First, we observe
+the SO model performance. The SO models are the finetuned
+Fig. 3. Performance of M2DS2 (blue line) for the LG → CV setting, when
+reducing the amount of available target samples to 50%, 25%, and 10% of
+the original dataset (horizontal axis). SO performance is indicated with the
+orange line. Vertical axis: WER, Horizontal Axis: target audio percentage
+(100% → 0%)
+models from Table IV, evaluated in out-of-domain settings.
+We see that out-of-domain evaluation results in a large perfor-
+mance hit, e.g., while in the CV9 → CV9 in-domain setting
+we achieve 29.33 WER, in the CV9 → HP out-of-domain
+setting we get 69.55 WER. This confirms that for real-world
+ASR tasks, multi-domain evaluation is of essence. Second, we
+observe that in most adaptation scenarios both CPT and PSL
+fail to surpass the SO (unadapted) baseline. In the case of CPT,
+we hypothesize that is due to the relatively data constrained
+version of our setting. In the best-case scenario, we have 99
+hours of available target domain audio, which is not enough
+to perform a discrete CPT stage. Note that most of works in
+the literature use ∼ 1000 hours of target audio for CPT. In
+the case of PSL, the poor performance is due to the quality
+of the silver labels created by the seed model. While the
+performance would improve with more elaborate approaches
+(e.g., confidence filtering), in challenging adaptation scenarios
+PSL approaches are limited by the SO model’s performance.
+Lastly, we observe that M2DS2 is the only approach among
+our baselines that manages to achieve a positive RAI in most
+adaptation scenarios, by consistently outperforming the SO
+baseline by significant margins. This is exaggerated when
+we include a LM during inference. One exception in this
+pattern is the HP → LG scenario, where the SO baseline
+achieves the best performance. We attribute this to the fact that
+we performed minimal hyper-parameter tuning during model
+development.
+
+65
+60
+WER
+55
+50
+100%
+90%
+80%
+70%
+60%
+50%
+40%
+30%
+20%
+10%
+0%
+Percentage of In-Domain Audio Data9
+A. The sample efficiency of M2DS2
+One key observation in the literature, and in our experiments
+is that CPT requires a large amount of un-transcribed target
+domain audio. This raises the question, can we leverage self-
+supervision for domain adaptation in data constrained settings?
+In Fig. 3 we evaluate the performance of M2DS2, when
+we reduce the amount of target domain audio. Specifically
+we focus on the scenario of LG → CV. The full training
+corpus of CV contains 12 hours of audio. We train M2DS2
+with 50%, 25% and 10% of the available samples, or 6, 3
+and 1.2 hours of audio respectively, and plot the resulting
+WER on the target (CV) test set. In all cases, the full source
+(LG) training corpus is used. We observe that M2DS2 achieves
+lower WER than the SO baseline, even with only 3 hours of
+target domain audio. While CPT can suffer from catastrophic
+forgetting, as most multi-stage training approaches, M2DS2
+avoids this issue, being a single-stage approach with a mixed
+task-specific and self-supervised objective. This provides a
+promising avenue for adaptation, when collection of in-domain
+recordings is expensive, or infeasible.
+(a) Only target domain self-supervision
+(b) Target and source domain self-supervision
+Fig. 4. T-SNE scatter plots of code vectors extracted from M2DS2 without
+source domain self-supervision (top) and with source domain self-supervision
+(bottom) for LG (red) and CV (teal)
+B. The importance of Multi-Domain Self-Supervision
+In Section III-B we argue that it is essential to include both
+source and target domain data for the self-supervised objective
+of M2DS2. To illustrate the effect of this approach, we train
+two versions of M2DS2 for the LG → CV scenario. For the
+TABLE VI
+LANGUAGE ADAPTATION OF THE M2DS2 LG → CV MODEL, USING
+BIASED AND AUGMENTED LMS. WE USE THE VARIANT OF THE MODEL
+TRAINED WITH 3 HOURS OF IN-DOMAIN AUDIO. WE VARY THE AMOUNT
+OF IN-DOMAIN TEXT DATA FROM 752K TOKENS TO 38K TOKENS.
+Biased LM
+Augmented LM
+100%
+11.22
+12.84
+50%
+15.13
+15.05
+25%
+20.84
+16.64
+10%
+27.75
+18.47
+5%
+33.04
+19.31
+Baseline (M2DS2 + Generic LM)
+20.7
+Fig. 5. Language-only adaptation for LG → HP using the SO model finetuned
+on LG. In-domain text data range from 11M tokens (left) to 110K tokens
+(right). Blue/dashed: Baseline with generic LM. Purple/circles: Biased LM.
+Orange/diamonds: Augmented LM.
+first version we set α = 0.01, while for the second we set
+α = 0, removing the second term of Eq. (4). We extract the
+code vectors for the first 100 samples of both LG and CV, and
+flatten them across the time steps , resulting to 60000 × 768
+code vectors corresponding to individual timesteps. We plot
+these code vectors using T-SNE [74] in Fig. 4 for both models.
+We see that when we do not include the source domain self-
+supervision, the code vector space collapses in a few tight
+clusters, and most audio segments correspond to just a few
+code vectors. This is a visual clue that indicates the mode
+collapse problem. When we include the source domain term,
+we see that the that the code vector space has more structure,
+and coverage of the space is more complete, both for CV
+(target domain) and LG (source domain). Experimentally we
+train M2DS2 with α = 0 for all source / target domain pairs
+and we find that the mode collapse is destructive for target
+domain performance. During our experiments we got WER in
+the range 80−99, indicating failure to converge to acceptable
+solutions across all scenarios. The simple inclusion of both
+source and target domain self supervision stabilizes training,
+avoids mode collapse and leads to successful unsupervised
+adaptation between domains.
+VIII. UNSUPERVISED AND WEAKLY SUPERVISED
+LANGUAGE ADAPTATION
+When small amounts of in-domain textual data are avail-
+able, simple N-gram LM adaptation techniques can be very
+effective. In this brief set of experiments, we first explore
+the unsupervised language adaptation setting, where no in-
+
+.
+:
+:·
+.
+！
+.
+.
+.0
+C
+·
+.
+.
+.
+.
+.
+.
+008
+:
+.80
+·
+.
+o43
+41
+39
+37
+WER
+35
+33
+31
+29
+100%
+90%
+80%
+70%
+60%
+50%
+40%
+30%
+20%
+10%
+0%
+Percentage of In-Domain Text Data10
+TABLE VII
+CLOSING THE GAP BETWEEN SO TRAINING AND FULLY SUPERVISED
+TRAINING FOR THE LG → CV ADAPTATION SCENARIO USING M2DS2,
+WITH VARYING AMOUNTS OF AVAILABLE UNPAIRED IN-DOMAIN AUDIO
+AND TEXT. (U): UNSUPERVISED ACOUSTIC OR LANGUAGE ADAPTATION.
+(W): WEAKLY SUPERVISED ADAPTATION.
+Method
+#Audio (h)
+#Tokens
+LM
+WER
+SO (U)
+-
+-
+N/A
+59.57
+M2DS2 (U)
+3
+-
+N/A
+57.31
+M2DS2 (U)
+12
+-
+N/A
+51.31
+SO (U)
+-
+-
+Generic
+25.96
+SO (U)
+-
+38, 632
+Augmented
+24.67
+SO (U)
+-
+751, 953
+Augmented
+20.46
+M2DS2 (U)
+3
+-
+Generic
+20.7
+M2DS2 (U)
+12
+-
+Generic
+17.3
+M2DS2 (W)
+3
+38, 632
+Augmented
+19.31
+M2DS2 (W)
+12
+38, 632
+Augmented
+16.29
+M2DS2 (W)
+3
+751, 953
+Augmented
+12.84
+M2DS2 (W)
+12
+751, 953
+Augmented
+10.61
+Supervised
+12
+751, 953
+Generic
+9.52
+Supervised
+12
+751, 953
+Augmented
+7.94
+domain audio is used, and then we relax the problem to
+the weakly supervised setting, where M2DS2 is combined
+with the adapted N-Gram LMs. These settings are described
+in Sections II-A2 and II-A3 respectively. We explore two
+approaches for LM adaptation: biased LMs, and in-domain
+data augmentation. To create biased LMs, we train a 4-gram
+LM on the available in-domain data. Then we replace the
+generic LM trained on GGC. For LM data augmentation we
+follow a perplexity filtering approach similar to [71]. We first
+train a biased LM using available target domain text, and
+then use it to calculate the perplexity of each line in the
+GGC corpus. We keep the 10% of the lines with the lowest
+perplexity. Then we train a 4-gram LM on the augmented “in-
+domain” corpus and use it for inference.
+Fig. 5 shows the performance of the SO LG → HP model
+with biased and augmented LMs, as we reduce the amount
+of available in-domain text data from 100% to 1% of the
+in-domain transcriptions (11B tokens to 110K tokens respec-
+tively). As a baseline we include the LG → HP SO model in
+combination with the generic LM trained on GGC. We observe
+that the use of biased LMs can lead to successful adaptation,
+when an adequate amount of in-domain text data is available.
+On the other hand the LM augmentation approach results to
+successful augmentation, even with very small amounts of in-
+domain text.
+In Table VI we see the results of LM adaptation, combined
+with the M2DS2 LG → CV model. To demonstrate the sample
+efficiency of the approach, we use the variant that was trained
+using only 25% of the target domain audio (3 hours). We
+compare with M2DS2 combined with the 4-gram GGC LM for
+inference. We draw similar conclusions, i.e., use of biased LMs
+performs well for sufficient text data. When we use augmented
+LMs we can leverage very small amounts of in-domain text.
+IX. DISCUSSION & CONCLUSIONS
+In this work, we have explored Unsupervised and Weakly
+Supervised Domain Adaptation of ASR systems in the con-
+text of an under-resourced language, i.e., Greek. We focus
+on domain adaptation through in-domain self-supervision for
+XLSR-53, a state-of-the-art multilingual ASR model. Specif-
+ically, we adopt a mixed task and self-supervised objective,
+inspired from NLP, and show that using only in-domain self-
+supervision can lead to mode collapse of the representa-
+tions created by the contrastive loss of XLSR-53. Therefore,
+we propose the use of mixed task and multi-domain self-
+supervision, M2DS2, where the contrastive loss leverages both
+the source and target domain audio data. For evaluation we
+create and release HParl, the largest to-date public corpus
+of transcribed Greek speech (120 hours), collected from the
+Greek Parliamentary Proceedings. HParl is combined with two
+other popular Greek speech corpora, i.e., Logotypografia and
+CommonVoice, for multi-domain evaluation.
+In our experiments, we find that while most UDA baselines
+fail in our low-resource setting, the proposed mixed task
+and multi-domain self-supervised finetuning strategy yields
+significant improvements for the majority of adaptation sce-
+narios. Furthermore, we focus our ablations on showcasing
+the sample efficiency of the proposed finetuning strategy,
+and demonstrating the necessity of including both source
+and target domain data for self-supervision. Finally, we show
+that M2DS2 can be combined with simple language model
+adaptation techniques in a relaxed weakly supervised setting,
+where we achieve significant performance improvements with
+a few hours of in-domain audio and a small, unpaired in-
+domain text corpus.
+More concretely, in Table VII we present a summary of
+the discussed unsupervised and weakly supervised adaptation
+combinations, for different amounts of available in-domain
+audio and text. Note that for the weakly supervised scenarios,
+the in-domain audio and text are unpaired. We see, that when
+no in-domain data are available, including an n-gram LM
+trained on large corpora is recommended. Furthermore, when
+in-domain audio is available, following a mixed multi-domain
+finetuning strategy using M2DS2 can yield significant WER
+reductions, even for a few hours of audio. When small amounts
+of in-domain text is available, using a corpus augmentation
+strategy, e.g., perplexity filtering, can produce adapted LMs
+and yield small improvements to the final WER. In the case
+of sufficient amounts of unpaired in-domain text and audio,
+independent adaptation of XLSR-53 using the audio data and
+the n-gram LM using the text data can yield comparable
+performance with a fully supervised finetuning pipeline.
+X. FUTURE WORK
+In the future we plan to explore the effectiveness of the
+proposed adaptation strategy for other languages, and different
+adaptation settings, e.g., accent or cross-lingual adaptation.
+Of special interest is the investigation of the effectiveness
+of our approach for endagered languages, e.g., Pomak. Fur-
+thermore, we plan to explore the combination of in-domain
+self-supervision, when combined with other popular UDA
+techniques, e.g., teacher student models, adversarial learning,
+and data augmentation approaches. On the language adaptation
+side, we plan to explore multi-resolution learning, which has
+
+11
+shown promise for ASR [75], and investigate more elaborate
+end-to-end weakly supervised adaptation methods. Finally, we
+plan to expand our study in a multimodal setting, where both
+audio and video are available, e.g., lip reading.
+REFERENCES
+[1] Mingsheng Long, Yue Cao, Jianmin Wang, and Michael Jordan, “Learn-
+ing transferable features with deep adaptation networks,” in Proc. ICML.
+PMLR, 2015, pp. 97–105.
+[2] Yaroslav Ganin, Evgeniya Ustinova, Hana Ajakan, Pascal Germain,
+Hugo Larochelle, Franc¸ois Laviolette, Mario Marchand, and Victor
+Lempitsky, “Domain-adversarial training of neural networks,” J. Mach.
+Learn. Res., vol. 17, no. 1, pp. 2096–2030, jan 2016.
+[3] Peter Bell, Joachim Fainberg, Ondrej Klejch, Jinyu Li, Steve Renals,
+and Pawel Swietojanski,
+“Adaptation algorithms for neural network-
+based speech recognition: An overview,” IEEE Open Journal of Signal
+Processing, vol. 2, pp. 33–66, 2021.
+[4] Jinyu Li, Michael L. Seltzer, Xi Wang, Rui Zhao, and Yifan Gong,
+“Large-scale domain adaptation via teacher-student learning,” in Proc
+Interspeech, 2017.
+[5] Vimal Manohar, Pegah Ghahremani, Daniel Povey, and Sanjeev Khu-
+danpur, “A teacher-student learning approach for unsupervised domain
+adaptation of sequence-trained asr models,” in Proc. Spoken Language
+Technology Workshop (SLT), 2018.
+[6] Yusuke Shinohara, “Adversarial Multi-Task Learning of Deep Neural
+Networks for Robust Speech Recognition,” in Proc. Interspeech 2016,
+2016, pp. 2369–2372.
+[7] Zhong Meng, Jinyu Li, Zhuo Chen, Yang Zhao, Vadim Mazalov,
+Yifan Gong, and Biing-Hwang Juang, “Speaker-invariant training via
+adversarial learning,” in Proc. ICASSP. 2018, IEEE.
+[8] Sining Sun, Ching-Feng Yeh, Mei-Yuh Hwang, Mari Ostendorf, and Lei
+Xie, “Domain adversarial training for accented speech recognition,” in
+Proc. ICASSP, 2018, pp. 4854–4858.
+[9] Anoop C S, Prathosh A P, and A G Ramakrishnan,
+“Unsupervised
+domain adaptation schemes for building asr in low-resource languages,”
+in Automatic Speech Recognition and Understanding Workshop (ASRU),
+2021, pp. 342–349.
+[10] Taichi Asami et al., “Domain adaptation of dnn acoustic models using
+knowledge distillation,” in Proc. ICASSP. IEEE, 2017, pp. 5185–5189.
+[11] Karol Nowakowski, Michal Ptaszynski, Kyoko Murasaki, and Jagna
+Nieuwa˙zny,
+“Adapting multilingual speech representation model for
+a new, underresourced language through multilingual fine-tuning and
+continued pretraining,”
+Information Processing & Management, vol.
+60, no. 2, pp. 103148, 2023.
+[12] Sadaoki Furui, “A training procedure for isolated word recognition sys-
+tems,” IEEE Transactions on Acoustics, Speech, and Signal Processing,
+vol. 28, no. 2, pp. 129–136, 1980.
+[13] Yajie Miao, Hao Zhang, and Florian Metze, “Towards speaker adaptive
+training of deep neural network acoustic models,” in Proc. Interspeech,
+2014, pp. 2189–2193.
+[14] Sree HK Parthasarathi e al., “fmllr based feature-space speaker adapta-
+tion of dnn acoustic models,” in Proc. Interspeech, 2015, pp. 3630–3634.
+[15] Vishwa Gupta, Patrick Kenny, Pierre Ouellet, and Themos Stafylakis,
+“I-vector-based speaker adaptation of deep neural networks for french
+broadcast audio transcription,” in Proc. ICASSP, 2014.
+[16] Hans-G¨unter Hirsch and David Pearce,
+“The aurora experimental
+framework for the performance evaluation of speech recognition systems
+under noisy conditions,”
+in ASR2000-Automatic speech recognition:
+challenges for the new Millenium ISCA tutorial and research workshop
+(ITRW), 2000.
+[17] Yanmin Qian, Tian Tan, and Dong Yu,
+“An investigation into using
+parallel data for far-field speech recognition,” in Proc. ICASSP, 2016.
+[18] William Chan, Navdeep Jaitly, Quoc Le, and Oriol Vinyals, “Listen,
+attend and spell: A neural network for large vocabulary conversational
+speech recognition,” in Proc. ICASSP, 2016, pp. 4960–4964.
+[19] Alex Graves, “Sequence transduction with recurrent neural networks,”
+CoRR, vol. abs/1211.3711, 2012.
+[20] Alexei Baevski, Yuhao Zhou, Abdelrahman Mohamed, and Michael
+Auli, “wav2vec 2.0: A framework for self-supervised learning of speech
+representations,” Advances in Neural Information Processing Systems,
+vol. 33, pp. 12449–12460, 2020.
+[21] Alexis Conneau et al.,
+“Unsupervised Cross-Lingual Representation
+Learning for Speech Recognition,” in Proc. Interspeech, 2021, pp. 2426–
+2430.
+[22] Pavel Denisov, Ngoc Thang Vu, and Marc Ferras Font, “Unsupervised
+domain adaptation by adversarial learning for robust speech recognition,”
+in Speech Communication, 2018, pp. 1–5.
+[23] Dongseong Hwang, Ananya Misra, Zhouyuan Huo, Nikhil Siddhartha,
+Shefali Garg, David Qiu, Khe Chai Sim, Trevor Strohman, Franc¸oise
+Beaufays, and Yanzhang He, “Large-scale asr domain adaptation using
+self- and semi-supervised learning,” in NeurIPS, 2022, pp. 6627–6631.
+[24] Wei-Ning Hsu, Anuroop Sriram, Alexei Baevski, Tatiana Likhoma-
+nenko, Qiantong Xu, Vineel Pratap, Jacob Kahn, Ann Lee, Ronan
+Collobert, Gabriel Synnaeve, and Michael Auli, “Robust wav2vec 2.0:
+Analyzing Domain Shift in Self-Supervised Pre-Training,”
+in Proc.
+Interspeech, 2021, pp. 721–725.
+[25] Sameer Khurana, Niko Moritz, Takaaki Hori, and Jonathan Le Roux,
+“Unsupervised domain adaptation for speech recognition via uncertainty
+driven self-training,” Proc. ICASSP, pp. 6553–6557, 2020.
+[26] Sankaran Panchapagesan, Daniel S. Park, Chung-Cheng Chiu, Yuan
+Shangguan, Qiao Liang, and Alexander Gruenstein, “Efficient knowl-
+edge distillation for rnn-transducer models,” in Proc ICASSP, 2021, pp.
+5639–5643.
+[27] Anmol Gulati et al, “Conformer: Convolution-augmented transformer
+for speech recognition,” Proc. Interspeech, pp. 5036–5040, 2020.
+[28] Hasim Sak, Andrew W. Senior, and Franc¸oise Beaufays, “Long short-
+term memory recurrent neural network architectures for large scale
+acoustic modeling,” in Proc. Interspeech, 2014, pp. 338–342.
+[29] Taichi Asami, Ryo Masumura, Yoshikazu Yamaguchi, Hirokazu Masa-
+taki, and Yushi Aono, “Domain adaptation of dnn acoustic models using
+knowledge distillation,” in Proc. ICASSP, 2017, pp. 5185–5189.
+[30] Takuya Yoshioka, Nobutaka Ito, Marc Delcroix, Atsunori Ogawa,
+Keisuke Kinoshita, Masakiyo Fujimoto, Chengzhu Yu, Wojciech J.
+Fabian, Miquel Espi, Takuya Higuchi, Shoko Araki, and Tomohiro
+Nakatani, “The ntt chime-3 system: Advances in speech enhancement
+and recognition for mobile multi-microphone devices,” in Workshop on
+Automatic Speech Recognition and Understanding (ASRU), 2015, pp.
+436–443.
+[31] Bhuvana Ramabhadran, Brian Farris, Isabel Leal, Manasa Prasad, Neeraj
+Gaur, Parisa Haghani, Pedro Jose Moreno Mengibar, and Yun Zhu,
+“Self-adaptive distillation for multilingual speech recognition: Leverag-
+ing student independence,” in Interspeech 2021, 2021.
+[32] Yanzhang He, Tara N. Sainath, Rohit Prabhavalkar, Ian McGraw, Raziel
+Alvarez, Ding Zhao, David Rybach, Anjuli Kannan, Yonghui Wu,
+Ruoming Pang, Qiao Liang, Deepti Bhatia, Yuan Shangguan, Bo Li,
+Golan Pundak, Khe Chai Sim, Tom Bagby, Shuo-yiin Chang, Kanishka
+Rao, and Alexander Gruenstein, “Streaming end-to-end speech recog-
+nition for mobile devices,” in Proc. ICASSP, 2019, pp. 6381–6385.
+[33] Dmitriy Serdyuk, Kartik Audhkhasi, Philemon Brakel, Bhuvana Ramab-
+hadran, Samuel Thomas, and Yoshua Bengio, “Invariant representations
+for noisy speech recognition,” CoRR, 2016.
+[34] Sining Sun, Binbin Zhang, Lei Xie, and Yanning Zhang, “An unsuper-
+vised deep domain adaptation approach for robust speech recognition,”
+Neurocomputing, vol. 257, pp. 79–87, 2017,
+Machine Learning and
+Signal Processing for Big Multimedia Analysis.
+[35] Vijayaditya Peddinti, Daniel Povey, and Sanjeev Khudanpur, “A time
+delay neural network architecture for efficient modeling of long temporal
+contexts,” in Proc. Interspeech, 2015, pp. 3214–3218.
+[36] Daniel Povey et al., “The kaldi speech recognition toolkit,” in Proc.
+ASRU Workshop. 2011, IEEE.
+[37] Seyedmahdad Mirsamadi and John H.L. Hansen,
+“Multi-domain ad-
+versarial training of neural network acoustic models for distant speech
+recognition,” Speech Communication, vol. 106, pp. 21–30, 2019.
+[38] Jan K Chorowski, Dzmitry Bahdanau, Dmitriy Serdyuk, Kyunghyun
+Cho, and Yoshua Bengio, “Attention-based models for speech recog-
+nition,” Advances in neural information processing systems, vol. 28,
+2015.
+[39] Hu Hu, Xuesong Yang, Zeynab Raeesy, Jinxi Guo, Gokce Keskin, Harish
+Arsikere, Ariya Rastrow, Andreas Stolcke, and Roland Maas, “redat:
+Accent-invariant representation for end-to-end asr by domain adversarial
+training with relabeling,” in Proc. ICASSP, 2021.
+[40] Aditay Tripathi, Aanchan Mohan, Saket Anand, and Maneesh Singh,
+“Adversarial learning of raw speech features for domain invariant speech
+recognition,” in Proc. ICASSP, 2018, pp. 5959–5963.
+[41] Konstantinos Bousmalis, George Trigeorgis, Nathan Silberman, Dilip
+Krishnan, and Dumitru Erhan,
+“Domain separation networks,”
+in
+Proc. NIPS, Red Hook, NY, USA, 2016, NIPS’16, p. 343–351, Curran
+Associates Inc.
+[42] Han Zhu, Gaofeng Cheng, Jindong Wang, Wenxin Hou, Pengyuan
+Zhang, and Yonghong Yan, “Boosting cross-domain speech recognition
+with self-supervision,” arXiv preprint arXiv:2206.09783, 2022.
+
+12
+[43] Jounghee Kim and Pilsung Kang, “K-Wav2vec 2.0: Automatic Speech
+Recognition based on Joint Decoding of Graphemes and Syllables,” in
+Proc. Interspeech, 2022, pp. 4945–4949.
+[44] Mitchell DeHaven and Jayadev Billa, “Improving low-resource speech
+recognition with pretrained speech models: Continued pretraining vs.
+semi-supervised training,” arXiv preprint arXiv:2207.00659, 2022.
+[45] Constantinos Karouzos, Georgios Paraskevopoulos, and Alexandros
+Potamianos, “UDALM: Unsupervised domain adaptation through lan-
+guage modeling,” in Proc. Association for Computational Linguistics:
+Human Language Technologies, Online, June 2021, pp. 2579–2590,
+Association for Computational Linguistics.
+[46] Alex Graves, Santiago Fern´andez, Faustino Gomez, and J¨urgen Schmid-
+huber, “Connectionist temporal classification: Labelling unsegmented
+sequence data with recurrent neural networks,” in Proc. of the 23rd Int.
+Conf. on Machine Learning, New York, NY, USA, 2006, Proc. ICML,
+p. 369–376, Association for Computing Machinery.
+[47] H. Scudder, “Probability of error of some adaptive pattern-recognition
+machines,” IEEE Transactions on Information Theory, vol. 11, no. 3,
+pp. 363–371, 1965.
+[48] David Yarowsky,
+“Unsupervised word sense disambiguation rivaling
+supervised methods,” in Annu. Meeting of the Association for Compu-
+tational Linguistics, 1995.
+[49] Ellen Riloff and Janyce Wiebe, “Learning extraction patterns for sub-
+jective expressions,” in Conf. Empirical Methods in Natural Language
+Processing, 2003, pp. 105–112.
+[50] Dongseong Hwang, Khe Chai Sim, Yu Zhang, and Trevor Strohman,
+“Comparison of soft and hard target rnn-t distillation for large-scale
+asr,” 2022.
+[51] Jacob Kahn, Ann Lee, and Awni Hannun, “Self-training for end-to-end
+speech recognition,” in Proc. ICASSP, 2020, pp. 7084–7088.
+[52] Daniel S. Park et al, “Improved noisy student training for automatic
+speech recognition,” in Proc. Interspeech, 2020.
+[53] Yu Zhang, James Qin, Daniel S. Park, Wei Han, Chung-Cheng Chiu,
+Ruoming Pang, Quoc V. Le, and Yonghui Wu, “Pushing the limits of
+semi-supervised learning for automatic speech recognition,” 2020.
+[54] Daniel S. Park, William Chan, Yu Zhang, Chung-Cheng Chiu, Barret
+Zoph, Ekin D. Cubuk, and Quoc V. Le, “SpecAugment: A Simple Data
+Augmentation Method for Automatic Speech Recognition,”
+in Proc.
+Interspeech, 2019, pp. 2613–2617.
+[55] Yaroslav Ganin and Victor Lempitsky, “Unsupervised domain adaptation
+by backpropagation,” in Proc. ICML. 2015, ICML’15, p. 1180–1189,
+JMLR.org.
+[56] Douglas B Paul and Janet Baker, “The design for the wall street journal-
+based csr corpus,” in Speech and Natural Language: Proc. of a Workshop
+Held at Harriman, New York, February 23-26, 1992, 1992.
+[57] Siu-Kei Au Yeung and Man-Hung Siu,
+“Improved performance of
+aurora 4 using htk and unsupervised mllr adaptation,” in Conf. Spoken
+Language Processing, 2004.
+[58] Suchin Gururangan, Ana Marasovi´c, Swabha Swayamdipta, Kyle Lo,
+Iz Beltagy, Doug Downey, and Noah A. Smith, “Don’t stop pretraining:
+Adapt language models to domains and tasks,” in Proc. of the 58th Annu.
+Meeting of the Association for Computational Linguistics, Online, July
+2020, pp. 8342–8360, Association for Computational Linguistics.
+[59] Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova,
+“BERT: pre-training of deep bidirectional transformers for language
+understanding,” CoRR, vol. abs/1810.04805, 2018.
+[60] Herve Jegou, Matthijs Douze, and Cordelia Schmid, “Product quantiza-
+tion for nearest neighbor search,” IEEE transactions on pattern analysis
+and machine intelligence, vol. 33, no. 1, pp. 117–128, 2010.
+[61] Eric Jang, Shixiang Gu, and Ben Poole, “Categorical reparametrization
+with gumbel-softmax,” in Proc. ICLR, Apr. 2017.
+[62] Vassil Panayotov, “Librispeech: an asr corpus based on public domain
+audio books,” in Proc. ICASSP. IEEE, 2015, pp. 5206–5210.
+[63] Aimilios Chalamandaris et al., “All greek to me! an automatic greeklish
+to greek transliteration system,” in Proc. LREC, 2006.
+[64] Carsten Meyer and Hauke Schramm, “Boosting hmm acoustic models
+in large vocabulary speech recognition,” Speech Communication, vol.
+48, no. 5, pp. 532–548, 2006.
+[65] Vimal Manohar, Daniel Povey, and Sanjeev Khudanpur,
+“Jhu kaldi
+system for arabic mgb-3 asr challenge using diarization, audio-transcript
+alignment and transfer learning,” in Proc. ASRU Workshop, 2017, pp.
+346–352.
+[66] Vassilios Digalakis et al., “Large vocabulary continuous speech recog-
+nition in greek: corpus and an automatic dictation system,” in Proc.
+Eurospeech, 2003, pp. 1565–1568.
+[67] T.F. Smith and M.S. Waterman, “Identification of common molecular
+subsequences,” Journal of Molecular Biology, vol. 147, no. 1, pp. 195–
+197, 1981.
+[68] Rosana Ardila et al., “Common voice: A massively-multilingual speech
+corpus,” in Proc. LREC, 2020, pp. 4218–4222.
+[69] Ilya Loshchilov and Frank Hutter, “Decoupled weight decay regulariza-
+tion,” in Proc. ICLR, 2018.
+[70] Alex Graves, Santiago Fern´andez, Faustino Gomez, and J¨urgen Schmid-
+huber,
+“Connectionist temporal classification: labelling unsegmented
+sequence data with recurrent neural networks,” in Proc. ICML, 2006,
+pp. 369–376.
+[71] Guillaume Wenzek, Marie-Anne Lachaux, Alexis Conneau, Vishrav
+Chaudhary, Francisco Guzm´an, Armand Joulin, and ´Edouard Grave,
+“Ccnet: Extracting high quality monolingual datasets from web crawl
+data,” in Proc. of the 12th Language Resources and Evaluation Conf.,
+2020, pp. 4003–4012.
+[72] Nick Hatzigeorgiu, Maria Gavrilidou, Stelios Piperidis, George Carayan-
+nis,
+Anastasia
+Papakostopoulou,
+Athanassia
+Spiliotopoulou,
+Anna
+Vacalopoulou, Penny Labropoulou, Elena Mantzari, Harris Papageor-
+giou, et al.,
+“Design and implementation of the online ilsp greek
+corpus.,” in LREC, 2000.
+[73] Kenneth Heafield, “Kenlm: Faster and smaller language model queries,”
+in Proc. of the 6th workshop on statistical machine translation, 2011,
+pp. 187–197.
+[74] Laurens Van der Maaten and Geoffrey Hinton, “Visualizing data using
+t-sne.,” Journal of machine learning research, vol. 9, no. 11, 2008.
+[75] Georgios Paraskevopoulos, Srinivas Parthasarathy, Aparna Khare, and
+Shiva Sundaram, “Multimodal and multiresolution speech recognition
+with transformers,” in Proc. of the 58th Annu. Meeting of the Association
+for Computational Linguistics, 2020, pp. 2381–2387.
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf,len=950
+page_content='1  Sample-Efficient Unsupervised Domain Adaptation  of Speech Recognition Systems:  A case study for Modern Greek  Georgios Paraskevopoulos Student Member, IEEE, Theodoros Kouzelis, Georgios Rouvalis, Athanasios  Katsamanis Member, IEEE, Vassilis Katsouros Member, IEEE, Alexandros Potamianos Fellow, IEEE  Abstract—Modern speech recognition systems exhibits rapid  performance degradation under domain shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' This issue is  especially prevalent in data-scarce settings, such as low-resource  languages, where diversity of training data is limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' In this work  we propose M2DS2, a simple and sample-efficient finetuning  strategy for large pretrained speech models, based on mixed  source and target domain self-supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' We find that including  source domain self-supervision stabilizes training and avoids  mode collapse of the latent representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' For evaluation, we  collect HParl, a 120 hour speech corpus for Greek, consisting  of plenary sessions in the Greek Parliament.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' We merge HParl  with two popular Greek corpora to create GREC-MD, a test-  bed for multi-domain evaluation of Greek ASR systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' In our  experiments we find that, while other Unsupervised Domain  Adaptation baselines fail in this resource-constrained environ-  ment, M2DS2 yields significant improvements for cross-domain  adaptation, even when a only a few hours of in-domain audio  are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' When we relax the problem in a weakly supervised  setting, we find that independent adaptation for audio using  M2DS2 and language using simple LM augmentation techniques  is particularly effective, yielding word error rates comparable to  the fully supervised baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  Index Terms—Unsupervised Domain Adaptation, Automatic  Speech Recognition, Multi-Domain Evaluation, Greek Speech  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' INTRODUCTION  Automatic Speech recognition (ASR) models have matured  to the point where they can enable commercial, real-world  applications, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=', voice assistants, dictation systems, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=', thus  being one of machine learning’s success stories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' However,  the performance of ASR systems rapidly deteriorates when  the test data domain differs significantly from the training  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Domain mismatches can be caused by differences in  the recording conditions, such as environmental noise, room  reverberation, speaker and accent variability, or shifts in the  target vocabulary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' These issues are extenuated in the case of  low-resource languages, where diversity in the training data  is limited due to poor availability of high-quality transcribed  audio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Therefore, specialized domain adaptation approaches  need to be employed when operating under domain-shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  Unsupervised Domain Adaptation (UDA) methods are of  special interest, as they do not rely on expensive annotation  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Paraskevopoulos is with the Graduate School of ECE, National Technical  University of Athens, Athens, Greece  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Paraskevopoulos, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Kouzelis, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Rouvalis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Katsamanis, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Katsouros  are with the Institute for Speech and Language Processing, Athena Research  Center, Athens, Greece  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Potamianos is with the Faculty of ECE, National Technical University  of Athens, Athens, Greece  of domain-specific data for supervised in-domain training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  In contrast to supervised approaches, where the existence of  labeled data would allow to train domain-specific models,  UDA methods aim to leverage data in the absense of labels  to improve system performance in the domain of interest [1],  [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' In the context of speech recognition the importance of  UDA is extenuated, as the transcription and alignment pro-  cess is especially expensive and time-consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Adaptation  methods have been explored since the early days of ASR,  at different levels of the system and different deployment  settings [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' UDA has been used to improve the robustness  of ASR on a variety of recording conditions including far-  field speech, environmental noise and reverberation [4], [5],  [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Furthermore, UDA has been used for speaker adaptation,  and to improve performance under speaker, gender and accent  variability [7], [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' UDA has also been employed for multilin-  gual and cross-lingual ASR, in order to improve ASR models  for low-resource languages [9], adapt to different dialects [10],  and even train speech recognition systems for endangered  languages [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  Classical speech adaptation techniques involve feature-  based techniques, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=', speaker normalization [12], feature-  based approaches [13]–[15], or multi-condition training [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  Generally, traditional approaches require some knowledge  about the target domain, and the domain mismatch, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=',  regarding the noise and reverberation variability [17], and  require specific engineering for each adaptation scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  Modern ASR pipelines, increasingly rely on end-to-end  neural networks, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=', [18], [19], or large pretrained models  with self-supervised objectives [20], [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' The key approaches  employed for UDA of end-to-end ASR models can be grouped  in three categories, namely, teacher-student learning [10],  domain adversarial training [22], and target domain self-  supervision [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' The benefit of these techniques is that they  do not require any special knowledge about the source or  the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' This makes end-to-end UDA approaches  versatile and able to be utilized in a larger array of adaptation  scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' In particular, adaptation through self-supervision  has been shown to be a robust, simple and efficient technique  for adaptation of state-of-the-art speech models [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  Here, we leverage in-domain self-supervision to propose  the Mixed Multi-Domain Self-Supervision (M2DS2) finetun-  ing strategy, enabling sample-efficient domain adaptation of  wav2vec2 [20] based speech recognition models, even when  available in-domain data are scarce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Our key contributions are  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='00304v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='CL]  31 Dec 2022  2  TABLE I  SUMMARY OF RELATED WORKS ON UNSUPERVISED DOMAIN ADAPTATION FOR ASR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  Work  Method  Model  Adaptation Setting  Language  [23],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' [25],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' [26]  Teacher-Student  Hard and soft labels  Conformer RNN-T [27]  Transformer CTC  RNN-T [19]  News speech,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Voice search,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Far-field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  Telephony,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' YouTube  English  [4],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' [5]  Teacher-Student  Soft labels  TDNN-LSTM [28]  Noise,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Far-field  English  [29]  Teacher-Student  Hard and soft labels  NiN-CNN [30]  Dialects  Children speech  Japanese  [31]  Teacher-Student  Soft labels  Streaming RNN-T [32]  Multilingual  English,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  Brazilian Portuguese,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  Russian,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Turkish,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  Nordic/Germanic  [6],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' [33],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' [34]  Domain Adversarial Training  TDNN Kaldi [35],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' [36]  DNN-HMM  DNN-HMM  Noise,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Channel  English  [37]  Domain Adversarial Training  RNN-CTC [38]  Far-field  English  [8],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' [39]  Domain Adversarial Training  TDNN Kaldi  RNN-T  Accent  Mandarin  [7],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' [40]  Domain Adversarial Training  DNN-HMM  CNN-DNN  Speaker,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Gender,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  Accent  English  [9]  Domain Adversarial Training  DSN [41]  Multilingual  Hindi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Sanskri  [24],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' [42]  Continual Pre-Training  wav2vec2 [20]  Audiobooks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Accents,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  Ted Talks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Telephony,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  Crowd-sourced,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Parlamentary speech  English  [43]  Continual Pre-Training  wav2vec2  Cross-lingual  Korean  [11],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' [44]  Continual Pre-Training  XLSR-53 [21]  wav2vec2  Low resource languages  Ainu  Georgian,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Somali,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  Tagalog,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Farsi  organized as follows:  1) Inspired by recent advances on UDA for Natural Lan-  guage Processing systems [45],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' we propose a finetuning  strategy for speech models,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' where the self-supervised  objective is based on a contrastive loss in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  Contrary to prior works, who leverage only in-domain  self-supervision, we find that in this contrastive setting  this leads to mode-collapse of the latent representations,  and mixed source and target domain self-supervision  is essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' We demonstrate this empirically in Sec-  tion VII-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  2) We collect and curate HParl, the largest publicly avail-  able1 speech corpus for Greek, collected from plenary  sessions in the Greek Parliament between 2018 and  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' We establish a data collection, pre-processing  and alignment pipeline that can be used for continuous  data integration, as the parliamentary proceedings get  regularly uploaded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' We provide a detailed description of  our data collection process and the dataset statistics in  Section IV-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' HParl is merged in Section IV with two  popular Greek corpora (Logotypografia and Common-  Voice) to create GREC-MD, a testbed for multi-domain  evaluation of ASR systems in Greek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  3) We demonstrate that, while other baselines fail at UDA  in our resource-constrained setting, M2DS2 can improve  model performance in the target domain in multiple  adaptation scenarios in Section VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Specifical emphasis  is given in the sample efficiency of our approach in Sec-  1We plan to release this version of HParl under the CC BY-NC 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='0 license  upon publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' The other corpora used in this work are available through  their respective distributors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  tion VII-A, where we demonstrate successful adaptation  even when we reduce the available in-domain data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  4) When we relax the problem to a weakly supervised  adaptation setting, where some in-domain text is avail-  able but the pairing between audio and text is unknown,  we find that M2DS2 can be effectively combined with  simple N-gram adaptation techniques to get compara-  ble performance with the fully supervised baseline in  Section VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Furthermore we find that a simple text  augmentation approach, based on perplexity filtering of a  large corpus can produce strong adaptation results, even  for small amounts of in-domain text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  Additionally, we provide a formulation of the UDA problem  for ASR in Section II-A and link prior works to this formu-  lation in Sections II-B, II-C and II-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' We provide detailed  experimental settings for reproducibility in Section V, and  an upper-bound estimation for UDA performance with fully  supervised finetuning in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' BACKGROUND  We start by formally defining the Unsupervised Domain  Adaptation (UDA) problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Initially, we formulate the prob-  lem in a classification setting and then we extend it for  speech recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' We then provide an overview of different  adaptation approaches in the literature, and link each approach  to the UDA problem formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Table I presents a summary  of the key adaptation settings and applications that are ex-  plored in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' We see, that a relatively small amount  of methods, and their variants, is used to address multiple  real-world ASR problems, for example, cross-lingual, accent,  speaker and noise adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Furthermore, while the majority  3  of the works focus on the English language, there is an effort  to explore other popular languages, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=', Mandarin, and under-  resourced languages, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=', Ainu, Somali etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Problem Definition  Formally, the problem of UDA can be defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  Let X ⊆ Rn be a real-valued space that consists of n-  dimentional feature vectors x ∈ X, and Y a finite set of  labels y ∈ Y , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=', Y = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' , L}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Furthermore, assume  two different distributions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=', the source domain distribution  S(x, y) and the target domain distribution T (x, y), defined on  the cartesian product X × Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  The goal is to train a model that learns a mapping between  feature vectors xT to their respective labels yT for samples  drawn from the target distribution (xT , yT ) ∼ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  At training time we have access to samples from the source  distribution S(x, y) and the marginalized target distribution  T (x), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=', no target labels are provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' We define the training  dataset D as the concatenation of the source and target training  sets, D = (DS, DT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' DS and DT are defined as sequences of  tuples, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=',  DS = {(xi, yi) | (xi, yi) ∼ S(x, y), 1 ≤ i ≤ N}  DT = {(xi, ∅) | xi ∼ T (x), 1 ≤ i ≤ M},  (1)  where we draw N samples from S(x, y) and M samples  from T (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Finally, we augment tuples in D with a domain  indicator function:  D = {(xi, y′  i, 1i) | 1 ≤ i ≤ N + M}  1i =  �  0  if xi ∼ S(x),  1  if xi ∼ T (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  y′  i =  �  yi  if xi ∼ S(x),  ∅  if xi ∼ T (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  (2)  1) Unsupervised (Acoustic) Adaptation for ASR: The above  definition can be directly extended in the case of speech  recognition, with some modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' In detail, we modify  the feature space X, to be the set of (finite) sequences of  real-valued feature vectors (xk)k∈N\\{∞} ∈ X ⊆ (Rn)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  Furthermore, the label space Y is modified to be the set  of sequences (ym)m∈N\\{∞}, where Y  = ({1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' , L})∗  contains finite-length sequences over a finite lexicon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' For  CTC training we make the assumption that k > m for any  sample (xk, ym), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=', feature sequences are longer than their  respective label sequences [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' The rest of the definitions need  no modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  2) Unsupervised (Language) Adaptation for ASR: Adapta-  tion for ASR systems can also be performed at the language  level, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=', the label space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' In this setting, we assume that  the target domain samples are drawn from the marginalized  target distribution T (y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' The target dataset DT now consists of  tuples in the form (∅, yi), where yi is the label word sequence  (ym)m∈N\\{∞} for the i-th sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  3) Weakly supervised Adaptation for ASR: The last setting  we explore is the case were both audio and language in-  domain samples are available, but the mapping between them  is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' This situation can be encountered in real-world  settings, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=', in the case in-domain audio and text are collected  independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' For example consider the case where audio  clips from news casts are collected, along with contemporary  newspaper articles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Another example is the case where long  audio clips alongside with transcriptions are available, but no  fine-grained time alignments2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' In this case the target domain  samples are drawn independently from the marginalized dis-  tributions T (x) and T (y), and the target dataset DT consists  of tuples in the form (xi, ∅) and (∅, yi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Teacher-Student Models  Teacher-Student learning or self-training, is one of the  earliest methods in semi-supervised learning [47]–[49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' The  key idea is to reduce the problem of unsupervised learning  of the task at hand in the target domain to a supervised one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  The general methodology is to train a teacher model gS using  the labeled data in the source domain DS, and then use this  for inference on the target domain to produce pseudolabels  ˆyi = gS(xi), xi ∼ T (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' The target domain dataset DT is  augmented with these silver labels, to contain tuples (xi, ˆyi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  Finally, a student model gT is trained in a supervised fashion,  using the augmented DT or a combination of DS and DT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  This process is usually repeated, with the student model  serving as the teacher model for the next iteration, until no  further improvement is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' More recently, soft target  Teacher-Student learning has been explored for ASR [26],  [31], [50], where the KL divergence between the teacher and  student output label distributions is used as the loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  Being trained only on the source domain data the teacher  model is susceptible to error propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Filtering is a com-  monly used technique to achieve the right balance between  the size of the target domain used for training the student  model and the noise in the pseudolabels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Confidence scoring  based on the likelihood is usually applied, discarding those  utterances for which the hypothesized labels are untrustworthy  [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' In [25] dropout is used to measure the model uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  The agreement between model predictions with and without  dropout are used for confidence scoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' In [23] a multi-task  training objective with a confidence loss is applied to minimise  the binary cross entropy between the estimated confidence and  the binary target sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' In order to learn more robust and  generalizable features from the teacher model, Noisy Student  Training (NST) has been proposed in [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' The teacher models  generates pseudolabels for DT while the student models are  trained on a heavily augmented version of DT [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' In [52],  [53] the augmentation of the input target data is performed  with SpecAugment [54], while in [29] a spectrum frequency  augmentation is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  In [4] Teacher-Student learning with soft labels is introduced  for ASR to tackle noisy, far-field, and children speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' In  2While a fully supervised in-domain dataset can be constructed in this  case using long / forced alignment methods, this is not a focal point for the  experimental part of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  4  [5], this approach is extended for LF-MMI based models and  used for noisy, far-field and bandwidth adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' In [29] a  weighted sum of hard and soft target cross entropy losses  is used for Japanese dialects and children speech adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  Ramabhadran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' [31] propose a self-adaptive distillation,  and a method for distilling from multiple teachers that is  applied across several multilingual ASR systems for different  language groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' A comparison between soft and hard targets  for RNN-T models [19] showed that soft targets perform better  when both the teacher and student models have the same  architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Otherwise, hard targets are superior [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Domain Adversarial Training  Domain Adversarial Training (DAT) was initially introduced  for image classification [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' The key idea is to train a  model that learns deep features that solve the task at hand  in the source domain, while being invariant with respect  to the domain shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Concretely, the model is trained end-  to-end using a combination of the supervised task loss Lt,  learned on DS, and the domain discrimination loss La, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=',  L = Lt − αLa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' The loss La is binary cross-entropy, trained  for domain discrimination using the tuples (xi, 1i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Notice  the − sign in the loss indicates adversarial learning, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=', the  model should learn features that cannot discriminate between  domains, while solving the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  In [6] DAT is employed for noise adaptation on a noise  corrupted version of WSJ [56] as the target dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Using the  Aurora-4 [57] dataset which has labels associated to the noise  type, Serdyuk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' [33] train an adversarial noise classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' In  [8] and [39] DAT is utilized for accent adaptation for Mandarin  and English respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Anoop C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' [9] propose DAT,  to address the scarcity of data in low-resource languages which  share a common acoustic space with a high-resource language,  namely Sanskrit and Hindi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' They empirically demonstrate the  effectiveness of adversarial training, presenting experiments  with and without the reversal of the domain classification loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Leveraging In-domain Self-supervision  These lines of work have roots in Natural Language Pro-  cessing tasks [45], [58], and explore domain adaptation by  leveraging the in-domain data DT for self-supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  The core focus is domain adaptation of large pre-trained  models, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=', [59], and self-supervision is achieved by use  of the pre-training self-supervised loss Ls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' This process can  either take part in stages, via continual pre-training [58], or by  constructing a multitask objective L = Lt + αLs, as in [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  Continual Pre-Training (CPT) has been explored for adap-  tation of ASR models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Robust wav2vec2 [24] explores the  effectiveness of CPT for domain adaptation, indicating the  importance of utilizing unlabeled in-domain data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' In CASTLE  [42], CPT is combined with an online pseudolabeling strategy  for domain adaptation of wav2vec2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Cross-dataset evaluation  for popular English speech corpora indicates that CPT helps  to reduce the error rate in the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' In [43] and [11]  CPT is utilized for cross-lingual adaptation of wav2vec2 for  Korean and Ainu respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Notably for Ainu, which is an  endagered language, CPT has resulted in significant system  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Target-domain adaptation through self-supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' In the left we see  the general pre-training stage of XLSR-53 using the self-supervised loss Ls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  General pre-training is performed on 56, 000 hours of audio in 53 languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  In the right, we see the proposed domain-adaptive finetuning stage, where the  speech recognition task is learned using transcribed source domain data, while  adaptation to the target domain is performed by including the self-supervised  loss over (audio-only) source and target domain data  improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' DeHaven and Jayadev [44] compare CPT and  pseudolabeling for adapting XLSR-53 to four under-resourced  languages, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=', Georgian, Somali, Tagalog and Farsi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' They find  that both approaches yield similar improvements, with CPT  being the more computationally efficient approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  While CPT yields significant improvements in a variety of  tasks, one common theme in these works is the assumption  of hundreds or thousands of hours of available in-domain  data, mostly from online resources, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=', YouTube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' This can be  infeasible when we consider more niche adaptation settings,  or possible privacy concerns, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=', how would one collect  1000 hours of psychotherapy sessions in Greek?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' In this work,  we explore domain adaptation methods in a more resource-  constrained environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' DOMAIN ADAPTATION THROUGH MULTI-DOMAIN  SELF-SUPERVISION  The proposed approach is based on end-to-end adaptation of  a large pre-trained speech model during the finetuning phase,  by including in-domain self-supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' We extend UDALM  [45], that has shown promise for NLP tasks, for adaptation of  wav2vec2 based acoustic models, and specifically XLSR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' We  focus on the problem of UDA in the context of a low-resource  language, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=', Greek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' The key finding of our exploration is  that straight-forward extension of UDALM, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=', by using only  target domain self-supervision, underperforms in this setting,  and use of both source and target domain data is essential for  successful adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' In this section, first, we will present  a quick overview of the XLSR-53 training procedure, and  then we are going to outline the proposed domain adaptation  approach, which is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' XLSR-53  XLSR-53 [21] is a massively pre-trained speech model,  trained on 56, 000 hours of multilingual speech, covering 53  languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' The model is based on wav2vec2 [20], which is  composed of a multi-layer convolutional feature encoder, that  General Pretraining  Finetuning  Ls  LCTC  Ls  Masked Transformer  Masked Transformer  XLSR  XLSR  MLS, CommonVoice and BABEL  Source Domain  Target Domain  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='0000 of speech data from 53 languages5  TABLE II  THE GREC-MD CORPUS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' WE CAN SEE THE DURATION OF EACH SPLIT IN H O U R S:M I N U T E S:S E C O N D S FORMAT, AS WELL AS THE NUMBER OF  SPEAKERS FOR EACH OF THE SUB-CORPORA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  Dataset  Domain  Speakers  Train  Dev  Test  Total Duration  HParl  Public (political) speech  387  99:31:41  9:03:33  11:12:28  119:47:42  CV  Crowd-sourced speech  325  12:16:17  1:57:44  1:59:19  16:13:20  Logotypografia  News casts  125  51:58:45  9:08:35  8:59:22  70:06:42  Total 713  163:46:43  20:09:52  22:11:44  206:08:19  extracts audio features zt from the raw audio, and a trans-  former context encoder that maps the latent audio features to  the output hidden states ct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Each latent feature zt corresponds  to 25 ms of audio with stride 20 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' A contrastive objective Lc  is used for pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' For this, product quantization [60] is  applied to the features zt, and then a discrete approximation of  zt is obtained by sampling from a Gumbel-softmax distribution  [61], to obtain discrete code vectors qt, organized into G = 2  codebooks with V  = 320 vocabulary entries each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' The  contrastive loss aims to identify the correct code vector for  a given time step, among a set of distractors Qt, obtained  through negative sampling from other timesteps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' To avoid  mode collapse, a diversity loss Ld is included by maximizing  the entropy over the averaged softmax distribution over the  code vector entries ¯pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' The total loss is:  Ls = −log  es(zt,qt)  �  ˜q∼Qt es(zt,˜q)  �  ��  �  Contrastive Loss  Diversity Loss  �  ��  �  − 1  GV  G  �  g=1  V  �  v=1  ¯pg,vlog(¯pg,v)  (3)  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Domain Adaptive finetuning for Contrastive Learning of  Speech Representations  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 1 shows the proposed finetuning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' The key  intuition is that we want the model to synergistically learn  the task at hand (in our case ASR), while being adapted to  the target domain by in-domain self-supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' In the left  we see the general pre-training stage of XLSR-53, which is  pre-trained on 56K hours of multilingual audio corpora using  the contrastive pre-training objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' In the right we see the  proposed finetuning stage, which is inspired by [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  During finetuning we form a mixed objective function:  L = LCT C(xs, ys) + αLs(xs) + βLs(xt),  (4)  where (xs, ys) ∼ S(x, y), xt ∼ T (x), LCT C is the CTC  objective function, optimized using transcribed source domain  data, and Ls is the contrastive loss from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' We scale the  contribution of each term using hyper-parameters α and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  Note that contrary to [45], who use only in-domain self-  supervision, we leverage both source and target domain sam-  ples for the mixed self-supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' We find that this is essen-  tial in our case to avoid mode collapse, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=', the model using  only a few of the available discrete code vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Simultaneous  self-supervision on both the source and target data alleviates  mode collapse by anchoring the target code vector space to  have a similar structure as the source code vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  Hence we refer to this approach as Mixed Multi-Domain  Self-Supervision (M2DS2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' THE GREC-MD CORPUS  For our experiments we compose a speech corpus for the  Greek language, that is suitable for multi- and cross-domain  evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' The GREC-MD corpus contains 206 hours of  Greek speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Audio is segmented into individual utterances  and each utterance is paired with its corresponding tran-  scription.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Table II summarizes the included sub-corpora, as  well as the train, development and test splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' The dataset is  constructed with three core principles in mind:  1) Data Volume: We collect the largest publicly available  speech recognition corpus for the Greek language, able  to scale to hundreds of hours of transcribed audio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  2) Temporal Relevance: Language changes over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' We  aim at an up-to-date corpus that encompasses the latest  terms and topics that appear in daily speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  3) Multi-Domain Evaluation: Single domain evaluation  can lead to misleading estimations of the expected  performance for ASR models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' For example, state-of-  the-art ASR models [27] achieve under 5% Word Error  Rate (WER) on Librispeech [62] test sets, but this is  an over-estimation of system performance in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  This is extenuated when considering different acoustic  conditions or terminology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' We consider multi-domain  evaluation essential when developing and deploying  real-world ASR models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  To satisfy the first two points, we collect data from a public,  continuously updated resource, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=', the Hellenic Parliament  Proceedings, where recordings of the parliamentary sessions  are regularly uploaded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' The benefit of using this resource is the  straight-forward collection of a continuously growing, multi-  speaker corpus of transcribed audio that is always up-to-date,  as the parliamentary discussions revolve around current affairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  We refer to this corpus as HParl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' For the multi-domain evalua-  tion, we merge HParl with two publicly available corpora, that  have different acoustic and language characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' We refer  to the merged, multi-domain corpus as GREC-MD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' In this  Section, we will describe the collection and curation process  of HParl, and present the relevant statistics for the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  TABLE III  PLENARY SESSIONS INCLUDED IN HPARL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' THE HOURS COLUMN REFERS  TO THE RAW (UNSEGMENTED) HOURS OF COLLECTED AUDIO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  Start date  End date  #Sessions  Hours  15-02-2022  01-03-2022  10  55  18-01-2019  01-02-2019  10  52  28-03-2019  10-05-2019  20  108  10-12-2018  21-12-2018  10  88  6  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Overview of the Hellenic Parliament Chamber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' The chamber has an  amphitheatrical shape and can accomodate approximately 400 − 450 people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  The positions of the key speakers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=', current speaker and the parliament  president are annotated in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Collection and Curation of HParl  Modern technological advances allow for more direct gov-  ernment transparency, through the commodification of storage  and internet speeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' In this spirit, the records of plenary ses-  sions of the Hellenic Parliament are made publicly available,  for direct access through a webpage3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' The available video  recordings date back to 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' For each plenary session, a  video recording is uploaded, along with a full transcription  that is recorded verbatim, and in real time by the parlia-  ment secretaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' For the creation of HParl, we build a web-  crawler that can traverse and download the video recordings,  along with the transcriptions from the official website.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' The  collection process is parallelized over multiple threads, and  parameterized by a range of dates and, optionally, a target  corpus size in GB or in hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' For this version of HParl, we  collect the plenary sessions in four date ranges, as described in  Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' The majority of the collected sessions are from 2019,  but we also include sessions from 2018 and 2022 to include  coverage of different topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' The individual components of the  HParl curation pipeline are: Audio Pre-processing, Text Pre-  processing, Alignment, Post-processing, and dataset Splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  1) Audio Pre-processing: Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 2 shows the layout of the  Hellenic Parliament Chamber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Plenary sessions mainly take  place in this room, or in the secondary House Chamber that  has similar setup but is smaller in size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Because of the room  and microphone characteristics, the captured audio in the  video streams contains reverberation, due to sound reflections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  We employ a light preprocessing pipeline, by passing the  input video streams through FFmpeg, and converting them to  monophonic, lossless audio format at 16000 Hz sampling rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  The resulting audio is not passed through any de-reverberation  or speech enhancement software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' The resulting audio files have  a minimum, average and maximum duration of 6 minutes, 6  hours and 16 hours respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  2) Text Pre-processing: The text files contain full, word-  by-word transcription of the speeches and questions asked by  members of the audience, as well as extra annotations made  by the parliament secretaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Some annotations are relevant,  3https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='hellenicparliament.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='gr/en/  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=', the speaker name, while others are plain text descriptions  of events happening during the session and need to be filtered  out (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=', “The session is interrupted for a 15 minute break”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  We use a rule-based system, based on regular expressions,  that filters the unnecessary information, keeping only the  transcriptions and the speaker names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' The speaker labels are  created by transliterating their names and roles from Greek  to Greeklish using the “All Greek to Me!”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' tool [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Text is  lower-cased and normalized to remove multiple whitespaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  The result is a text file containing the raw transcriptions, and  a mapping from speaker labels to their respective text parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  3) Aligment and Segmentation: The primary challenge of  exploiting the plenary sessions for ASR purposes is the length  of the plenary recordings, as their durations vary from 6  minutes to 16 hours in length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' However, data samples used to  train ASR are generally less than 30 seconds long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Computa-  tional challenges have limited the length of training utterances  for HMM-GMM models [64], and continue to do so in the  contemporary neural network models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Therefore, we need to  segment the sessions into smaller pieces more suitable for ASR  training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' A second challenge is posed by mismatches between  audio and transcripts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Parliamentary proceedings do not fully  capture everything that is said during the parliamentary ses-  sions, and do not account for speech disfluencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  In order to obtain smaller, clean segments, that are suit-  able for ASR training we follow the segmentation procedure  proposed by [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Initially the raw recordings are segmented  into  30 second segments and the transcriptions are split  into smaller segments of approximately 1000 words called  documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Each segment is decoded using a seed acoustic  model trained on the Logotypografia corpus [66] and a 4-  gram biased LM trained on the corresponding transcription  of each recording.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' The best path transcript of each segment  is obtained and paired with the best matching document via  TF-IDF similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Finally each hypothesis is aligned with the  transcription using Smith-Waterman alignment [67] to select  the best matching sub-sequence of words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' The above method  yields a list of text utterances, with their corresponding start  and end times in the source audio files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' The procedure yields  120 hours of useable segmented utterances out of the original  303 hours of raw audio, or a ratio of 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='6%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  4) Post-processing: After the segments are extracted, we  filter out extremely short segments (less than 2 words).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  Moreover, the iterative alignment algorithm may replace some  intermediate words with a <spoken-noise> tag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' When this  tag is inserted, we match the surrounding text with the raw  transcriptions and re-insert the missing words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Furthermore,  we match each segment to its corresponding speaker label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  Segments without a speaker label are discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Lastly, speak-  ers are associated to their gender based on name suffixes, using  a simple, Greek language-specific, rule: Speaker names which  end in a(α), h(η), w(ω) or is(ις) are classified as female, while  the rest as male.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' We format the segments, speaker and gender  mappings in the standard folder structure used by the Kaldi  speech recognition toolkit [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  5) Data Splitting: We provide an official train - devel-  opment - test split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' The development set contains 3 plenary  sessions, one from 2018, one from 2019 and one from 2022,  Current Speaker  Parliament President7  resulting to 9 hours of segmented speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Similarly, the test  set contains one session from each year, resulting to 11 hours  of segmented speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' The rest 99 hours of segmented speech  are assigned to the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Including corpora from different domains  We merge HParl with two publicly available corpora to  create GREC-MD for multi-domain evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  1) Common Voice: Common Voice (CV) [68] is a crowd-  sourced, multi-lingual corpus of dictated speech, created by  Mozilla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' The data collection is performed by use of a web  app or an iPhone app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Contributors are presented with a  prompt and are asked to read it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' The prompts are taken from  public domain sources, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=', books, wikipedia, user submitted  prompts and other public corpora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' The maximum prompt  length is 15 words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' A rating system is built into the plat-  form, where contributors can upvote or downvote submitted  <audio,transcript> pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' A pair is considered valid, if  it receives two upvotes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Speaker independent train, develop-  ment and test splits are provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' The dataset is open to the  research community, released under a permisFsive Creative  Commons license (CC0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' In this work, we use version 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='0  of CV, accessed on April 27, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' We keep only the valid  utterances, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=', 16 hours of speech from 325 contributors  (19 − 49 years old, 67% male / 23% female).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  2) Logotypografia: Logotypografia [66] is one of the first  corpora for Large Vocabulary Continuous Speech Recognition  in Greek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' The dataset contains 33, 136 newscast utterances, or  72 hours of speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' The utterances were collected from 125  speakers (55 male, 70 female), who were staff of the popular  “Eleftherotypia” newspaper in Greece, under varied acoustic  conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Approximately one third of the utterances were  collected in a sound proof room, one third in a quiet room and  the last third in an office room.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' The average utterance duration  is 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='8 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' The transcriptions contain several speech and  non-speech events (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=', <cough>), lower-cased Greek words  and stress marks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Numbers are expanded to full words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' We  use the whole dataset, and perform light preprocessing in  the transcriptions, by discarding the annotated events and  punctuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  We hence refer to each dataset by the abbreviations: HParl:  HP, CommonVoice: CV, Logotypografia: LG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' EXPERIMENTAL SETTINGS  For our experiments we use the following hyper-parameter  settings, unless explicitly stated otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' For model training,  we use AdamW optimizer [69] with learning rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='0003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' We  apply warmup for the first 10% of the maximum training  steps, and a linear learning rate decay after that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Models  are finetuned for a maximum of 10000 steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' For speech  recognition training, we make use of the Connectionist Tem-  poral Classification (CTC) loss [70], optimized using the  available transcribed data in each scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Validation runs  every 500 steps on the development set, and early stopping  is employed on the development CTC loss with patience 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  Batch size is set to 8 during finetuning for all scenarios,  except for M2DS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' In the case of M2DS2 we create mixed  batches of size 12, containing 4 transcribed source domain  samples and 8 unlabeled target domain samples and train  for 10, 000 CTC updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' For memory reasons we split the  mixed batches in mini-batches of 4 and interleave them during  model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Gradients are accumulated over 3 interleaved  batches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' For the self-supervised objective, we create masks  of maximum timestep length 10, with masking probability  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' We weigh the contributions of the source and target  domain contrastive objectives, and bring them to the same  order of magnitude as the CTC loss, by setting α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='01 and  β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' The convolutional feature encoder is kept frozen  for all experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Our code is based on the huggingface 4  implementation of XLSR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' For all experiments we resample  the audio files to 16 kHz and downsample to single channel  audio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' We exclude utterances in the training set that are longer  than 12 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' All experiments are run on a single NVIDIA  RTX 3090 GPU, with mixed precision training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  For the Language model training, we create a large corpus  for the Greek language using a subset of the Greek part of CC-  Net [71] (approximately 11 billion tokens) and combine it with  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='5 billion tokens from the Greek version of Wikipedia and the  Hellenic National Corpus (HNC) [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' During preprocessing,  we remove all punctuation and accents, deduplicate lines and  convert all letters to lowercase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' We will refer to this corpus as  the Generic Greek Corpus (GGC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' We train a 4-gram language  model on GGC using KenLM [73] and prune bigrams, trigrams  and four-grams with counts less than 3, 5 and 7 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  We incorporate the n-gram LMs at inference time using the  pyctcdecode framework5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' We use language model rescoring  over a beam search decoder with 13 beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  The evaluation metric is the Word Error Rate (WER) over  the target test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' For assessing the adaptation effectiveness we  also report the relative WER improvement over the unadapted  baseline in appropriate scenarios, which is defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  We refer to this metric as Relative Adaptation Improvement  (RAI) for the rest of this paper:  RAI = −WERadapted − WERunadapted  WERunadapted  × 100%  (5)  The minus sign is included, so that RAI takes negative  values when the adaptation fails, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=', when WERunadapted <  WERadapted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  TABLE IV  ASR PERFORMANCE OF XLSR-53 OVER THE THREE CORPORA FOR FULLY  SUPERVISED IN-DOMAIN FINETUING (WER)  Dataset  LM  No LM  4g GGC  HP  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='21  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='64  CV  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='33  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='52  LG  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='94  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='45  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' SUPERVISED IN-DOMAIN TRAINING  In the first set of experiments, we explore the performance  of supervised finetuning of XLSR-53 for each domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' This  4https://huggingface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='co/docs/transformers/  5https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='com/kensho-technologies/pyctcdecode  8  TABLE V  M2DS2 PERFORMANCE USING GREEDY DECODING FOR UDA BETWEEN HP, CV, AND LG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' A → B INDICATES THAT A IS THE SOURCE DOMAIN AND B IS  THE TARGET DOMAIN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' (G) INDICATES GREEDY DECODING.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' (LM) INDICATES BEAM SEARCH WITH LM RESCORING.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' WE REPORT THE WER ON THE  TARGET TEST SET, AS WELL AS THE RAI (%) OVER THE SO (UNADAPTED) BASELINE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' WER: LOWER IS BETTER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' RAI: HIGHER IS BETTER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  Method  SO (G)  CPT (G)  PSL (G)  M2DS2 (G)  SO (LM)  CPT (LM)  PSL (LM)  M2DS2 (LM)  Setting  WER  WER  RAI  WER  RAI  WER  RAI  WER  WER  RAI  WER  RAI  WER  RAI  HP → CV  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='9  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='68  −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='8  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='2  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='95  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='3  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='26  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='44  −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='7  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='24  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='0  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='35  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='4  HP → LG  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='65  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='63  −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='2  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='68  −18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='6  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='99  −21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='3  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='34  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='27  −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='4  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='32  −29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='6  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='58  −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='4  LG → CV  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='57  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='43  −13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='4  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='90  −39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='8  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='31  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='4  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='96  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='51  −21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='4  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='05  −100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='5  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='30  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='4  LG → HP  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='13  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='51  −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='7  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='46  −15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='0  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='09  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='3  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='48  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='58  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='3  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='36  −44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='1  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='36  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='4  CV → LG  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='55  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='12  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='3  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='34  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='6  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='40  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='8  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='80  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='40  −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='2  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='68  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='2  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='93  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='3  CV → HP  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='72  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='83  −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='4  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='05  −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='4  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='70  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='9  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='09  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='18  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='2  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='82  −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='2  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='88  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='6  will give an upper bound estimation for UDA performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  We finetune XLSR-53 on CV, HP and LG (separately) and  perform in-domain evaluation on the respective test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  Results are summarized in Table IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' The first row indicates the  performance of greedy decoding, while in the second row we  report the performance of the beam search decoder, rescored  using the scores of the 4-gram GGC language model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' We  observe that the greedy decoding performance is under 30  WER for both HP and CV, while for LG we achieve ∼ 32  WER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' This makes sense, as LG is the most diverse dataset,  with respect to the included acoustic conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Furthermore,  we observe that the incorporation of a language model results  in an impressive WER reduction on CV, followed by HP and  then LG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' While CV includes relatively simple phrases with  common vocabulary, HP and LG contain more specialized  terminology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' UNSUPERVISED DOMAIN ADAPTATION USING  IN-DOMAIN AUDIO  Here, we evaluate the effectiveness of M2DS2 for UDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  We compare with three baselines:  1) Source Only Training (SO): We perform supervised  finetuning of XLSR-53 (CTC) using only the source-  domain data, and run decoding on the target domain  test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' No in-domain data are used for adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  2) Continual Pre-Training (CPT): We perform a pre-  training phase using the loss in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' (3) on the target  domain train set, to create adapted versions of XLSR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  Pre-training is run for 20000 steps with batch size  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Only the audio is used, without transcriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' The  adapted checkpoints are then finetuned by use of CTC  loss on the source domain transcribed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Evaluation  is performed on the target test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  3) Pseudolabeling (PSL): We finetune XLSR-53 using the  source domain data with CTC loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Then we run infer-  ence on the source model, to extract silver transcriptions  for the target domain training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' We use the silver  transcriptions for supervised finetuning on the target  domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  In Table V we compare M2DS2 with the SO, CPT and  PSL baselines for six adaptation scenarios, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=', cross dataset  evaluation between the three datasets in GREC-MD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' The left  half corresponds to greedy decoding, while for the right half  we use the 4-gram LM trained on GGC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' First, we observe  the SO model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' The SO models are the finetuned  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Performance of M2DS2 (blue line) for the LG → CV setting, when  reducing the amount of available target samples to 50%, 25%, and 10% of  the original dataset (horizontal axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' SO performance is indicated with the  orange line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Vertical axis: WER, Horizontal Axis: target audio percentage  (100% → 0%)  models from Table IV, evaluated in out-of-domain settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  We see that out-of-domain evaluation results in a large perfor-  mance hit, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=', while in the CV9 → CV9 in-domain setting  we achieve 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='33 WER, in the CV9 → HP out-of-domain  setting we get 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='55 WER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' This confirms that for real-world  ASR tasks, multi-domain evaluation is of essence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Second, we  observe that in most adaptation scenarios both CPT and PSL  fail to surpass the SO (unadapted) baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' In the case of CPT,  we hypothesize that is due to the relatively data constrained  version of our setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' In the best-case scenario, we have 99  hours of available target domain audio, which is not enough  to perform a discrete CPT stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Note that most of works in  the literature use ∼ 1000 hours of target audio for CPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' In  the case of PSL, the poor performance is due to the quality  of the silver labels created by the seed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' While the  performance would improve with more elaborate approaches  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=', confidence filtering), in challenging adaptation scenarios  PSL approaches are limited by the SO model’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  Lastly, we observe that M2DS2 is the only approach among  our baselines that manages to achieve a positive RAI in most  adaptation scenarios, by consistently outperforming the SO  baseline by significant margins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' This is exaggerated when  we include a LM during inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' One exception in this  pattern is the HP → LG scenario, where the SO baseline  achieves the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' We attribute this to the fact that  we performed minimal hyper-parameter tuning during model  development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  65  60  WER  55  50  100%  90%  80%  70%  60%  50%  40%  30%  20%  10%  0%  Percentage of In-Domain Audio Data9  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' The sample efficiency of M2DS2  One key observation in the literature, and in our experiments  is that CPT requires a large amount of un-transcribed target  domain audio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' This raises the question, can we leverage self-  supervision for domain adaptation in data constrained settings?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 3 we evaluate the performance of M2DS2, when  we reduce the amount of target domain audio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Specifically  we focus on the scenario of LG → CV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' The full training  corpus of CV contains 12 hours of audio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' We train M2DS2  with 50%, 25% and 10% of the available samples, or 6, 3  and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='2 hours of audio respectively, and plot the resulting  WER on the target (CV) test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' In all cases, the full source  (LG) training corpus is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' We observe that M2DS2 achieves  lower WER than the SO baseline, even with only 3 hours of  target domain audio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' While CPT can suffer from catastrophic  forgetting, as most multi-stage training approaches, M2DS2  avoids this issue, being a single-stage approach with a mixed  task-specific and self-supervised objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' This provides a  promising avenue for adaptation, when collection of in-domain  recordings is expensive, or infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  (a) Only target domain self-supervision  (b) Target and source domain self-supervision  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' T-SNE scatter plots of code vectors extracted from M2DS2 without  source domain self-supervision (top) and with source domain self-supervision  (bottom) for LG (red) and CV (teal)  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' The importance of Multi-Domain Self-Supervision  In Section III-B we argue that it is essential to include both  source and target domain data for the self-supervised objective  of M2DS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' To illustrate the effect of this approach, we train  two versions of M2DS2 for the LG → CV scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' For the  TABLE VI  LANGUAGE ADAPTATION OF THE M2DS2 LG → CV MODEL, USING  BIASED AND AUGMENTED LMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' WE USE THE VARIANT OF THE MODEL  TRAINED WITH 3 HOURS OF IN-DOMAIN AUDIO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' WE VARY THE AMOUNT  OF IN-DOMAIN TEXT DATA FROM 752K TOKENS TO 38K TOKENS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  Biased LM  Augmented LM  100%  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='22  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='84  50%  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='13  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='05  25%  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='84  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='64  10%  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='75  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='47  5%  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='04  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='31  Baseline (M2DS2 + Generic LM)  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='7  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Language-only adaptation for LG → HP using the SO model finetuned  on LG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' In-domain text data range from 11M tokens (left) to 110K tokens  (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Blue/dashed: Baseline with generic LM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Purple/circles: Biased LM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  Orange/diamonds: Augmented LM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  first version we set α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='01, while for the second we set  α = 0, removing the second term of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' We extract the  code vectors for the first 100 samples of both LG and CV, and  flatten them across the time steps , resulting to 60000 × 768  code vectors corresponding to individual timesteps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' We plot  these code vectors using T-SNE [74] in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 4 for both models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  We see that when we do not include the source domain self-  supervision, the code vector space collapses in a few tight  clusters, and most audio segments correspond to just a few  code vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' This is a visual clue that indicates the mode  collapse problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' When we include the source domain term,  we see that the that the code vector space has more structure,  and coverage of the space is more complete, both for CV  (target domain) and LG (source domain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Experimentally we  train M2DS2 with α = 0 for all source / target domain pairs  and we find that the mode collapse is destructive for target  domain performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' During our experiments we got WER in  the range 80−99, indicating failure to converge to acceptable  solutions across all scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' The simple inclusion of both  source and target domain self supervision stabilizes training,  avoids mode collapse and leads to successful unsupervised  adaptation between domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' UNSUPERVISED AND WEAKLY SUPERVISED  LANGUAGE ADAPTATION  When small amounts of in-domain textual data are avail-  able, simple N-gram LM adaptation techniques can be very  effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' In this brief set of experiments, we first explore  the unsupervised language adaptation setting, where no in-  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  :  :·  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  ！' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='0  C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  008  :  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='80 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  o43  41  39  37  WER  35  33  31  29  100%  90%  80%  70%  60%  50%  40%  30%  20%  10%  0%  Percentage of In-Domain Text Data10  TABLE VII  CLOSING THE GAP BETWEEN SO TRAINING AND FULLY SUPERVISED  TRAINING FOR THE LG → CV ADAPTATION SCENARIO USING M2DS2,  WITH VARYING AMOUNTS OF AVAILABLE UNPAIRED IN-DOMAIN AUDIO  AND TEXT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' (U): UNSUPERVISED ACOUSTIC OR LANGUAGE ADAPTATION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  (W): WEAKLY SUPERVISED ADAPTATION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  Method  #Audio (h)  #Tokens  LM  WER  SO (U) N/A  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='57  M2DS2 (U)  3 N/A  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='31  M2DS2 (U)  12 N/A  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='31  SO (U) Generic  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='96  SO (U) 38, 632  Augmented  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='67  SO (U) 751, 953  Augmented  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='46  M2DS2 (U)  3 Generic  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='7  M2DS2 (U)  12 Generic  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='3  M2DS2 (W)  3  38, 632  Augmented  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='31  M2DS2 (W)  12  38, 632  Augmented  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='29  M2DS2 (W)  3  751, 953  Augmented  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='84  M2DS2 (W)  12  751, 953  Augmented  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='61  Supervised  12  751, 953  Generic  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='52  Supervised  12  751, 953  Augmented  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='94  domain audio is used, and then we relax the problem to  the weakly supervised setting, where M2DS2 is combined  with the adapted N-Gram LMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' These settings are described  in Sections II-A2 and II-A3 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' We explore two  approaches for LM adaptation: biased LMs, and in-domain  data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' To create biased LMs, we train a 4-gram  LM on the available in-domain data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Then we replace the  generic LM trained on GGC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' For LM data augmentation we  follow a perplexity filtering approach similar to [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' We first  train a biased LM using available target domain text, and  then use it to calculate the perplexity of each line in the  GGC corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' We keep the 10% of the lines with the lowest  perplexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Then we train a 4-gram LM on the augmented “in-  domain” corpus and use it for inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 5 shows the performance of the SO LG → HP model  with biased and augmented LMs, as we reduce the amount  of available in-domain text data from 100% to 1% of the  in-domain transcriptions (11B tokens to 110K tokens respec-  tively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' As a baseline we include the LG → HP SO model in  combination with the generic LM trained on GGC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' We observe  that the use of biased LMs can lead to successful adaptation,  when an adequate amount of in-domain text data is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  On the other hand the LM augmentation approach results to  successful augmentation, even with very small amounts of in-  domain text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  In Table VI we see the results of LM adaptation, combined  with the M2DS2 LG → CV model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' To demonstrate the sample  efficiency of the approach, we use the variant that was trained  using only 25% of the target domain audio (3 hours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' We  compare with M2DS2 combined with the 4-gram GGC LM for  inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' We draw similar conclusions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=', use of biased LMs  performs well for sufficient text data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' When we use augmented  LMs we can leverage very small amounts of in-domain text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' DISCUSSION & CONCLUSIONS  In this work, we have explored Unsupervised and Weakly  Supervised Domain Adaptation of ASR systems in the con-  text of an under-resourced language, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=', Greek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' We focus  on domain adaptation through in-domain self-supervision for  XLSR-53, a state-of-the-art multilingual ASR model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Specif-  ically, we adopt a mixed task and self-supervised objective,  inspired from NLP, and show that using only in-domain self-  supervision can lead to mode collapse of the representa-  tions created by the contrastive loss of XLSR-53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Therefore,  we propose the use of mixed task and multi-domain self-  supervision, M2DS2, where the contrastive loss leverages both  the source and target domain audio data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' For evaluation we  create and release HParl, the largest to-date public corpus  of transcribed Greek speech (120 hours), collected from the  Greek Parliamentary Proceedings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' HParl is combined with two  other popular Greek speech corpora, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=', Logotypografia and  CommonVoice, for multi-domain evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  In our experiments, we find that while most UDA baselines  fail in our low-resource setting, the proposed mixed task  and multi-domain self-supervised finetuning strategy yields  significant improvements for the majority of adaptation sce-  narios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Furthermore, we focus our ablations on showcasing  the sample efficiency of the proposed finetuning strategy,  and demonstrating the necessity of including both source  and target domain data for self-supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Finally, we show  that M2DS2 can be combined with simple language model  adaptation techniques in a relaxed weakly supervised setting,  where we achieve significant performance improvements with  a few hours of in-domain audio and a small, unpaired in-  domain text corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  More concretely, in Table VII we present a summary of  the discussed unsupervised and weakly supervised adaptation  combinations, for different amounts of available in-domain  audio and text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Note that for the weakly supervised scenarios,  the in-domain audio and text are unpaired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' We see, that when  no in-domain data are available, including an n-gram LM  trained on large corpora is recommended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Furthermore, when  in-domain audio is available, following a mixed multi-domain  finetuning strategy using M2DS2 can yield significant WER  reductions, even for a few hours of audio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' When small amounts  of in-domain text is available, using a corpus augmentation  strategy, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=', perplexity filtering, can produce adapted LMs  and yield small improvements to the final WER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' In the case  of sufficient amounts of unpaired in-domain text and audio,  independent adaptation of XLSR-53 using the audio data and  the n-gram LM using the text data can yield comparable  performance with a fully supervised finetuning pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' FUTURE WORK  In the future we plan to explore the effectiveness of the  proposed adaptation strategy for other languages, and different  adaptation settings, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=', accent or cross-lingual adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  Of special interest is the investigation of the effectiveness  of our approach for endagered languages, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=', Pomak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Fur-  thermore, we plan to explore the combination of in-domain  self-supervision, when combined with other popular UDA  techniques, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=', teacher student models, adversarial learning,  and data augmentation approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' On the language adaptation  side, we plan to explore multi-resolution learning, which has  11  shown promise for ASR [75], and investigate more elaborate  end-to-end weakly supervised adaptation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Finally, we  plan to expand our study in a multimodal setting, where both  audio and video are available, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=', lip reading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  REFERENCES  [1] Mingsheng Long, Yue Cao, Jianmin Wang, and Michael Jordan, “Learn-  ing transferable features with deep adaptation networks,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' ICML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  PMLR, 2015, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 97–105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [2] Yaroslav Ganin, Evgeniya Ustinova, Hana Ajakan, Pascal Germain,  Hugo Larochelle, Franc¸ois Laviolette, Mario Marchand, and Victor  Lempitsky, “Domain-adversarial training of neural networks,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Mach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  Learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 17, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 2096–2030, jan 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [3] Peter Bell, Joachim Fainberg, Ondrej Klejch, Jinyu Li, Steve Renals,  and Pawel Swietojanski,  “Adaptation algorithms for neural network-  based speech recognition: An overview,” IEEE Open Journal of Signal  Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 33–66, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [4] Jinyu Li, Michael L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Seltzer, Xi Wang, Rui Zhao, and Yifan Gong,  “Large-scale domain adaptation via teacher-student learning,” in Proc  Interspeech, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [5] Vimal Manohar, Pegah Ghahremani, Daniel Povey, and Sanjeev Khu-  danpur, “A teacher-student learning approach for unsupervised domain  adaptation of sequence-trained asr models,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Spoken Language  Technology Workshop (SLT), 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [6] Yusuke Shinohara, “Adversarial Multi-Task Learning of Deep Neural  Networks for Robust Speech Recognition,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Interspeech 2016,  2016, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 2369–2372.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [7] Zhong Meng, Jinyu Li, Zhuo Chen, Yang Zhao, Vadim Mazalov,  Yifan Gong, and Biing-Hwang Juang, “Speaker-invariant training via  adversarial learning,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' ICASSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 2018, IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [8] Sining Sun, Ching-Feng Yeh, Mei-Yuh Hwang, Mari Ostendorf, and Lei  Xie, “Domain adversarial training for accented speech recognition,” in  Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' ICASSP, 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 4854–4858.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [9] Anoop C S, Prathosh A P, and A G Ramakrishnan,  “Unsupervised  domain adaptation schemes for building asr in low-resource languages,”  in Automatic Speech Recognition and Understanding Workshop (ASRU),  2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 342–349.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [10] Taichi Asami et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=', “Domain adaptation of dnn acoustic models using  knowledge distillation,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' ICASSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' IEEE, 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 5185–5189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [11] Karol Nowakowski, Michal Ptaszynski, Kyoko Murasaki, and Jagna  Nieuwa˙zny,  “Adapting multilingual speech representation model for  a new, underresourced language through multilingual fine-tuning and  continued pretraining,”  Information Processing & Management, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  60, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 103148, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [12] Sadaoki Furui, “A training procedure for isolated word recognition sys-  tems,” IEEE Transactions on Acoustics, Speech, and Signal Processing,  vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 28, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 129–136, 1980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [13] Yajie Miao, Hao Zhang, and Florian Metze, “Towards speaker adaptive  training of deep neural network acoustic models,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Interspeech,  2014, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 2189–2193.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [14] Sree HK Parthasarathi e al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=', “fmllr based feature-space speaker adapta-  tion of dnn acoustic models,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Interspeech, 2015, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 3630–3634.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [15] Vishwa Gupta, Patrick Kenny, Pierre Ouellet, and Themos Stafylakis,  “I-vector-based speaker adaptation of deep neural networks for french  broadcast audio transcription,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' ICASSP, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [16] Hans-G¨unter Hirsch and David Pearce,  “The aurora experimental  framework for the performance evaluation of speech recognition systems  under noisy conditions,”  in ASR2000-Automatic speech recognition:  challenges for the new Millenium ISCA tutorial and research workshop  (ITRW), 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [17] Yanmin Qian, Tian Tan, and Dong Yu,  “An investigation into using  parallel data for far-field speech recognition,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' ICASSP, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [18] William Chan, Navdeep Jaitly, Quoc Le, and Oriol Vinyals, “Listen,  attend and spell: A neural network for large vocabulary conversational  speech recognition,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' ICASSP, 2016, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 4960–4964.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [19] Alex Graves, “Sequence transduction with recurrent neural networks,”  CoRR, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' abs/1211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='3711, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [20] Alexei Baevski, Yuhao Zhou, Abdelrahman Mohamed, and Michael  Auli, “wav2vec 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='0: A framework for self-supervised learning of speech  representations,” Advances in Neural Information Processing Systems,  vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 33, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 12449–12460, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [21] Alexis Conneau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=',  “Unsupervised Cross-Lingual Representation  Learning for Speech Recognition,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Interspeech, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 2426–  2430.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [22] Pavel Denisov, Ngoc Thang Vu, and Marc Ferras Font, “Unsupervised  domain adaptation by adversarial learning for robust speech recognition,”  in Speech Communication, 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 1–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [23] Dongseong Hwang, Ananya Misra, Zhouyuan Huo, Nikhil Siddhartha,  Shefali Garg, David Qiu, Khe Chai Sim, Trevor Strohman, Franc¸oise  Beaufays, and Yanzhang He, “Large-scale asr domain adaptation using  self- and semi-supervised learning,” in NeurIPS, 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 6627–6631.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [24] Wei-Ning Hsu, Anuroop Sriram, Alexei Baevski, Tatiana Likhoma-  nenko, Qiantong Xu, Vineel Pratap, Jacob Kahn, Ann Lee, Ronan  Collobert, Gabriel Synnaeve, and Michael Auli, “Robust wav2vec 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='0:  Analyzing Domain Shift in Self-Supervised Pre-Training,”  in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  Interspeech, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 721–725.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [25] Sameer Khurana, Niko Moritz, Takaaki Hori, and Jonathan Le Roux,  “Unsupervised domain adaptation for speech recognition via uncertainty  driven self-training,” Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' ICASSP, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 6553–6557, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [26] Sankaran Panchapagesan, Daniel S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Park, Chung-Cheng Chiu, Yuan  Shangguan, Qiao Liang, and Alexander Gruenstein, “Efficient knowl-  edge distillation for rnn-transducer models,” in Proc ICASSP, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  5639–5643.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [27] Anmol Gulati et al, “Conformer: Convolution-augmented transformer  for speech recognition,” Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Interspeech, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 5036–5040, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [28] Hasim Sak, Andrew W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Senior, and Franc¸oise Beaufays, “Long short-  term memory recurrent neural network architectures for large scale  acoustic modeling,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Interspeech, 2014, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 338–342.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [29] Taichi Asami, Ryo Masumura, Yoshikazu Yamaguchi, Hirokazu Masa-  taki, and Yushi Aono, “Domain adaptation of dnn acoustic models using  knowledge distillation,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' ICASSP, 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 5185–5189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [30] Takuya Yoshioka, Nobutaka Ito, Marc Delcroix, Atsunori Ogawa,  Keisuke Kinoshita, Masakiyo Fujimoto, Chengzhu Yu, Wojciech J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  Fabian, Miquel Espi, Takuya Higuchi, Shoko Araki, and Tomohiro  Nakatani, “The ntt chime-3 system: Advances in speech enhancement  and recognition for mobile multi-microphone devices,” in Workshop on  Automatic Speech Recognition and Understanding (ASRU), 2015, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  436–443.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [31] Bhuvana Ramabhadran, Brian Farris, Isabel Leal, Manasa Prasad, Neeraj  Gaur, Parisa Haghani, Pedro Jose Moreno Mengibar, and Yun Zhu,  “Self-adaptive distillation for multilingual speech recognition: Leverag-  ing student independence,” in Interspeech 2021, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [32] Yanzhang He, Tara N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Sainath, Rohit Prabhavalkar, Ian McGraw, Raziel  Alvarez, Ding Zhao, David Rybach, Anjuli Kannan, Yonghui Wu,  Ruoming Pang, Qiao Liang, Deepti Bhatia, Yuan Shangguan, Bo Li,  Golan Pundak, Khe Chai Sim, Tom Bagby, Shuo-yiin Chang, Kanishka  Rao, and Alexander Gruenstein, “Streaming end-to-end speech recog-  nition for mobile devices,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' ICASSP, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 6381–6385.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [33] Dmitriy Serdyuk, Kartik Audhkhasi, Philemon Brakel, Bhuvana Ramab-  hadran, Samuel Thomas, and Yoshua Bengio, “Invariant representations  for noisy speech recognition,” CoRR, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [34] Sining Sun, Binbin Zhang, Lei Xie, and Yanning Zhang, “An unsuper-  vised deep domain adaptation approach for robust speech recognition,”  Neurocomputing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 257, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 79–87, 2017,  Machine Learning and  Signal Processing for Big Multimedia Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [35] Vijayaditya Peddinti, Daniel Povey, and Sanjeev Khudanpur, “A time  delay neural network architecture for efficient modeling of long temporal  contexts,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Interspeech, 2015, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 3214–3218.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [36] Daniel Povey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=', “The kaldi speech recognition toolkit,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  ASRU Workshop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 2011, IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [37] Seyedmahdad Mirsamadi and John H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Hansen,  “Multi-domain ad-  versarial training of neural network acoustic models for distant speech  recognition,” Speech Communication, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 106, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 21–30, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [38] Jan K Chorowski, Dzmitry Bahdanau, Dmitriy Serdyuk, Kyunghyun  Cho, and Yoshua Bengio, “Attention-based models for speech recog-  nition,” Advances in neural information processing systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 28,  2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [39] Hu Hu, Xuesong Yang, Zeynab Raeesy, Jinxi Guo, Gokce Keskin, Harish  Arsikere, Ariya Rastrow, Andreas Stolcke, and Roland Maas, “redat:  Accent-invariant representation for end-to-end asr by domain adversarial  training with relabeling,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' ICASSP, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [40] Aditay Tripathi, Aanchan Mohan, Saket Anand, and Maneesh Singh,  “Adversarial learning of raw speech features for domain invariant speech  recognition,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' ICASSP, 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 5959–5963.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [41] Konstantinos Bousmalis, George Trigeorgis, Nathan Silberman, Dilip  Krishnan, and Dumitru Erhan,  “Domain separation networks,”  in  Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' NIPS, Red Hook, NY, USA, 2016, NIPS’16, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 343–351, Curran  Associates Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [42] Han Zhu, Gaofeng Cheng, Jindong Wang, Wenxin Hou, Pengyuan  Zhang, and Yonghong Yan, “Boosting cross-domain speech recognition  with self-supervision,” arXiv preprint arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='09783, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  12  [43] Jounghee Kim and Pilsung Kang, “K-Wav2vec 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='0: Automatic Speech  Recognition based on Joint Decoding of Graphemes and Syllables,” in  Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Interspeech, 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 4945–4949.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [44] Mitchell DeHaven and Jayadev Billa, “Improving low-resource speech  recognition with pretrained speech models: Continued pretraining vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  semi-supervised training,” arXiv preprint arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='00659, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [45] Constantinos Karouzos, Georgios Paraskevopoulos, and Alexandros  Potamianos, “UDALM: Unsupervised domain adaptation through lan-  guage modeling,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Association for Computational Linguistics:  Human Language Technologies, Online, June 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 2579–2590,  Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [46] Alex Graves, Santiago Fern´andez, Faustino Gomez, and J¨urgen Schmid-  huber, “Connectionist temporal classification: Labelling unsegmented  sequence data with recurrent neural networks,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' of the 23rd Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' on Machine Learning, New York, NY, USA, 2006, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' ICML,  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 369–376, Association for Computing Machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [47] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Scudder, “Probability of error of some adaptive pattern-recognition  machines,” IEEE Transactions on Information Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 11, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 3,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 363–371, 1965.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [48] David Yarowsky,  “Unsupervised word sense disambiguation rivaling  supervised methods,” in Annu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Meeting of the Association for Compu-  tational Linguistics, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [49] Ellen Riloff and Janyce Wiebe, “Learning extraction patterns for sub-  jective expressions,” in Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Empirical Methods in Natural Language  Processing, 2003, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 105–112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [50] Dongseong Hwang, Khe Chai Sim, Yu Zhang, and Trevor Strohman,  “Comparison of soft and hard target rnn-t distillation for large-scale  asr,” 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [51] Jacob Kahn, Ann Lee, and Awni Hannun, “Self-training for end-to-end  speech recognition,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' ICASSP, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 7084–7088.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [52] Daniel S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Park et al, “Improved noisy student training for automatic  speech recognition,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Interspeech, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [53] Yu Zhang, James Qin, Daniel S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Park, Wei Han, Chung-Cheng Chiu,  Ruoming Pang, Quoc V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Le, and Yonghui Wu, “Pushing the limits of  semi-supervised learning for automatic speech recognition,” 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [54] Daniel S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Park, William Chan, Yu Zhang, Chung-Cheng Chiu, Barret  Zoph, Ekin D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Cubuk, and Quoc V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Le, “SpecAugment: A Simple Data  Augmentation Method for Automatic Speech Recognition,”  in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  Interspeech, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 2613–2617.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [55] Yaroslav Ganin and Victor Lempitsky, “Unsupervised domain adaptation  by backpropagation,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' ICML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 2015, ICML’15, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 1180–1189,  JMLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [56] Douglas B Paul and Janet Baker, “The design for the wall street journal-  based csr corpus,” in Speech and Natural Language: Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' of a Workshop  Held at Harriman, New York, February 23-26, 1992, 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [57] Siu-Kei Au Yeung and Man-Hung Siu,  “Improved performance of  aurora 4 using htk and unsupervised mllr adaptation,” in Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Spoken  Language Processing, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [58] Suchin Gururangan, Ana Marasovi´c, Swabha Swayamdipta, Kyle Lo,  Iz Beltagy, Doug Downey, and Noah A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Smith, “Don’t stop pretraining:  Adapt language models to domains and tasks,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' of the 58th Annu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  Meeting of the Association for Computational Linguistics, Online, July  2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 8342–8360, Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [59] Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova,  “BERT: pre-training of deep bidirectional transformers for language  understanding,” CoRR, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' abs/1810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='04805, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [60] Herve Jegou, Matthijs Douze, and Cordelia Schmid, “Product quantiza-  tion for nearest neighbor search,” IEEE transactions on pattern analysis  and machine intelligence, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 33, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 117–128, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [61] Eric Jang, Shixiang Gu, and Ben Poole, “Categorical reparametrization  with gumbel-softmax,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' ICLR, Apr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [62] Vassil Panayotov, “Librispeech: an asr corpus based on public domain  audio books,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' ICASSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' IEEE, 2015, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 5206–5210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [63] Aimilios Chalamandaris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=', “All greek to me!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' an automatic greeklish  to greek transliteration system,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' LREC, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [64] Carsten Meyer and Hauke Schramm, “Boosting hmm acoustic models  in large vocabulary speech recognition,” Speech Communication, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  48, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 532–548, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [65] Vimal Manohar, Daniel Povey, and Sanjeev Khudanpur,  “Jhu kaldi  system for arabic mgb-3 asr challenge using diarization, audio-transcript  alignment and transfer learning,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' ASRU Workshop, 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  346–352.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [66] Vassilios Digalakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=', “Large vocabulary continuous speech recog-  nition in greek: corpus and an automatic dictation system,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  Eurospeech, 2003, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 1565–1568.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [67] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Smith and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Waterman, “Identification of common molecular  subsequences,” Journal of Molecular Biology, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 147, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 195–  197, 1981.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [68] Rosana Ardila et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=', “Common voice: A massively-multilingual speech  corpus,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' LREC, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 4218–4222.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [69] Ilya Loshchilov and Frank Hutter, “Decoupled weight decay regulariza-  tion,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' ICLR, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [70] Alex Graves, Santiago Fern´andez, Faustino Gomez, and J¨urgen Schmid-  huber,  “Connectionist temporal classification: labelling unsegmented  sequence data with recurrent neural networks,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' ICML, 2006,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 369–376.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [71] Guillaume Wenzek, Marie-Anne Lachaux, Alexis Conneau, Vishrav  Chaudhary, Francisco Guzm´an, Armand Joulin, and ´Edouard Grave,  “Ccnet: Extracting high quality monolingual datasets from web crawl  data,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' of the 12th Language Resources and Evaluation Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=',  2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 4003–4012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [72] Nick Hatzigeorgiu, Maria Gavrilidou, Stelios Piperidis, George Carayan-  nis,  Anastasia  Papakostopoulou,  Athanassia  Spiliotopoulou,  Anna  Vacalopoulou, Penny Labropoulou, Elena Mantzari, Harris Papageor-  giou, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=',  “Design and implementation of the online ilsp greek  corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=',” in LREC, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [73] Kenneth Heafield, “Kenlm: Faster and smaller language model queries,”  in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' of the 6th workshop on statistical machine translation, 2011,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 187–197.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [74] Laurens Van der Maaten and Geoffrey Hinton, “Visualizing data using  t-sne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=',” Journal of machine learning research, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 9, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 11, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content='  [75] Georgios Paraskevopoulos, Srinivas Parthasarathy, Aparna Khare, and  Shiva Sundaram, “Multimodal and multiresolution speech recognition  with transformers,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' of the 58th Annu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' Meeting of the Association  for Computational Linguistics, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
+page_content=' 2381–2387.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf'}
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+MNRAS 000, 1–7 (2023)
+Preprint 30 January 2023
+Compiled using MNRAS LATEX style file v3.0
+Formation of asymmetric arms in barred galaxies
+P. Sánchez-Martín,1⋆ C. García-Gómez,2 J. J. Masdemont3 and M. Romero-Gómez4
+1 IUMA, Universidad de Zaragoza, Dept. de Matemática Aplicada, Pedro Cerbuna, 12, 50009 Zaragoza, Spain
+2D.E.I.M, Universitat Rovira i Virgili, Avd. Països Catalans, 26, 43007 Tarragona, Spain
+3IEEC & Universitat Politècnica de Catalunya, Dept. de Matemàtiques, Diagonal 647, E08028 Barcelona, Spain
+4Institut de Ciències del Cosmos (ICCUB), Universitat de Barcelona (IEEC-UB), Martí i Franquès 1, E08028 Barcelona, Spain
+Accepted XXX. Received YYY; in original form ZZZ
+ABSTRACT
+We establish a dynamical mechanism to explain the origin of the asymmetry between the arms observed in some barred disk
+galaxies, where one of the two arms emanating from the bar ends is very well defined, while the second one displays a ragged
+structure, extending between its ridge and the bar. To this purpose, we study the invariant manifolds associated to the Lyapunov
+periodic orbits around the unstable equilibrium points at the ends of the bar. Matter from the galaxy center is transported along
+these manifolds to the periphery, forming this way the spiral arms that emanate from the bar ends. If the mass distribution in
+the galaxy center is not homogeneous, because of an asymmetric bar with one side stronger than the other, or because of a
+non-centered bulge, the dynamics about the two unstable Lagrange points at the ends of the bar will not be symmetric as well.
+One of their invariant manifolds becomes more extended than the other, enclosing a smaller section and the escaping orbits on it
+are fewer and dispersed in a wider region. The result is a weaker arm, and more ragged than the one at the other end of the bar.
+Key words: galaxies: kinematics and dynamics – galaxies: structure – galaxies: spiral
+1 INTRODUCTION
+A high fraction of disk galaxies appear to be barred (74% to 85%
+in automatic classifications or 36% to 63% with alternative meth-
+ods (Lee et al. 2019)). These authors also suggest that strongly
+barred galaxies, classified as SBs, are preponderant in late-type
+galaxies. Most of these strongly barred galaxies appear to be asym-
+metric under a simple visual inspection. They usually present a
+strong bar and two spiral arms emanating from the bar ends, and
+frequently some inner rings as well. However, a detailed visual in-
+spection of these galaxies, available in the STScI Digitized Sky Sur-
+vey or in the Sloan Digital Sky Survey (SDSS) (Bundy et al. 2015),
+reveals that some barred galaxies exhibit important asymmetries in
+their spiral structure. While one of the two spiral arms is long and
+strongly defined, the second arm shows a ragged structure, and the
+matter is distributed irregularly between the ridge of the spiral arm
+and the bar. A non-symmetric distribution of matter in the galaxy
+disc can be a natural outcome, since disc galaxies may be formed
+through a combination of secular evolution and violent events, in-
+cluding smooth accretion, disc instabilities and minor and major
+mergers (e.g. Tonini et al. 2016). In Fig. 1 we show some examples
+of galaxies showing asymmetric discs.
+In this paper we study the dynamic response of an asymmetric
+mass distribution on the orbital structure of barred galaxies, mod-
+elling the asymmetry of the central parts by a slightly off-centered
+bulge which, viewing the galaxy as a whole, represents a simple but
+realistic model of an asymmetric bar. In this scenario, the Lagrange
+⋆ E-mail: patricias@unizar.es
+Figure 1. Images of some strongly barred galaxies showing the asymmetric
+spiral arm patterns in the R-filter from the STScI Digitized Sky Survey. In the
+top row we show the galaxies PGC 05849 (NGC 0613), PGC 12412 (NGC
+1300) and in the bottom one PGC 15941 (NGC 1672) and PGC 54849 (NGC
+5921).
+invariant points located at the bar ends will present also an asym-
+metric orbital structure, as initially shown in Colin & Athanassoula
+(1989). The new step in this work is to additionally study the unsta-
+ble periodic orbits around these Lagrange points and the associated
+unstable manifolds which provide escape routes for orbits which
+can transport material from the central regions to the outer parts of
+the discs (Romero-Gómez et al. 2006; Sánchez-Martín et al. 2016).
+© 2023 The Authors
+arXiv:2301.11385v1  [astro-ph.GA]  26 Jan 2023
+
+.PGC 05849(R)
+PGC 12412 (R)
+PGC 15941 (R)
+PGC 54849 (R)2
+P. Sánchez-Martín et al.
+These unstable manifolds are the backbones of the spiral arms ema-
+nating from the bar ends. In the case of asymmetric Lagrange points,
+however, the unstable manifolds display also important differences.
+The orbits following one of these manifolds are close together and
+can explain the presence of a strong spiral arm. On the other hand,
+the second manifold has a more open orbital structure, with its es-
+caping orbits dispersed in a wider zone, causing the ragged struc-
+ture of the second arm. Thus, the orbital structure of the unstable
+Lagrange points is able to explain the asymmetric phenomenology
+present in some of the strongly barred galaxies.
+The paper is organized as follows: in Section 2 we quantify the
+level of asymmetry in asymmetric barred galaxies using a two-
+dimensional Fourier transform method. In Sections 3 and 4 we de-
+scribe the asymmetric barred galaxy model used in this work and
+show the orbital analysis and invariant manifolds, respectively. Dif-
+ferent approaches are summarized and conclusions are given in Sec-
+tion 5.
+2 ANALYSIS OF THE BAR ASYMMETRY
+In this paper, we model the galactic asymmetric mass distribution
+using a classical galactic model with three symmetric components
+(disc, bar and bulge) but displacing slightly the bulge from the
+galaxy centre. This results in a total mass distribution biased to-
+wards one side of the bar. Many barred galaxies show some kind
+of asymmetries in their inner mass distribution with one side of the
+bar stronger than the other. An example is the galaxy NGC 1300,
+studied by Patsis et al. (2010). They generated a numerical model
+of the potential using K images of the galaxy. The resultant model
+was clearly asymmetric, containing a bar with a stronger arm. This
+model was used to study the orbits associated with this asymmetric
+barred structure. Many barred galaxies show similar asymmetries.
+Some of these galaxies are shown in Fig. 1. In order to quantify
+these asymmetries, we can analyze in detail the mass distribution of
+one of the galaxies showing this phenomenology, namely the galaxy
+PGC 70419 (NGC 7479). For this purpose, we use a galaxy image
+from the SDSS survey in the infrared z-filter. Assuming a constant
+M/L ratio, the light distribution is a good tracer of the mass distribu-
+tion in the disk. The image is previously cleaned from background
+stars and then deprojected using the FFT method (García-Gómez et
+al. 2004) giving a position angle (PA) of 38◦ and an inclination angle
+of 45◦.
+The image is then decomposed in its Fourier components using a
+technique first introduced by Considère & Athanassoula (1982); Iye
+et al. (1982), and further developed by García-Gómez et al. (2017).
+For deprojected image of the galaxy I(u,θ), where u = ln(r), we cal-
+culate the two-dimensional Fourier transform defined as
+A(p,m) =
+� umax
+umin
+� 2π
+0
+I(u,θ)ei(pu+mθ) dθdu
+(1)
+Where m is the azimuthal frequency, associated with the multiplicity
+of the structures, i.e., the number of arms while p is the radial fre-
+quency, associated to the pitch angle of the structure i, through the
+relation
+p = −
+m
+tan(i)
+(2)
+In this way, the m = 1 spectrum contains the spiral components with
+no symmetry, the m = 2 spectrum the components with a periodicity
+of π radians or bisymmetric signals, and so on for the rest of the m
+frequencies. Each of the azimuthal components m = 1,2,... can be
+further decomposed in its radial components using a Gaussian fit to
+the modulus and keeping the phase constant as follows:
+| A(p,m) |=
+Ng
+�
+j=1
+C j exp−(p− pj)2
+2σ2
+j
+.
+(3)
+In this relation, pj represents the central frequency of the Gaussian,
+σj its dispersion, and C j its amplitude. The number of Gaussians
+used in each fit, Ng, will depend on the complexity of the spectrum.
+In the upper panel of Fig. 2 we show the deprojected SDSS galaxy
+image of NGC 7479 using the z-filter. Note that one of its arms is
+very pronounced, while the opposite arm appears diffuse. On the left
+of the middle and lower panels of this figure we show the modulus
+of the Fourier spectrum of the m = 1 component which is associ-
+ated to the asymmetries in the light distribution. The modulus of the
+Fourier components are normalized to the modulus of the stronger
+component, which in this case is the m = 2 containing the bisymmet-
+ric signals of the strong bar and the spiral arms. The m = 1 spectrum
+shown here contains the spiral components responsible for the asym-
+metries in the mass distribution. The relative low values of the m = 1
+in this scale shows that the mass asymmetry is a second order effect
+for this galaxy. In red we superpose two of the Gaussian components
+into which this signal is decomposed. These Gaussian components
+can be transformed back to obtain the density distribution associated
+to each particular spiral mode. The density distributions of these spi-
+ral components are presented in the respective right panels in green,
+superposed on the galaxy image. The isocontours come from the
+normalized density corresponding to the Gaussian components and
+show that the asymmetries are related to the southern end of the
+bar and the strong arm emanating from its end. This shows that the
+galaxy has an asymmetric mass distribution, biased to the southern
+part of the galaxy.
+3 CHARACTERISTICS OF THE GALACTIC MODEL
+The equations of motion of the classical galactic model (see e.g.
+Pfenniger 1984; Skokos et al. 2002; Romero-Gómez et al. 2006;
+Sánchez-Martín et al. 2016) describe the movement of a particle in
+a gravitational potential φ . In the rotating frame, the equations of
+motion are described by
+¨r = −2(Ωp × ˙r)−Ωp ×(Ωp ×r)−∇φ,
+(4)
+where r = (x, y, z) is the position of the particle, φ is the total grav-
+itational potential of the system and Ωp is the angular velocity of
+the bar around the z-axis, Ωp = (0,0,Ω). The origin of the reference
+frame is located at the center of mass of the system and the frame is
+aligned with the main axis of the bar.
+The potential model φ used in this paper is a combination of three
+analytical components: an axisymmetric Miyamoto-Nagai disc with
+potential φd (Miyamoto & Nagai 1975), an ellipsoid Ferrers bar with
+potential φb (Ferrers 1877) and a bulge structure represented by a
+Plummer spheroid potential φbl (Plummer 1911). The total potential
+is the addition of these three components, φ = φd +φb +φbl.
+The disc potential is described by the equation
+φd = −
+GMd
+�
+R2 +(A+
+�
+B2 +z2)2
+,
+(5)
+were R2 = x2 + y2 is the cylindrical coordinate radius of the poten-
+tial in the disc plane, and z is the vertical distance over the disk
+component. The parameters G, Md, A and B denote the gravitational
+MNRAS 000, 1–7 (2023)
+
+Formation of asymmetric arms in barred galaxies
+3
+Figure 2. Analysis of the m = 1 component of the SDSS galaxy image of
+NGC 7479 in the z-filter. The upper panel shows the deprojected image of the
+galaxy. On the left of the middle and lower panels we show the modulus of
+the m = 1 component of the Fourier spectrum, and with red-dashed lines two
+Gaussian components fitted to this modulus. In the respective right panels,
+we show in green the density distribution associated to each of these single
+components superimposed on the galaxy image.
+constant, the disc mass and the shape of the disc, respectively. Tak-
+ing A = 0 the potential becomes the Plummer potential. The bar is
+modelled by an ellipsoid with density function
+ρ =
+�
+ρ0(1−m2)nh,
+m ≤ 1
+0,
+m > 1
+(6)
+where m2 = x2/a2 + y2/b2 + z2/c2, a (semi-major axis), b (interme-
+diate axis) and c (semi-minor axis) determine the shape of the bar,
+nh is the homogeneity degree of the mass distribution (nh = 2 in our
+work) and ρ0 is the density at the origin (ρ0 = 105
+32π
+GMb
+abc if nh = 2,
+where Mb is the bar mass).
+The unit of length considered is the kpc, the time unit is ut =
+2 × 106 yr, Ω is in [ut]−1, and the mass unit is uM = 2 × 1011M⊙,
+where M⊙ denotes the mass of the Sun. G stands for the gravitational
+constant.
+In our model, we select a disc radius of A = 3 kpc, height B = 1 kpc
+and mass giving a value of GMd = 0.52 kpc3/u2
+t . The dimensions of
+the bar are a = 6 kpc, b = 1.5 kpc, c = 0.4 kpc, and its mass is such
+that GMb = 0.4 kpc3/u2
+t . The Plummer bulge with a radius B = 1 kpc
+and GMbl is set to have around 15% of the mass of the bar GMb, in
+order that G(Md + Mb + Mbl) = 1. The bar pattern speed is fixed as
+Ω = 0.0633 [ut]−1 (∼ 30.97 km/s/kpc).
+In the rotating reference frame aligned with the main axis of the
+bar, the equations of motion given in Eq.(4) are written as the fol-
+lowing dynamical system:
+���������
+¨x = 2Ω ˙y+Ω2 x−φx
+¨y = −2Ω ˙x+Ω2 y−φy
+¨z = −φz .
+(7)
+The Jacobi first integral of Eq.(4) (which can be regarded as the
+energy in the rotating frame) is given by
+CJ(x,y,z, ˙x, ˙y, ˙z) = −(˙x2 + ˙y2 + ˙z2)+Ω2 (x2 +y2)−2φ,
+(8)
+and the effective potential is defined by φeff = φ− 1
+2 Ω2 (x2 +y2).
+The goal of this paper is to analyze which is the effect of an asym-
+metric distribution of mass in the central parts of the galaxy on the
+external spiral structure. To introduce this asymmetry, we displace
+the bulge potential along the x-axis (main axis of the bar) towards the
+equilibrium point placed at the right end of the bar. The bulge center
+is then located at (xd,0,0) using several values for the displacement
+xd (in kpc), namely: (0,0,0), (0.5,0,0), (1,0,0) and (1.5,0,0). We
+move the center of our potential model to the resulting center of
+mass of the system. The plot of the equal density contours of the
+resulting models are shown in Fig. 3. Note that the displacement of
+the bulge along the main axis of the bar creates an asymmetry in the
+central part of the model and around the libration points L1 and L2.
+The rotation curves of the model, defined as V2
+rot = r dφ
+dr , where the
+potential is φ = φd +φb +φbl with the selected values of the parame-
+ters and for the above explained positions of the bulge displacement,
+are shown in Fig. 4. The resulting rotation curve is reasonably flat in
+the outer parts, and displays only minor differences in the position
+of the maximum.
+4 DYNAMICS OF THE MODELS
+The solutions of ∇φeff = 0 in rotating coordinates give five La-
+grangian equilibrium points of the model (Li, i = 1,...,5). Points L1
+and L2 are linearly unstable points and lie on the x-axis at the ends of
+the bar. Point L3 is linearly stable and it is placed on the origin of co-
+ordinates in the case of a symmetric model. Points L4 and L5 are also
+linearly stable and located out of the x-axis. A detailed explanation
+of the dynamics around these points can be found in Athanassoula
+et al. (1983); Romero-Gómez et al. (2006); Sánchez-Martín et al.
+(2016).
+The regions where φeff > CJ are forbidden regions for a star of en-
+ergy CJ. In the plane, these regions are delimited by the zero velocity
+curves, which are defined by the level surfaces φeff = CJ intersected
+with z = 0. In Fig. 5 we show the zero velocity curves corresponding
+to an energy slightly above of that of the equilibrium point, CJ,Li +δ,
+for the models with off-centered bulges in Fig. 4, i.e., setting the val-
+ues xd = 0, 0.5, 1, 1.5. These zero velocity curves limit the inner and
+outer regions in the galaxy. In the symmetric case (xd = 0), the en-
+ergy of both equilibrium points L1 and L2 is the same, CJ,L1 = CJ,L2.
+For the asymmetric cases, as xd grows CJ,L2 becomes smaller than
+CJ,L1, which makes the zero velocity curves related to L1 more open
+at the opposite point.
+Particular attention is given in this work to the unstable points
+L1 and L2. They are surrounded by planar and vertical families of
+Lyapunov periodic orbits, which are unstable in the neighborhood of
+the equilibrium point. The relevant family for the transport of matter
+between the inner and outer regions of the galaxy is the planar family
+(Romero-Gómez et al. 2009). Fig. 6 shows the (x, y) projection of
+the planar family for the models with xd = 0 (solid black line) and
+xd = 1.5 (dotted red line). The family around L2 is displayed at the
+left panel and that around L1 at the right one. In the symmetric case
+(xd = 0) both families coincide, in the asymmetric one (xd = 1.5) the
+family around L2 is smaller than the one around L1.
+These critical points are characterized by the superposition of a
+saddle and two harmonic oscillations in the rotating frame. Conse-
+quently, for a given Jacobi constant, stable and unstable invariant
+MNRAS 000, 1–7 (2023)
+
+PGC70419
+ (z)
+NGC 7479
+m=1
+A=0.11
+g=
+0.57
+A
+m=1
+A=
+0.09
+1.86
+=6
+0.86
+A
+-10
+0
+10
+p4
+P. Sánchez-Martín et al.
+-5
+0
+5
+x
+-5
+0
+5
+y
+-5
+0
+5
+x
+-5
+0
+5
+y
+-5
+0
+5
+x
+-5
+0
+5
+y
+-5
+0
+5
+x
+-5
+0
+5
+y
+Figure 3. Isodensity curves of the potential φ = φd +φb +φbl. Equilibrium points of the system are marked with a cross. The Ferrers bar and the Plummer bulge
+are outlined by dotted black curves. From left to right: Bulge centered at (0,0,0), (0.5,0,0), (1,0,0) and at (1.5,0,0).
+0
+2
+4
+6
+8
+10
+R [kpc]
+0
+50
+100
+150
+Vrot  [km/s]
+Figure 4. Rotation curve of the potential φ = φd + φb + φbl for the bulge
+centered at (0,0,0) (in blue), (0.5,0,0) (in red), (1,0,0) (in magenta) and
+(1.5,0,0) (in green).
+manifolds emanate from the periodic Lyapunov orbit around each
+point. The stable manifold is defined as the set of orbits that asymp-
+totically tend to the periodic orbit forward in time, and the unstable
+manifold consists of those orbits which depart asymptotically from
+the periodic orbit. These latter manifold drive the escape orbits that
+are responsible for the visible trajectories, in the form of arms and
+rings. Fig. 7 shows the invariant manifolds associated to L1 and L2
+for the set of models where xd = 0, 0.5, 1, 1.5. The effect of an asym-
+metric mass distribution, modelled by the displacement of the bulge,
+makes the exterior manifold that emanates from L2 to differ from the
+invariant manifold associated with L1.
+The transit orbits trapped inside the manifolds are in charge of
+the transfer of matter, from the inner to the outer regions delimited
+by the zero velocity curves (Gidea & Masdemont 2007; Romero-
+Gómez et al. 2006; Sánchez-Martín et al. 2018). As the dynamics of
+our system (4) takes place in a four dimensional phase space when
+we consider orbits with z = ˙z = 0 (the plane z = 0 is invariant), the
+intersection of the trajectories of the inner branch of the stable in-
+variant manifold of a Lyapunov orbit with the hyperplane S defined
+by the section x = 0 in phase space gives a closed curve in the (y, ˙y)
+projection. Given a pair (y, ˙y) and an energy level, this lets us to de-
+fine a state on S by selecting (0,y,0, ˙x, ˙y,0), where ˙x is determined
+by the fixed energy level of the Lyapunov orbit and the orientation
+of the crossing, taking into account Eq. (8),
+˙x =
+�
+−˙y2 +Ω2y2 −2φ(0,y,0)−(CJ,Li +δ),
+(9)
+with CJ,Li +δ the energy of the Lyapunov orbit around the point Li,
+i = 1,2, slightly above of that of the equilibrium point. The forward
+in time integration of initial conditions corresponding to (y, ˙y) points
+inside the closed curve establishes the trajectories of the particles
+confined inside the invariant manifold that transit from the inner to
+the outer region.
+This procedure enables us to quantify the amount of matter trans-
+ferred inside each invariant manifold associated to any energy level
+and, consequently, from each spiral arm of the galaxy. Fig. 8 repre-
+sents the (y, ˙y) projection of the intersection with the hyperplane S
+of the invariant manifolds arising from three orbits with different Ja-
+cobi constants in the Lyapunov family around L2 (top), and the same
+corresponding orbits for L1 (bottom). The three closed curves are
+displayed with different colors, from red to yellow, according to the
+increasing Jacobi constant. The initial conditions located inside each
+of the closed curves are marked with crosses of the same color as the
+curve. From left to right, the figure displays the (y, ˙y) projection for
+the models with bulge center ranging from (0,0,0) to (1.5,0,0). The
+main feature to notice in this figure is the narrowing and stretching
+of the curves as the bulge moves away from the L2 equilibrium point.
+This constriction marks the difference between initial conditions for
+the escape orbits related with L2 and those related to L1. The initial
+conditions emanating from near L2 are more extended along the y-
+axis, resulting in a dispersion in space of the orbits enclosed by the
+manifold. As we integrate forward in time these initial conditions,
+we obtain fewer escape orbits, and dispersed in a wider region, in
+comparison to those emanating near L1. So, the spiral arm defined
+by these orbits becomes more spread out and less bright. The result
+of these integrations is exhibited in Fig. 9 where it can be appreci-
+ated how the orbits inside the invariant manifolds develop the arms.
+The orbits corresponding to the L1 point are more concentrated as
+the density of the bar increases in the region close to L1.
+5 DISCUSSION AND CONCLUSIONS
+The goal of this work is to analyze the relation between the asym-
+metric arms observed in certain barred galaxies and the mass distri-
+bution in the central part of the galaxy. We propose and show that
+there is a strong correlation between their asymmetries. The mod-
+els used to study this fact consist of a superposition of a bar and an
+off-centered bulge. The displacements of the bulge along the main
+axis of the bar introduce an increasing asymmetry in the central part
+of the model. When the center of mass of the galaxy is displaced
+along the major axis of the bar, the zero velocity curves of the ef-
+fective potential become asymmetric: the one at the opposite side
+of the dispacement becomes more open (see Fig. 5). We show that
+this asymmetry between the zero velocity curves in their opening
+is carried to the orbits trapped by invariant manifolds around the
+Lagrangian equilibrium points L1 and L2. This orbits become asym-
+MNRAS 000, 1–7 (2023)
+
+Formation of asymmetric arms in barred galaxies
+5
+-5
+0
+5
+x
+-5
+0
+5
+y
+-5
+0
+5
+x
+-5
+0
+5
+y
+-5
+0
+5
+x
+-5
+0
+5
+y
+-5
+0
+5
+x
+-5
+0
+5
+y
+Figure 5. Zero velocity curves for a Jacobi constant slightly above to that of L1 (blue) and for one slightly above to that of L2 (red). Equilibrium points of
+the system are marked with a cross. The Ferrers bar and the Plummer bulge are outlined by dotted black curves. From left to right: Bulge centered at (0,0,0),
+(0.5,0,0), (1,0,0) and at (1.5,0,0).
+-7
+-6.5
+-6
+-5.5
+x
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+y
+5.5
+6
+6.5
+7
+x
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+y
+Figure 6. Lyapunov family of periodic orbits around Li, i = 1, 2, for a range
+of values of the Jacobi constant in (CJ,Li, CJ,Li +10−4). Solid black line: (x, y)
+projection of the family around L2 (left) and L1 (right) for the symmetric
+model with bulge centered at (0,0,0), CJ,Li = −0.2338, i = 1, 2. Dotted red
+line: (x, y) projection of the family around L2 (left) and L1 (right) for the
+asymmetric model with bulge centered at (1.5,0,0), CJ,L2 = −0.2358, CJ,L1 =
+−0.2331.
+metric as well, with the arm at the opposite side of the displacement
+wider and more spread out. We quantify this effect by computing the
+fraction of orbits trapped by the manifolds. The procedure consists
+in intersecting the manifold with a hyperplane and, inside the result-
+ing closed curve in the (y, ˙y) projection, obtaining the set of points
+with the same energy of the manifold as initial conditions that char-
+acterize the escaping orbits.
+Indeed, the above closed curve turns out to be the most relevant
+feature in order to predict the asymmetry of the arms. When the
+bulge is displaced along the main axis of the bar, causing an asym-
+metry in the density distribution of the model, the closed curves
+around the equilibrium point at the end of the less dense side of
+the bar, narrow. This leads to a smaller measure of states in phase
+space that become initial conditions for escaping orbits. Moreover,
+these closed curves are more stretched, making the distribution of
+the points inside them more spread out in space. Both aspects are
+responsible for the resulting ragged and dispersed spiral arms in the
+less dense end of the bar, while the arm associated with the denser
+end becomes brighter and well defined.
+The dynamics of off-centered bars have been studied using ana-
+lytical models (Colin & Athanassoula 1989), showing how the La-
+grangian points vary as a function of the displacement with respect
+to the center of mass of the galaxy. Łokas (2021) digs into the
+barred galaxies in the IllustrisTNG simulations to check how com-
+mon asymmetric and off-centered bars are and study its possible ori-
+gin, concluding that asymmetric bars are persistent in time and this
+asymmetric may be due to the interaction with a companion galaxy
+or due to the disc itself being asymmetric. In either case, no further
+development has been proposed to link the asymmetry in the bar
+with the one-armed dominant spiral structure.
+As shown in Fig. 1, this is a quite common phenomenon in barred
+galaxies. In particular, the dynamics of NGC 1300 have been studied
+in detail by (Patsis et al. 2010), including an orbital analysis. The iso-
+contours of a smooth K-band image (see their Fig. 1) show a clear
+asymmetric bar distribution leading to an asymmetry in the spiral
+arms, which is reproduced by the different orbital models. Other ex-
+amples of one armed dominated galaxies with an asymmetric bar
+may be: NGC 4027 (Phookun et al. 1992), the density contours
+show an asymmetric and off-centered bar leading to a mass distri-
+bution dominated by an m = 1 mode, though a weak counter-part is
+also clear; the Large Magellanic Cloud (de Vaucouleurs & Freeman
+1972), showing a rotational asymmetry, which is confirmed by more
+recent studies (e.g. Jiménez-Arranz et al. 2022; Niederhofer et al.
+2022, and references therein).
+To sum up, we show that asymmetric arms are a common feature
+in barred galaxies and that there is a clear correlation between arm
+asymmetry and the displacement of the center of mass caused by an
+asymmetric central density distribution. Escaping orbits trapped in
+the invariant manifolds of an asymmetric bar distribution are asym-
+metric and the limiting case would be to have only one armed barred
+spiral.
+ACKNOWLEDGEMENTS
+P.S.M. thanks the Spanish Ministry of Economy grants PID2020-
+117066GB-I00
+and
+PID2021-123968NB-I00.
+J.J.M.
+thanks
+MINECO-FEDER for the grant PID2021-123968NB-I00 and
+the Catalan government grant 2017SGR-1049. M.R.G. thanks
+the Spanish Ministry of Science grant MICIU/FEDER RTI2018-
+095076-B-C21, and the Institute of Cosmos Sciences University
+of Barcelona (ICCUB, Unidad de Excelencia "María de Maeztu")
+grant CEX2019-000918-M.
+Funding for the Sloan Digital Sky Survey IV has been provided
+by the Alfred P. Sloan Foundation, the U.S. Department of En-
+ergy Office of Science, and the Participating Institutions. SDSS
+(www.sdss.org) acknowledges support and resources from the Cen-
+ter for High-Performance Computing at the University of Utah.
+DATA AVAILABILITY
+The data underlying this article are available in the article.
+MNRAS 000, 1–7 (2023)
+
+6
+P. Sánchez-Martín et al.
+-10
+0
+10
+x
+-10
+-5
+0
+5
+10
+y
+-10
+0
+10
+x
+-10
+-5
+0
+5
+10
+y
+-10
+0
+10
+x
+-10
+-5
+0
+5
+10
+y
+-10
+0
+10
+x
+-10
+-5
+0
+5
+10
+y
+Figure 7. Unstable invariant manifolds associated to the Lyapunov periodic orbits of L1 and L2. The position of the equilibrium points is marked with crosses.
+The bar and the bulge are outlined by dotted black curves. The reference system is marked with a solid black line and the center of the bar with a dotted magenta
+line. From left to right: Bulge centered at (0,0,0), (0.5,0,0), (1,0,0) and at (1.5,0,0).
+1.6
+1.8
+2
+2.2
+2.4
+2.6
+2.8
+y
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+1.6
+1.8
+2
+2.2
+2.4
+2.6
+y
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+1.2
+1.4
+1.6
+1.8
+2
+2.2
+y
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+y
+0.1
+0.15
+0.2
+0.25
+0.3
+-2.6
+-2.4
+-2.2
+-2
+y
+-0.06
+-0.04
+-0.02
+0
+-2.8
+-2.6
+-2.4
+-2.2
+y
+-0.06
+-0.04
+-0.02
+0
+-3
+-2.8
+-2.6
+-2.4
+y
+-0.06
+-0.04
+-0.02
+0
+-3
+-2.8
+-2.6
+-2.4
+y
+-0.06
+-0.04
+-0.02
+0
+Figure 8. (y, ˙y) projection of the intersection of the plane S with the stable manifolds associated to three orbits with different Jacobi constant of the Lyapunov
+family around L2 (top) and L1 (bottom). The intersection of each manifold is in a color, from red to yellow, according to the energy of the manifold. Initial
+conditions distributed inside each curve are marked with a cross with the same color as the curve. From left to right: Bulge centered at (0,0,0), (0.5,0,0), (1,0,0)
+and (1.5,0,0). Note that the axis limits are different but their scale and range length are constant in each row.
+Figure 9. Orbits resulting from the integration of the initial conditions of Fig. 8. Equilibrium points marked in red. Bar and bulge outlined by dotted black
+curves. The reference system is marked with a solid black line and the center of the bar with a dotted magenta line. From left to right: Bulge centered at (0,0,0),
+(0.5,0,0), (1,0,0) and at (1.5,0,0).
+REFERENCES
+Athanassoula E., Bienaymé O., Martinet L., Pfenniger D., 1983, A&A, 127,
+349.
+Bundy K. et al., 2015, ApJ, 798, 7.
+Colin J., Athanassoula E., 1989, A&A, 214, 99.
+Considère S., Athanassoula E., 1982, A&A, 111, 28.
+de Vaucouleurs G., Freeman K. C., 1972, Vistas Astron., 14, 163.
+Ferrers N. M., 1877, Q. J. Pure Appl. Math., 14, 1.
+García-Gómez C., Barberà C., Athanassoula E., Bosma A., Whyte L., 2004,
+A&A, 421, 595.
+García-Gómez C., Athanassoula E., Barberà C., Bosma A., 2017, A&A, 601,
+A132.
+Gidea M., Masdemont J. J., 2007, Int. J. of bifurcation and chaos, 17, 1151.
+Iye M., Okamura S., Hamabe M., Watanabe M., 1982, ApJ, 256, 103.
+Jiménez-Arranz Ó. et al., 2022, preprint (arXiv:2210.01728).
+Lee Y. H., Ann H. B., Park M. G., 2019, ApJ, 872, 97.
+Łokas E. L., 2021, A&A, 655, A97.
+Miyamoto M., Nagai R. 1975, PASJ, 27, 533.
+Niederhofer F. et al., 2022, MNRAS, 512, 5423.
+Patsis P.A., Kalapotharakos C., Grosbol P., 2010, MNRAS, 408, 22.
+Pfenniger D., 1984, A&A, 134, 373.
+Phookun B., Mundy L.G., Teuben P.J., Wainscoat R.J., 1992, ApJ, 400, 516.
+Plummer H. C., 1911, MNRAS, 71, 460.
+Romero-Gómez M., Masdemont J. J., Athanassoula E., García-Gómez C.,
+2006, A&A, 453, 39.
+Romero-Gómez M., Masdemont J. J., García-Gómez C., Athanassoula E.,
+MNRAS 000, 1–7 (2023)
+
+10
+5
+0
+-5
+-10
+-10
+0
+10
+X10
+5
+0
+-5
+-10
+-10
+0
+10
+X10
+5
+0
+-5
+-10
+-10
+0
+10
+X10
+5
+0
+-5
+-10
+-10
+0
+10
+XFormation of asymmetric arms in barred galaxies
+7
+2009, CNSNS, 14, 4123.
+Sánchez-Martín P., Romero-Gómez M., Masdemont J. J., 2016, A&A, 588,
+A76.
+Sánchez-Martín P., Masdemont J. J., Romero-Gómez M., 2018, A&A, 618,
+A72.
+Skokos Ch., Patsis P. A., Athanassoula E., 2002, MNRAS, 333, 847.
+Tonini C., Mutch S. J., Croton D. J., Wyithe J. S. B., 2016, MNRAS, 459,
+4109.
+This paper has been typeset from a TEX/LATEX file prepared by the author.
+MNRAS 000, 1–7 (2023)
+
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+page_content=' Països Catalans, 26, 43007 Tarragona, Spain  3IEEC & Universitat Politècnica de Catalunya, Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' de Matemàtiques, Diagonal 647, E08028 Barcelona, Spain  4Institut de Ciències del Cosmos (ICCUB), Universitat de Barcelona (IEEC-UB), Martí i Franquès 1, E08028 Barcelona, Spain  Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' in original form ZZZ  ABSTRACT  We establish a dynamical mechanism to explain the origin of the asymmetry between the arms observed in some barred disk  galaxies, where one of the two arms emanating from the bar ends is very well defined, while the second one displays a ragged  structure, extending between its ridge and the bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' To this purpose, we study the invariant manifolds associated to the Lyapunov  periodic orbits around the unstable equilibrium points at the ends of the bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' Matter from the galaxy center is transported along  these manifolds to the periphery, forming this way the spiral arms that emanate from the bar ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' If the mass distribution in  the galaxy center is not homogeneous, because of an asymmetric bar with one side stronger than the other, or because of a  non-centered bulge, the dynamics about the two unstable Lagrange points at the ends of the bar will not be symmetric as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  One of their invariant manifolds becomes more extended than the other, enclosing a smaller section and the escaping orbits on it  are fewer and dispersed in a wider region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' The result is a weaker arm, and more ragged than the one at the other end of the bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  Key words: galaxies: kinematics and dynamics – galaxies: structure – galaxies: spiral  1 INTRODUCTION  A high fraction of disk galaxies appear to be barred (74% to 85%  in automatic classifications or 36% to 63% with alternative meth-  ods (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' 2019)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' These authors also suggest that strongly  barred galaxies, classified as SBs, are preponderant in late-type  galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' Most of these strongly barred galaxies appear to be asym-  metric under a simple visual inspection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' They usually present a  strong bar and two spiral arms emanating from the bar ends, and  frequently some inner rings as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' However, a detailed visual in-  spection of these galaxies, available in the STScI Digitized Sky Sur-  vey or in the Sloan Digital Sky Survey (SDSS) (Bundy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' 2015),  reveals that some barred galaxies exhibit important asymmetries in  their spiral structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' While one of the two spiral arms is long and  strongly defined, the second arm shows a ragged structure, and the  matter is distributed irregularly between the ridge of the spiral arm  and the bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' A non-symmetric distribution of matter in the galaxy  disc can be a natural outcome, since disc galaxies may be formed  through a combination of secular evolution and violent events, in-  cluding smooth accretion, disc instabilities and minor and major  mergers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' Tonini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' 1 we show some examples  of galaxies showing asymmetric discs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  In this paper we study the dynamic response of an asymmetric  mass distribution on the orbital structure of barred galaxies, mod-  elling the asymmetry of the central parts by a slightly off-centered  bulge which, viewing the galaxy as a whole, represents a simple but  realistic model of an asymmetric bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' In this scenario, the Lagrange  ⋆ E-mail: patricias@unizar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='es  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' Images of some strongly barred galaxies showing the asymmetric  spiral arm patterns in the R-filter from the STScI Digitized Sky Survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' In the  top row we show the galaxies PGC 05849 (NGC 0613), PGC 12412 (NGC  1300) and in the bottom one PGC 15941 (NGC 1672) and PGC 54849 (NGC  5921).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  invariant points located at the bar ends will present also an asym-  metric orbital structure, as initially shown in Colin & Athanassoula  (1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' The new step in this work is to additionally study the unsta-  ble periodic orbits around these Lagrange points and the associated  unstable manifolds which provide escape routes for orbits which  can transport material from the central regions to the outer parts of  the discs (Romero-Gómez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' Sánchez-Martín et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  © 2023 The Authors  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='11385v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='GA]  26 Jan 2023  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='PGC 05849(R)  PGC 12412 (R)  PGC 15941 (R)  PGC 54849 (R)2  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' Sánchez-Martín et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  These unstable manifolds are the backbones of the spiral arms ema-  nating from the bar ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' In the case of asymmetric Lagrange points,  however, the unstable manifolds display also important differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  The orbits following one of these manifolds are close together and  can explain the presence of a strong spiral arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' On the other hand,  the second manifold has a more open orbital structure, with its es-  caping orbits dispersed in a wider zone, causing the ragged struc-  ture of the second arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' Thus, the orbital structure of the unstable  Lagrange points is able to explain the asymmetric phenomenology  present in some of the strongly barred galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  The paper is organized as follows: in Section 2 we quantify the  level of asymmetry in asymmetric barred galaxies using a two-  dimensional Fourier transform method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' In Sections 3 and 4 we de-  scribe the asymmetric barred galaxy model used in this work and  show the orbital analysis and invariant manifolds, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' Dif-  ferent approaches are summarized and conclusions are given in Sec-  tion 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  2 ANALYSIS OF THE BAR ASYMMETRY  In this paper, we model the galactic asymmetric mass distribution  using a classical galactic model with three symmetric components  (disc, bar and bulge) but displacing slightly the bulge from the  galaxy centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' This results in a total mass distribution biased to-  wards one side of the bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' Many barred galaxies show some kind  of asymmetries in their inner mass distribution with one side of the  bar stronger than the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' An example is the galaxy NGC 1300,  studied by Patsis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' They generated a numerical model  of the potential using K images of the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' The resultant model  was clearly asymmetric, containing a bar with a stronger arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' This  model was used to study the orbits associated with this asymmetric  barred structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' Many barred galaxies show similar asymmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  Some of these galaxies are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' In order to quantify  these asymmetries, we can analyze in detail the mass distribution of  one of the galaxies showing this phenomenology, namely the galaxy  PGC 70419 (NGC 7479).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' For this purpose, we use a galaxy image  from the SDSS survey in the infrared z-filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' Assuming a constant  M/L ratio, the light distribution is a good tracer of the mass distribu-  tion in the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' The image is previously cleaned from background  stars and then deprojected using the FFT method (García-Gómez et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' 2004) giving a position angle (PA) of 38◦ and an inclination angle  of 45◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  The image is then decomposed in its Fourier components using a  technique first introduced by Considère & Athanassoula (1982);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' Iye  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' (1982), and further developed by García-Gómez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  For deprojected image of the galaxy I(u,θ), where u = ln(r), we cal-  culate the two-dimensional Fourier transform defined as  A(p,m) =  � umax  umin  � 2π  0  I(u,θ)ei(pu+mθ) dθdu  (1)  Where m is the azimuthal frequency, associated with the multiplicity  of the structures, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', the number of arms while p is the radial fre-  quency, associated to the pitch angle of the structure i, through the  relation  p = −  m  tan(i)  (2)  In this way, the m = 1 spectrum contains the spiral components with  no symmetry, the m = 2 spectrum the components with a periodicity  of π radians or bisymmetric signals, and so on for the rest of the m  frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' Each of the azimuthal components m = 1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' can be  further decomposed in its radial components using a Gaussian fit to  the modulus and keeping the phase constant as follows:  | A(p,m) |=  Ng  �  j=1  C j exp−(p− pj)2  2σ2  j  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  (3)  In this relation, pj represents the central frequency of the Gaussian,  σj its dispersion, and C j its amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' The number of Gaussians  used in each fit, Ng, will depend on the complexity of the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  In the upper panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' 2 we show the deprojected SDSS galaxy  image of NGC 7479 using the z-filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' Note that one of its arms is  very pronounced, while the opposite arm appears diffuse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' On the left  of the middle and lower panels of this figure we show the modulus  of the Fourier spectrum of the m = 1 component which is associ-  ated to the asymmetries in the light distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' The modulus of the  Fourier components are normalized to the modulus of the stronger  component, which in this case is the m = 2 containing the bisymmet-  ric signals of the strong bar and the spiral arms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' The m = 1 spectrum  shown here contains the spiral components responsible for the asym-  metries in the mass distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' The relative low values of the m = 1  in this scale shows that the mass asymmetry is a second order effect  for this galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' In red we superpose two of the Gaussian components  into which this signal is decomposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' These Gaussian components  can be transformed back to obtain the density distribution associated  to each particular spiral mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' The density distributions of these spi-  ral components are presented in the respective right panels in green,  superposed on the galaxy image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' The isocontours come from the  normalized density corresponding to the Gaussian components and  show that the asymmetries are related to the southern end of the  bar and the strong arm emanating from its end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' This shows that the  galaxy has an asymmetric mass distribution, biased to the southern  part of the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  3 CHARACTERISTICS OF THE GALACTIC MODEL  The equations of motion of the classical galactic model (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  Pfenniger 1984;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' Skokos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' Romero-Gómez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  Sánchez-Martín et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' 2016) describe the movement of a particle in  a gravitational potential φ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' In the rotating frame, the equations of  motion are described by  ¨r = −2(Ωp × ˙r)−Ωp ×(Ωp ×r)−∇φ,  (4)  where r = (x, y, z) is the position of the particle, φ is the total grav-  itational potential of the system and Ωp is the angular velocity of  the bar around the z-axis, Ωp = (0,0,Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' The origin of the reference  frame is located at the center of mass of the system and the frame is  aligned with the main axis of the bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  The potential model φ used in this paper is a combination of three  analytical components: an axisymmetric Miyamoto-Nagai disc with  potential φd (Miyamoto & Nagai 1975), an ellipsoid Ferrers bar with  potential φb (Ferrers 1877) and a bulge structure represented by a  Plummer spheroid potential φbl (Plummer 1911).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' The total potential  is the addition of these three components, φ = φd +φb +φbl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  The disc potential is described by the equation  φd = −  GMd  �  R2 +(A+  �  B2 +z2)2  ,  (5)  were R2 = x2 + y2 is the cylindrical coordinate radius of the poten-  tial in the disc plane, and z is the vertical distance over the disk  component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' The parameters G, Md, A and B denote the gravitational  MNRAS 000, 1–7 (2023)  Formation of asymmetric arms in barred galaxies  3  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' Analysis of the m = 1 component of the SDSS galaxy image of  NGC 7479 in the z-filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' The upper panel shows the deprojected image of the  galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' On the left of the middle and lower panels we show the modulus of  the m = 1 component of the Fourier spectrum, and with red-dashed lines two  Gaussian components fitted to this modulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' In the respective right panels,  we show in green the density distribution associated to each of these single  components superimposed on the galaxy image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  constant, the disc mass and the shape of the disc, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' Tak-  ing A = 0 the potential becomes the Plummer potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' The bar is  modelled by an ellipsoid with density function  ρ =  �  ρ0(1−m2)nh,  m ≤ 1  0,  m > 1  (6)  where m2 = x2/a2 + y2/b2 + z2/c2, a (semi-major axis), b (interme-  diate axis) and c (semi-minor axis) determine the shape of the bar,  nh is the homogeneity degree of the mass distribution (nh = 2 in our  work) and ρ0 is the density at the origin (ρ0 = 105  32π  GMb  abc if nh = 2,  where Mb is the bar mass).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  The unit of length considered is the kpc, the time unit is ut =  2 × 106 yr, Ω is in [ut]−1, and the mass unit is uM = 2 × 1011M⊙,  where M⊙ denotes the mass of the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' G stands for the gravitational  constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  In our model, we select a disc radius of A = 3 kpc, height B = 1 kpc  and mass giving a value of GMd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='52 kpc3/u2  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' The dimensions of  the bar are a = 6 kpc, b = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='5 kpc, c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='4 kpc, and its mass is such  that GMb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='4 kpc3/u2  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' The Plummer bulge with a radius B = 1 kpc  and GMbl is set to have around 15% of the mass of the bar GMb, in  order that G(Md + Mb + Mbl) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' The bar pattern speed is fixed as  Ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='0633 [ut]−1 (∼ 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='97 km/s/kpc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  In the rotating reference frame aligned with the main axis of the  bar, the equations of motion given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' (4) are written as the fol-  lowing dynamical system:  ���������  ¨x = 2Ω ˙y+Ω2 x−φx  ¨y = −2Ω ˙x+Ω2 y−φy  ¨z = −φz .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  (7)  The Jacobi first integral of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' (4) (which can be regarded as the  energy in the rotating frame) is given by  CJ(x,y,z, ˙x, ˙y, ˙z) = −(˙x2 + ˙y2 + ˙z2)+Ω2 (x2 +y2)−2φ,  (8)  and the effective potential is defined by φeff = φ− 1  2 Ω2 (x2 +y2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  The goal of this paper is to analyze which is the effect of an asym-  metric distribution of mass in the central parts of the galaxy on the  external spiral structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' To introduce this asymmetry, we displace  the bulge potential along the x-axis (main axis of the bar) towards the  equilibrium point placed at the right end of the bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' The bulge center  is then located at (xd,0,0) using several values for the displacement  xd (in kpc), namely: (0,0,0), (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='5,0,0), (1,0,0) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='5,0,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' We  move the center of our potential model to the resulting center of  mass of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' The plot of the equal density contours of the  resulting models are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' Note that the displacement of  the bulge along the main axis of the bar creates an asymmetry in the  central part of the model and around the libration points L1 and L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  The rotation curves of the model, defined as V2  rot = r dφ  dr , where the  potential is φ = φd +φb +φbl with the selected values of the parame-  ters and for the above explained positions of the bulge displacement,  are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' The resulting rotation curve is reasonably flat in  the outer parts, and displays only minor differences in the position  of the maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  4 DYNAMICS OF THE MODELS  The solutions of ∇φeff = 0 in rotating coordinates give five La-  grangian equilibrium points of the model (Li, i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=',5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' Points L1  and L2 are linearly unstable points and lie on the x-axis at the ends of  the bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' Point L3 is linearly stable and it is placed on the origin of co-  ordinates in the case of a symmetric model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' Points L4 and L5 are also  linearly stable and located out of the x-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' A detailed explanation  of the dynamics around these points can be found in Athanassoula  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' (1983);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' Romero-Gómez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' (2006);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' Sánchez-Martín et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  The regions where φeff > CJ are forbidden regions for a star of en-  ergy CJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' In the plane, these regions are delimited by the zero velocity  curves, which are defined by the level surfaces φeff = CJ intersected  with z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' 5 we show the zero velocity curves corresponding  to an energy slightly above of that of the equilibrium point, CJ,Li +δ,  for the models with off-centered bulges in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' 4, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', setting the val-  ues xd = 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='5, 1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' These zero velocity curves limit the inner and  outer regions in the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' In the symmetric case (xd = 0), the en-  ergy of both equilibrium points L1 and L2 is the same, CJ,L1 = CJ,L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  For the asymmetric cases, as xd grows CJ,L2 becomes smaller than  CJ,L1, which makes the zero velocity curves related to L1 more open  at the opposite point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  Particular attention is given in this work to the unstable points  L1 and L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' They are surrounded by planar and vertical families of  Lyapunov periodic orbits, which are unstable in the neighborhood of  the equilibrium point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' The relevant family for the transport of matter  between the inner and outer regions of the galaxy is the planar family  (Romero-Gómez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' 6 shows the (x, y) projection of  the planar family for the models with xd = 0 (solid black line) and  xd = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='5 (dotted red line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' The family around L2 is displayed at the  left panel and that around L1 at the right one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' In the symmetric case  (xd = 0) both families coincide, in the asymmetric one (xd = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='5) the  family around L2 is smaller than the one around L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  These critical points are characterized by the superposition of a  saddle and two harmonic oscillations in the rotating frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' Conse-  quently, for a given Jacobi constant, stable and unstable invariant  MNRAS 000, 1–7 (2023)  PGC70419  (z)  NGC 7479  m=1  A=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='11  g=  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='57  A  m=1  A=  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='09  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='86  =6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='86  A  10  0  10  p4  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' Sánchez-Martín et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  5  0  5  x  5  0  5  y  5  0  5  x  5  0  5  y  5  0  5  x  5  0  5  y  5  0  5  x  5  0  5  y  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' Isodensity curves of the potential φ = φd +φb +φbl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' Equilibrium points of the system are marked with a cross.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' The Ferrers bar and the Plummer bulge  are outlined by dotted black curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' From left to right: Bulge centered at (0,0,0), (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='5,0,0), (1,0,0) and at (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='5,0,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  0  2  4  6  8  10  R [kpc]  0  50  100  150  Vrot  [km/s]  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' Rotation curve of the potential φ = φd + φb + φbl for the bulge  centered at (0,0,0) (in blue), (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='5,0,0) (in red), (1,0,0) (in magenta) and  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='5,0,0) (in green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  manifolds emanate from the periodic Lyapunov orbit around each  point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' The stable manifold is defined as the set of orbits that asymp-  totically tend to the periodic orbit forward in time, and the unstable  manifold consists of those orbits which depart asymptotically from  the periodic orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' These latter manifold drive the escape orbits that  are responsible for the visible trajectories, in the form of arms and  rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' 7 shows the invariant manifolds associated to L1 and L2  for the set of models where xd = 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='5, 1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' The effect of an asym-  metric mass distribution, modelled by the displacement of the bulge,  makes the exterior manifold that emanates from L2 to differ from the  invariant manifold associated with L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  The transit orbits trapped inside the manifolds are in charge of  the transfer of matter, from the inner to the outer regions delimited  by the zero velocity curves (Gidea & Masdemont 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' Romero-  Gómez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' Sánchez-Martín et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' As the dynamics of  our system (4) takes place in a four dimensional phase space when  we consider orbits with z = ˙z = 0 (the plane z = 0 is invariant), the  intersection of the trajectories of the inner branch of the stable in-  variant manifold of a Lyapunov orbit with the hyperplane S defined  by the section x = 0 in phase space gives a closed curve in the (y, ˙y)  projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' Given a pair (y, ˙y) and an energy level, this lets us to de-  fine a state on S by selecting (0,y,0, ˙x, ˙y,0), where ˙x is determined  by the fixed energy level of the Lyapunov orbit and the orientation  of the crossing, taking into account Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' (8),  ˙x =  �  −˙y2 +Ω2y2 −2φ(0,y,0)−(CJ,Li +δ),  (9)  with CJ,Li +δ the energy of the Lyapunov orbit around the point Li,  i = 1,2, slightly above of that of the equilibrium point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' The forward  in time integration of initial conditions corresponding to (y, ˙y) points  inside the closed curve establishes the trajectories of the particles  confined inside the invariant manifold that transit from the inner to  the outer region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  This procedure enables us to quantify the amount of matter trans-  ferred inside each invariant manifold associated to any energy level  and, consequently, from each spiral arm of the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' 8 repre-  sents the (y, ˙y) projection of the intersection with the hyperplane S  of the invariant manifolds arising from three orbits with different Ja-  cobi constants in the Lyapunov family around L2 (top), and the same  corresponding orbits for L1 (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' The three closed curves are  displayed with different colors, from red to yellow, according to the  increasing Jacobi constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' The initial conditions located inside each  of the closed curves are marked with crosses of the same color as the  curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' From left to right, the figure displays the (y, ˙y) projection for  the models with bulge center ranging from (0,0,0) to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='5,0,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' The  main feature to notice in this figure is the narrowing and stretching  of the curves as the bulge moves away from the L2 equilibrium point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  This constriction marks the difference between initial conditions for  the escape orbits related with L2 and those related to L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' The initial  conditions emanating from near L2 are more extended along the y-  axis, resulting in a dispersion in space of the orbits enclosed by the  manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' As we integrate forward in time these initial conditions,  we obtain fewer escape orbits, and dispersed in a wider region, in  comparison to those emanating near L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' So, the spiral arm defined  by these orbits becomes more spread out and less bright.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' The result  of these integrations is exhibited in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' 9 where it can be appreci-  ated how the orbits inside the invariant manifolds develop the arms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  The orbits corresponding to the L1 point are more concentrated as  the density of the bar increases in the region close to L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  5 DISCUSSION AND CONCLUSIONS  The goal of this work is to analyze the relation between the asym-  metric arms observed in certain barred galaxies and the mass distri-  bution in the central part of the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' We propose and show that  there is a strong correlation between their asymmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' The mod-  els used to study this fact consist of a superposition of a bar and an  off-centered bulge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' The displacements of the bulge along the main  axis of the bar introduce an increasing asymmetry in the central part  of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' When the center of mass of the galaxy is displaced  along the major axis of the bar, the zero velocity curves of the ef-  fective potential become asymmetric: the one at the opposite side  of the dispacement becomes more open (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' We show that  this asymmetry between the zero velocity curves in their opening  is carried to the orbits trapped by invariant manifolds around the  Lagrangian equilibrium points L1 and L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' This orbits become asym-  MNRAS 000, 1–7 (2023)  Formation of asymmetric arms in barred galaxies  5  5  0  5  x  5  0  5  y  5  0  5  x  5  0  5  y  5  0  5  x  5  0  5  y  5  0  5  x  5  0  5  y  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' Zero velocity curves for a Jacobi constant slightly above to that of L1 (blue) and for one slightly above to that of L2 (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' Equilibrium points of  the system are marked with a cross.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' The Ferrers bar and the Plummer bulge are outlined by dotted black curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' From left to right: Bulge centered at (0,0,0),  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='5,0,0), (1,0,0) and at (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='5,0,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  7  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='5  6  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='5  x  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='4  y  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='5  6  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='5  7  x  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='4  y  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' Lyapunov family of periodic orbits around Li, i = 1, 2, for a range  of values of the Jacobi constant in (CJ,Li, CJ,Li +10−4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' Solid black line: (x, y)  projection of the family around L2 (left) and L1 (right) for the symmetric  model with bulge centered at (0,0,0), CJ,Li = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='2338, i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' Dotted red  line: (x, y) projection of the family around L2 (left) and L1 (right) for the  asymmetric model with bulge centered at (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='5,0,0), CJ,L2 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='2358, CJ,L1 =  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='2331.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  metric as well, with the arm at the opposite side of the displacement  wider and more spread out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' We quantify this effect by computing the  fraction of orbits trapped by the manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' The procedure consists  in intersecting the manifold with a hyperplane and, inside the result-  ing closed curve in the (y, ˙y) projection, obtaining the set of points  with the same energy of the manifold as initial conditions that char-  acterize the escaping orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  Indeed, the above closed curve turns out to be the most relevant  feature in order to predict the asymmetry of the arms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' When the  bulge is displaced along the main axis of the bar, causing an asym-  metry in the density distribution of the model, the closed curves  around the equilibrium point at the end of the less dense side of  the bar, narrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' This leads to a smaller measure of states in phase  space that become initial conditions for escaping orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' Moreover,  these closed curves are more stretched, making the distribution of  the points inside them more spread out in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' Both aspects are  responsible for the resulting ragged and dispersed spiral arms in the  less dense end of the bar, while the arm associated with the denser  end becomes brighter and well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  The dynamics of off-centered bars have been studied using ana-  lytical models (Colin & Athanassoula 1989), showing how the La-  grangian points vary as a function of the displacement with respect  to the center of mass of the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' Łokas (2021) digs into the  barred galaxies in the IllustrisTNG simulations to check how com-  mon asymmetric and off-centered bars are and study its possible ori-  gin, concluding that asymmetric bars are persistent in time and this  asymmetric may be due to the interaction with a companion galaxy  or due to the disc itself being asymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' In either case, no further  development has been proposed to link the asymmetry in the bar  with the one-armed dominant spiral structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' 1, this is a quite common phenomenon in barred  galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' In particular, the dynamics of NGC 1300 have been studied  in detail by (Patsis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' 2010), including an orbital analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' The iso-  contours of a smooth K-band image (see their Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' 1) show a clear  asymmetric bar distribution leading to an asymmetry in the spiral  arms, which is reproduced by the different orbital models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' Other ex-  amples of one armed dominated galaxies with an asymmetric bar  may be: NGC 4027 (Phookun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' 1992), the density contours  show an asymmetric and off-centered bar leading to a mass distri-  bution dominated by an m = 1 mode, though a weak counter-part is  also clear;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' the Large Magellanic Cloud (de Vaucouleurs & Freeman  1972), showing a rotational asymmetry, which is confirmed by more  recent studies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' Jiménez-Arranz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' Niederhofer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  2022, and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  To sum up, we show that asymmetric arms are a common feature  in barred galaxies and that there is a clear correlation between arm  asymmetry and the displacement of the center of mass caused by an  asymmetric central density distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' Escaping orbits trapped in  the invariant manifolds of an asymmetric bar distribution are asym-  metric and the limiting case would be to have only one armed barred  spiral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  ACKNOWLEDGEMENTS  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' thanks the Spanish Ministry of Economy grants PID2020-  117066GB-I00  and  PID2021-123968NB-I00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  thanks  MINECO-FEDER for the grant PID2021-123968NB-I00 and  the Catalan government grant 2017SGR-1049.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' thanks  the Spanish Ministry of Science grant MICIU/FEDER RTI2018-  095076-B-C21, and the Institute of Cosmos Sciences University  of Barcelona (ICCUB, Unidad de Excelencia "María de Maeztu")  grant CEX2019-000918-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  Funding for the Sloan Digital Sky Survey IV has been provided  by the Alfred P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' Sloan Foundation, the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' Department of En-  ergy Office of Science, and the Participating Institutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' SDSS  (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='sdss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='org) acknowledges support and resources from the Cen-  ter for High-Performance Computing at the University of Utah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  DATA AVAILABILITY  The data underlying this article are available in the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  MNRAS 000, 1–7 (2023)  6  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' Sánchez-Martín et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  10  0  10  x  10  5  0  5  10  y  10  0  10  x  10  5  0  5  10  y  10  0  10  x  10  5  0  5  10  y  10  0  10  x  10  5  0  5  10  y  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' Unstable invariant manifolds associated to the Lyapunov periodic orbits of L1 and L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' The position of the equilibrium points is marked with crosses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  The bar and the bulge are outlined by dotted black curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
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+page_content='02  0  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' (y, ˙y) projection of the intersection of the plane S with the stable manifolds associated to three orbits with different Jacobi constant of the Lyapunov  family around L2 (top) and L1 (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' The intersection of each manifold is in a color, from red to yellow, according to the energy of the manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' Initial  conditions distributed inside each curve are marked with a cross with the same color as the curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' From left to right: Bulge centered at (0,0,0), (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='5,0,0), (1,0,0)  and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='5,0,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' Note that the axis limits are different but their scale and range length are constant in each row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' Orbits resulting from the integration of the initial conditions of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' Equilibrium points marked in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' Bar and bulge outlined by dotted black  curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' The reference system is marked with a solid black line and the center of the bar with a dotted magenta line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' From left to right: Bulge centered at (0,0,0),  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='5,0,0), (1,0,0) and at (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='5,0,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  REFERENCES  Athanassoula E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', Bienaymé O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', Martinet L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', Pfenniger D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', 1983, A&A, 127,  349.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  Bundy K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', 2015, ApJ, 798, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  Colin J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', Athanassoula E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', 1989, A&A, 214, 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  Considère S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', Athanassoula E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', 1982, A&A, 111, 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  de Vaucouleurs G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', Freeman K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', 1972, Vistas Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', 14, 163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  Ferrers N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', 1877, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' Pure Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', 14, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  García-Gómez C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', Barberà C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', Athanassoula E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', Bosma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', Whyte L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', 2004,  A&A, 421, 595.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  García-Gómez C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', Athanassoula E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', Barberà C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', Bosma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', 2017, A&A, 601,  A132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  Gidea M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', Masdemont J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', 2007, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' of bifurcation and chaos, 17, 1151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  Iye M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', Okamura S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', Hamabe M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', Watanabe M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', 1982, ApJ, 256, 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  Jiménez-Arranz Ó.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', 2022, preprint (arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='01728).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  Lee Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', Ann H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', Park M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', 2019, ApJ, 872, 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  Łokas E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', 2021, A&A, 655, A97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  Miyamoto M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', Nagai R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' 1975, PASJ, 27, 533.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  Niederhofer F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', 2022, MNRAS, 512, 5423.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  Patsis P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', Kalapotharakos C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', Grosbol P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', 2010, MNRAS, 408, 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  Pfenniger D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', 1984, A&A, 134, 373.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  Phookun B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', Mundy L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', Teuben P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', Wainscoat R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', 1992, ApJ, 400, 516.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  Plummer H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', 1911, MNRAS, 71, 460.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  Romero-Gómez M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', Masdemont J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', Athanassoula E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', García-Gómez C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=',  2006, A&A, 453, 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  Romero-Gómez M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', Masdemont J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', García-Gómez C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', Athanassoula E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=',  MNRAS 000, 1–7 (2023)  10  5  0  5  10  10  0  10  X10  5  0  5  10  10  0  10  X10  5  0  5  10  10  0  10  X10  5  0  5  10  10  0  10  XFormation of asymmetric arms in barred galaxies  7  2009, CNSNS, 14, 4123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  Sánchez-Martín P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', Romero-Gómez M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', Masdemont J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', 2016, A&A, 588,  A76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  Sánchez-Martín P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', Masdemont J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', Romero-Gómez M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', 2018, A&A, 618,  A72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  Skokos Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', Patsis P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', Athanassoula E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', 2002, MNRAS, 333, 847.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  Tonini C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', Mutch S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', Croton D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', Wyithe J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content=', 2016, MNRAS, 459,  4109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  This paper has been typeset from a TEX/LATEX file prepared by the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
+page_content='  MNRAS 000, 1–7 (2023)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FIT4oBgHgl3EQf4Su_/content/2301.11385v1.pdf'}
diff --git a/1NE4T4oBgHgl3EQfaAwh/content/tmp_files/2301.05060v1.pdf.txt b/1NE4T4oBgHgl3EQfaAwh/content/tmp_files/2301.05060v1.pdf.txt
new file mode 100644
index 0000000000000000000000000000000000000000..6d4336eebc9d812cd9d96ad42fe0cbc55411fe31
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@@ -0,0 +1,634 @@
+arXiv:2301.05060v1  [cs.CR]  12 Jan 2023
+Evaluating the Fork-Awareness of Coverage-Guided Fuzzers
+Marcello Maugeri1
+a, Cristian Daniele2
+b, Giampaolo Bella1
+c, and Erik Poll2
+d
+1Department of Maths and Computer Science, University of Catania, Catania, Italy
+2Department of Digital Security, Radboud University, Nijmegen, The Netherlands
+marcello.maugeri@phd.unict.it, cristian.daniele@ru.nl, giampaolo.bella@unict.it, erikpoll@cs.ru.nl
+Keywords:
+Fuzzing, Fork, Security Testing, Software Security
+Abstract:
+Fuzz testing (or fuzzing) is an effective technique used to find security vulnerabilities. It consists of feeding a
+software under test with malformed inputs, waiting for a weird system behaviour (often a crash of the system).
+Over the years, different approaches have been developed, and among the most popular lies the coverage-based
+one. It relies on the instrumentation of the system to generate inputs able to cover as much code as possible.
+The success of this approach is also due to its usability as fuzzing techniques research approaches that do
+not require (or only partial require) human interactions. Despite the efforts, devising a fully-automated fuzzer
+still seems to be a challenging task. Target systems may be very complex; they may integrate cryptographic
+primitives, compute and verify check-sums and employ forks to enhance the system security, achieve better
+performances or manage different connections at the same time. This paper introduces the fork-awareness
+property to express the fuzzer ability to manage systems using forks. This property is leveraged to evaluate 14
+of the most widely coverage-guided fuzzers and highlight how current fuzzers are ineffective against systems
+using forks.
+1
+Introduction
+In the last years, plenty of fuzzers have been devel-
+oped to deal with sophisticated software and nowa-
+days it is extremely common that network systems
+employ forks to deal with different connections at the
+same time. This leads to 1) the need to devise accurate
+and ad-hoc fuzzers and 2) the need to evaluate these
+fuzzer according to their ability to cope with such ad-
+vanced systems.
+Unfortunately,
+as
+pointed
+out
+in
+[Hazimeh et al., 2020],
+it is not easy to bench-
+mark all of them since the fuzzers are very different
+from each other. Metzman et al. faced this problem
+by devising FuzzBench [Metzman et al., 2021], an
+open-source service for the evaluations of state-
+less fuzzers.
+Later, Natella and Pham presented
+ProFuzzBench
+[Natella and Pham, 2021],
+which
+similarly to FuzzBench provides a service to evaluate
+stateful fuzzers.
+Although FuzzBench includes a sample of real
+word programs and ProFuzzBench includes dif-
+a
+https://orcid.org/0000-0002-6585-5494
+b
+https://orcid.org/0000-0001-7435-4176
+c
+https://orcid.org/0000-0002-7615-8643
+d
+https://orcid.org/0000-0003-4635-187X
+ferent network systems (i.e.
+systems that often
+employ forks to deal with multiple connections
+[Tanenbaum, 2009]), they do not evaluate the ability
+of the fuzzers to cope with programs that use forks.
+Despite forks representing the only way to create
+a new process [Tanenbaum, 2009], experimental re-
+sults have shown that current fuzzers cannot deal with
+forked processes.
+The existing approach merely relies on code mod-
+ifications to remove the forks. Unfortunately, this ap-
+proach goes against the willingness to reduce manual
+work and improve automation during a fuzzing cam-
+paign. [Boehme et al., 2021].
+In this work, we explore and classify the limita-
+tions current fuzzers exhibit in front of forking pro-
+grams.
+In summary, this paper:
+1. devises a novel property capturing the ability of
+fuzzers to deal with forks appropriately;
+2. evaluates 14 coverage-guided fuzzers based on
+this property;
+3. proposes possible improvements to the current
+state-of-the-art and future directions.
+The paper is organised as follows. Section 2 de-
+scribes the relevant background, Section 3 presents
+
+our contributions to knowledge, Section 4 shows the
+existing approaches that try to cope with the fork
+problem and, eventually, Section 5 discuss the results
+and propose possible future directions.
+2
+Background
+2.1
+Fuzz testing
+Fuzzing is an automated testing technique pioneered
+by Miller et al. [Miller et al., 1990] in 1990 to test
+UNIX utilities. As outlined in Figure 1, coverage-
+guided fuzzing is composed at least of seed selection,
+input generation and system execution.
+1) Seeds selection. The user must provide some
+input messages (seeds) representative of some usual
+inputs for the system.
+2) Input generation. The core of every fuzzer is
+the generation of slightly malformed input messages
+to forward to the software under test. A fuzzer is as
+efficient as the generated inputs are able to break the
+system. According to the approach used to generate
+the messages, the fuzzers may be classified into:
+• dumb: generate random strings (as the first fuzzer
+[Miller et al., 1995] did);
+• dumb mutational: blindly mutate seed messages
+provided by the user;
+• grammar-based: leverage the grammar of the sys-
+tem to craft the input messages;
+• smart mutational (often called evolutionary): re-
+quire a sample of inputs and leverage feedback
+mechanisms to craft system-tailored messages.
+An example of feedback mechanisms is the code
+coverage feedback, explored in Section 2.2.
+3) System execution. Each execution of the fuzzer
+involves three components:
+• Bugs detector: it reports eventual bugs. The ma-
+jority of the bugs detectors only report crashes,
+however for many systems, also a weird deviation
+from the happy flow of the protocol may represent
+significant security issues;
+• Hangs detector:
+it detects program execution
+hangs;
+• Code coverage detector: as further explained in
+Section 2.2, the code coverage represents one of
+the feedbacks the fuzzer leverages to improve the
+quality of the input messages.
+2.2
+Coverage-Guided Fuzzing
+Smart mutational fuzzers use feedback mecha-
+nisms to steer the generation of the messages.
+Different
+types
+of
+feedback
+mechanisms
+exist
+[Shahid et al., 2011], and often different terms are
+used to express the same idea. To avoid further noise,
+in this work we use the term code coverage to express
+the lines of code that are reached by a specific mes-
+sage.
+Code coverage fuzzers need to recompile the code
+with ad-hoc compilers (e.g. the AFL compiler) to in-
+strument the code and obtain run-time information.
+AFL [Zalewski, 2017], for example, instruments
+the code to fill a bitmap that represents the lines of
+the code covered by the inputs.
+Later, it uses this bitmap to assign a higher score
+to messages able to explore previously unseen lines of
+code.
+Start
+Seeds selection I
+Input
+generation
+System execution
+Hangs
+detector
+Bugs
+detector
+Code
+coverage
+detector
+Report O
+End
+Figure 1: Coverage-guided fuzzing process
+
+2.3
+Inter-Process Communication
+Operating systems provide system calls to perform
+different tasks (e.g. writing and reading files, access-
+ing hardware services, creating and executing new
+processes). On UNIX systems, new processes are cre-
+ated by using the fork system call [Tanenbaum, 2009].
+In short, the first process, called parent process, gen-
+erates a clone, called child process, that is an exact
+copy of the parent process. After the fork, file de-
+scriptors and registers are duplicated, thus a change
+in one of the processes does not affect the other one.
+Also, the parent and child process will follow sepa-
+rate execution paths.
+3
+Our contribution
+This paper aims to understand how the state-of-the-
+art coverage-guided fuzzers deal with software under
+tests containing forks.
+It was not obvious to come up with a way to com-
+pare and contrast the various tools.
+We devised a
+novel property, the fork awareness, that must be sat-
+isfied when a fuzzer deals with forks effectively and
+efficiently. As we shall see below, fork awareness
+rests upon three aspects representing the ability to
+deal with child processes.
+Also, we evaluate the novel property over the
+most widely used fuzzers from two benchmark frame-
+works, reaching a total of 14 evaluated tools, 11
+drawn from FuzzBench and 3 from ProFuzzBench.
+3.1
+Fork-awareness
+Abstractly, fork awareness insists that every fuzzer
+should address the child process as the parent one.
+During the system execution, the system monitor
+should detect bugs or hangs regardless of their loca-
+tion and the coverage should be measured also in the
+child process. This is formalised through Definition 1.
+Definition 1. A coverage-guided fuzzer is fork-aware
+if it can detect bugs and hangs and measure coverage
+in the same way for both the child and the parent’s
+branch.
+The three aspects in this definition are called:
+[C.1] Child bugs detection: any anomaly is reported
+also if it occurs in child processes;
+[C.2] Child hangs detection: any infinite hang is re-
+ported also if it occurs in child processes;
+[C.3] Child code coverage: code coverage is measured
+also for child processes.
+3.2
+Example challenges
+We wrote three simple C programs to use as chal-
+lenges for the fuzzers, namely to test whether the
+fuzzers satisfy the aspects given above.
+a) Bugs detection challenge:
+1
+if(fork()==0){ //Child process
+2
+raise(SIGSEGV); //Simulated crash
+3
+} else { //Parent process
+4
+wait(NULL); //Waiting child
+5
+//termination
+6
+}
+The snippet sends a SIGSEGV signal to simulate
+a bug in the child process. This signal is used to
+report a segmentation fault, i.e. a memory access
+violation, which is common in programs written
+in low-level languages. The fuzzer must detect
+this bug also after the parent’s termination.
+b) Hangs detection challenge:
+1
+if(fork()==0){ //Child process
+2
+while(1){ ; } //Simulation of
+3
+//blocking code
+4
+}
+The snippet simulates an infinite loop in the child
+process. The fuzzers must report processes still
+in execution after the loop and must kill child
+processes at the end of the fuzzing campaign,
+avoiding pending process executions.
+c) Code coverage challenge:
+1
+pid_t pid = fork();
+2
+if(pid==0){ //Child process
+3
+if(data %2 == 0){ do_something(); }
+4
+else { do_something(); }
+5
+if(data %3 == 0){ do_something(); }
+6
+else { do_something(); }
+7
+if(data %5 == 0){ do_something(); }
+8
+else { do_something(); }
+9
+if(data %7 == 0){ do_something(); }
+10
+else { do_something(); }
+11
+}
+12 else { //Parent process
+13
+wait(NULL); //Waiting child
+14
+//termination
+15
+}
+This snippet simulates a child with several branches.
+A fuzzer must cover and consider every child’s
+branches.
+We run the 14 fuzzers over these challenges and
+organised the results in Table 1. We noticed that none
+of the fuzzers succeeded through all three challenges.
+
+Fuzzer
+Based on
+Monitor technique
+Bugs
+Detection
+(C1)
+Hangs
+Detection
+(C2)
+Code
+coverage
+(C3)
+AFL [Zalewski, 2017]
+-
+POSIX signals
+×
+×
+✓
+AFL++ [Fioraldi et al., 2020]
+AFL
+POSIX signals
+×
+×
+✓
+AFLFast [Bohme et al., 2017]
+AFL
+POSIX signals
+×
+×
+✓
+AFLSmart [Pham et al., 2021]
+AFL
+POSIX signals
+×
+×
+✓
+Eclipser [Choi et al., 2019]
+AFL
+POSIX signals
+×
+×
+✓
+FairFuzz [Lemieux and Sen, 2018]
+AFL
+POSIX signals
+×
+×
+✓
+lafintel [Besler and Frederic, 2016]
+AFL
+POSIX signals
+×
+×
+✓
+AFLnwe1
+AFL
+POSIX signals
+×
+×
+✓
+AFLNet [Pham et al., 2020]
+AFL
+POSIX signals
+×
+×
+✓
+MOpt-AFL [Lyu et al., 2019]
+AFL
+POSIX signals
+×
+×
+✓
+StateAFL [Natella, 2022]
+- AFL
+- AFLNet
+POSIX signals
+×
+×
+✓
+LibFuzzer2
+-
+- UBSAN
+- ASAN
+- MSAN
+✓
+×
+✓
+Entropic [Bohme et al., 2020]
+LibFuzzer
+- UBSAN
+- ASAN
+- MSAN
+✓
+×
+✓
+Honggfuzz3
+-
+ptrace (Linux)
+✓
+×
+✓
+Table 1: Coverage guided fuzzers evaluation
+3.3
+Testbed
+We decided to analyse only the coverage-guided
+fuzzers present in FuzzBench [Metzman et al., 2021]
+and ProFuzzBench [Natella and Pham, 2021] even
+though the property applies to every coverage-guided
+fuzzer. All fuzzers were executed on an Ubuntu 20.04
+server machine and all our source codes are freely
+available online4 so that our experiments are fully re-
+producible.
+3.4
+Fuzzers evaluation
+We run all selected fuzzers against our three example
+challenges. Table 1 summarises our findings.
+All the fuzzers based on AFL use POSIX signals
+and a bitmap respectively to report bugs and keep
+track of the code coverage.
+As shown in the Table 1, while the bitmaps are
+able to keep track of the child’s code coverage, bugs
+triggered in the child’s processes are not detected
+since AFL catches signals from the main process
+only, as pointed out in the documentation5. The only
+fuzzers able to detect bugs in the child process are
+LibFuzzer6, Entropic [Bohme et al., 2020] and Hong-
+4https://github.com/marcellomaugeri/forks-break-afl
+5https://github.com/google/AFL/blob/master/README.md
+6https://llvm.org/docs/LibFuzzer.html
+fuzz7, as discussed in more detail below:
+• LibFuzzer 8 and Entropic [Bohme et al., 2020]
+employ a set of sanitizers9 to report bugs. These
+mechanisms make the fuzzers able to find the
+bug in Challenge 1 and measure the different
+code paths in Challenge 3, thereby satisfying chal-
+lenges C.1 and C.3, as seen above. Unfortunately,
+challenge C.2 is not satisfied since the fuzzer can-
+not detect hangs in the child process.
+• Honggfuzz supports different software/hardware
+feedback mechanisms and a low-level interface to
+monitor targets.
+When executed on Linux ma-
+chines, Honggfuzz uses the ptrace system call to
+manage processes.
+This mechanism allows the
+fuzzer to capture a wide range of signals.
+As
+shown in Table 1, the use of ptrace (along with
+the SanitizerCoverage) allows the fuzzer to detect
+bugs and to consider coverage also in the child
+process. Unfortunately, neither this mechanism is
+able to detect hangs in the child process.
+In summary, while all selected fuzzers detect the
+code coverage (C3), none detect hangs (C2) and only
+a few detect bugs (C1) in the child process. The eval-
+uation underlines that:
+7https://honggfuzz.dev/
+8https://llvm.org/docs/LibFuzzer.html
+9AddressSanitizer,
+UndefinedBehaviorSanitizer
+and
+MemorySanitizer
+
+• Loops detection challenge is the most difficult be-
+cause fuzzers do not wait for all the child pro-
+cesses but only for the main one;
+• Code coverage challenge is the easiest because
+the instrumentation allows measuring coverage
+from the execution, regardless of the process in-
+volved;
+• Bug detection challenge depends on the technique
+used to observe bugs, as well as the use of sanitis-
+ers.
+We interpret this general outcome as a clear call for
+future research and developments.
+4
+Existing solutions
+Nowadays the only solutions to fuzz programs that
+use forks are manually modifying the code or break-
+ing the multi-process nature of the system (by em-
+ploying tools like defork10) in order to get rid of the
+forks.
+Unfortunately, making modifications to the code,
+as pointed out in the AFLNet documentation 11, to
+remove all the forks is a challenging and error-prone
+task and break the multi-process nature of the system
+often leads to weird system behaviours. The only so-
+lution, therefore, remains to modify the fuzzers.
+5
+Conclusions
+This paper analyses the fork awareness of the
+coverage-guided fuzzers using three different aspects.
+The analysis conducted on 14 well-known fuzzers
+highlights that while is it clear how important is to
+handle multi-process programs, the majority of the
+fuzzers overlook the problem. 11 of 14 fuzzers are
+not able to detect bugs in the child process. The intu-
+ition behind these outcomes is related to the way these
+fuzzers detect bugs. All the AFL-derived fuzzers use
+signals (SIGSEGV, SIGABRT, etc) to detect bugs and
+this mechanism misses bugs in child processes. We
+noticed that dealing with forks is not the only problem
+and other issues may be related to the IPC scheduling.
+For example, the IPC may influence the success of
+the fuzzing process since some bugs may be triggered
+only after a specific process schedule and only after
+access to a particular cell of memory. We believe this
+paper represents a first step towards the devising of
+fuzzers aware of the eventual multiprocess nature of
+10https://github.com/zardus/preeny/blob/master/src/defork.c
+11https://github.com/aflnet/aflnet
+the software. The first step to achieve this goal might
+be the implementation of a loop detector at an early
+stage, e.g. by leveraging a dynamic library to keep
+track of all process identifiers of forked processes. To
+summarise, this work not only provides the first con-
+crete way to evaluate the fuzzers according to their
+fork awareness but sheds light for the first time on a
+class of problems that have been ignored until now,
+showing interesting future directions.
+REFERENCES
+Besler
+and
+Frederic
+(2016).
+Circumventing
+fuzzing roadblocks with compiler transforma-
+tions. https://lafintel.wordpress.com.
+Boehme, M., Cadar, C., and Roychoudhury, A.
+(2021).
+Fuzzing: Challenges and reflections.
+IEEE Softw., 38(3):79–86.
+Bohme, M., Manes, V. J., and Cha, S. K. (2020).
+Boosting fuzzer efficiency: An information the-
+oretic perspective. In Proceedings of the 28th
+ACM Joint Meeting on European Software En-
+gineering Conference and Symposium on the
+Foundations of Software Engineering, pages
+678–689.
+Bohme, M., Pham, V.-T., and Roychoudhury, A.
+(2017).
+Coverage-based greybox fuzzing as
+markov chain. IEEE Transactions on Software
+Engineering, 45(5):489–506.
+Choi, J., Jang, J., Han, C., and Cha, S. K. (2019).
+Grey-box concolic testing on binary code.
+In
+2019 IEEE/ACM 41st International Conference
+on Software Engineering (ICSE), pages 736–
+747. IEEE.
+Fioraldi, A., Maier, D., Eißfeldt, H., and Heuse, M.
+(2020). Afl++: Combining incremental steps of
+fuzzing research. In 14th USENIX Workshop on
+Offensive Technologies (WOOT 20).
+Hazimeh, A., Herrera, A., and Payer, M. (2020).
+Magma:
+A ground-truth fuzzing benchmark.
+Proceedings of the ACM on Measurement and
+Analysis of Computing Systems, 4(3):1–29.
+Lemieux, C. and Sen, K. (2018). Fairfuzz: A tar-
+geted mutation strategy for increasing greybox
+fuzz testing coverage. In Proceedings of the 33rd
+ACM/IEEE International Conference on Auto-
+mated Software Engineering, pages 475–485.
+Lyu, C., Ji, S., Zhang, C., Li, Y., Lee, W.-H., Song,
+Y., and Beyah, R. (2019).
+{MOPT}: Opti-
+mized mutation scheduling for fuzzers. In 28th
+USENIX Security Symposium (USENIX Security
+19), pages 1949–1966.
+
+Metzman, J., Szekeres, L., Maurice Romain Si-
+mon, L., Trevelin Sprabery, R., and Arya, A.
+(2021).
+FuzzBench: An Open Fuzzer Bench-
+marking Platform and Service. In Proceedings
+of the 29th ACM Joint Meeting on European
+Software Engineering Conference and Sympo-
+sium on the Foundations of Software Engineer-
+ing, ESEC/FSE 2021, pages 1393–1403, New
+York, NY, USA. Association for Computing Ma-
+chinery.
+Miller, B. P., Fredriksen, L., and So, B. (1990). An
+empirical study of the reliability of unix utilities.
+Communications of the ACM, 33(12):32–44.
+Miller, B. P., Koski, D., Lee, C. P., Maganty, V.,
+Murthy, R., Natarajan, A., and Steidl, J. (1995).
+Fuzz revisited: A re-examination of the reliabil-
+ity of unix utilities and services. Technical re-
+port, University of Wisconsin-Madison Depart-
+ment of Computer Sciences.
+Natella, R. (2022).
+Stateafl: Greybox fuzzing for
+stateful network servers. Empirical Software En-
+gineering, 27(7):191.
+Natella, R. and Pham, V. T. (2021). Profuzzbench:
+A benchmark for stateful protocol fuzzing. In
+ISSTA 2021 - Proceedings of the 30th ACM
+SIGSOFT International Symposium on Software
+Testing and Analysis, pages 662–665. Associa-
+tion for Computing Machinery, Inc.
+Pham, V.-T., Bohme, M., and Roychoudhury, A.
+(2020). Aflnet: a greybox fuzzer for network
+protocols. In 2020 IEEE 13th International Con-
+ference on Software Testing, Validation and Ver-
+ification (ICST), pages 460–465. IEEE.
+Pham, V.-T., B¨ohme, M., Santosa, A. E., C˘aciulescu,
+A. R., and Roychoudhury, A. (2021).
+Smart
+greybox fuzzing. IEEE Transactions on Software
+Engineering, 47(9):1980–1997.
+Shahid, M., Ibrahim, S., and Mahrin, M. N. (2011).
+A study on test coverage in software test-
+ing. Advanced Informatics School (AIS), Univer-
+siti Teknologi Malaysia, International Campus,
+Jalan Semarak, Kuala Lumpur, Malaysia.
+Tanenbaum, A. (2009). Modern operating systems.
+Pearson Education, Inc.,.
+Zalewski,
+M.
+(2017).
+American
+fuzzy
+lop.
+https://lcamtuf.coredump.cx/afl/.
+
+This figure "orcid.png" is available in "png"� format from:
+http://arxiv.org/ps/2301.05060v1
+
+ScileP
+TeSS
+ScienceandTechnologyPublications
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+page_content='CR]  12 Jan 2023  Evaluating the Fork-Awareness of Coverage-Guided Fuzzers  Marcello Maugeri1  a, Cristian Daniele2  b, Giampaolo Bella1  c, and Erik Poll2  d  1Department of Maths and Computer Science, University of Catania, Catania, Italy  2Department of Digital Security, Radboud University, Nijmegen, The Netherlands  marcello.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+page_content='it, cristian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+page_content='nl, giampaolo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+page_content='it, erikpoll@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='ru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='nl  Keywords:  Fuzzing, Fork, Security Testing, Software Security  Abstract:  Fuzz testing (or fuzzing) is an effective technique used to find security vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' It consists of feeding a  software under test with malformed inputs, waiting for a weird system behaviour (often a crash of the system).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Over the years, different approaches have been developed, and among the most popular lies the coverage-based  one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' It relies on the instrumentation of the system to generate inputs able to cover as much code as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  The success of this approach is also due to its usability as fuzzing techniques research approaches that do  not require (or only partial require) human interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' Despite the efforts, devising a fully-automated fuzzer  still seems to be a challenging task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' Target systems may be very complex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' they may integrate cryptographic  primitives, compute and verify check-sums and employ forks to enhance the system security, achieve better  performances or manage different connections at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' This paper introduces the fork-awareness  property to express the fuzzer ability to manage systems using forks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' This property is leveraged to evaluate 14  of the most widely coverage-guided fuzzers and highlight how current fuzzers are ineffective against systems  using forks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  1  Introduction  In the last years, plenty of fuzzers have been devel-  oped to deal with sophisticated software and nowa-  days it is extremely common that network systems  employ forks to deal with different connections at the  same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' This leads to 1) the need to devise accurate  and ad-hoc fuzzers and 2) the need to evaluate these  fuzzer according to their ability to cope with such ad-  vanced systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Unfortunately,  as  pointed  out  in  [Hazimeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', 2020],  it is not easy to bench-  mark all of them since the fuzzers are very different  from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' Metzman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' faced this problem  by devising FuzzBench [Metzman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', 2021], an  open-source service for the evaluations of state-  less fuzzers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Later, Natella and Pham presented  ProFuzzBench  [Natella and Pham, 2021],  which  similarly to FuzzBench provides a service to evaluate  stateful fuzzers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Although FuzzBench includes a sample of real  word programs and ProFuzzBench includes dif-  a  https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='org/0000-0002-6585-5494  b  https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='org/0000-0001-7435-4176  c  https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='org/0000-0002-7615-8643  d  https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='org/0000-0003-4635-187X  ferent network systems (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  systems that often  employ forks to deal with multiple connections  [Tanenbaum, 2009]), they do not evaluate the ability  of the fuzzers to cope with programs that use forks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Despite forks representing the only way to create  a new process [Tanenbaum, 2009], experimental re-  sults have shown that current fuzzers cannot deal with  forked processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  The existing approach merely relies on code mod-  ifications to remove the forks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' Unfortunately, this ap-  proach goes against the willingness to reduce manual  work and improve automation during a fuzzing cam-  paign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' [Boehme et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  In this work, we explore and classify the limita-  tions current fuzzers exhibit in front of forking pro-  grams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  In summary, this paper:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' devises a novel property capturing the ability of  fuzzers to deal with forks appropriately;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' evaluates 14 coverage-guided fuzzers based on  this property;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' proposes possible improvements to the current  state-of-the-art and future directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  The paper is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' Section 2 de-  scribes the relevant background, Section 3 presents  our contributions to knowledge, Section 4 shows the  existing approaches that try to cope with the fork  problem and, eventually, Section 5 discuss the results  and propose possible future directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  2  Background  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='1  Fuzz testing  Fuzzing is an automated testing technique pioneered  by Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' [Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', 1990] in 1990 to test  UNIX utilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' As outlined in Figure 1, coverage-  guided fuzzing is composed at least of seed selection,  input generation and system execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  1) Seeds selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' The user must provide some  input messages (seeds) representative of some usual  inputs for the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  2) Input generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' The core of every fuzzer is  the generation of slightly malformed input messages  to forward to the software under test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' A fuzzer is as  efficient as the generated inputs are able to break the  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' According to the approach used to generate  the messages, the fuzzers may be classified into:  dumb: generate random strings (as the first fuzzer  [Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', 1995] did);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  dumb mutational: blindly mutate seed messages  provided by the user;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  grammar-based: leverage the grammar of the sys-  tem to craft the input messages;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  smart mutational (often called evolutionary): re-  quire a sample of inputs and leverage feedback  mechanisms to craft system-tailored messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  An example of feedback mechanisms is the code  coverage feedback, explored in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  3) System execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' Each execution of the fuzzer  involves three components:  Bugs detector: it reports eventual bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' The ma-  jority of the bugs detectors only report crashes,  however for many systems, also a weird deviation  from the happy flow of the protocol may represent  significant security issues;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Hangs detector:  it detects program execution  hangs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Code coverage detector: as further explained in  Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='2, the code coverage represents one of  the feedbacks the fuzzer leverages to improve the  quality of the input messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='2  Coverage-Guided Fuzzing  Smart mutational fuzzers use feedback mecha-  nisms to steer the generation of the messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Different  types  of  feedback  mechanisms  exist  [Shahid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', 2011], and often different terms are  used to express the same idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' To avoid further noise,  in this work we use the term code coverage to express  the lines of code that are reached by a specific mes-  sage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Code coverage fuzzers need to recompile the code  with ad-hoc compilers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' the AFL compiler) to in-  strument the code and obtain run-time information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  AFL [Zalewski, 2017], for example, instruments  the code to fill a bitmap that represents the lines of  the code covered by the inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Later, it uses this bitmap to assign a higher score  to messages able to explore previously unseen lines of  code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Start  Seeds selection I  Input  generation  System execution  Hangs  detector  Bugs  detector  Code  coverage  detector  Report O  End  Figure 1: Coverage-guided fuzzing process  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='3  Inter-Process Communication  Operating systems provide system calls to perform  different tasks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' writing and reading files, access-  ing hardware services, creating and executing new  processes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' On UNIX systems, new processes are cre-  ated by using the fork system call [Tanenbaum, 2009].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  In short, the first process, called parent process, gen-  erates a clone, called child process, that is an exact  copy of the parent process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' After the fork, file de-  scriptors and registers are duplicated, thus a change  in one of the processes does not affect the other one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Also, the parent and child process will follow sepa-  rate execution paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  3  Our contribution  This paper aims to understand how the state-of-the-  art coverage-guided fuzzers deal with software under  tests containing forks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  It was not obvious to come up with a way to com-  pare and contrast the various tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  We devised a  novel property, the fork awareness, that must be sat-  isfied when a fuzzer deals with forks effectively and  efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' As we shall see below, fork awareness  rests upon three aspects representing the ability to  deal with child processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Also, we evaluate the novel property over the  most widely used fuzzers from two benchmark frame-  works, reaching a total of 14 evaluated tools, 11  drawn from FuzzBench and 3 from ProFuzzBench.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='1  Fork-awareness  Abstractly, fork awareness insists that every fuzzer  should address the child process as the parent one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  During the system execution, the system monitor  should detect bugs or hangs regardless of their loca-  tion and the coverage should be measured also in the  child process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' This is formalised through Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' A coverage-guided fuzzer is fork-aware  if it can detect bugs and hangs and measure coverage  in the same way for both the child and the parent’s  branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  The three aspects in this definition are called:  [C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='1] Child bugs detection: any anomaly is reported  also if it occurs in child processes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  [C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='2] Child hangs detection: any infinite hang is re-  ported also if it occurs in child processes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  [C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='3] Child code coverage: code coverage is measured  also for child processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='2  Example challenges  We wrote three simple C programs to use as chal-  lenges for the fuzzers, namely to test whether the  fuzzers satisfy the aspects given above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  a) Bugs detection challenge:  1  if(fork()==0){ //Child process  2  raise(SIGSEGV);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' //Simulated crash  3  } else { //Parent process  4  wait(NULL);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' //Waiting child  5  //termination  6  }  The snippet sends a SIGSEGV signal to simulate  a bug in the child process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' This signal is used to  report a segmentation fault, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' a memory access  violation, which is common in programs written  in low-level languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' The fuzzer must detect  this bug also after the parent’s termination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  b) Hangs detection challenge:  1  if(fork()==0){ //Child process  2  while(1){ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' } //Simulation of  3  //blocking code  4  }  The snippet simulates an infinite loop in the child  process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' The fuzzers must report processes still  in execution after the loop and must kill child  processes at the end of the fuzzing campaign,  avoiding pending process executions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  c) Code coverage challenge:  1  pid_t pid = fork();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  2  if(pid==0){ //Child process  3  if(data %2 == 0){ do_something();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' }  4  else { do_something();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' }  5  if(data %3 == 0){ do_something();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' }  6  else { do_something();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' }  7  if(data %5 == 0){ do_something();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' }  8  else { do_something();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' }  9  if(data %7 == 0){ do_something();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' }  10  else { do_something();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' }  11  }  12 else { //Parent process  13  wait(NULL);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' //Waiting child  14  //termination  15  }  This snippet simulates a child with several branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  A fuzzer must cover and consider every child’s  branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  We run the 14 fuzzers over these challenges and  organised the results in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' We noticed that none  of the fuzzers succeeded through all three challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Fuzzer  Based on  Monitor technique  Bugs  Detection  (C1)  Hangs  Detection  (C2)  Code  coverage  (C3)  AFL [Zalewski, 2017] POSIX signals  ×  ×  ✓  AFL++ [Fioraldi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', 2020]  AFL  POSIX signals  ×  ×  ✓  AFLFast [Bohme et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', 2017]  AFL  POSIX signals  ×  ×  ✓  AFLSmart [Pham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', 2021]  AFL  POSIX signals  ×  ×  ✓  Eclipser [Choi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', 2019]  AFL  POSIX signals  ×  ×  ✓  FairFuzz [Lemieux and Sen, 2018]  AFL  POSIX signals  ×  ×  ✓  lafintel [Besler and Frederic, 2016]  AFL  POSIX signals  ×  ×  ✓  AFLnwe1  AFL  POSIX signals  ×  ×  ✓  AFLNet [Pham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', 2020]  AFL  POSIX signals  ×  ×  ✓  MOpt-AFL [Lyu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', 2019]  AFL  POSIX signals  ×  ×  ✓  StateAFL [Natella, 2022]  AFL  AFLNet  POSIX signals  ×  ×  ✓  LibFuzzer2 UBSAN  ASAN  MSAN  ✓  ×  ✓  Entropic [Bohme et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', 2020]  LibFuzzer  UBSAN  ASAN  MSAN  ✓  ×  ✓  Honggfuzz3 ptrace (Linux)  ✓  ×  ✓  Table 1: Coverage guided fuzzers evaluation  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='3  Testbed  We decided to analyse only the coverage-guided  fuzzers present in FuzzBench [Metzman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', 2021]  and ProFuzzBench [Natella and Pham, 2021] even  though the property applies to every coverage-guided  fuzzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' All fuzzers were executed on an Ubuntu 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='04  server machine and all our source codes are freely  available online4 so that our experiments are fully re-  producible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='4  Fuzzers evaluation  We run all selected fuzzers against our three example  challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' Table 1 summarises our findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  All the fuzzers based on AFL use POSIX signals  and a bitmap respectively to report bugs and keep  track of the code coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  As shown in the Table 1, while the bitmaps are  able to keep track of the child’s code coverage, bugs  triggered in the child’s processes are not detected  since AFL catches signals from the main process  only, as pointed out in the documentation5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' The only  fuzzers able to detect bugs in the child process are  LibFuzzer6, Entropic [Bohme et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', 2020] and Hong-  4https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='com/marcellomaugeri/forks-break-afl  5https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='com/google/AFL/blob/master/README.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='md  6https://llvm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='org/docs/LibFuzzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='html  fuzz7, as discussed in more detail below:  LibFuzzer 8 and Entropic [Bohme et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', 2020]  employ a set of sanitizers9 to report bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' These  mechanisms make the fuzzers able to find the  bug in Challenge 1 and measure the different  code paths in Challenge 3, thereby satisfying chal-  lenges C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='1 and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='3, as seen above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' Unfortunately,  challenge C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='2 is not satisfied since the fuzzer can-  not detect hangs in the child process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Honggfuzz supports different software/hardware  feedback mechanisms and a low-level interface to  monitor targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  When executed on Linux ma-  chines, Honggfuzz uses the ptrace system call to  manage processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  This mechanism allows the  fuzzer to capture a wide range of signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  As  shown in Table 1, the use of ptrace (along with  the SanitizerCoverage) allows the fuzzer to detect  bugs and to consider coverage also in the child  process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' Unfortunately, neither this mechanism is  able to detect hangs in the child process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  In summary, while all selected fuzzers detect the  code coverage (C3), none detect hangs (C2) and only  a few detect bugs (C1) in the child process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' The eval-  uation underlines that:  7https://honggfuzz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='dev/  8https://llvm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='org/docs/LibFuzzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='html  9AddressSanitizer,  UndefinedBehaviorSanitizer  and  MemorySanitizer  Loops detection challenge is the most difficult be-  cause fuzzers do not wait for all the child pro-  cesses but only for the main one;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Code coverage challenge is the easiest because  the instrumentation allows measuring coverage  from the execution, regardless of the process in-  volved;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Bug detection challenge depends on the technique  used to observe bugs, as well as the use of sanitis-  ers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  We interpret this general outcome as a clear call for  future research and developments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  4  Existing solutions  Nowadays the only solutions to fuzz programs that  use forks are manually modifying the code or break-  ing the multi-process nature of the system (by em-  ploying tools like defork10) in order to get rid of the  forks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Unfortunately, making modifications to the code,  as pointed out in the AFLNet documentation 11, to  remove all the forks is a challenging and error-prone  task and break the multi-process nature of the system  often leads to weird system behaviours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' The only so-  lution, therefore, remains to modify the fuzzers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  5  Conclusions  This paper analyses the fork awareness of the  coverage-guided fuzzers using three different aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  The analysis conducted on 14 well-known fuzzers  highlights that while is it clear how important is to  handle multi-process programs, the majority of the  fuzzers overlook the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' 11 of 14 fuzzers are  not able to detect bugs in the child process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' The intu-  ition behind these outcomes is related to the way these  fuzzers detect bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' All the AFL-derived fuzzers use  signals (SIGSEGV, SIGABRT, etc) to detect bugs and  this mechanism misses bugs in child processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' We  noticed that dealing with forks is not the only problem  and other issues may be related to the IPC scheduling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  For example, the IPC may influence the success of  the fuzzing process since some bugs may be triggered  only after a specific process schedule and only after  access to a particular cell of memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' We believe this  paper represents a first step towards the devising of  fuzzers aware of the eventual multiprocess nature of  10https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='com/zardus/preeny/blob/master/src/defork.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='c  11https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='com/aflnet/aflnet  the software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' The first step to achieve this goal might  be the implementation of a loop detector at an early  stage, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' by leveraging a dynamic library to keep  track of all process identifiers of forked processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' To  summarise, this work not only provides the first con-  crete way to evaluate the fuzzers according to their  fork awareness but sheds light for the first time on a  class of problems that have been ignored until now,  showing interesting future directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  REFERENCES  Besler  and  Frederic  (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Circumventing  fuzzing roadblocks with compiler transforma-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' https://lafintel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='wordpress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Boehme, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', Cadar, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', and Roychoudhury, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Fuzzing: Challenges and reflections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  IEEE Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', 38(3):79–86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Bohme, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', Manes, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', and Cha, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Boosting fuzzer efficiency: An information the-  oretic perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' In Proceedings of the 28th  ACM Joint Meeting on European Software En-  gineering Conference and Symposium on the  Foundations of Software Engineering, pages  678–689.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Bohme, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', Pham, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', and Roychoudhury, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Coverage-based greybox fuzzing as  markov chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' IEEE Transactions on Software  Engineering, 45(5):489–506.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Choi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', Jang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', Han, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', and Cha, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Grey-box concolic testing on binary code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  In  2019 IEEE/ACM 41st International Conference  on Software Engineering (ICSE), pages 736–  747.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Fioraldi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', Maier, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', Eißfeldt, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', and Heuse, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' Afl++: Combining incremental steps of  fuzzing research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' In 14th USENIX Workshop on  Offensive Technologies (WOOT 20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Hazimeh, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', Herrera, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', and Payer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Magma:  A ground-truth fuzzing benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Proceedings of the ACM on Measurement and  Analysis of Computing Systems, 4(3):1–29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Lemieux, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' and Sen, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' Fairfuzz: A tar-  geted mutation strategy for increasing greybox  fuzz testing coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' In Proceedings of the 33rd  ACM/IEEE International Conference on Auto-  mated Software Engineering, pages 475–485.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Lyu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', Ji, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', Zhang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', Lee, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', Song,  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', and Beyah, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  {MOPT}: Opti-  mized mutation scheduling for fuzzers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' In 28th  USENIX Security Symposium (USENIX Security  19), pages 1949–1966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Metzman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', Szekeres, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', Maurice Romain Si-  mon, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', Trevelin Sprabery, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', and Arya, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  FuzzBench: An Open Fuzzer Bench-  marking Platform and Service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' In Proceedings  of the 29th ACM Joint Meeting on European  Software Engineering Conference and Sympo-  sium on the Foundations of Software Engineer-  ing, ESEC/FSE 2021, pages 1393–1403, New  York, NY, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' Association for Computing Ma-  chinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Miller, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', Fredriksen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', and So, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' (1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' An  empirical study of the reliability of unix utilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Communications of the ACM, 33(12):32–44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Miller, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', Koski, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', Lee, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', Maganty, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=',  Murthy, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', Natarajan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', and Steidl, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Fuzz revisited: A re-examination of the reliabil-  ity of unix utilities and services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' Technical re-  port, University of Wisconsin-Madison Depart-  ment of Computer Sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Natella, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Stateafl: Greybox fuzzing for  stateful network servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' Empirical Software En-  gineering, 27(7):191.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Natella, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' and Pham, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' Profuzzbench:  A benchmark for stateful protocol fuzzing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' In  ISSTA 2021 - Proceedings of the 30th ACM  SIGSOFT International Symposium on Software  Testing and Analysis, pages 662–665.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' Associa-  tion for Computing Machinery, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Pham, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', Bohme, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', and Roychoudhury, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' Aflnet: a greybox fuzzer for network  protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' In 2020 IEEE 13th International Con-  ference on Software Testing, Validation and Ver-  ification (ICST), pages 460–465.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Pham, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', B¨ohme, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', Santosa, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', C˘aciulescu,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', and Roychoudhury, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Smart  greybox fuzzing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' IEEE Transactions on Software  Engineering, 47(9):1980–1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Shahid, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', Ibrahim, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', and Mahrin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  A study on test coverage in software test-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+arXiv:2301.08420v1  [math.PR]  20 Jan 2023
+Convergence in Wasserstein Distance for
+Empirical Measures of Non-Symmetric
+Subordinated Diffusion Processes∗
+Feng-Yu Wang
+Center for Applied Mathematics, Tianjin University, Tianjin 300072, China
+Department of Mathematics, Swansea University, Bay Campus, SA1 8EN, United Kingdom
+wangfy@tju.edu.cn
+January 23, 2023
+Abstract
+By using the spectrum of the underlying symmetric diffusion operator, the convergence
+in Wasserstein distance is characterized for the empirical measure of non-symmetric sub-
+ordinated diffusion processes in an abstract framework.
+In particular, let µ(dx) := eV (x)dx be a probability measure on an n-dimensional com-
+pact connected Riemannian manifold M, let Wp be the Lp-Wasserstein distance induced
+by the Riemannian distance, and let µt be the empirical measure of the diffusion process
+Xt generated by L := ∆ + ∇V + Z, where Z is a µ-divergence free vector field on M.
+When t → ∞, we derive the sharp convergence rate of Wp(µt, µ) for p ≥ 1, as well as the
+precise limit of tW2(µt, µ)2 for n ≤ 3:
+lim
+t→∞ E
+����tW2(µt, µ)2 −
+∞
+�
+i=1
+1
+tλi
+� � t
+0
+φi(Xs)ds
+�2����
+q
+= 0,
+q ∈
+�
+1,
+n
+2(n − 2)+
+�
+,
+where {λi}i≥1 are all non-trivial eigenvalues of −ˆL := −(∆+∇V ) with unit eigenfunctions
+{φi}i≥1 ⊂ L2(µ). Consequently, for n ≤ 3 we have
+lim
+t→∞ E[tW2(µt, µ)2] =
+∞
+�
+i=1
+2
+λ2
+i
+�
+1 − 1
+λi
+µ
+�
+|∇L−1(Zφi)|2��
+∈ (0, ∞),
+which provides a precise characterization on the physical observation that a divergence
+free perturbation Z ̸= 0 accelerates the convergence of the symmetric system. Moreover,
+for n ∈ {2, 3, 4} a critical phenomenon appears to the convergence of E[Wp(µt, µ)2q]
+1
+q with
+the critical rate t−1 log t as t → ∞.
+These results are included by our main results established in an abstract framework for
+subordinated diffusion processes, which are further illustrated by some other examples.
+∗Supported in part by NNSFC (11921001) and the National Key R&D Program of China (No.
+2022YFA1006000, 2020YFA0712900).
+1
+
+AMS subject Classification: 60B05, 60B10.
+Keywords: Empirical measure, Wasserstein distance, non-symmetric diffusion process, subor-
+dination.
+1
+Introduction
+In statistical physics, the empirical measure is a fundamental object to simulate the stationary
+distribution (Gibbs measure). Since the ground breaking series work [12] where the celebrated
+Donsker-Varadhan larger deviation principle was developed, the long time behavior of empirical
+measures has become a key research topic in the study of Markov processes, see [39, 40] for
+criteria on the central limit theorem and large deviations for hyperbounded Markov processes.
+On the other hand, the Wasserstein distance is an intrinsic object in the theory of optimal
+transport and calculus in Wasserstein space, see [1, 28] and references therein. So, it is crucial
+and interesting to study the convergence in Wasserstein distance for the empirical measure of
+Markov processes.
+Moreover, it was observed in [16] that a divergence-free perturbation to symmetric stochastic
+systems may accelerate the algorithm of Gibbs measures. This has been confirmed in several
+papers for the convergence of Markov semigroups to stationary distributions, see [17, 19, 20]
+and references therein. It is interesting to provide a sharp characterization on the acceleration
+for the convergence of empirical measures in Wasserstein distance.
+In recent years, the sharp convergence rate in the second moment of L2-Wasserstein distance
+has been derived in [33, 34, 35, 36, 38] for empirical measures of symmetric diffusions.
+In
+particular, in lower dimensions the precise limit is explicitly formulated by using eigenvalues
+and eigenfunctions of the generator. These results have been extended to subordinated processes
+in [37, 22, 23, 24] and the fractional Brownian motion on torus in [18], see also [13] for the
+study of McKean-Vlasov SDEs.
+Based on the above backgrounds, this paper investigates the convergence of empirical mea-
+sures for non-symmetric subordinated diffusion processes in an abstract framework, describes
+the acceleration of the convergence for divergence-free perturbations to symmetric systems, and
+illustrates the main results by typical examples.
+To figure out a clear picture of the our general results (see Section 2 for details), in the
+following we only consider non-symmetric diffusion processes on a compact manifold. See Sec-
+tion 5 for applications of the general results to three more examples including the subordinated
+conditional diffusion process, the subordinated Wright-Fisher process, and the subordinated
+subelliptic diffusion process on SU(2).
+Let M be an n-dimensional compact connected Riemannian manifold possibly with a bound-
+ary ∂M, let P be the space of all probability measures on M, let ρ be the Riemannian distance,
+and for any p ≥ 1, let Wp be the Lp-Wasserstein distance induced by ρ, cf. (2.1) below.
+Let µ(dx) := eV (x)dx ∈ P, where V ∈ C2(M) and dx is the volume measure on M, and let
+2
+
+Z be a C1-vector field with divµZ = 0, i.e.
+µ(Zf) :=
+�
+M
+⟨Z, ∇f⟩dµ = 0,
+f ∈ C1(M).
+It is well known that the spectrum of ˆL := ∆ + ∇V (with Neumann boundary if ∂M exists) is
+discrete and all eigenvalues {λi}i≥0 of −ˆL listed in the increasing order counting multiplicities
+satisfy
+c1i
+n
+2 ≤ λi ≤ c2λ
+n
+2
+i ,
+i ≥ 0
+for some constants c1, c2 > 0, see for instance [9]. Let {φi}i≥0 with φ0 ≡ 1 being the corre-
+sponding unitary eigenfunctions in L2(µ).
+Let Xt be the diffusion process on M generated by
+L := ∆ + ∇V + Z,
+with reflecting boundary if ∂M exists. We consider the empirical measure
+µt := 1
+t
+� t
+0
+δXsds,
+t > 0,
+where δXs is the Dirac measure at Xs. By the central limit theorem (see [39]), for any
+f ∈ L2
+0(µ) :=
+�
+f ∈ L2(µ) : µ(f) = 0
+�
+,
+we have
+(1.1)
+lim
+t→∞
+√
+tµt(f) = lim
+t→∞
+1
+√
+t
+� t
+0
+f(Xs)ds = N(0, V(f)) in law,
+where N(0, V(f)) is the centered normal distribution with variance
+(1.2)
+V(f) :=
+� ∞
+0
+µ(fPsf)ds = µ(|∇L−1f|2).
+For any k, R ≥ 1, let
+(1.3)
+Pk,R
+µ
+:=
+�
+ν ∈ P : dν = hdµ, ∥h∥k ≤ R
+�
+,
+where ∥ · ∥k is the norm in Lk(µ). For any ν ∈ P, let Eν be the expectation for the diffusion
+process Xt with initial distribution ν.
+We first consider the long time behavior of W2(µt, µ). The following result shows that when
+n ≤ 3 and ∂M is either empty or convex, tW2(µt, µ)2 behaves as
+Ξ(t) :=
+∞
+�
+i=1
+1
+λi
+|ψi(t)|2,
+where ψi(t) :=
+1
+√
+t
+� t
+0 φi(Xs)ds (i ≥ 1, t > 0), so that uniformly in ν ∈ P, tEν[W2(µt, µ)2]
+converges to
+(1.4)
+ηZ :=
+∞
+�
+i=1
+2V(φi)
+λi
+=
+∞
+�
+i=1
+2
+λ2
+i
+�
+1 − 1
+λi
+V(Zφi)
+�
+,
+where the second equality follows from Lemma 4.2 below.
+3
+
+Theorem 1.1. There exists a constant κ ≥ 1 with κ = 1 when ∂M is either empty or convex,
+such that the following assertions hold.
+(1) When n ≤ 2, for any q ∈ [1, ∞),
+(1.5)
+lim
+t→∞ sup
+ν∈P
+E���
+tW2(µt, µ)2 − Ξ(t)
+�+ +
+�
+Ξ(t) − κtW2(µt, µ)2�+��q�
+= 0,
+so that when ∂M is either empty or convex,
+(1.6)
+lim
+t→∞ sup
+ν∈P
+Eν���tW2(µt, µ)2 − Ξ(t)|q�
+= 0,
+q ∈ [1, ∞).
+(2) When n = 3, for any R ∈ [1, ∞), k ∈ ( 3
+2, ∞] and q ∈ [1, 3
+2),
+(1.7)
+lim
+t→∞ sup
+ν∈Pk,R
+µ
+E���
+tW2(µt, µ)2 − Ξ(t)
+�+ +
+�
+Ξ(t) − κtW2(µt, µ)2�+��q�
+= 0,
+so that when ∂M is either empty or convex,
+(1.8)
+lim
+t→∞ sup
+ν∈Pk,R
+µ
+Eν���tW2(µt, µ)2 − Ξ(t)|q�
+= 0.
+(3) For n ≤ 3 we have ηZ ∈ (0, ∞) and
+(1.9)
+lim
+t→∞ sup
+ν∈P
+��
+tEν[W2(µt, µ)2] − ηZ
+�+ +
+�
+ηZ − κtEν[W2(µt, µ)2]
+�+�
+= 0.
+In particular, when ∂M is either empty or convex,
+(1.10)
+lim
+t→∞ sup
+ν∈P
+��tEν[W2(µt, µ)2] − ηZ
+�� = 0.
+(4) For n = 4, there exist constants c1, c2, t0 > 0 such that
+(1.11) c1
+t log(1 + t) ≤ inf
+ν∈P Eν[W2(µt, µ)2] ≤ sup
+ν∈P
+Eν[W2(µt, µ)2] ≤ c2
+t log(1 + t),
+t ≥ t0.
+(5) For n ≥ 5, there exist constants c1, c2, t0 > 0 such that
+c1t−
+2
+d−2 ≤ inf
+ν∈P
+�
+Eν[W1(µt, µ)]
+�2 ≤ sup
+ν∈P
+Eν[W2(µt, µ)2] ≤ c2t−
+2
+d−2,
+t ≥ 1.
+(1.12)
+Remark 1.1.
+(1) By (1.4) we have ηZ < η0 for Z ̸= 0, so (1.9) and (1.10) provide a precise
+characterization on the observation of [16] that a divergence-free perturbation Z accelerates
+the convergence.
+(2) When Z = 0 (i.e. in the symmetric case), (1.9) and (1.10) have been presented in
+[38] with a central limit theorem on tW2(µt, µ)2, which are covered by Theorem 1.1. The Lq-
+convergence (1.5) and (1.6) appear here for the first time, which are new also in the symmetric
+case.
+4
+
+(3) In the symmetric case (i.e. Z = 0), estimates (1.9), (1.12) and the upper bound in (1.11)
+have been presented in [38]. Other estimates are new also in the symmetric case. Moreover, it
+is proved in [38] that for M being the 4-dimensional torus and L = ∆, there exists a constant
+c > 0 such that
+inf
+ν∈P
+�
+Eν[W1(µt, µ)]
+�2 ≥ ct−1 log(1 + t),
+t ≥ 1.
+We hope that this estimate also holds for general non-symmetric diffusions on 4-dimensional
+compact manifolds, such that the lower bound estimate in (1.11) is strengthened with W1
+replacing W2.
+In the next result, we estimate
+�
+E[Wp(µt, µ)2q]
+� 1
+q for all p, q ∈ [1, ∞). Besides the critical
+phenomenon for the convergence rate in Theorem 1.1 with (p, q) = (2, 1) and the critical rate
+t−1 log t in dimension n = 4, the critical rate also appears to dimensions n = 2, 3 with different
+(p, q).
+Theorem 1.2. There exist c, t0 ∈ (0, ∞) and κ : [2, ∞) × [1, ∞) → (0, ∞), such that the
+following assertions hold.
+(1) When n = 1, for any (p, q) ∈ [2, ∞) × [1, ∞),
+(1.13)
+c
+t ≤ inf
+ν∈P
+�
+Eν[W1(µt, µ)]
+�2 ≤ sup
+ν∈P
+�
+Eν[Wp(µt, µ)2q]
+� 1
+q ≤ κp,q
+t ,
+t ≥ t0.
+(2) Let n = 2. Then (1.13) holds for any p ∈ [2, ∞) and q ∈ [1,
+p
+p−2). Next, for any p ∈ (2, ∞)
+and q =
+p
+p−2,
+(1.14)
+sup
+ν∈P
+�
+Eν[Wp(µt, µ)2q]
+� 1
+q ≤ κp,q
+t
+log(1 + t),
+t ≥ t0.
+Finally, for any p ∈ (2, ∞) and q ∈ (
+p
+p−2, ∞),
+(1.15)
+sup
+ν∈P
+�
+Eν[Wp(µt, µ)2q]
+� 1
+q ≤ κp,qt−
+2pq
+3npq−2pq−2nq−np ,
+t ≥ 1.
+(3) Let n = 3. Then (1.13) holds for any p ∈ [2, 3) and q ∈ [1,
+3p
+5p−6); (1.14) holds for p ∈ [2, 3)
+and q =
+3p
+5p−6; and (1.15) holds for any p ∈ [2, ∞) and q ∈ (
+3p
+5p−6), ∞).
+(4) Let n = 4. Then (1.15) holds for (2, 1) ̸= (p, q) ∈ [2, ∞) × [1, ∞).
+(5) When n ≥ 5, (1.15) holds for any (p, q) ∈ [2, ∞) × [1, ∞).
+The remainder of the paper is organized as follows. In Section 2, we state our main results
+for non-symmetric subordinated diffusion processes in an abstract framework. In Sections 3
+and 4, we prove the main results on upper and lower bound estimates respectively. In Section 5,
+we apply the main results to some concrete models, where the result for the first model covers
+Theorems 1.1 and 1.2 as direct consequences.
+5
+
+2
+Main results
+We first introduce the framework of the study, then state the main results on the Wasserstein
+distance of the empirical measures for subordinated non-symmetric diffusion processes.
+2.1
+The framework
+We first introduce the sate space of diffusion processes. Let (M, ρ) be a length space, let P
+be the set of all probability measures on M, let Bb(M) be the class of bounded measurable
+functions on M, and let Cb,L(M) be the set of all bounded Lipschitz continuous functions on
+M. For any p ∈ [1, ∞), the Lp-Wasserstein distance is defined as
+(2.1)
+Wp(ν1, ν2) :=
+inf
+π∈C (ν,γ)
+� �
+M×M
+ρ(x, y)pπ(dx, dy)
+� 1
+p
+,
+ν1, ν2 ∈ P,
+where P is the space of all probability measures on M, and C (ν1, ν2) is the set of all couplings
+for ν1 and ν2.
+Next, we introduce the reversible Makrov process ˆXt on M with unique invariant probability
+measure µ ∈ P having full support. The Markov semigroup ˆPt is formulated as
+ˆPtf(x) = Ex[f( ˆXt)],
+t ≥ 0, x ∈ M, f ∈ Bb(M),
+where Ex stands for the expectation for the underlying Markov process starting at point x. In
+general, for any ν ∈ P, Eν is the expectation for the Markov process with initial distribution
+ν. Let ( ˆE , D( ˆE )) and (ˆL, D(ˆL)) be the associated symmetric Dirichlet form and self-adjoint
+generator in L2(µ). We assume that Cb,L(M) is a dense subset of D( ˆE ) under the ˆE1-norm
+∥f∥ ˆE1 :=
+�
+µ(f 2) + ˆE (f, f),
+where µ(f 2) :=
+�
+M f 2dµ, and that
+ˆE (f, g) =
+�
+M
+Γ(f, g)dµ,
+f, g ∈ Cb,L(M)
+holds for a symmetric local square field (champ de carr´e)
+Γ : Cb,L(M) × Cb,L(M) → Bb(M),
+such that for any f, g, h ∈ Cb,L and φ ∈ C1
+b (R), we have
+�
+Γ(f, f)(x) = |∇f(x)| := lim sup
+y→x
+|f(y) − f(x)|
+ρ(x, y)
+,
+x ∈ M,
+and the chain rule
+Γ(fg, h) = fΓ(g, h) + gΓ(f, h),
+Γ(φ(f), h) = φ′(f)Γ(f, h).
+6
+
+We also assume that ˆL satisfies the chain rule
+ˆLΦ(f) = Φ′(f)ˆLf + Φ′′(f)|∇f|2,
+f ∈ D(ˆL) ∩ Cb,l, Φ ∈ C2(R).
+For any q ≥ p ∈ [1, ∞], let ∥ · ∥p be the norm in Lp(µ), and let ∥ · ∥p→q the operator norm from
+Lp(µ) to Lq(µ). Throughout the paper, we simply denote µ(f) =
+�
+M fdµ for f ∈ L1(µ).
+Moreover, we make a non-symmetric perturbation to the symmetric diffusion process by
+adding a µ-divergence free drift Z to the generator. More precisely, let
+Z : Cb,L(M) → Bb(M)
+be a bounded vector field with divµZ = 0, i.e.
+Z(fg) = fZg + gZf,
+Z(φ(f)) = φ′(f)Zf
+holds for f, g ∈ Cb,L(M) and φ ∈ C1(R), and
+∥Z∥∞ := inf
+�
+K ≥ 0 : |Zf| ≤ K|∇f|, f ∈ Cb,L(M)
+�
+< ∞,
+µ(Zf) :=
+�
+M
+Zfdµ = 0,
+f ∈ Cb,L(M).
+Consequently, Z uniquely extends to a bounded linear operator from D( ˆE ) to L2(µ) with
+(2.2)
+µ(Zf) = 0,
+f ∈ D( ˆE ),
+and
+E (f, g) := ˆE (f, g) + µ(fZg),
+f, g ∈ D(E ) = D( ˆE )
+is a (non-symmetric) conservative Dirichlet form with generator
+L := ˆL + Z,
+D(L) = D(ˆL),
+which satisfies the chain rule
+LΦ(f) = Φ′(f)Lf + Φ′′(f)|∇f|2,
+f ∈ D(ˆL) ∩ Cb,l, Φ ∈ C2(R).
+Assume that L generates a unique diffusion process Xt on M, such that the associated Markov
+semigroup is given by
+Ptf(x) = Ex[f(Xt)],
+x ∈ M, t ≥ 0, f ∈ Bb(M).
+By Duhamel’s formula,
+(2.3)
+Ptf = ˆPtf +
+� t
+0
+Ps{Z ˆPt−sf}ds,
+f ∈ D( ˆE ), t ≥ 0.
+Finally, we make subordination of the diffusion process Xt. Let B be the set of Bernstein
+functions B satisfying B(0) = 0, B(r) > 0 for r > 0 and
+(2.4)
+� ∞
+0
+e− 1
+2 B(r)dr < ∞.
+7
+
+For each B ∈ B, there exists a unique stable process SB
+t on [0, ∞) with Laplace transform
+(2.5)
+E[e−λSB
+t ] = e−B(λ)t,
+t, λ ≥ 0.
+Let SB
+t be independent of Xt. We consider the subordinated diffusion process
+XB
+t := XSB
+t ,
+t ≥ 0,
+and study the convergence to µ in Wp (p ≥ 1) for the empirical measure
+µB
+t := 1
+t
+� t
+0
+δXB
+s ds,
+t > 0.
+We will mainly consider α-stable type time change for α ∈ [0, 1], i.e. the Bernstein function
+B is in the classes
+Bα :=
+�
+B ∈ B : lim inf
+λ→∞ B(λ)λ−α > 0
+�
+,
+Bα :=
+�
+B ∈ B : lim sup
+λ→∞
+B(λ)λ−α < ∞
+�
+.
+2.2
+Upper bound estimates
+We make the following assumption, where (2.6) implies that the spectrum of −ˆL is discrete
+and all eigenvalues {λi}i≥0 listed in the increasing order counting multiplicities satisfy
+λi ≥ ci
+2
+d,
+i ∈ Z+
+for some constant c > 0, where λ1 ≥ λ, see for instance see [11]. In general, λi may increase
+faster than λ
+2
+d
+i , see for instance Subsection 5.2 where d = n + 2 but λi ∼ i
+2
+n, we make the
+additional assumption (2.7).
+(A1) There exist constants c, λ > 0, d ≥ d′ ≥ 1 and a map k : (1, ∞) → (0, ∞) such that
+(2.6)
+∥ ˆPt − µ∥1→∞ ≤ c(1 ∧ t)− d
+2 e−λt,
+t > 0,
+(2.7)
+λi ≥ ci
+2
+d′ ,
+i ∈ Z+,
+(2.8)
+|∇ ˆPtf| ≤ k(p)( ˆPt|∇f|p)
+1
+p,
+t ∈ [0, 1], p ∈ (1, ∞), f ∈ Cb,L(M).
+We will also need the following condition on the continuity of ˆXt.
+(A2) For any p ∈ [1, ∞) there exists a constant c(p) > 0 such that
+(2.9)
+Eµ�
+ρ( ˆX0, ˆXt)p�
+≤ c(p)t
+p
+2,
+t ∈ [0, 1].
+8
+
+Let {φi}i≥0 with φ0 ≡ 1 be the unitary eigenfunctions for {λi}i≥0, i.e.
+(2.10)
+ˆLφi = −λiφi,
+ˆPtφi = e−λitφi,
+µ(φiφj) = 1{i=j},
+i, j ∈ Z+, t ≥ 0.
+Let
+(2.11)
+ΞB(t) :=
+∞
+�
+i=1
+ψB
+i (t)2
+λi
+,
+ψB
+i (t) := 1
+√
+t
+� t
+0
+φi(XB
+s )ds,
+t > 0, i ∈ N.
+Let Pk,R be in (1.3). Finally, let
+(2.12)
+qα := 1 +
+2(1 + α) − d′
+(2d + d′ − 2 − 2α)+,
+and denote the integer parts of q ∈ [1, ∞) by
+i(q) := sup
+�
+i ∈ N : i ≤ q
+�
+.
+The first main result of the paper is the following.
+Theorem 2.1. Assume (A1), (A2) and let B ∈ Bα for some α ∈ [0, 1] with d′ < 2(1 + α).
+(1) If qα >
+d
+2α and
+α > α(d, d′) := 1
+4
+��
+(2 + d − d′)2 + 4d(d + d′ − 2) + d′ − d − 2
+�
+,
+then
+(2.13)
+lim
+t→∞ sup
+ν∈P
+E���
+�
+tW2(µB
+t , µ)2 − ΞB(t)
+�+���
+q�
+= 0,
+q ∈ [1, qα).
+(2) For any q ∈ [1, qα) and k ∈ (
+d
+2αi(q), ∞] ∩ [1, ∞], where we set (
+d
+2αi(q), ∞] = {∞} if α = 0,
+(2.14)
+lim
+t→∞ sup
+ν∈Pk,R
+E���
+�
+tW2(µB
+t , µ)2 − ΞB(t)
+�+���
+q�
+= 0,
+R ∈ (0, ∞).
+To estimate E[W2(µB
+t , µ)2], we let
+ηB
+Z :=
+∞
+�
+i=1
+2VB(φi)
+λi
+,
+VB(φi) :=
+� ∞
+0
+µ(φiP B
+s φi)ds.
+Theorem 2.2. Assume (A1) and (A2). Let q ∈ [1, ∞) and B ∈ Bα for some α ∈ [0, 1].
+(1) If d′ < 2(1 + α), then ηB
+Z < ∞ and
+(2.15)
+lim sup
+t→∞
+sup
+ν∈P
+tEν[W2(µB
+t , µ)2] ≤ ηB
+Z .
+9
+
+(2) Let d′ ≥ 2(1 + α). Then there exists a constant c > 0 such that for any t ≥ 1,
+(2.16)
+sup
+ν∈P
+Eν[W2(µB
+t , µ)2] ≤
+�
+ct−1 log(1 + t),
+if d′ = 2(1 + α),
+ct−
+2
+d′−2α ,
+if d′ > 2(1 + α).
+To estimate Wp(µB
+t , µ), we will use the Lp-boundedness of the Riesz transform ∇(a0 − ˆL)− 1
+2
+for some a0 ≥ 0. According to [3], together with the non-degeneracy condition, the volume
+doubling condition and the scaled Poincar´e inequality, (2.8) implies
+(2.17)
+∥∇(a0 − ˆL)− 1
+2∥p < ∞ for some a0 ∈ [0, ∞) and all p ∈ (2, ∞).
+Under assumption (A1), let
+γα,p,q := d′
+2 + d
+2
+�
+2 − p−1 − q−1�
+− α − 1,
+p, q ∈ [1, ∞), α ∈ [0, 1].
+Theorem 2.3. Assume (A1) and (A2). Let B ∈ Bα for some α ∈ [0, 1].
+(1) If γα,p,q < 0, then there exists a constant c > 0 such that
+(2.18)
+sup
+ν∈P
+�
+Eν[W2p(µB
+t , µ)2q� 1
+q ≤ ct−1,
+t ≥ 1.
+(2) If γα,p,q ≥ 0, then for any γ > γα,p,q, there exists a constant c > 0 such that
+(2.19)
+sup
+ν∈P
+�
+Eν[W2p(µB
+t , µ)2q]
+� 1
+q ≤ ct−
+1
+1+γ ,
+t ≥ 1.
+(3) Let (2.17) hold. If γα,p,q ≥ 0, then there exists a constant c > 0 such that for any t ≥ 1,
+(2.20)
+sup
+ν∈P
+�
+Eν[W2p(µB
+t , µ)2q]
+� 1
+q ≤
+�
+ct−1 log(1 + t),
+if γα,p,q = 0,
+ct
+−
+1
+1+γα,p,q ,
+if γα,p,q > 0.
+2.3
+Lower bound estimate
+To derive sharp lower bound for E[W2(µB
+t , µ)2], we make the following assumption.
+(B) (M, ρ) is a geodesic space, there exist constants θ, K > 0 and m ≥ 1 such that
+(2.21)
+|∇Ptef|2 ≤ (Ptef)Pt(|∇f|2ef) + Ktθ∥∇f∥2
+∞(Pte2mf)
+1
+m,
+t ∈ [0, 1], f ∈ Cb,L(M),
+and there exists a function h ∈ C([0, 1]; [1, ∞)) such that
+(2.22)
+W2(ν ˆPr, µ)2 ≤ h(r)W2(ν, µ)2,
+ν ∈ P, r ∈ [0, 1].
+When M is a Riemannian manifold without boundary or with convex boundary, if the
+Bakry-Emery curvature of ˆL is bounded below by a constant −K/2, then (B) holds for m = 1
+and h(r) = eKr, see for instance [32, Theorem 2.3.3(2′)(9)] or [26].
+10
+
+Theorem 2.4. Assume (A1) and (B). Let B ∈ Bα for some α ∈ [0, 1] with d′ < 2(1 + α).
+(1) If α > α(d, d′) and qα >
+d
+2α, then
+lim
+t→∞ sup
+ν∈P
+E���
+�
+th(0)W2(µB
+t , µ)2 − ΞB(t)
+�−���
+q�
+= 0,
+q ∈ [1, qα).
+(2) For any q ∈ [1, qα) and k ∈ (
+d
+2αi(q), ∞] ∩ [1, ∞), where we set (
+d
+2αi(q), ∞] = {∞} if α = 0,
+we have
+lim
+t→∞ sup
+ν∈Pk,R
+E���
+�
+th(0)W2(µB
+t , µ)2 − ΞB(t)
+�−���
+q�
+= 0,
+q ∈ [1, qα), R ∈ [1, ∞).
+(3) We have
+lim inf
+t→∞ sup
+ν∈P
+tEν[W2(µB
+t , µ)2] ≥ h(0)−1ηB
+Z .
+The next result manages the critical case where the convergence rate of E[W2(µB
+t , µ)2] is at
+most t−1 log t, correspondingly to (2.16) on the upper bound estimate.
+Theorem 2.5. Assume (A1), (A2), (B) and that
+(2.23)
+k′i
+2
+d′ ≤ λi ≤ ki
+2
+d′ ,
+i ∈ N
+holds for some constants k, k′ > 0. Let B ∈ Bα for some α ∈ [0, 1] such that d′ = 2(1 + α).
+Then there exist constants c, t0 > 0 such that
+(2.24)
+inf
+ν∈P Eν[W2(µB
+t , µ)2] ≥ ct−1 log(1 + t),
+t ≥ t0.
+Finally, we consider the lower bound estimate on W1.
+Theorem 2.6. Let B ∈ B. Then the following assertions hold.
+(1) Assume (2.6), (2.8) and that the completion ¯
+M of M is a Polish space. Then there exist
+constants c, t0 > 0 such that
+(2.25)
+inf
+ν∈P Eν[W1(µB
+t , µ)] ≥ ct− 1
+2,
+t ≥ t0.
+(2) Assume that (2.9) holds for p = 1, and there exists constants k, d′′ > 0 such that
+(2.26)
+sup
+x∈M
+µ(B(x, r)) ≤ krd′′,
+r ≥ 0,
+where B(x, r) := {y ∈ M : ρ(x, y) ≤ r}. If B ∈ Bα for some α ∈ [0, 1] with d′′ > 2(1+α),
+then there exist constants c, t0 > 0 such that
+(2.27)
+inf
+ν∈P Eν[W1(µB
+t , µ)] ≥ ct−
+1
+d′′−2α,
+t ≥ t0.
+11
+
+3
+Proofs of Theorems 2.1-2.3
+For a density function f with respect to µ, let (fµ)(A) :=
+�
+A fdµ for measurable A ⊂ M.
+Recall that for any probability density functions f, f1, f2 ∈ L2(µ), we have
+(3.1)
+W2(fµ, µ)2 ≤
+�
+M
+|∇ˆL−1(f − 1)|2
+M (f)
+dµ,
+M (f) := 1{f>0}
+f − 1
+log f ,
+(3.2)
+Wp(f1µ, f2µ)p ≤ pp
+�
+M
+|∇ˆL−1(f1 − f2)|p
+f p−1
+2
+dµ.
+These estimates have been presented in [2] and [21] respectively by using the Kantorovich dual
+formula and properties on Hamilton-Jacobi equations, which are available when (M, ρ) is a
+length space as we assumed, see [28].
+Since the empirical measure µB
+t is singular with respect to µ, to apply these estimates we
+make the following regularization of µB
+t :
+(3.3)
+µB
+t,r := f B
+t,rµ,
+f B
+t,r := 1 +
+∞
+�
+i=1
+e−λirψB
+i (t),
+t, r > 0,
+where ψB
+i (t) :=
+1
+√
+t
+� t
+0 φi(XB
+s )ds. Letting ν ˆPr being the distribution of ˆXr with initial distribu-
+tion ν, by (2.10) and the spectral representation
+ˆpr(x, y) = 1 +
+∞
+�
+i=1
+e−λirφi(x)φi(y)
+for the heat kernel of ˆPr with respect to µ, we have
+(3.4)
+µB
+t,r = µB
+t ˆPr,
+t, r > 0.
+So, (3.1) implies
+(3.5)
+W2(µB
+t,r, µ)2 ≤
+�
+M
+|∇ˆL−1(f B
+t,r − 1)|2
+M (f B
+t,r)
+dµ,
+t, r > 0.
+According to (2.6), we have f B
+t,r → 1 as t → ∞. When f B
+t,r → 1 fast enough, by (2.10) and
+(3.3), for large enough t we may bound tW2(µB
+t,r, µ)2 by
+(3.6)
+ΞB
+r (t) := tµ(|∇ˆL−1(f B
+t,r − 1)|2) =
+∞
+�
+i=1
+e−2λir
+λi
+ψB
+i (t)2,
+t, r > 0.
+On the other hand, by (3.2) we have
+(3.7)
+Wp(µB
+t,r, µ)p ≤ ppµ
+�
+|∇ˆL−1(f B
+t,r − 1)|p�
+,
+t, r > 0, p ∈ [1, ∞).
+With the above observations, in the following we present some lemmas to estimate ΞB
+r (t),
+µ
+�
+|∇ˆL−1(f B
+t,r − 1)|p�
+and Wp(µB
+t,r, µB
+t ), which will be used to prove Theorems 2.1-2.3.
+12
+
+3.1
+Some lemmas
+To apply (3.7), we need estimate ∥∇ˆL−1(f B
+t,r − 1)∥p, see (3.12) below. To this and also for
+later use, we first estimate ∥P B
+t − µ∥p→q and ∥PtZ∥2p for q ≥ p ≥ 1, where P B
+t
+is the Markov
+semigroup for the subordinated diffusion process XB
+t given by
+(3.8)
+P B
+t f(x) := Ex[f(XB
+t )] = E[PSB
+t f(x)],
+t ≥ 0, x ∈ M, f ∈ Bb(M).
+Lemma 3.1. Assume (2.6). Then the following assertions hold.
+(1) Let B ∈ Bα for some α ∈ (0, 1]. There exists a constant k > 0 such that
+(3.9)
+∥P B
+t − µ∥p→q ≤ k(1 ∧ t)− d(q−p)
+2pqα e−λt,
+t > 0, q ≥ p ∈ [1, ∞].
+(2) If (2.8) holds, then for any p ∈ [1, ∞) there exists a constant c(p) > 0 such that
+(3.10)
+∥∇Ptf∥p ≤ c(p)(1 ∧ t)− 1
+2e−λt∥f∥p,
+t > 0, f ∈ Cb,L(M),
+(3.11)
+∥Pt(Zf)∥2p ≤ c(p)∥Z∥∞t− 1
+2e−λt∥f∥2p,
+f ∈ D( ˆE ) ∩ L2p(µ), t > 0.
+Moreover, for any κ ∈ (0, 1
+2) there exists a constant c(p, κ) > 0 such that
+(3.12)
+∥∇ˆL−1f∥2p ≤ c(p, κ)∥(−ˆL)
+d(p−1)
+4p
+−κf∥2,
+µ(f) = 0.
+Proof. (a) We will use some known results on functional inequalities. Firstly, λ1 > 0 is equiva-
+lent to the Poincar´e inequality
+(3.13)
+µ(f 2) ≤ 1
+λ1
+ˆE (f, f),
+f ∈ D( ˆE ), µ(f) = 0.
+By (2.2) we have E (f, f) = ˆE (f, f). So, (3.13) implies
+(3.14)
+∥Pt − µ∥2 ≤ e−λ1t,
+t ≥ 0.
+Next, according to [30, Theorem 3.3.14 and 3.3.15], (2.6) implies the super Poincar´e inequality
+(3.15)
+µ(f 2) ≤ r ˆE (f, f) + c1(1 + r− d
+2)µ(|f|)2,
+r > 0, f ∈ D( ˆE )
+for some constant c1 > 0, which further yields
+(3.16)
+∥Pt∥1→∞ ≤ c2(1 ∧ t)− d
+2,
+t > 0
+for some constant c2 > 0. Noting that B ∈ Bα implies
+(3.17)
+B(λ) ≥ k1{λ≥λ1}λα
+13
+
+for some constant k > 0, by (2.5) we find a constant c3 > 0 such that
+E[(SB
+t )−d] = E
+� ∞
+0 λd−1e−λSB
+t dλ
+� ∞
+0 λd−1e−λdλ
+≤ c3(1 ∧ t)− d
+α,
+t > 0.
+Combining this with (2.5), (3.8), (3.14) and (3.16), we find constants c4, c5 ≥ 1 such that
+∥P B
+t − µ∥1→∞ ≤ E[∥PSB
+t − µ∥1→∞] ≤ c4E
+��
+1 + (SB
+t )− d
+2�
+e−λ1SB
+t
+�
+≤ 2c4
+�
+E[1 + (SB
+t )−d]
+� 1
+2�
+E[e−2λ1SB
+t ]
+� 1
+2 ≤ c5(1 ∧ t)− d
+2α e−B(2λ1)t/2,
+t > 0.
+By the interpolation theorem, this and ∥P B
+t − µ∥p ≤ 2 for p ∈ [1, ∞] imply (3.9) for some
+constants c, λ > 0. In particular, for B(r) = r and Z = 0 or Z ̸= 0, (3.9) implies to
+(3.18)
+∥ ˆPt − µ∥p→q ∨ ∥Pt − µ∥p→q ≤ kt− d(q−p)
+2pq e−λt,
+t > 0, q ≥ p ≥ 1.
+(b) To prove (3.10), we first prove that for some decreasing c : (1, ∞) → (0, ∞),
+(3.19)
+|∇ ˆPtf| ≤ c1(p)
+√
+t ( ˆPt|f|p)
+1
+p,
+t ∈ (0, 1], f ∈ Cb,L(M).
+By H¨older’s inequality and using f = f + − f −, it suffices to prove for p ∈ (1, 2] and f ≥ 0.
+Moreover, by first using f + ε replacing f for ε > 0 then letting ε ↓ 0, we may and do assume
+that inf f > 0.
+By (2.8) we have ˆPt−sf ∈ D(ˆL) ∩ Cb,L(M) for s ∈ [0, t). Then by the chain rule we obtain
+d
+ds
+ˆPs( ˆPt−sf)pds = ˆPs ˆL( ˆPt−sf)p − ˆPs
+�
+p( ˆPt−sf)p−1ˆL ˆPt−sf
+�
+= p(p − 1) ˆPs
+�
+( ˆPt−sf)p−2|∇ ˆPt−sf|2�
+,
+s ∈ [0, t).
+So, for p ∈ (1, 2], we have
+I := ˆPtf p − ( ˆPtf)p =
+� t
+0
+d
+ds
+ˆPs( ˆPt−sf)pds
+= p(p − 1)
+� t
+0
+ˆPs
+�
+( ˆPt−sf)p−2|∇ ˆPt−sf|2�
+ds.
+(3.20)
+By the H¨older and Jensen inequalities, we obtain
+� ˆPs|∇ ˆPt−sf|p� 2
+p ≤
+� ˆPs{(Pt−sf)p−2|∇ ˆPt−sf|2}
+�� ˆPs( ˆPt−sf)p� 2−p
+p
+≤
+� ˆPs{(Pt−sf)p−2|∇ ˆPt−sf|2}
+�� ˆPtf p� 2−p
+p .
+Combining this with (3.20) and (2.8) where we may assume that k(p) is decreasing in p due to
+Jensen’s inequality, we find increasing C : (1, ∞) → (0, ∞) such that
+I ≥ p(p − 1)
+� t
+0
+� ˆPs|∇ ˆPt−sf|p� 2
+p( ˆPtf p)
+p−2
+p
+14
+
+≥ C(p)
+� t
+0
+|∇ ˆPtf|2( ˆPtf p)
+p−2
+p ds = C(p)t|∇ ˆPtf|2( ˆPtf p)
+p−2
+p .
+This implies (3.19) for some decreasing c : (1, ∞) → (0, ∞).
+Next, we intend to prove that for some constant c > 0,
+(3.21)
+∥∇Ptf∥∞ ≤ c∥f∥∞t− 1
+2,
+t ∈ (0, 1], f ∈ Bb(M).
+For any x ̸= y ∈ M, let
+ht(x, y) :=
+sup
+∥g∥∞≤1
+|Ptg(x) − Ptg(y)|
+ρ(x, y)
+,
+t ≥ 0.
+By (2.3) and (3.19), we find a constant c1 > 0 such that
+ht(x, y) ≤ c1t− 1
+2 +
+� t
+0
+c1(t − s)− 1
+2hs(x, y)ds,
+t ∈ (0, 1].
+By the generalized Gronwall inequality, see [41], this implies (3.21).
+Moreover, by (3.19), the Lp-contraction of Pt and ˆPt, and the Duhamel’s formula
+Ptf = ˆPtf +
+� t
+0
+ˆPs(ZPt−sf)ds,
+we obtain
+∥∇Ptf∥p ≤ ∥∇ ˆPtf∥p +
+� t
+0
+∥∇ ˆPs(ZPt−sf)∥pds
+≤ c(p)t− 1
+2∥f∥p +
+� t
+0
+c(p)∥Z||∞s− 1
+2∥∇Pt−sf∥pds,
+t > 0.
+When f ∈ Bb(M), by (3.21) and the generalized Gronwall inequality, this imply (3.10) for
+t ∈ (0, 1].
+Finally, by (3.18) for p = q such that
+∥Pt − µ∥p ≤ ke−λt.
+Combining this with the semigroup property and (3.10) for t ∈ (0, 1], for any t > 1 we have
+∥∇Ptf∥p = ∥∇P1(Pt−1f − 1)∥p ≤ c(p)∥Pt−1(f − 1)∥p ≤ c(p)ke−λ(t−1)∥f∥p.
+So, (3.10) also holds for t > 1.
+(c) Let P ∗
+t be the L2(µ)-adjoint operator of Pt. By (2.2), P ∗
+t is the diffusion semigroup
+generated by L∗ := ˆL − Z, and satisfies
+(3.22)
+P ∗
+t g = ˆPtg −
+� t
+0
+ˆPs(ZP ∗
+t−sg)ds,
+t > 0,
+g ∈ L2(µ).
+15
+
+Let g ∈ D( ˆE ) with ∥g∥
+2p
+2p−1 ≤ 1. We have
+∥∇P ∗
+t g∥2
+2p
+2p−1 ≤ ∥∇P ∗
+t g∥2
+2 = ˆE (P ∗
+t g, P ∗
+t g) ≤ ˆE (g, g) < ∞,
+t ≥ 0,
+so that (2.2), (3.10) and (3.22) yield that for some constant c1 > 0,
+∥∇P ∗
+t g∥
+2p
+2p−1 ≤ c1t− 1
+2 + c1
+� t
+0
+s− 1
+2∥∇P ∗
+t−sg∥
+2p
+2p−1ds < ∞,
+t ∈ (0, 1].
+By the generalized Gronwall inequality, see [41], we find a constant c2 > 0 such that
+sup
+g∈D( ˆ
+E ),∥g∥
+2p
+2p−1
+≤1
+∥∇P ∗
+t g∥
+2p
+2p−1 ≤ c2t− 1
+2,
+t ∈ (0, 1].
+Combining this with the semigroup property and (3.18), we find a constant c2 > 0 such that
+sup
+g∈D( ˆ
+E ),∥g∥
+2p
+2p−1
+≤1
+∥∇P ∗
+t g∥
+2p
+2p−1 ≤ c3t− 1
+2e−λt,
+t > 0.
+Therefore, by (2.2), for any f ∈ D( ˆE ) ∩ L2p(µ) we have
+∥Pt(Zf)∥2p =
+sup
+g∈D( ˆ
+E ),∥g∥
+2p
+2p−1
+≤1
+��µ((P ∗
+t g)(Zf))
+��
+=
+sup
+g∈D( ˆE ),∥g∥
+2p
+2p−1
+≤1
+��µ(f(ZP ∗
+t g))
+�� ≤ c3∥Z∥∞t− 1
+2e−λt∥f∥2p,
+t > 0.
+Therefore, (3.11) holds for some constants c(p), λ > 0.
+(d) Noting that
+(−ˆL)−(1+ d(p−1)
+4p
+−κ) =
+1
+Γ(1 + d(p−1)
+4p
+− κ)
+� ∞
+0
+s
+d(p−1)
+4p
+−κ ˆPsds,
+by (2.6) and (3.10), we find constants c1, c2, c3 > 0 such that
+∥∇ˆL−1f∥2p = ∥∇(−ˆL)−(1+ d(p−1)
+4p
+−κ)(−ˆL)
+d(p−1)
+4p
+−κf∥2p
+≤
+1
+Γ(1 + d(p−1)
+4p
+− κ)
+� ∞
+0
+(1 ∧ s)
+d(p−1)
+4p
+−κ∥∇ ˆPs/2{ ˆPs/2(−ˆL)
+d(p−1)
+4p
+−κf}∥2pds
+≤ c1
+� ∞
+0
+(1 ∧ s)
+d(p−1)
+4p
+−κ− 1
+2e−λs/2∥ ˆPs/2(−ˆL)
+d(p−1)
+4p
+−κf∥2pds
+≤ c2
+� ∞
+0
+(1 ∧ s)−(κ+ 1
+2 )e−λs∥(−ˆL)
+d(p−1)
+4p
+−κf∥2ds ≤ c3∥(−ˆL)
+d(p−1)
+4p
+−κf∥2,
+where the last step is due to 1
+2 + κ < 1. Thus, (3.12) holds.
+16
+
+Next, we present some consequence of (A1) and (A2).
+Lemma 3.2. We have the following assertions.
+(1) If (2.6) holds, then (M, ρ) is bounded, i.e.
+(3.23)
+D := sup
+x,y∈M
+ρ(x, y) < ∞.
+(2) If (2.9) holds, then for any p, q ∈ [1, ∞), there exists a constant c > 0 such that
+(3.24)
+�
+Eµ[W2p(µB
+t , µB
+t,r)2q]
+� 1
+q ≤ cr,
+r ∈ (0, 1].
+(3) If (2.8) and (2.21) hold, then there exist constants κ0, κ1 > 0 such that
+|∇Ptef|2 ≤ (Ptef)Pt(|∇f|2ef) + κ1tθ∥∇f∥2
+∞(Ptef)2,
+for t ∈ [0, 1], f ∈ Cb,L(M) with t∥∇f∥2
+∞ ≤ κ0.
+(3.25)
+Proof. (1) According to [30, Theorem 3.3.15(2)], (2.6) implies the super Poincar´e inequality
+µ(f 2) ≤ r ˆE (f, f) + (1 + r− d
+2 )µ(|f|)2,
+r > 0, f ∈ D( ˆE ),
+which further implies (3.23) due to [30, Theorem 3.3.20].
+(2) By Jensen’s inequality, we only need to prove (3.24) for q ≥ p ≥ 1. Recall that δXB
+s ˆPr is
+the distribution of ˆXr with initial value XB
+s , we have
+π := 1
+t
+� t
+0
+�
+δXB
+s × (δXB
+s ˆPr)
+�
+ds ∈ C (µB
+t , µB
+t,r),
+so that
+W2p(µB
+t , µB
+t,r)2p ≤ 1
+t
+� t
+0
+Ex[ρ(x, ˆXr)2p]
+��
+x=XB
+s ds.
+Noting that Eµ =
+�
+M Exµ(dx) and ˆXt is stationary with initial distribution µ, by combining
+this with Jensen’s inequality and (2.9), we obtain
+Eµ[W2p(µB
+t , µB
+t,r)2q] ≤ Eµ
+�1
+t
+� t
+0
+Ex[ρ(x, ˆXr)2q]
+��
+x=XB
+s ds
+�
+= 1
+t
+� t
+0
+Eµ[ρ( ˆX0, ˆXr)2q]ds ≤ c(2q)rq,
+t > 0, r ∈ (0, 1].
+So, (3.24) holds.
+(3) Let f ∈ Cb,L(M). By (3.21), we have (Pt−sef)2m ∈ D(L) ∩ Cb,L(M) for s ∈ [0, t), so that
+the chain rule implies
+d
+dsPs(Pt−sef)2m = PsL(Pt−sef)2m − Ps
+�
+2m(Pt−sef)2m−1LPt−sef�
+17
+
+= 2m(2m − 1)(Pt−sef)2m−2|∇Pt−sef|2,
+s ∈ [0, t).
+By combining this with (2.8) for p = 2 and Jensen’s inequality, we find a constant c1 > 0 such
+that
+Pte2mf − (Ptef)2m =
+� t
+0
+d
+dsPs(Pt−sef)2mds
+=
+� t
+0
+Ps
+�
+2m(2m − 1)(Pt−sef)2m−2|∇Pt−sef|2�
+ds
+≤ c1∥∇f∥2
+∞
+� t
+0
+Ps
+�
+(Pt−sef)2m−2Pt−se2f�
+ds ≤ c1t∥∇f∥2
+∞Pte2mf.
+Taking κ0 =
+1
+2c2 such that t∥∇f∥2
+∞ ≤ κ0 implies c1t∥∇f∥2
+∞ ≤ 1
+2, we derive
+(3.26)
+Pte2mf ≤ 2(Ptef)2m.
+Combining this with (2.21) we obtain (3.25) for some constant κ1 > 0.
+Noting that (2.10) and (3.3) imply
+(3.27)
+∥(−ˆL)β(f B
+t,r − 1)∥2
+2 = 1
+t
+∞
+�
+i=1
+λ2β
+i e−2λirψB
+i (t)2,
+r, t > 0, β ∈ R,
+to bound Eν[Wp(µB
+t,r, µ)2q] from above using (3.7) and (3.12), we estimate Eν[|ψB
+i |2q] as follows.
+Lemma 3.3. Assume (2.6). Then the following assertions hold.
+(1) For any q ∈ [1, ∞) there exists a constant c(q) > 0 such that
+(3.28)
+sup
+t>0 Eν[|ψB
+i (t)|2q] ≤ c(q)∥h∥∞λ
+d(q−1)
+2
+i
+B(λi)−q,
+i ∈ N, ν = hµ.
+(2) Let B ∈ Bα for some α ∈ (0, 1]. For any q ∈ [1, ∞) and k ∈ (
+d
+2αi(q), ∞] ∩ [1, ∞], there
+exists a constant c(q, k) > 0 such that
+(3.29)
+Eν[|ψB
+i (t)|2q] ≤ c(q, k)∥h∥k(1 ∧ t)−
+d
+2αk λ
+d(q−1)
+2
+−qα
+i
+,
+i ∈ N, ν = hµ, t > 0.
+Moreover, if i(q) >
+d
+2α, then there exists a constant c(q) > 0 such that
+(3.30)
+sup
+ν∈P
+Eν[|ψB
+i (t)|2q] ≤ c(q)(1 ∧ t)− d
+2α λ
+d(q−1)
+2
+−qα
+i
+,
+i ∈ N, t > 0.
+Proof. (1) We first prove the following estimate for some constants k1, k2 > 0:
+(3.31)
+∥P B
+t φi − e−B(λi)tφi∥2q ≤ k1∥Z∥∞λ
+d(q−1)
+4q
+−1
+i
+e−k2t,
+t > 0, i ∈ N, q ∈ [1, ∞].
+18
+
+By (2.10) and (3.18), we find constants c1, c2 > 0 such that
+(3.32)
+∥φi∥2q = inf
+s>0 ∥ ˆPsφi∥2qe−λis ≤ c1 inf
+s∈(0,1] s− d(2q−2)
+8q
+e−λis ≤ c2λ
+d(q−1)
+4q
+i
+,
+i ∈ N, q ∈ [1, ∞].
+By (2.3), (3.8) and (2.10), we obtain
+(3.33)
+P B
+t φi = E
+�
+e−λiSB
+t φi +
+� SB
+t
+0
+e−λi(SB
+t −s)Ps(Zφi) ds
+�
+,
+t > 0.
+Combining this with (2.5), (3.11) and (3.32), we derive
+∥P B
+t φi − e−B(λi)tφi∥2q ≤ c(q)∥Z∥∞∥φi∥2qλ
+d(q−1)
+4q
+−1
+i
+E
+� SB
+t
+0
+e−λi(SB
+t −s)(1 ∧ s)− 1
+2e−λsds
+≤ c2c(q)∥Z∥∞λ
+d(q−1)
+4q
+i
+E
+� SB
+t
+0
+e−λi(SB
+t −s)(1 ∧ s)− 1
+2e−λsds.
+(3.34)
+Noting that λi ≥ λ1 ≥ λ implies
+−λi(SB
+t − s) − λs ≤ −λi
+2 (SB
+t − s) − λ
+2(SB
+t − s) − λs = −λi
+2 (SB
+t − s) − λ
+2SB
+t − λ
+2s,
+by the FKG inequality, we find a constant c3 > 0 such that
+� SB
+t
+0
+e−λi(SB
+t −s)(1 ∧ s)− 1
+2e−λsds ≤ e−λSB
+t /2
+� SB
+t
+0
+e−λi(SB
+t −s)/2(1 ∧ s)− 1
+2e−λs/2ds
+≤ e−λSB
+t /2
+� � SB
+t
+0
+e−λi(SB
+t −s)/2ds
+� 1
+SB
+t
+� SB
+t
+0
+(1 ∧ s)− 1
+2e−λsds
+≤ c3
+λi
+e−λSB
+t /2(1 ∧ SB
+t )− 1
+2.
+(3.35)
+Moreover, by (2.5), the increasing property of B and
+� ∞
+0 e− 1
+2 B(λ)dλ < ∞ in the definition of B,
+we find constants c4, c5, c6 > 0 such that
+E
+�
+(1 ∧ SB
+t )− 1
+2e−λSB
+t /2�
+≤ E
+��
+1 + c2
+� ∞
+0
+u− 1
+2e−uSB
+t du
+�
+e−λSB
+t /2
+�
+= e−B(λ/2)t + c4
+� ∞
+0
+u− 1
+2e−B(u+λ/2)tdu
+≤ e−B(λ/2)t + c4
+� ∞
+0
+u− 1
+2e− 1
+2 B(u)− 1
+2B(λ/2)tdu ≤ c5e−c6t.
+(3.36)
+Combining this with (3.34) and (3.35), we obtain (3.31).
+Next, we prove (3.28) for q ∈ N. By [37, (2.14)] for f = φi, we find a constant k0 > 0 such
+that
+(3.37)
+Eν�
+|ψB
+i (t)|2q�
+≤ k0
+�1
+t
+� t
+0
+ds1
+� s1
+0
+�
+Eν[|φiP B
+s1−sφi|q](XB
+s )
+� 1
+q ds
+�q
+.
+19
+
+By ν = hµ and the Markov property, we obtain
+Eν�
+|φiP B
+s1−sφi|q(XB
+s )
+�
+= µ
+�
+hP B
+s |φiP B
+s1−sφi|q�
+≤ ∥h∥k
+��P B
+s |φiP B
+s1−sφi|q��
+k
+k−1
+≤ ∥h∥k∥P B
+s ∥1→
+k
+k−1∥φiP B
+s1−sφi∥q
+q ≤ ∥h∥k∥P B
+s ∥1→
+k
+k−1∥φi∥q
+2q∥P B
+s1−sφi∥q
+2q,
+k ∈ [1, ∞].
+(3.38)
+Taking k = ∞ and combining with (3.31), (3.32) and (3.37), we find constants k1, k2 > 0 such
+that
+Eν�
+|ψB
+i (t)|2q�
+≤ k1∥h∥∞λ
+d(q−1)
+2
+i
+�1
+t
+� t
+0
+ds1
+� s1
+0
+�
+e−B(λi)(s1−s) + λ−1
+i e−k2(s1−s)�
+ds
+�q
+.
+Noting that B(λi) ≤ cλi for some constant c > 0 and all i ∈ N, this implies (3.28) for q ∈ N.
+Finally, for any q ∈ (1, ∞), let i(q) be the integer part of q. By (3.28) for i(q) and 1 + i(q)
+replacing q which have just been proved, and using H¨older’s inequality, we find a constant
+c(q) > 0 such that
+Eν�
+|ψB
+i (t)|2q�
+≤
+�
+Eν�
+|ψB
+i (t)|2i(q)��i(q)+1−q�
+Eν�
+|ψB
+i (t)|2+2i(q)��q−i(q)
+≤ c(q)∥h∥∞λ
+1
+2{d(i(q)−1)(i(q)+1−q)+di(q)(q−i(q))}
+i
+B(λi)−i(q)(i(q)+1−q)−(1+i(q))(q−i(q))
+= c(q)∥h∥∞λ
+d(q−1)
+2
+i
+B(λi)−q,
+t > 0, i ∈ N.
+(3.39)
+Then (3.28) is proved.
+(2) By the same reason leading to (3.39), we only need to prove (3.29) and (3.30) for q ∈ N
+so that i(q) = q.
+Let k ∈ ( d
+2αq, ∞] ∩ [1, ∞].
+By (3.17), (3.31), (3.37) and (3.38), we find
+constants c1, c2 > 0 such that
+Eν�
+|ψB
+i (t)|2q�
+≤ c1∥h∥kλ
+d(q−1)
+2
+i
+�1
+t
+� t
+0
+ds1
+� s1
+0
+(1 ∧ s)−
+d
+2kqα�
+e−c2λα
+i (s1−s) + λ−1
+i e−c2(s1−s)�
+ds
+�q
+.
+(3.40)
+By the FKG inequality, we find a constant c3 > 0 such that
+� s1
+0
+(1 ∧ s)−
+d
+2kqα �
+e−c2λα
+i (s1−s) + λ−1
+i e−c2(s1−s)�
+ds
+≤
+� 1
+s1
+� s1
+0
+(1 ∧ s)−
+d
+2kqα ds
+� � s1
+0
+�
+e−c2λα
+i (s1−s) + λ−1
+i e−c2(s1−s)�
+ds
+≤ c3(1 ∧ s1)−
+d
+2kqαλ−α
+i
+,
+t > 0, i ∈ N.
+(3.41)
+This together with (3.40) yields
+Eν�
+|ψB
+i (t)|2q�
+≤ c∥h∥k(1 ∧ t)−
+d
+2αk λ
+d(q−1)
+2
+−qα
+i
+,
+t > 0, i ∈ N
+for some constant c > 0. Therefore, (3.29) holds for q ∈ N.
+20
+
+It remains to prove (3.30) for
+d
+2α < q ∈ N. By (3.9), (3.31), (3.32) and (3.17), we find
+constants c1, c2 > 0 such that
+sup
+ν∈P
+�
+Eν�
+|φiP B
+s1−sφi|q(XB
+s )
+�� 1
+q = sup
+ν∈P
+�
+ν
+�
+P B
+s |φiP B
+s1−sφi|q�� 1
+q
+≤ ∥P B
+s ∥
+1
+q
+1→∞∥φiP B
+s1−sφi∥q ≤ c1(1 ∧ s)−
+d
+2qα ∥φi∥2q∥P B
+s1−sφi∥2q
+≤ c1(1 ∧ s)−
+d
+2qα λ
+d(q−1)
+2q
+i
+�
+e−c2λα
+i (s1−s) + λ−1
+i e−c2(s1−s)�
+.
+Combining this with (3.37) and (3.41) for k = 1, we get (3.30) for md,α ≤ q ∈ N.
+We are now ready to estimate Eν[∥(−ˆL)β(f B
+t,r − 1)∥2q
+2 ] for β ∈ R and q ∈ [1, ∞). More
+generally, to apply (2.2) we estimate Eν[∥(a0 − ˆL)
+1
+2(−ˆL)β− 1
+2(f B
+t,r − 1)∥2q
+2 ].
+Lemma 3.4. Assume (A1), (A2). Let a0 ∈ [0, ∞), q ∈ [1, ∞), β ∈ R, and B ∈ Bα for some
+α ∈ [0, 1].
+(1) For any k ∈ (
+d
+2αi(q), ∞] ∩ [1, ∞] where k = ∞ if α = 0, for any R ∈ [1, ∞), there exists a
+constant c > 0 such that
+sup
+t≥1,ν∈Pk,R
+tqEν���(a0 − ˆL)
+1
+2(−ˆL)β− 1
+2(f B
+t,r − 1)
+��2q
+2
+�
+≤ c
+�
+r−(2β+ d′
+2 + d(q−1)
+2q
+−α)+ + 1{α=2β+ d′
+2 + d(q−1)
+2q
+} log(1 + r−1)
+�q,
+r ∈ (0, 1].
+(3.42)
+(2) If i(q) >
+d
+2α, then there exists a constant c > 0 such that
+sup
+t≥1,ν∈P
+tqEν���(a0 − ˆL)
+1
+2(−ˆL)β− 1
+2(f B
+t,r − 1)
+��2q
+2
+�
+≤ c
+�
+r−(2β+ d′
+2 + d(q−1)
+2q
+−α)+ + 1{α=2β+ d′
+2 + d(q−1)
+2q
+} log(1 + r−1)
+�q,
+r ∈ (0, 1].
+(3.43)
+Proof. By (3.17), (3.27), H¨older’s inequality and (3.29), we find a constant c1 > 0 such that
+I :=
+sup
+t≥1,ν∈Pk,R
+tqEν���(a0 − ˆL)
+1
+2(−ˆL)β− 1
+2(f B
+t,r − 1)
+��2q
+2
+�
+=
+sup
+t≥1,ν∈Pk,R
+Eν
+�����
+∞
+�
+i=1
+�λi + a0
+λi
+�
+λ2β
+i e−2λirψB
+i (t)2
+����
+q�
+≤
+�λ1 + a0
+λ1
+�q
+sup
+t≥1,ν∈Pk,R
+� ∞
+�
+i=1
+λθ
+i e−2λir�q−1
+∞
+�
+i=1
+λ2qβ−θ(q−1)
+i
+e−2λirEν[|ψB
+i (t)|2q]
+≤ c1
+�
+∞
+�
+i=1
+λθ
+ie−2λir�q−1
+∞
+�
+i=1
+λ
+2qβ−θ(q−1)+ d(q−1)
+2
+−qα
+i
+e−2λir,
+θ ∈ R.
+Taking
+(3.44)
+θ := 2β + d(q − 1)
+2q
+− α,
+21
+
+so that θ = 2qβ − θ(q − 1) + d(q−1)
+2
+− qα, and noting that
+λθ+
+i e−λir ≤ sup
+s>0
+sθ+e−sr ≤ cr−θ+,
+r ∈ (0, 1]
+holds for some constant c > 0 depending on θ+, we find a constant c2 > 0 such that
+I ≤ c1
+�
+∞
+�
+i=1
+λθ
+i e−2λir�q
+≤ c2r−qθ+�
+∞
+�
+i=1
+λ−θ−
+i
+e−λir�q
+.
+On the other hand, by (2.7) and the integral transform t = rs
+2
+d′ , we find constants c3, c4, c5 > 0
+such that
+∞
+�
+i=1
+λ−θ−
+i
+e−λir ≤ c3
+� ∞
+1
+s− 2θ−
+d′ e−c3rs
+2
+d′ ds
+= c3d′
+2
+� ∞
+r
+rθ−− d′
+2 t
+d′
+2 −θ−−1e−c3tdt
+≤ c5
+�
+r−( d′
+2 −θ−)+ + 1{ d′
+2 =θ−} log(1 + r−1)
+�
+,
+r ∈ (0, 1].
+(3.45)
+Thus, we find a constant c6 > 0 such that
+I ≤ c6r−qθ+�
+r−( d′
+2 −θ−)+ + 1{ d′
+2 =θ−} log(1 + r−1)
+�q
+= c6
+�
+r−( d′
+2 +θ)+ + 1{ d′
+2 +θ=0} log(1 + r−1)
+�q,
+r ∈ (0, 1].
+This together with (3.44) implies (3.42).
+When i(q) >
+d
+2α, (3.43) can be proved in the same way by using (3.30) replacing (3.29).
+The next lemmas shows that the limit of E[ΞB
+r (t)] as t → ∞ is
+(3.46)
+ηB
+Z,r :=
+∞
+�
+i=1
+2e−2λir
+λi
+VB(φi),
+r > 0.
+Lemma 3.5. Assume (A1), and let R ∈ [1, ∞).
+(1) There exists a constant c > 0 such that
+(3.47)
+ηB
+Z,r ≤
+∞
+�
+i=1
+ce−2rλi
+λiB(λi),
+r > 0.
+Consequently, ηB
+Z < ∞ provided �∞
+i=1
+1
+λiB(λi) < ∞.
+22
+
+(2) There exists a constant c > 0 such that
+(3.48)
+sup
+ν∈P∞,R
+��Eν[ΞB
+r (t)] − ηB
+Z,r
+�� ≤ c
+t
+∞
+�
+i=1
+e−2λir
+λiB(λi),
+t, r > 0.
+Consequently, when �∞
+i=1
+1
+λiB(λi) < ∞, there exists a constant c′ > 0 such that
+(3.49)
+sup
+ν∈P2,R
+Eν[ΞB(t)] ≤ ηB
+Z + c′
+t < ∞,
+t > 0.
+(3) Let B ∈ Bα for some α ∈ [0, 1]. For any k >
+d
+2α, there exists a constant c > 0 such that
+(3.50)
+sup
+t≥1,ν∈Pk,R
+Eν[ΞB
+r (t)] ≤ c2
+�
+r−( d′
+2 −1−α)+ + 1{d′=2(1+α)} log(1 + r−1)
+�
+,
+r ∈ (0, 1].
+Proof. (1) By (3.31) for q = 1, and noting that B(λi) ≤ c0λi for some constant c0 > 0, we find
+constants c1, c2, c3 > 0 such that
+VB(φi) :=
+� ∞
+0
+µ(φiPtφi)dt ≤ c1
+� ∞
+0
+(e−B(λi)t + λ−1
+i e−c2t)dt ≤
+c3
+B(λi),
+i ≥ 1.
+This together with (3.46) implies (3.47). By the dominated convergence theorem with r → 0,
+the claimed consequence follows from (3.47).
+(2) By (3.3) and the Markov property, we obtain
+Eν[ψB
+i (t)2] = 2
+t
+� t
+0
+ds1
+� s1
+0
+Eν[φi(XB
+s1)φi(XB
+s )]ds
+= 2
+t
+� t
+0
+ds1
+� s1
+0
+ν
+�
+P B
+s {φiP B
+s1−sφi}
+�
+ds.
+(3.51)
+Since ν ∈ P∞,R implies ν = hµ with ∥h − 1∥∞ ≤ R + 1, by (2.3), (3.40) and (3.9), we find
+constants c1, c2 > 0 such that
+sup
+ν∈P∞,R
+��ν
+�
+P B
+s {φiP B
+s1−sφi}
+�
+− µ(φiP B
+s1−sφ)
+�� =
+sup
+ν∈P∞,R
+��µ
+�
+{P B
+s )∗h − 1}φiP B
+s1−sφ)
+��
+≤ c1e−c2s∥φiP B
+s1−sφi∥1 ≤ c1e−c2s∥P B
+s1−sφi∥2 ≤ c1e−c2s�
+e−B(λi)(s1−s) + λ−1
+i e−c2(s1−s)�
+.
+Combining this with (3.51) and that B(λi) ≤ c0λi for some constant c0 > 0, we find a constant
+c3 > 0 such that
+sup
+ν∈P∞,R
+����tEν[ΞB
+r (t)] −
+∞
+�
+i=1
+2e−2λir
+λit
+� t
+0
+ds1
+� s1
+0
+µ(φiP B
+s1−sφi)ds
+����
+≤
+∞
+�
+i=1
+2e−2λir
+λit
+� t
+0
+ds1
+� s1
+0
+c1e−c2s�
+e−B(λi)(s1−s) + λ−1
+i e−c2(s1−s)�
+ds
+≤ c3
+t
+∞
+�
+i=1
+e−2λir
+λiB(λi),
+t, r > 0.
+(3.52)
+23
+
+Similarly, by (3.31) for q = 1, ∥φi∥2 = 1 and B(λi) ≤ c0λi, we find constants c4, c5, c6 > 0 such
+that
+����
+� s1
+0
+µ
+�
+φiP B
+s1−sφi
+�
+ds − VB(φi)
+���� ≤
+� ∞
+s1
+��µ
+�
+φiP B
+s φi
+��� ≤
+� ∞
+s1
+∥P B
+s φi∥2ds
+≤ c4
+� ∞
+s1
+�
+e−B(λi)s + λ−1
+i e−c5s�
+ds ≤ c6e−c5s1
+B(λi) ,
+s1 > 0, i ∈ N.
+This together with (3.52) implies (3.48) for some constant c > 0.
+(3) Let k >
+d
+2α. By (3.29) for q = 1, and (3.45) for θ− = 1 + α, we find constants c1, c2 > 0
+such that
+sup
+t≥1,ν∈Pk,R
+Eν[ΞB
+r (t)] =
+sup
+t≥1,ν∈Pk,R
+∞
+�
+i=1
+e−2λir
+λi
+Eν[ψB
+i (t)2]
+≤ c1
+∞
+�
+i=1
+e−2λir
+λ1+α
+i
+≤ c2
+�
+r−( d′
+2 −1−α)+ + 1{d′=2(1+α)} log(1 + r−1)
+�
+,
+r ∈ (0, 1].
+Then the proof is finished.
+Finally, to get rid of the term M (f B
+t,r) from (3.5), we present one more lemma.
+Lemma 3.6. Assume (A1) and (A2). Let B ∈ Bα for some α ∈ [0, 1] such that d′ < 2(1 + α).
+Then the following assertions hold.
+(1) There exists a constant c > 0 and σ ∈ (0, 1) such that
+(3.53)
+Eµ[µ(|f B
+t,r − 1|2)] ≤ ct−1r−σ,
+t ≥ 1, r ∈ (0, 1].
+(2) There exists a constant γ > 1 such that
+(3.54)
+lim
+t→∞ Eµ�
+µ
+�
+|M (ft,t−γ)−1 − 1|q��
+= 0,
+q ∈ [1, ∞).
+(3) We have qα > 1 and
+(3.55)
+sup
+t,r>0 tqEµ�
+µ
+�
+|∇ˆL−1(f B
+t,r − 1)|2q��
+< ∞,
+q ∈ [1, qα).
+Proof. (1) By (3.3), (3.28) and (3.17), we find constants c1, c2 > 0 such that
+tEµ[µ(|f B
+t,r − 1|2)] ≤ c1
+∞
+�
+i=1
+e−2λirEµ[ψB
+i (t)2] ≤ c2
+∞
+�
+i=1
+e−2λirλ−α
+i
+,
+t, r > 0.
+Since d′ < 2(1 + α) implies d′
+2 − α < 1, combining this with (3.45) we derive (3.53) for any
+σ ∈ ( d′
+2 − α, 1).
+24
+
+(2) Let σ ∈ (0, 1) be in (3.53) and take θ ∈ (0, q−1(1 − σ)). According to [37, Lemma 3.2],
+for any η ∈ (0, 1) and δ(η) := |(1 − η)− 1
+2 −
+2
+η+2|, we have
+Eµ�
+|M (ft,r(y))−1 − 1|q�
+≤ δ(η) + (1 + θ−1r− θ
+2)qEµ[1{|fB
+t,r(y)−1|>η}],
+t ≥ 1, r ∈ (0, 1], y ∈ M.
+Next, by (3.53) and Chebyshev’s inequality, we obtain
+�
+M
+Eµ[1{|fB
+t,r(y)−1|>η}]µ(dy) ≤ η−2Eµ[µ(|f B
+t,r − 1|2)] ≤ cη−2t−1r−σ.
+Putting these two estimates together, we find a function C : (0, 1) → (0, ∞) such that
+Eµ�
+µ
+�
+|M (ft,r)−1 − 1|q��
+≤ δ(η) + C(η)t−1r−(σ+ θ
+2 ),
+t ≥ 1, r, η ∈ (0, 1).
+Noting that θ ∈ (0, q−1(1 − σ)) implies θ′ := σ + θ
+2 ∈ (0, 1), by taking r = t−γ for γ ∈ (1, 1
+θ′),
+and letting first t → ∞ then η → 0, we derive (3.54).
+(3) By (2.12), d′ < 2(1 + α) implies qα > 1. For any q ∈ [1, qα), we have
+d(q − 1)
+2q
+− 1 < α < −d′
+2 − d(q − 1)
+2q
+.
+So, there exists κ ∈ (0, 1
+2) such that β := d(q−1)
+4q
+− κ satisfies
+(3.56)
+2β < α − d′
+2 − d(q − 1)
+2q
+.
+By (3.12), (3.3), (3.17) and (3.42), we find constants c1, c2 > 0 such that
+tqEµ�
+µ
+�
+|∇ˆL−1(f B
+t,r − 1)|2q��
+≤ c1tqEµ�
+∥(−ˆL)
+d(q−1)
+4q
+−κ(f B
+t,r − 1)∥2q
+2
+�
+≤ c2r−(2β+ d′
+2 + d(q−1)
+2q
+−α)+ = c2,
+where the last step follows from (3.56). Then (3.55) holds.
+3.2
+Proof of Theorem 2.1
+We first consider the stationary case where the initial distribution is the invariant measure µ,
+then extend to more general setting by using an approximation argument. To this end, we need
+the following further modification of the empirical measure:
+(3.57)
+d˜µB
+t,r = ˜f B
+t,rdµ,
+˜f B
+t,r := (1 − r)f B
+t,r + r,
+r ∈ (0, 1], t > 0.
+By (3.2), we have
+(3.58)
+W2(µB
+t,r, ˜µB
+t,r)2 ≤ 4rΞB
+r (t),
+t > 0, r ∈ (0, 1].
+25
+
+Proposition 3.7. Assume (A1) and (A2). Let B ∈ Bα for some α ∈ [0, 1] such that d′ <
+2(1 + α). Then
+(3.59)
+lim
+t→∞ Eµ���{tW2(µB
+t , µ)2 − ΞB(t)}+��q�
+= 0,
+q ∈ [1, qα).
+Proof. Let t ≥ 1 and take rt = t−γ for γ > 1 in (3.54). By (3.6) and (3.57), we have
+tµ(|∇ˆL−1( ˜ft,rt − 1)|2) = (1 − rt)2ΞB
+rt(t),
+so that (3.5), (3.55) and (3.54) yield
+lim
+t→∞ Eµ���{tW2(˜µB
+t,rt, µ)2 − (1 − rt)2ΞB
+rt(t)}+��q�
+= lim
+t→∞ Eµ���{tW2(˜µB
+t,rt, µ)2 − tµ(|∇ˆL−1( ˜ft,rt − 1)|2)}+��q�
+≤ lim
+t→∞ tqEµ��
+µ
+�
+|∇ˆL−1( ˜f B
+t,rt − 1)|2|M ( ˜f B
+t,rt)−1 − 1|
+��q�
+≤ lim
+t→∞ tqEµ�
+µ
+�
+|∇ˆL−1( ˜ft,rt − 1)|2q|M ( ˜f B
+t,rt)−1 − 1|q��
+≤ lim
+t→∞
+�
+tq′Eµ�
+µ
+�
+|∇ˆL−1( ˜ft,rt − 1)|2q′��� q
+q′ �
+Eµ�
+µ
+�
+|M ( ˜f B
+t,rt)−1 − 1|
+qq′
+q′−q ��� q′−q
+q′
+= 0,
+q′ ∈ (q, qα).
+Noting that (2.11) and (3.6) imply
+ΞB(t) ≥ ΞB
+rt(t) ≥ (1 − rt)2ΞB
+rt(t),
+we derive
+(3.60)
+lim
+t→∞ Eµ���{tW2(˜µB
+t,rt, µ)2 − ΞB(t)}+��q�
+= 0,
+q ∈ [1, qα).
+On the other hand, noting that q ∈ [1,
+d
+(d+d′−2−2α)+ ) implies 1 + α − d′
+2 − d(q−1)
+2q
+< 0, by (3.6),
+(3.7) and (3.42), we obtain
+sup
+r>0,t≥1
+4−qtqEµ[W2(µB
+t,r, µ)2q] ≤
+sup
+r>0,t≥1
+tqEµ�
+∥(−ˆL)− 1
+2(f B
+t,r − 1)∥2q
+2
+�
+=
+sup
+r>0,t≥1
+Eµ�
+ΞB
+r (t)q�
+< ∞,
+1 ≤ q <
+d
+(d + d′ − 2 − 2α)+.
+(3.61)
+Since q ∈ [1, qα) implies q ∈ [1,
+d
+(d+d′−2−2α)+ ), by combining this with (3.24), (3.58), the triangle
+inequality and rt = t−γ with γ > 1, we find a constant c > 0 such that
+lim
+t→∞ tqEµ[W2(µB
+t , ˜µB
+t,rt)2q]
+≤ 2q lim
+t→∞ tqEµ[W2(µB
+t,rt, ˜µB
+t,rt)2q + W2(µB
+t,rt, µB
+t )2q]
+≤ c lim
+t→∞(rtt)q�
+1 + Eµ[ΞB
+rt(t)q]
+�
+= 0.
+(3.62)
+This together with (3.60) and (3.61) implies (3.59).
+26
+
+Next, we consider arbitrary initial distribution ν ∈ P. Let
+νε := ν ˆPε,
+ε ∈ (0, 1).
+By (2.6), there exists a constant c > 0 such that
+(3.63)
+νε ≤ cε− d
+2αµ,
+Eνε ≤ cε− d
+2αEµ,
+ε ∈ (0, 1).
+Let
+(3.64)
+µB,ε
+t
+:= 1
+t
+� t+ε
+ε
+δXB
+s ds,
+t, ε > 0.
+By the Markov property, µB,ε
+t
+is the empirical measure with initial distribution νε, so that for
+any nonnegative measurable function F on P,
+(3.65)
+Eν[F(µB,ε
+t
+)] = Eνε[F(µB
+t )],
+t, ε > 0.
+To estimate W2(µB,ε
+t
+, µ), we take
+(3.66)
+ΞB,ε(t) :=
+∞
+�
+i=1
+1
+λi
+ψB,ε
+i
+(t)2,
+ψB,ε
+i
+(t) := 1
+√
+t
+� t+ε
+ε
+φi(XB
+s )ds.
+We have the following result.
+Proposition 3.8. Assume (A1) and (A2). Let B ∈ Bα for some α ∈ [0, 1].
+(1) If α > 0 such that d + d′ < 2 + 4α, then for any q ∈ [1,
+d
+(d+d′−2−2α)+ ),
+(3.67)
+lim
+ε↓0
+sup
+t≥1,ν∈P
+Eν�
+|tW2(µB
+t , µ)2 − tW2(µB,ε
+t
+, µ)2|2q�
+= 0.
+(2) If α > 0 and d + d′ ≥ 2 + 4α, then for any k > d+d′−2−2α
+2α
+and q ∈ [1,
+d
+(d+d′−2−2α)+ ),
+(3.68)
+lim
+ε↓0
+sup
+t≥1,ν∈Pk,R
+Eν�
+|tW2(µB
+t , µ)2 − tW2(µB,ε
+t
+, µ)2|2q�
+= 0,
+R ∈ [1, ∞).
+(3) For any q ∈ [1, ∞), k = ∞ or k ∈ (
+d
+2αi(q), ∞] ∩ [1, ∞], there exists a constant c > 0 such
+that
+(3.69)
+sup
+ν∈Pk,R
+Eν�
+|ψB,ε
+i
+(t)2 − ψB
+i (t)2|q�
+≤ cRε
+q
+2t− q
+2λ
+d(q−1)
+2
+−qα
+i
+,
+i ∈ N, t ≥ 1, ε ∈ (0, 1).
+Moreover, if i(q) >
+d
+2α, then there exists a constant c > 0 such that
+(3.70)
+sup
+ν∈P
+Eν�
+|ψB,ε
+i
+(t)2 − ψB
+i (t)2|q�
+≤ cε
+q
+2t− q
+2λ
+d(q−1)
+2
+−qα
+i
+,
+i ∈ N, t ≥ 1, ε ∈ (0, 1).
+27
+
+Proof. (1) By d + d′ < 2 + 4α, we have
+d
+2α <
+d
+(d+d′−2−2α)+. So, it suffices to prove for 1 ≤ q ∈
+( d
+2α,
+d
+(d+d′−2−2α)+ ). For t > ε ≥ 0, we have
+π := 1
+t
+� t
+ε
+δ(XB
+s ,XB
+s )ds + 1
+t
+� ε
+0
+δ(XB
+s ,XB
+t+s)ds ∈ C (µB
+t , µB,ε
+t
+).
+So, (3.23) implies
+(3.71)
+Wp(µB
+t , µB,ε
+t
+)p ≤ 1
+t
+� ε
+0
+ρ(XB
+s , XB
+s+t)pds ≤ εDp
+t ,
+t > ε ≥ 0, p ∈ [1, ∞).
+On the other hand, by (3.2), (3.61), (3.63) and (3.65), we find a map
+c :
+�
+1,
+d
+(d + d′ − 2 − 2α)+
+�
+→ (0, ∞)
+such that
+sup
+t≥1,ν∈P
+tqEν[W2(µB,ε
+t
+, µ)2q] =
+sup
+t≥1,r∈(0,1],ν∈P
+tqEνε[W2(µB
+t,r, µ)2q]
+≤ 4q
+sup
+t≥1,r∈(0,1],ν∈P
+Eνε[ΞB
+r (t)q] ≤ c(q)ε− d
+2α ,
+ε ∈ (0, 1), q ∈
+�
+1,
+d
+(d + d′ − 2 − 2α)+
+�
+.
+(3.72)
+Combining this with (3.71), (3.63) and q >
+d
+2α, we find constants c1, c2 > 0 such that
+lim
+ε↓0
+sup
+t≥1,ν∈P
+Eν�
+|tW2(µB
+t , µ)2 − tW2(µB,ε
+t
+, µ)2|2q�
+≤ lim
+ε↓0
+sup
+t≥1,ν∈P
+Eν�
+|tW2(µB
+t , µB,ε
+t
+)2 + 2tW2(µB
+t , µB,ε
+t
+)W2(µB,ε
+t
+, µ)|2q�
+≤ lim
+ε↓0
+sup
+t≥1,ν∈P
+c1
+�
+ε2q + εq sup
+ν∈P
+Eνε[W2(µB
+t , µ)2q]
+�
+≤ lim
+ε↓0 c2εq− d
+2α = 0.
+So, (3.67) holds.
+(2) Let α > 0, d + d′ ≥ 2 + 4α and k > d+d′−2−2α
+2α
+. It suffices to prove (3.68) for 1 ≤ q ∈
+( d
+2αk,
+d
+(d+d′−2−2α)+ ). By the same reason leading to (3.63), we find a constant c > 0 such that
+sup
+ν∈Pk,R
+Eνε ≤ cε−
+d
+2αk Eµ,
+ε ∈ (0, 1).
+Hence, as shown above, we derive
+lim
+ε↓0
+sup
+t≥1,ν∈Pk,R
+Eν�
+|tW2(µB
+t , µ)2 − tW2(µB,ε
+t
+, µ)2|2q�
+≤ lim
+ε↓0 c2εq−
+d
+2αk = 0
+since q >
+d
+2αk.
+28
+
+(3) Let ψB
+i and ψB,ε
+i
+be in (2.11) and (3.66). We have
+��ψB
+i (t)2 − ψB,ε
+i
+(t)2��
+= 1
+t
+����
+� ε
+0
+�
+φi(XB
+t+s) − φi(XB
+s )
+�
+ds
+����
+����
+� t
+0
+�
+φi(XB
+s+ε) + φi(XB
+s )
+�
+ds
+����
+≤
+√ε
+√
+t
+�
+|ψB,t
+i
+(ε) + |ψB
+i (ε)|
+��
+|ψB,ε
+i
+(t) + |ψB
+i (t)|
+�
+.
+Since (3.65) implies {νt : ν ∈ P, t ≥ 1} ⊂ P∞,R for some R > 0, (3.28) and (3.17) yield
+sup
+ε>0,t≥1,ν∈P
+Eν[ψB,t
+i
+(ε)2q] ≤ c1λ
+d(q−1)
+2
+−qα
+i
+Eµ
+for some constant c1 > 0. Moreover, by (3.28) for k = ∞, (3.29) for α > 0 and k ∈ (
+d
+2αi(q), ∞] ∩
+[1, ∞], and the fact that ν ∈ Pk,R implies νε ∈ Pk,R for ε > 0, we find a constant c2 > 0 such
+that
+(3.73)
+sup
+ε>0,t≥1,ν∈Pk,R
+Eν[ψB,ε
+i
+(t)2q + ψB
+i (ε)2q] ≤ c2λ
+d(q−1)
+2
+−qα
+i
+.
+Combining these estimate we derive (3.69).
+Finally, when i(q) >
+d
+2α, (3.70) can be proved in the same way by using (3.30) in place of
+(3.29).
+We are now ready to prove Theorem 2.1.
+Proof of Theorem 2.1. (1) It suffices to prove for q ∈ ( d
+2α, qα). By α > α(d, d′), we have
+d
+2α <
+2(1 + α) − d′
+(d + d′ − 2 − 2α)+.
+So, either i(q) >
+d
+2α or i(q) <
+2+2α−d′
+(d+d′−2−2α)+ . Below we consider these two situations respectively.
+(1a) Let i(q) >
+d
+2α. By (3.65), (3.63) and (3.60), we obtain
+lim
+t→∞ sup
+ν∈P
+E���
+�
+tW2(µB,ε
+t
+, µ)2 − ΞB,ε(t)
+�+���
+q�
+= lim
+t→∞ sup
+ν∈P
+Eνε����
+�
+tW2(µB
+t , µ)2 − ΞB(t)
+�+���
+q�
+≤ lim
+t→∞ cε− d
+2α Eµ����
+�
+tW2(µB
+t , µ)2 − ΞB(t)
+�+���
+q�
+= 0,
+ε ∈ (0, 1).
+(3.74)
+Next, by (2.11) and (3.66),
+|ΞB(t) − ΞB,ε(t)| ≤
+∞
+�
+i=1
+1
+λi
+|ψB,ε
+i
+(t)2 − ψB
+i (t)2|.
+29
+
+Combining this with (3.70), when i(q) >
+d
+2α we find a constant k1 > 0 such that
+sup
+t≥1,ν∈P
+t
+q
+2Eν�
+|ΞB(t) − ΞB,ε(t)|q�
+≤
+� ∞
+�
+i=1
+λ−θ
+i
+�q−1
+∞
+�
+i=1
+λθ(q−1)−q
+i
+sup
+t≥1,ν∈P
+t
+q
+2Eν�
+|ψB,ε
+i
+(t)2 − ψB
+i (t)2|q�
+≤ k1ε
+q
+2
+�
+∞
+�
+i=1
+λ−θ
+i
+�q−1
+∞
+�
+i=1
+λ
+θ(q−1)+ d(q−1)
+2
+−q−αq
+i
+,
+θ ∈ R, ε ∈ (0, 1).
+Taking
+(3.75)
+θ = 1 + α − d(q − 1)
+2q
+such that −θ = θ(q − 1) + d(q−1)
+2
+− q − αq, we arrived at
+(3.76)
+sup
+t≥1,ε∈(0,1),ν∈P
+ε− q
+2t
+q
+2Eν�
+|ΞB(t) − ΞB,ε(t)|q�
+≤ k1
+�
+∞
+�
+i=1
+λ−θ
+i
+�q
+.
+Noting that (2.12), (3.75) and q < qα imply 2θ
+d′ > 1, by (2.7) we find a constant k2 > 0 such
+that
+∞
+�
+i=1
+λ−θ
+i
+≤ k2
+∞
+�
+i=1
+i− 2θ
+d′ < ∞.
+Thus,
+(3.77)
+sup
+t≥1,ε∈(0,1),ν∈P
+ε− q
+2t
+q
+2Eν�
+|ΞB(t) − ΞB,ε(t)|q�
+< ∞.
+Consequently,
+lim
+t→∞
+sup
+ε∈(0,1),ν∈P
+Eν�
+|ΞB(t) − ΞB,ε(t)|q�
+= 0.
+Combining this with (3.67) and (3.74), we derive (2.13).
+(1b) Let i(q) <
+2+2α−d′
+(d+d′−2−2α)+ . Since q >
+d
+2α implies i(q) + 1 >
+d
+2α, by (3.30) and H¨older’s
+inequality, we find a constant c1 > 0 such that
+sup
+t≥1,ν∈P
+Eν�
+|ψB
+i (t)|2q�
+≤
+sup
+t≥1,ν∈P
+�
+Eν�
+|ψB
+i (t)|2{i(q)+1}��
+q
+i(q)+1
+≤ c1λ
+dqi(q)
+2(i(q)+1) −qα
+i
+,
+i ∈ N.
+By the calculations leading to (3.76), we derive the same estimate for
+θ := 1 + α −
+di(q)
+2(i(q) + 1).
+We have θ > d′
+2 due to i(q) <
+2+2α−d′
+(d+d′−2−2α)+. Hence, (3.77) holds, which together with (3.67) and
+(3.74) imply (2.13).
+30
+
+(2) Let α ∈ (0, 1], q ∈ [1, qα) and k ∈ (
+d
+2αi(q), ∞] ∩ [1, ∞]. By using (3.69) in place of (3.70),
+the proof of (3.77) implies
+sup
+t≥1,ε∈(0,1),ν∈Pk,R
+ε− q
+2t
+q
+2Eν�
+|ΞB(t) − ΞB,ε(t)|q�
+< ∞,
+R ∈ [1, ∞),
+so that
+lim
+t→∞
+sup
+ε∈(0,1),ν∈Pk,R
+Eν�
+|ΞB(t) − ΞB,ε(t)|q�
+= 0,
+R ∈ [1, ∞).
+This together with (3.67) and (3.74) implies (2.14).
+When k = ∞, (2.14) follows from (3.59) and that Eν ≤ ∥h∥∞Eµ for ν = hµ.
+3.3
+Proof of Theorem 2.2
+(1) Let d′ < 2(1 + α). By Lemma 3.5, we have ηB
+Z < ∞ and (3.49) holds. Combining this with
+(3.59) and Eν ≤ ∥h∥∞Eµ for ν = hµ, we obtain
+(3.78)
+lim sup
+t→∞
+sup
+ν∈P∞,R
+tEν�
+W2(µB
+t , µ)2�
+≤ lim sup
+t→∞
+sup
+ν∈P∞,R
+Eν�
+ΞB(t)
+�
+≤ ηB
+Z ,
+R ∈ [1, ∞).
+On the other hand, by (3.63), for any ε ∈ (0, 1) there exists R ∈ [1, ∞) such that νε ∈ P∞,R
+holds for all ν ∈ P. So, (3.78) together with (3.65) implies
+(3.79)
+lim sup
+t→∞
+sup
+ν∈P
+Eν�
+W2(µB,ε
+t
+, µ)2�
+= lim sup
+t→∞
+sup
+ν∈P
+Eνε�
+W2(µB
+t , µ)2�
+≤ ηB
+Z ,
+ε ∈ (0, 1).
+Combining this with (3.71) for p = 2 and using the triangle inequality, we find a constant c > 0
+such that
+lim sup
+t→∞
+sup
+ν∈P
+Eν�
+W2(µB
+t , µ)2�
+≤ lim sup
+t→∞
+sup
+ν∈P
+Eν�
+(1 + ε
+1
+2)W2(µB,ε
+t
+, µ)2 + (1 + ε− 1
+2)W2(µB,ε
+t
+, µB
+t )2�
+≤ (1 + ε
+1
+2)ηB
+Z + c(ε + ε
+1
+2),
+ε ∈ (0, 1).
+Letting ε ↓ 0 we obtain (2.15).
+(2) Let d′ ≥ 2(1 + α). By (3.6), (3.7) and (3.42), we find a constant c1 > 0 such that
+Eµ[W2(µB
+t,r, µ)2] ≤ 4
+t Eµ[µ(|∇ˆL−1(f B
+t,r − 1)|2)] = 4
+t Eµ[µ(|(−ˆL)− 1
+2(f B
+t,r − 1)|2)]
+≤ c1
+t
+�
+r1+α− d′
+2 + 1{d′=2+2α} log(1 + r−1)
+�
+,
+t ≥ 1, r ∈ (0, 1].
+(3.80)
+Combining this with (3.24) and the triangle inequality, we find a constant c2 > 0 such that
+Eµ[W2(µB
+t , µ)2] ≤ 2Eµ[W2(µB
+t,r, µ)2] + 2Eµ[W2(µB
+t,r, µB
+t )2]
+≤ c1
+t
+�
+r1+α− d′
+2 + 1{d′=2+2α} log(1 + r−1)
+�
++ c2r,
+t ≥ 1, r ∈ (0, 1].
+31
+
+Taking r = t−
+2
+d′−2α when d′ > 2 + 2α, and r = t−1 when d′ = 2 + 2α, we find a constant c3 > 0
+such that
+Eµ[W2(µB
+t , µ)2] ≤
+�
+c3t−1 log(1 + t),
+if d′ = 2(1 + α),
+c3t−
+2
+d′−2α,
+if d′ > 2(1 + α).
+By combining this with (3.63) and (3.65), we find a constant c4 > 0 such that
+sup
+ν∈P
+Eν[W2(µB,1
+t
+, µ)2] = sup
+ν∈P
+Eν1[W2(µB
+t , µ)2] ≤ c4Eµ[W2(µB
+t , µ)2]
+≤ c3c4
+�
+1{d′=2(1+α)}t−1 log(1 + t) + t−
+2
+d′−2α
+�
+,
+t ≥ 1.
+(3.81)
+Noting that the triangle inequality implies
+Eν[W2(µB
+t , µ)2] ≤ 2W2(µB,1
+t
+, µ)2 + 2W2(µB,1
+t
+, µB
+t )2,
+we deduce (2.16) from (3.71) and (3.81).
+3.4
+Proof of Theorem 2.3
+By approximating µB
+t using µB,1
+t
+as in (3.71) and (3.81), we only need to prove for ν = µ.
+By (3.7), (3.12) and (3.42), for any κ ∈ (0, 1
+2) we find constants c1, c2 > 0 such that for any
+t ≥ 1 and r ∈ (0, 1],
+�
+Eµ[W2p(µB
+t,r, µ)2q]
+� 1
+q ≤ c1
+�
+Eµ[∥(−ˆL)
+d(p−1)
+4p
+−κ(f B
+t,r − 1)∥2q
+2
+� 1
+q
+≤ c2
+t
+�
+r−
+�
+d(p−1)
+2p
+−2κ+ d′
+2 + d(q−1)
+2q
+−α
+�+
++ 1�
+d(p−1)
+2p
+−2κ+ d′
+2 + d(q−1)
+2q
+−α=0
+� log(1 + r−1)
+�
+.
+(3.82)
+Below we prove assertions (1)-(3) in Theorem 2.3 respectively.
+(1) Let γα,p,q < 0. We may take κ ∈ (0, 1
+2) such that
+d(p − 1)
+2p
+− 2κ + d′
+2 + d(q − 1)
+2q
+− α < 0,
+so that (3.82) implies
+�
+Eµ[W2p(µB
+t,r, µ)2q]
+� 1
+q ≤ c2
+t ,
+r ∈ (0, 1], t ≥ 1.
+By Fatou’s lemma for r → 0, we obtain (2.19) for ν = µ.
+(2) Let γα,p,q ≥ 0. For any γ > γα,p,q, we find κ ∈ (0, 1
+2) such that
+d(p − 1)
+2p
+− 2κ + d′
+2 + d(q − 1)
+2q
+− α ≤ γ,
+so that (3.82), (3.24) and the triangle inequality imply
+�
+Eµ[W2p(µB
+t , µ)2q]
+� 1
+q ≤ c1
+�
+t−1r−γ + r
+�
+,
+r ∈ (0, 1], t ≥ 1
+32
+
+for some constant c1 > 0. Taking r = t−
+1
+1+γ we obtain (2.20) for ν = µ.
+(3) Let (2.17) hold and γα,p,q ≥ 0. By (3.7) we find constants c1, c2 > 0 such that
+W2p(µB
+t,r, µ)2p ≤ c1∥∇(−ˆL)−1(f B
+t,r − 1)∥2p
+2p ≤ c2∥(a0 − ˆL)
+1
+2(−ˆL)−1(f B
+t,r − 1)∥2p
+2p.
+(3.83)
+On the other hand, by the Sobolev embedding theorem, (2.6) implies that for any constants
+k2 ≥ k1 > −∞ and q1 ≥ q2 ≥ 1 with
+(3.84)
+1
+q1
+= 1
+q2
++ k1 − k2
+d
+,
+there exists a constant C > 0 such that
+∥(−ˆL)
+k1
+2 f∥q1 ≤ C∥(−ˆL)
+k2
+2 f∥q2,
+µ(f) = 0.
+Taking
+k1 = −2,
+k2 = d(p − 1)
+2p
+− 2,
+q1 = 2p,
+q2 = 2
+such that (3.84) holds, we find a constant c2 > 0 such that
+∥(a0 − ˆL)
+1
+2(−ˆL)−1(f B
+t,r − 1)∥2p ≤ c2∥(a0 − ˆL)
+1
+2(−ˆL)
+d(p−1)
+4p
+−1f∥2.
+Combining this with (3.42) and (3.83), we find a constant c3 > 0 such that
+�
+Eµ[W2p(µB
+t,r, µ)2q]
+� 1
+q ≤ c3
+t
+�
+r−γα,p,q + 1{γα,p,q=0} log(1 + r−1)
+�
+,
+t ≥ 1, r ∈ (0, 1].
+By this together with (3.24) and the triangle inequality, we find a constant c4 > 0 such that
+�
+Eµ[W2p(µB
+t , µ)2q]
+� 1
+q ≤ c4
+�
+t−1r−γα,p,q + t−11{γα,p,q=0} log(1 + r−1) + r
+�
+,
+t ≥ 1, r ∈ (0, 1].
+Taking r = t
+−
+1
+1+γα,p,q we obtain (2.20).
+4
+Proofs of Theorems 2.4 and 2.5
+We will follow the line of [38] to estimate the lower bound of W2(µB
+t , µ) by using an idea of [1].
+For any f ∈ D(L) with ∥f∥∞ + ∥∇f∥∞ + ∥Lf∥∞ < ∞, let
+T σ
+t f := −σ log ˆP σ
+2 eσ−1f,
+σ > 0, t ∈ [0, 1].
+Lemma 4.1. Assume that (M, ρ) is a geodesic space. If (2.8) and (2.21) hold, then there exists
+constants k1, k2 > 0 such that ∥∇f∥2
+∞ ≤ k1σ implies
+T σ
+1 f(y) − f(x) ≤ 1
+2ρ(x, y)2 + σ∥ˆLf∥2
+∞ + k2σθ∥∇f∥2
+∞,
+µ(T σ
+1 f − f) ≤ 1
+2µ(|∇f|2) + k2σ−1∥∇f∥4
+∞.
+(4.1)
+33
+
+Proof. Without loss of generality, we assume θ ≤ 1
+4 such that in (3.25) we have θ ∧ 1
+2 = θ. Let
+k1 = 2κ0, then ∥∇f∥2
+∞ ≤ k1σ implies
+tσ
+2 ∥∇σ−1f∥2
+∞ ≤ 1
+2σ−1∥∇f∥2
+∞ ≤ κ0,
+t ∈ [0, 1],
+so that (3.26) holds. Combining (3.26) for m = 1 with (2.8) for σ−1f replacing f, we obtain
+|∇T σ
+t f|2 =
+σ2|∇ ˆP tσ
+2 eσ−1f|2
+( ˆP tσ
+2 eσ−1f)2
+≤
+(1 + k(2))σ2 ˆP tσ
+2 (|∇σ−1f|2e2σ−1f)
+( ˆP tσ
+2 eσ−1f)2
+≤
+(1 + k(2))∥∇f∥2
+∞ ˆP tσ
+2 (e2σ−1f)
+( ˆP tσ
+2 eσ−1f)2
+≤ 2(1 + k(2))∥∇f∥2
+∞ =: c1∥∇f∥2
+∞.
+(4.2)
+Next, by (3.25) with θ ≤ 1
+4, we find a constant c2 > 0 such that
+ˆLT σ
+t f = −
+σˆL ˆP tσ
+2 e−σ−1f
+ˆP tσ
+2 eσ−1f
++
+σ|∇ ˆP tσ
+2 eσ−1f|2
+( ˆP tσ
+2 eσ−1f)2
+=
+ˆP tσ
+2 {eσ−1f ˆLf}
+ˆP tσ
+2 eσ−1f
++
+σ|∇ ˆP tσ
+2 eσ−1f|2 − σ( ˆP tσ
+2 eσ−1f) ˆP tσ
+2 (|∇σ−1f|2eσ−1f)
+( ˆP tσ
+2 eσ−1f)2
+≤ ∥ˆLf∥∞ + c2σθ−1∥∇f∥2
+∞,
+σ, t ∈ (0, 1], ∥∇f∥2
+∞ ≤ k1σ.
+(4.3)
+Moreover, for any two points x, y ∈ M, let γ : [0, 1] → M be the minimal geodesic from x to y
+with
+|˙γt| := lim sup
+s→t
+ρ(γt, γs)
+|t − s| = ρ(x, y),
+a.e. t ∈ [0, 1].
+So,
+(4.4)
+lim sup
+s→t
+|f(γt) − f(γs)|
+|t − s|
+≤ |∇f(γt)|ρ(x, y),
+t ∈ [0, 1].
+By the Kolmogorov equation and the chain rule, we have
+(4.5)
+∂tT σ
+t f = −
+σ2 ˆL ˆP tσ
+2 eσ−1f
+2 ˆP tσ
+2 eeσ−1f
+= σ
+2
+ˆLT σ
+t f − 1
+2|∇T σ
+t f|2.
+This together with (4.3) and (4.4) yields
+d
+dtT σ
+t f(γt) = (∂tT σ
+t f)(γt) + d
+dtT σ
+s f(γt)
+���
+s=t
+≤ σ
+2
+ˆLT σ
+t f(γt) − 1
+2|∇T σ
+t f(γt)|2 + |∇T σ
+t f(γt)|ρ(x, y)
+≤ 1
+2
+�
+ρ(x, y)2 + σ
+2 ∥ˆLf∥∞ + c2σθ∥∇f∥2
+∞
+�
+,
+t ∈ [0, 1], ∥∇f∥2
+∞ ≤ k1σ.
+34
+
+Integrating over t ∈ [0, 1] and noting that T σ
+0 f = f, we derive the first inequality in (4.1).
+On the other hand, by T σ
+0 f = f and µ(ˆLT σ
+t f) = 0, we obtain
+µ(f − T σ
+1 f) = −
+�
+M
+dµ
+� 1
+0
+(∂tT σ
+t f)dt
+=
+� t
+0
+dt
+�
+M
+�1
+2|∇T σ
+t f|2 − σ
+2
+ˆLT σ
+t f
+�
+dµ = 1
+2
+� 1
+0
+µ(|∇T σ
+t f|2)dt.
+(4.6)
+Moreover, by (4.5) and the integration by parts formula, we obtain
+d
+dsµ(|∇T σ
+s f|2) = − d
+ds
+�
+M
+(T σ
+s f)ˆLT σ
+s fdµ
+= −
+�
+M
+(ˆLT σ
+s f)∂sT σ
+s fdµ −
+�
+M
+(T σ
+s f)ˆL(∂sT σ
+s f)dµ
+= −2
+�
+M
+(ˆLT σ
+s f)∂sT σ
+s f dµ = −2
+�
+M
+(ˆLT σ
+s f)
+�σ
+2
+ˆLT σ
+s f − 1
+2|∇T σ
+s f|2�
+dµ
+≤ 1
+4σ∥∇T σ
+s f∥4
+∞,
+s ∈ (0, 1], t ∈ [0, 1].
+This together with (4.2) implies
+µ(|∇T σ
+t f|2) − µ(|∇f|2) ≤ c2
+1
+4σ∥∇f∥4
+∞,
+t ∈ [0, 1], σ ∈ (0, 1], ∥∇f∥2
+∞ ≤ k1σ
+(4.7)
+Substituting this into (4.6), we derive the second estimate in (4.1).
+Proof of Theorem 2.4. Similarly to the proof of Theorem 2.1 using Proposition 3.7 and the
+approximation argument with Proposition 3.8, Theorem 2.4(1)(2) follow from
+(4.8)
+lim
+t→∞ Eµ���{th(0)W2(µB
+t , µ)2 − ΞB(t)}−��q�
+= 0,
+q ∈ [1, qα).
+Moreover, according to the proof of Theorem 2.2(1), Theorem 2.4(3) is implied by Theorem
+2.4(1)(2). So, it remains to verify (4.8). The main idea for the proof of (4.8) goes back to
+[1, 38], but we have to make suitable modifications for the present more general situation. Let
+ˆft,r = (−ˆL)−1(f B
+t,r − 1).
+Firstly, by (3.18), we find a constant c1 > 0 such that
+(4.9)
+∥ ˆft,r∥∞ ≤
+� ∞
+0
+∥ ˆPs(f B
+t,r − 1)∥∞ds ≤ c1∥f B
+t,r − 1∥∞
+� ∞
+0
+e−λ1sds = c1
+λ1
+∥f B
+t,r − 1∥∞.
+By (3.10) and (3.9), we find a constant c2 > 0 such that
+∥∇ ˆPsg∥∞ ≤ c2(s ∧ 1)− 1
+2e−λ1s∥g∥∞,
+s > 0, g ∈ Bb(M).
+Noting that ˆft,r := (−ˆL)−1(f B
+t,r − 1) =
+� ∞
+0
+ˆPs(f B
+t,r − 1)ds, we obtain
+∥∇ ˆft,r∥∞ ≤
+� ∞
+0
+∥∇ ˆPs(f B
+t,r − 1)∥∞ds ≤ c2∥f B
+t,r − 1∥∞
+� ∞
+0
+(1 ∧ s)− 1
+2e−λ1sds.
+35
+
+Combining this with (4.9) and |ˆL ˆft,r| = |f B
+t,r − 1|, we find a constant c > 0 such that
+(4.10)
+∥ˆL ˆft,r∥∞ + ∥ ˆft,r∥∞ + ∥∇ ˆft,r∥∞ ≤ c∥f B
+t,r − 1∥∞,
+t, r > 0.
+Next, let
+C1( ˆft,r, σ) := σ∥ˆL ˆft,r∥2
+∞ + k2σθ∥∇ ˆft,r∥2
+∞,
+C2( ˆft,r, σ) := k2σ−1∥∇ ˆft,r∥4
+∞,
+Bt,r(σ) := {∥f B
+t,r − 1∥2
+∞ ≤ k1c−1σ1+θ},
+σ ∈ (0, 1], t, r > 0.
+(4.11)
+By (4.1), the integration by parts formula, and the Kantorovich dual formula, we obtain
+C1( ˆft,r, σ) + 1
+2W2(µB
+t,r, µ)2 ≥ µB
+t,r( ˆft,r) − µ(T σ
+1 ˆft,r)
+= µ
+�
+(f B
+t,r − 1)(−ˆL)−1(f B
+t,r − 1)
+�
++ µ( ˆft,r) − µ(T σ
+1 ˆft,r)
+≥ µ(|∇ ˆft,r|2) − 1
+2µ(|∇ ˆft,r|2) − C2( ˆft,r, σ) = 1
+2tΞB
+r (t) − C2( ˆft,r, σ),
+σ ∈ (0, 1], t, r > 0.
+This together with (4.10) and (4.11) yields
+(4.12)
+1Bt,r(σ)
+�
+tW2(µB
+t,r, µ)2 − ΞB
+r (t)
+�
+≥ −c3tσ1+2θ,
+σ ∈ (0, 1], t, r > 0
+for some constant c3 > 0. On the other hand, by (2.6) we find a constant c4 > 0 such that
+∥f B
+t,r − 1∥2
+∞ = ∥ ˆP r
+2(f B
+t, r
+2 − 1)∥2
+∞ ≤ c4r− d
+2∥f B
+t, r
+2 − 1∥2
+2,
+r ∈ (0, 1],
+so that by (3.53) with 1 replacing σ ∈ (0, 1), there exists a constant c5 > 0 such that
+Eµ[∥f B
+t,r − 1∥2
+∞] ≤ c4r− d
+2 t−1
+∞
+�
+i=1
+e−λirEµ[ψB
+i (t)2]
+≤ c5r− d
+2 −1t−1,
+t ≥ 1, r ∈ (0, 1].
+Hence, we find a constant c6 > 0 such that
+(4.13)
+Pµ(Bt,r(σ)c) = Pµ(∥f B
+t,r − 1∥2
+∞ > k1c−1σ1+θ) ≤ c6r− d
+2 −1t−1σ−(1+θ),
+t ≥ 1, σ, r ∈ (0, 1].
+Taking σ = σt := t−
+1
+1+3θ/2 in (4.12) and (4.13), we arrive at
+lim sup
+t→∞
+Pµ�
+{tW2(µB
+t,r, µ)2 − ΞB
+r (t)}− ≥ ε
+�
+≤ lim sup
+t→∞
+Pµ(Bt,r(σt)c) = 0,
+ε > 0, r ∈ (0, 1].
+On the other hand, by B ∈ Bα, (3.28) and �∞
+i=1 λ−1−α
+i
+< ∞ due to (2.7) and d′ < 2(1 + α),
+we obtain
+lim
+r→0 sup
+t≥1
+Eµ�
+|ΞB
+r (t) − ΞB(t)|
+�
+≤ lim
+r→0
+∞
+�
+i=1
+1 − e−λir
+λi
+sup
+t≥1
+Eµ[ψB
+i (t)2]
+36
+
+≤ c(1) lim
+r→0
+∞
+�
+i=1
+�
+1 − e−λir�
+λ−1−α
+i
+= 0.
+Moreover, (2.22) and (3.61) imply
+lim
+r→0 sup
+t≥1
+Eµ[{th(0)W2(µB
+t , µ)2 − tW2(µB
+t,r, µ)2}−] = 0.
+Therefore, for any ε > 0,
+lim sup
+t→∞
+Pµ�
+{th(0)W2(µB
+t , µ)2 − ΞB(t)}− ≥ 3ε
+�
+≤ lim
+r→0 lim sup
+t→∞
+Pµ�
+{tW2(µB
+t,r, µ)2 − ΞB
+r (t)}− ≥ ε
+�
++ lim
+r→0 sup
+t≥1
+�
+Pµ�
+|ΞB(t) − ΞB
+r (t)| ≥ ε
+�
++ Pµ�
+{th(0)W2(µB
+t , µ)2 − tW2(µB
+t,r, µ)2}− ≥ ε
+��
+= 0.
+Combining this with (3.61) and applying the dominated convergence theorem, we prove (4.8).
+To prove Theorem 2.5, we need the following lemma.
+Lemma 4.2. Assume (2.6).
+(1) Let V = VB for B(λ) = λ. We have
+(4.14)
+V(φi) = λ−1
+i
+− λ−2
+i V(Zφi),
+i ≥ 1.
+(2) If (3.10) holds, then there exists a constant c > 0 such that
+(4.15)
+VB(φi) ≥
+1
+B(λi) − cλ
+− 3
+2
+i
+,
+i ≥ 1.
+Proof. (1) By ˆLφi = −λiφi and using the Kolmogorov equation, we obtain
+(4.16)
+Psφi = − 1
+λi
+Ps ˆLφi = − 1
+λi
+d
+dsPsφi + 1
+λi
+Ps(Zφi).
+This together with (2.2) and µ(φ2
+i ) = 1 implies
+� t
+0
+µ(φiPsφi)ds = − 1
+λi
+� t
+0
+� d
+dsµ(φiPsφi) − µ
+�
+φi(ZPsφi)
+��
+ds
+= 1
+λi
+�
+1 − µ(φiPtφi)
+�
+− 1
+λi
+� t
+0
+µ
+�
+{Zφi}Psφi
+�
+ds
+= 1
+λi
+�
+1 − µ(φiPtφi)
+�
++ 1
+λ2
+i
+� t
+0
+� d
+dsµ
+�
+{Zφi}Psφi
+�
+− µ
+�
+{Zφi}Ps{Zφi}
+��
+ds
+= 1
+λi
+�
+1 − µ(φiPtφi)
+�
++ 1
+λ2
+i
+µ({Zφi}Ptφi) − 1
+λ2
+i
+� t
+0
+µ
+�
+{Zφi}Ps{Zφi}
+�
+ds.
+(4.17)
+37
+
+By (1.2) and ∥Pt − µ∥2 ≤ e−λ1t, we may let t → ∞ to derive
+V(φi) = 1
+λi
+− 1
+λ2
+i
+V(Zφi),
+i ≥ 1.
+(2) Let (3.10) hold. By (2.10), (2.5) and (3.33) we obtain
+(4.18)
+VB(φi) :=
+� ∞
+0
+µ(φiP B
+t φi)dt =
+1
+B(λi) +
+� ∞
+0
+�
+E
+� SB
+t
+0
+e−λi(SB
+t −s)µ
+�
+φiPs(Zφi)
+�
+ds
+�
+dt.
+By (2.10) and (2.3) for (P ∗
+t , −Z) replacing (Pt, Z), we derive
+P ∗
+s φi = ˆPsφi −
+� s
+0
+P ∗
+r {Z ˆPs−rφi}dr = e−λisφi −
+� s
+0
+e−λi(s−r)P ∗
+r (Zφi)dr.
+This together with by (2.2), (3.11) and
+∥Zφi∥2
+2 ≤ ∥Z∥∞∥∇φi∥2
+2 = ∥Z∥2
+∞λi,
+due to (2.10), implies
+µ(φiPs(Zφi)) = µ((P ∗
+s φi)Zφi) = −
+� s
+0
+e−λi(s−r)µ((Zφi)Pr(Zφi))dr
+≤ c(1)λ
+1
+2
+i
+� s
+0
+e−λi(s−r)r− 1
+2e−λrdr.
+(4.19)
+By λ ≤ λ1 ≤ λi, we have
+−λi(s − r) − λr ≤ −(s − r)λi/2 − λs/2 − λr/2,
+so that by the FGK inequality, we find a constant a1 > 0 such that
+� s
+0
+e−λi(s−r)r− 1
+2e−λrdr ≤ e−λs/2
+� s
+0
+e−λi(s−r)/2r− 1
+2e−λr/2dr
+≤ e−λs/2
+� � s
+0
+e−λi(s−r)/2dr
+�1
+s
+� s
+0
+r− 1
+2e−λr/2dr
+≤ a1λ−1
+i e−λs/2(1 ∧ s)− 1
+2.
+(4.20)
+So, by the same reason leading to (4.20), this and (4.19) imply
+� SB
+t
+0
+e−λi(SB
+t −s)µ(φiPs(Zφi))ds ≤ c(1)a1λ
+− 1
+2
+i
+� SB
+t
+0
+e−λi(SB
+t −s)e−λs/2(1 ∧ s)− 1
+2ds
+≤ a2λ
+− 3
+2
+i
+(1 ∧ SB
+t )− 1
+2e−λSB
+t /4
+for some constant a2 > 0. Combining this with (3.36), we find a constant a3 > 0 such that
+� ∞
+0
+�
+E
+� SB
+t
+0
+e−λi(SB
+t −s)µ(φiPs(Zφi))ds
+�
+dt ≤ a3λ
+− 3
+2
+i
+.
+Therefore, (4.15) follows from (4.18).
+38
+
+Proof of Theorem 2.5. By (2.23), (4.15), B ∈ Bα, d′ = 2(1 + α) and
+5
+2(1+α) ≥ 5
+4 > 1, we find
+constants a1, a2, a3, a4 > 0 such that
+ηB
+Z,r =
+∞
+�
+i=1
+e−2λir VB(φi)
+λi
+≥
+∞
+�
+i=1
+e−2λir� a1
+λ1+α
+i
+− a2
+λ
+5
+2
+i
+�
+≥
+∞
+�
+i=1
+e−2rc2i
+1
+1+α �
+a1c−1−α
+2
+i−1 − a2c2
+1i−
+5
+2(1+α)
+�
+≥ a3 log(1 + r−1) − a4,
+r ∈ (0, 1].
+(4.21)
+Next, by (4.12), we obtain
+Eµ[W2(µB
+t,r, µ)2] ≥ Eµ[1BσW2(µB
+t,r, µ)2] ≥ t−1Eµ[1Bt,r(σ)ΞB
+r (t)] − c3σ1+2θ
+≥ t−1Eµ[ΞB
+r (t)] − t−1Eµ[1BcσΞB
+r (t)] − c3σ1+2θ,
+t ≥ 1, r, σ ∈ (0, 1].
+(4.22)
+By (3.48), (2.23) and d′ = 2(1 + α), we find constants k1, k2 > 0 such that
+Eµ[ΞB
+r (t)] =
+∞
+�
+i=1
+e−2λir
+λi
+Eµ[ψB
+i (t)2] ≥ ηB
+Z,r − k1t−1
+∞
+�
+i=1
+e−2rλiλ−(1+α)
+i
+≥ ηB
+Z,r − k1t−1
+∞
+�
+i=1
+e−2rc2i
+1
+1+α c−1−α
+2
+i−1 ≥ ηB
+Z,r − k2t−1 log(1 + r−1),
+t ≥ 0, r ∈ (0, 1].
+Combining this with (4.21) and (4.22), we derive
+Eµ[W2(µB
+t,r, µ)2] ≥ a3t−1 log(1 + r−1) − a4t−1 − k2t−2 log(1 + r−1)
+− t−1Eµ[1Bt,r(σ)cΞB
+r (t)] − c3σ1+2θ,
+t ≥ 1, r, σ ∈ (0, 1].
+(4.23)
+On the other hand, by (3.42) and (4.13), we find constants k3, k4 > 0 such that
+Eµ[ΞB
+r (t)2] = Ex[∥(−ˆL)− 1
+2(f B
+t,r − 1)∥4
+2] ≤ k3r−k4,
+Pµ(Bt,r(σ)c) ≤ k3r−k4t−1σ−(1+θ) t ≥ 1, r ∈ (0, 1], σ ∈ (0, 1],
+where θ > 0 is a constant. Thus,
+Eµ[1Bt,r(σ)cΞB
+r (t)] ≤
+�
+Pµ(Bt,r(σ)c)Eµ[ΞB
+r (t)2]] ≤ k3r−k4t− 1
+2σ− 1+θ
+2 ,
+t ≥ 1, r, σ ∈ (0, 1].
+Combining this with (4.23), we arrive at
+Eµ[W2(µB
+t,r, µ)2] ≥ a3t−1 log(1 + r−1) − a4t−1 − k2t−2 log(1 + r−1)
+− k3r−k4t− 3
+2σ− 1+θ
+2 − c3σ1+2θ,
+t ≥ 1, r, σ ∈ (0, 1].
+Taking σ = t−
+1
+1+2θ and r = t−
+θ
+2k4(1+2θ) , obtain
+Eµ[W2(µB
+t,r, µ)2] ≥
+a3
+1 + 2θt−1 log t − a4t−1 − k2t−2 log(1 + t) − (k3 + c3)t−1,
+t ≥ 1.
+Therefore, (2.24) holds for some constants c, t0 > 0.
+39
+
+Finally, to prove Theorem 2.6, we present one more lemma.
+Lemma 4.3. Let (E, ρ) be a Polish space. Let Xt be a continuous time Markov process on E
+such that the associated semigroup Pt satisfies
+(4.24)
+∥Pt − µ∥2 ≤ c1e−λ1t,
+t ≥ 0
+for some constants c1, λ1 > 0 and a probability measure µ on E. If there exists φ ∈ Cb,L(E)
+such that µ(φ) = 0 and
+V(φ) :=
+� ∞
+0
+µ(φPtφ)dt > 0,
+then there exist constants c, t0 > 0 such that µt := 1
+t
+� t
+0 δXsds satisfies
+(4.25)
+Eµ[W1(µt, µ)] ≥ ct− 1
+2,
+t ≥ t0.
+If M ⊂ E such that µ(M) = 1 and ∥Pt∥1→2 < ∞ for t > 0, then (4.25) holds for infν∈P Eν
+replacing Eµ, where P is the set of all probability measures on M.
+Proof. By [39, Theorem 2.1(c)], we have
+(4.26)
+lim
+t→∞
+√
+tEµ[|µt(φ)|] =
+�
+2πV(φ)
+�− 1
+2
+� ∞
+−∞
+re−
+r2
+2V(φ) dr.
+So, by the Kantorovich dual formula, there exist constants c, t0 > 0 such that
+√
+tEµ[W1(µt, µ)] ≥
+√
+tEµ[|µt(φ) − µ(φ)|] ≥ c,
+t ≥ t0.
+Next, let M ⊂ E such that µ(M) = 1 and ∥Pt∥1→2 < ∞ for t > 0. Then
+{νP1 : ν ∈ P} ⊂
+�
+ν ∈ P : ν = hµ, ∥h∥2 ≤ ∥P1∥1→2
+�
+,
+so that [39, Theorem 2.1(c)], ˆµt := 1
+t
+� t+1
+1
+δXsds satisfies
+lim
+t→∞ inf
+ν∈P
+√
+tEν[|ˆµt(φ)|] > 0.
+Noting that
+|ˆµt(φ) − µt(φ)| ≤ ∥φ∥∞t−1,
+t > 1,
+we obtain
+lim
+t→∞ inf
+ν∈P
+√
+tEν[|µt(φ)|] > 0.
+By the Kantorovich dual formula and µ(φ) = 0, this implies (4.25) for infν∈P replacing Eµ.
+40
+
+Proof of Theorem 2.6. (1) By (2.6) and (2.8), for any i ≥ 1 we have ∥φi∥∞ < ∞ and there
+exist constants c1, c2 > 0 such that
+∥∇φi∥∞ = eλi∥∇ ˆP1φi∥∞ ≤ c1eλi��(P1|∇φi|2)
+1
+2��
+∞
+≤ c2eλi∥∇φi∥2 = c2eλi�
+λi < ∞.
+So, φi ∈ Cb,L(M), and hence is uniquely extended to φi ∈ Cb,L(E) for E := ¯
+M. On the other
+hand, by (4.15) we have VB(φi) > 0 for large enough i. Moreover, (3.9) implies ∥P B
+t ∥1→2 < ∞
+for t > 0. So, the first assertion follows from Lemma 4.3 with E = ¯
+M.
+(2) Let d′′ > 2(1 + α). By (2.9) for p = 1 we find a constant c1 > 0 such that
+Eµ[ρ( ˆXt, ˆX0)] ≤ c1t
+1
+2,
+t ≥ 0.
+Combining this with (2.3) and (2.8), we find a constant c2 > 0 such that
+Eµ[ρ(Xt, X0)] =
+�
+M
+ˆPtρ(x, ·)(x)µ(dx)
+= Eµ[ρ( ˆXt, ˆX0)] +
+�
+M
+µ(dx)
+� t
+0
+Ps{Z ˆPt−sρ(x, ·)}(x)ds
+≤ c1t− 1
+2 + c2
+� t
+0
+(t − s)− 1
+2ds ≤ c3t
+1
+2,
+t ≥ 0.
+According to the proof of [37, Theorem 1.1(2)], this and (2.26) imply
+(4.27)
+Eµ[W1(µB
+t , µ)] ≥ c4t−
+1
+d′′−2α ,
+t ≥ t1
+for some constants c4, t1 > 0.
+Finally, by (3.9), we find a constant t2 > 0 such that ∥Pt2 − µ∥1→∞ ≤ 1
+2, so that
+νt2 := νP B
+t2 ≥ 1
+2µ,
+ν ∈ P.
+Let µB,t2
+t
+be in (3.64) for ε = t2. Then the Markov property and (4.27) yield
+inf
+ν∈P Eν[W1(µB,t2
+t
+, µ)] = inf
+ν∈P Eνt2[W1(µB
+t , µ)] ≥ 1
+2c4t−
+1
+d′′−2α ,
+t ≥ t1.
+Combining this with (3.71), the triangle inequality and d′′−2α > 2, we find constants c5, c, t0 >
+t1 such that
+inf
+ν∈P Eν[W1(µB
+t , µ)] ≥ 1
+2c4t−
+1
+d′′−2α − c5t−1 ≥ ct−
+1
+d′′−2α ,
+t ≥ t0.
+41
+
+5
+Some concrete models
+In this part, we apply our general results to the (reflecting) subordinated diffusion process on a
+compact manifold, the subordinated conditional diffusion process on a bounded open domain,
+the subordinated Wright-Fisher diffusion process, and subordinated subelliptic diffusion process
+on SU(2). It is possible to consider more general hypoelliptic diffusion processes studied in
+[6, 7] by using the generalized curvature-dimension condition.
+For simplicity, throughout this section, we take
+B ∈ Bα ∩ Bα for some α ∈ (0, 1].
+5.1
+Subordinated (Reflecting) diffusion process
+In this part, we consider the model stated in Introduction, for which all conditions in Theorems
+2.1-2.6 are satisfied for d = d′ = d′′ = n.
+Indeed, (2.6) and (2.23) are well known (see [9, 11]), (2.8) follows from [31, Lemma 2.1],
+and (2.22) with h(r) = κeKr is implied by [38, (3.36), (3.37)], where κ ≥ 1 and K ≥ 0 are
+constants with κ = 1 when ∂M is empty or convex. Moreover, the following lemma confirms
+other conditions.
+Lemma 5.1. (2.21), (A2) and (2.17) hold.
+Proof. (1) Let lt be the local time of ˆXt on ∂M if ∂M exists, and let lt = 0 otherwise. By [31,
+(2.1)] and the proof of [31, Lemma 2.1], there exist constants c1, K, δ > 0 such that
+(5.1)
+|∇ ˆPtf(x)| ≤ Ex[|∇f( ˆXt)|eKt+δlt],
+t ≥ 0, x ∈ M, f ∈ C1
+b (M),
+(5.2)
+sup
+x∈M
+Ex[l2
+t ] ≤ c1t,
+sup
+x∈M
+Ex[eλlt] < ∞,
+λ, t ≥ 0.
+By the Schwarz inequality, (5.1) implies
+|∇ ˆPtef(x)| ≤ ˆPt|∇ef|(x) + Ex�
+|∇ef( ˆXt)|(eKt+δlt − 1)
+�
+≤
+� ˆPtef(x)
+� 1
+2� ˆPt(|∇f|2ef)(x)
+� 1
+2 + ∥∇f∥∞( ˆPte2f)
+1
+2�
+Ex[e2Kt+2δlt − 1]
+� 1
+2.
+On the other hand, by (5.2) we find a constant c2 > 0 such that
+Ex[e2Kt+2δlt − 1] ≤ Ex[(2K + 2δlt)e2Kt+2δlt]
+≤
+�
+Ex[(2K + 2δlt)2]
+� 1
+2�
+Ex[e4Kt+4δlt]
+� 1
+2 ≤ c2t
+1
+2,
+t ∈ [0, 1].
+Therefore, (2.21) holds for θ = 1
+4 and m = 1.
+(2) When ∂M is empty or convex, we have
+⟨∇ρ( ˆX0, ·), N⟩( ˆXt)dlt ≤ 0,
+42
+
+where N is the inward unit normal vector on ∂M.
+On the other hand, by the Laplacian
+comparison theorem, there exits a constant c0 > 0 such that
+ˆLρ( ˆX0, ·)2 ≤ c0.
+So, by Itˆo’s formula, we find a constant c1 > 0 such that
+dρ( ˆX0, ˆXt)2 ≤ c1dt + 2
+√
+2ρ( ˆX0, ˆXt)dBt,
+where Bt is the one-dimensional Brownian motion. Thus, for any p ≥ 1, there exists a constant
+c(p) > 0 such that
+dρ( ˆX0, ˆXt)2p ≤ c(p)ρ( ˆX0, ˆXt)2(p−1)dt + dMt
+holds for some martingale Mt, so that
+E[ρ( ˆX0, ˆXt)2p] ≤ c(p)
+� t
+0
+E[ρ( ˆX0, ˆXs)2(p−1)]ds,
+t ≥ 0.
+Consequently, for p = 1 we get
+Ex[ρ( ˆX0, ˆXt)2] ≤ c(1)t,
+and by inducting in p ∈ N, we derive (2.9) for p ∈ 2N. Therefore, (2.9) holds for all p ≥ 1 due
+to Jensen’s inequality.
+When ∂M is non-convex, as explained in the proof of [32, Proposition 3.2.7], there exists a
+function 1 ≤ φ ∈ C∞
+b (M) such that ∂M is convex under the metric
+⟨·, ·⟩′ := φ−1⟨·, ·⟩,
+and ˆL = φ−2∆′ +Z′, where ∆′ is the Laplacian induced by the new metric and Z′ is a C1
+b vector
+field. Let ρ′ be the Riemannian distance induced by the new metric, we have ρ ≤ ∥φ∥∞ρ′, so
+that the above argument for convex ∂M leads to
+E[ρ( ˆX0, ˆXt)2p] ≤ ∥φ∥2p
+∞E[ρ′( ˆX0, ˆXt)2p] ≤ k(p)tp,
+t ∈ [0, 1].
+(3) To verify (2.17), we follow the line of [3]. As explained in the end of page 12 in [3],
+see also the proof of [3, Theorem 1.5], under (3.10), it remains to verify the volume doubling
+condition and scaled Poincar´e inequalities on balls. More precisely, we only need to find a
+distance ˜ρ and constants c1, c2, c3 > 0 such that
+c1˜ρ ≤ ρ ≤ c2˜ρ,
+and the balls ˜B(x, r) := {y ∈ M : ˜ρ(x, y) ≤ r} for all x ∈ M and r > 0 satisfy
+(5.3)
+µ( ˜B(x, 2r)) ≤ c3µ( ˜B(x, r)),
+(5.4)
+µ(1 ˜B(x,r)f 2) ≤ c3r2µ(1 ˜B(x,r)|∇f|2) + µ(1 ˜B(x,r)f)2,
+f ∈ C1
+b (M).
+Since for the present model we have ˜D := ∥˜ρ∥∞ < ∞ and krd ≤ µ( ˜B(x, r)) ≤ Krd for some
+constants K > k > 0 and all r ∈ [0, ˜D], (5.3) holds true. To verify (5.4), by the conformal
+43
+
+change of metric used in step (2), we may and do assume that ∂M is either empty or convex.
+In this case, there exists a constant r0 > 0 such that B(x, r) := {y ∈ M : ρ(x, y) ≤ r} is convex
+for all x ∈ M and r ∈ (0, r0]. Then we take ˜ρ := ρ ∧ r0, so that ˜B(x, r) is convex for all r > 0,
+since ˜B(x, r) = B(x, r) for r < r0 and ˜B(x, r) = M for r ≥ r0. Thus, (5.4) follows from [29,
+Theorem 1.4].
+According to the above observations, we conclude that all assertions in Theorems 2.1-2.6
+hold for d = d′ = d′′ = n, i.e.
+qα =
+n
+2(n − 1 − α)+,
+α(d, d′) = α(n) := 1
+2
+�
+1 + 2n(n − 1) − 2,
+γα,p,q = n
+2(3 − p−1 − q−1) − 1 − α,
+p, q ∈ [1, ∞).
+(5.5)
+Below we summarize these results, which in particular imply Theorems 1.1 and 1.2 stated in
+Introduction, by B(λ) = λ, (4.14) and
+V(Zφi) := µ((Zφi)(−L)−1(Zφi)) = µ(|∇L−1(Zφi)|2).
+Theorem 5.2. Let qα, α(n), γα,p,q for p, q ∈ [1, ∞) be in (5.5). Then the following assertions
+hold for some constant κ ∈ [1, ∞), where κ = 1 when ∂M is either empty or convex.
+(1) If n < 2(1 + α), qα >
+n
+2α and α > α(n), then
+lim
+t→∞ sup
+ν∈P
+Eν���{tW2(µB
+t , µ)2 − ΞB(t)}+ + {tκW2(µB
+t , µ)2 − ΞB(t)}−��q�
+= 0,
+q ∈ [1, qα).
+(2) If n < 2(1 + α), then for any q ∈ [1, qα) and k ∈ (
+d
+2αi(q), ∞] ∩ [1, ∞],
+lim
+t→∞ sup
+ν∈Pk,R
+Eν����{tW2(µB
+t , µ)2 − ΞB(t)}+ + {tκW2(µB
+t , µ)2 − ΞB(t)}−���
+q�
+= 0,
+R ∈ (0, ∞).
+(3) If n < 2(1 + α), then
+κ−1ηB
+Z ≤ lim inf
+t→∞
+inf
+ν∈P tEν[W2(µB
+t , µ)2] ≤ lim sup
+t→∞
+sup
+ν∈P
+tEν[W2(µB
+t , µ)2] ≤ ηB
+Z .
+(4) If n = 2(1 + α), then there exist constants c, t0 > 1 such that
+c−1t−1 log t ≤ Eν[W2(µB
+t , µ)2] ≤ ct−1 log t,
+t ≥ t0, ν ∈ P.
+(5) If n > 2(1 + α), then there exist constants c, t0 > 1 such that
+c−1t−
+2
+n−2α ≤
+�
+Eν[W1(µB
+t , µ)]
+�2 ≤ Eν[W2(µB
+t , µ)2] ≤ ct−
+2
+n−2α ,
+t ≥ t0.
+(6) If p, q ∈ [1, ∞) with n(3 − p−1 − q−1) < 2(1 + α), then there exist constants c, t0 > 1 such
+that
+c−1t−1 ≤
+�
+Eν[W1(µB
+t , µ)]
+�2 ≤
+�
+Eν[W2p(µB
+t , µ)2q]
+� 1
+q ≤ ct−1,
+t ≥ t0, ν ∈ P.
+44
+
+(7) If p, q ∈ [1, ∞) such that n(3 − p−1 − q−1) ≥ 2(1 + α), then there exists a constant c > 0
+such that for any t ≥ 1,
+sup
+ν∈P
+�
+Eν[W2p(µB
+t , µ)2q]
+� 1
+q ≤
+�
+ct−1 log(1 + t),
+if n(3 − p−1 − q−1) = 2(1 + α),
+ct
+−
+1
+1+γα,p,q ,
+if n(3 − p−1 − q−1) > 2(1 + α).
+5.2
+Subordinated conditional diffusion process
+Let M be a bounded connected C2 open domain in an n-dimensional complete Riemannian
+manifold, and let V ∈ C2
+b (M) be such that µ0(dx) := eV (x)dx is a probability measure on M.
+Consider the Dirichlet eigenproblem for ˆL0 := ∆ + ∇V in M:
+ˆL0hi = −θihi,
+hi|∂M = 0,
+i ≥ 0,
+where {θi}i≥0 are listed in the increasing order counting multiplicities, and {hi}i≥0 are the
+associated unitary eigenfunctions in L2(µ0) with h0 > 0. Let
+ˆL := ˆL0 + 2∇ log h0 = ∆ + ∇(V + 2 log h0),
+µ(dx) := h0(x)2µ0(dx).
+Then the diffusion process ˆXt generated by ˆL is non-explosive in M, whose distribution coincides
+with the conditional distribution of the ˆL0-diffusion process ˆX0
+t under the condition that
+τ := inf{t ≥ 0 : ˆX0
+t ∈ ∂M} = ∞,
+in the sense that for any T > 0 and any F ∈ Cb(C[0, T]; M)),
+E[F( ˆX[0,T])] = lim
+m→∞ E[F( ˆX[0,T]0)|τ > m].
+Let Z be a C1
+b -vector field on M satisfying (2.2).
+It is well known that {λi := θi −θ0}i≥0 are all eigenvalues of −ˆL with unitary eigenfunctions
+{φi := hih−1
+0 }i≥0, and that (2.6), (2.23) and (2.26) hold for
+d = n + 1,
+d′ = d′′ = n,
+see for instance [9, 25].
+Next, by [34, Lemma 4.6], (2.22) holds for h(r) = κeKr for some
+constants κ ≥ 1 and K ≥ 0, where κ = 1 when ∂M is convex. The following lemma confirms
+other conditions in Theorems 2.1-2.6, except (2.17) which is not yet verified.
+Lemma 5.3. For the present model, (2.8) and (A2) hold. When ∂M is convex or n ≤ 3, (2.21)
+is satisfied.
+Proof. As explained in the proof of [34, Lemma 4.6], if ∂M is convex then the Bakry-Emery
+curvature ˆL is bounded from below by a constant −K, so that
+|∇Ptf| ≤ eKtPt|∇f|,
+which implies (2.8) for k(p) = eK as well as (2.21) for θ = 1. So, in the following we only prove
+these conditions for non-convex ∂M, and to verify (A2).
+45
+
+(1) When ∂M is non-convex, let ρ′ be the Riemannian distance induced by ⟨·, ·⟩′ introduced
+in the proof of (5.1). According to the proof of [34, Lemma 4.6], for any x, y ∈ M, there exists
+a coupling ( ˆXt, ˆYt) of the diffusion process generated by ˆL starting from (x, y), such that for
+some constant c1 > 0 we have
+dρ′( ˆXt, ˆYt) ≤ c1ρ′( ˆXt, ˆYt)dt + dMt,
+where Mt is a martingale with d⟨M⟩t ≤ c1ρ′( ˆXt, ˆYt)dt. Thus, for any q ∈ (1, ∞), there exists a
+constant K(q) > 0 such that
+�
+E[ρ′( ˆXt, ˆYt)q]
+� 1
+q ≤ K(q)ρ′(x, y),
+t ∈ [0, 1].
+Therefore, by ρ′ ≤ ρ ≤ ∥φ∥∞ρ′,
+|∇ ˆPtf(x)| := lim sup
+y→x
+| ˆPtf(x) − ˆPtf(y)|
+ρ(x, y)
+≤ lim sup
+y→x
+E
+�|f( ˆXt) − f( ˆYt)|
+ρ′( ˆXt, ˆYt)
+· ρ′( ˆXt, ˆYt)
+ρ(x, y)
+�
+≤ lim sup
+y→x
+�
+E
+�|f( ˆXt) − f( ˆYt)|p
+ρ′( ˆXt, ˆYt)p
+�� 1
+p (E[ρ′( ˆXt, ˆYt)
+p
+p−1])
+p−1
+p
+ρ(x, y)
+≤ K
+�
+p
+p − 1
+�
+∥φ∥∞(Pt|∇f|p)
+1
+p.
+So, (2.8) holds.
+(2) Let ∂M be non-convex and n ≤ 3. Since ∇V ∈ C1
+b (M) and
+−Hesslog h0 = −Hessh0
+h0
++ ∇h0 ⊗ h0
+h2
+0
+≥ −∥∇2h0∥∞
+h0
+,
+there exists a constant c1 > 0 such that the Bakry-Emery curvature of ˆL is bounded below by
+− c1
+2h0, i.e.
+Γ2(g) := 1
+2L|∇g|2 − ⟨∇g, ∇ˆLg⟩ ≥ −c1
+2 h−1
+0 |∇g|2,
+g ∈ C(M).
+So, by (3.18) for d = n + 2, and applying Jensen’s inequality, we find a constant c2 > 0 such
+that
+|∇ ˆPtef|2 − ˆPt|∇ef|2 = −
+� t
+0
+d
+ds
+ˆPs|∇ ˆPt−sef|2ds
+≤ c1
+� t
+0
+ˆPs
+�
+h−1
+0 |∇ˆˆPt−sef|2�
+ds ≤ c1
+� t
+0
+( ˆPsh
+−
+m
+m−1
+0
+)
+m−1
+m ( ˆPs|∇ ˆPt−sef|2m)
+1
+mds
+≤ c2
+� t
+0
+s− (n+2)(m−1)
+2m
+µ(h
+−
+m
+m−1
+0
+)
+m−1
+m ( ˆPs|∇ ˆPt−sef|2m)
+1
+mds
+≤ c2∥∇f∥2
+∞( ˆPte2mf)
+1
+m
+� t
+0
+s− (n+2)(m−1)
+2m
+µ(h
+−
+m
+m−1
+0
+)
+m−1
+m ds.
+Noting that n + 2 ≤ 5 and µ(h−r
+0 ) = µ0(h2−r
+0
+) < ∞ for r < 3, for any m ∈ ( 3
+2, 5
+3), we have
+θ1 := 1 − (n + 2)(m − 1)
+2m
+∈ (0, 1),
+µ
+�
+h
+−
+m
+m−1
+0
+�
+< ∞,
+46
+
+so that for some constant c3 > 0 we have
+(5.6)
+|∇ ˆPtef|2 − ˆPt|∇ef|2 ≤ c3tθ1∥∇f∥2
+∞( ˆPte2m)
+1
+m.
+Similarly, by (2.8) and its consequence
+|∇ ˆPtg| ≤ c
+√
+t( ˆPt|∇g|2)
+1
+2,
+we find a constant c4 > 0 such that
+ˆPt|∇ef|2 − ( ˆPtef) ˆPt(|∇f|2ef) =
+� t
+0
+d
+ds
+ˆPt−s
+�
+( ˆPsef) ˆPs(|∇f|2ef)
+�
+ds
+= −
+� t
+0
+ˆPt−s⟨∇ ˆPsef, ∇ ˆPs(|∇f|2ef)⟩ds
+≤ c4∥∇f∥2
+∞
+� t
+0
+s− 1
+2 ˆPt−s( ˆPse2f)ds = 2c4t
+1
+2∥∇f∥2
+∞ ˆPte2f.
+This together with (5.6) implies (2.21) for θ = θ1 ∧ 1
+2.
+(3) It remains to verify (2.9). By the conformal change of metric as in the end of the proof
+of Lemma 5.1, we only consider the case where ∂M is convex, so that
+⟨N, ∇ρ(x, ·)⟩|∂M ≤ 0,
+x ∈ M,
+where N is the inward unit normal vector field of ∂M. Let ρ∂ be the distance to ∂M. It
+is well known (see for instance [25]) that ∇h0 is inward normal on the boundary and c1 :=
+∥h−1
+0 ρ∂∥∞ < ∞. So,
+(5.7)
+ρ∂ ≤ c1h0,
+⟨∇h0, ∇ρ(x, ·)⟩|∂M ≤ 0,
+x ∈ M.
+Moreover, by the Hessian comparison theorem, there exists a constant c2 > 0 such that
+(5.8)
+Hessρ(x,·)2(v, v) ≤ c2|v|2,
+x ∈ M, v ∈ TM.
+We intend to show that these two estimates imply
+(5.9)
+sup
+x,y∈M
+⟨∇ log h0, ∇ρ(x, ·)2⟩(y) ≤ c
+for some constant c > 0. To see this, for any x, y ∈ M, let z ∈ ∂M such that ρ(y, z) = ρ∂(y).
+Let
+γ : [0, 1] → M,
+γ0 = z,
+γ1 = y,
+|˙γ| = ρ∂(y)
+be the minimal geodesic from z to y. Let vs = ∇h0(γs). We have
+v0 = a0 ˙γ0,
+∥0→s v0 = a0 ˙γs,
+s ∈ [0, 1],
+where a0 := ⟨N, ∇h0⟩(z) and ∥0,s is the parallel displacement along the geodesic γs. Since
+h0 ∈ C2
+b (M), we find a constant c3 > 0 such that
+|v1 − a0 ˙γ1| = |v1− ∥0→1 v0| ≤ c3ρ∂(y).
+47
+
+Combining this with (5.7) and (5.8), and noting that |∇ρ2| ≤ 2∥ρ∥∞ < ∞, we find a constant
+c4 > 0 such that
+⟨∇h0, ∇ρ(x, ·)2⟩(y) = ⟨v1, ∇ρ(x, ·)2(γ1)⟩ ≤ a0⟨˙γ1, ∇ρ(x, ·)2(γ1)⟩ + c3ρ∂(y)
+= a0⟨˙γ0, ∇ρ(x, ·)2(γ0)⟩ + a0
+� 1
+0
+d
+ds⟨˙γs, ∇ρ(x, ·)2(γs)⟩ds + c3ρ∂(y)
+≤ a0
+� 1
+0
+Hessρ(x,·)2(˙γs, ˙γs)ds + c3ρ∂(y) ≤ a0c2ρ∂(y)2 + c3ρ∂(y) ≤ c4h0(y).
+Therefore, (5.9) holds for c = c4.
+By (5.9) and Itˆo’s formula, we obtain
+dρ( ˆX0, ·)2( ˆXt) ≤ cdt + 2
+√
+2ρ( ˆX0, ˆXt)dBt,
+where Bt is the one-dimensional Brownian motion. This implies (A2) as explained in the proof
+of Lemma 5.1(2).
+We now conclude that all assertions in Theorems 2.1-Theorem 2.5 hold, except Theorem
+2.2(4) where the condition (2.17) is to be verified for this model, for d = n + 2, d′ = d′′ = n and
+qα := 1 + 2(1 + α) − n
+4n + 2 − 2α ,
+α(d, d′) = ˜α(n) := 1
+2
+��
+4 + 2n(n + 2) − 2
+�
+,
+γα,p,q := n
+2 + n + 2
+2
+(2 − p−1 − q−1) − α − 1.
+(5.10)
+So, we have the following result.
+Theorem 5.4. Let ˆL := L0 + 2∇ log h0, and L = ˆL + Z for some C1
+b -vector field Z satisfying
+(2.2). The following assertions hold for qα, ˜a(n) and, γα,p,q in (5.10), and a constant κ ≥ 1 with
+κ = 1 when ∂M is convex.
+(1) If n < 2(1 + α), qα >
+n
+2α and α > ˜α(n), then
+lim
+t→∞ sup
+ν∈P
+Eν���{tW2(µB
+t , µ)2 − ΞB(t)}+ + {tκW2(µB
+t , µ)2 − ΞB(t)}−��q�
+= 0,
+q ∈ [1, qα).
+(2) If n < 2(1 + α), then for any q ∈ [1, qα) and k ∈ (
+d
+2αi(q), ∞] ∩ [1, ∞],
+lim
+t→∞ sup
+ν∈Pk,R
+Eν����{tW2(µB
+t , µ)2 − ΞB(t)}+ + {tκW2(µB
+t , µ)2 − ΞB(t)}−���
+q�
+= 0,
+R ∈ (0, ∞).
+(3) If n < 2(1 + α), then
+κ−1ηB
+Z ≤ lim inf
+t→∞
+inf
+ν∈P tEν[W2(µB
+t , µ)2] ≤ lim sup
+t→∞
+sup
+ν∈P
+tEν[W2(µB
+t , µ)2] ≤ ηB
+Z .
+48
+
+(4) Let n = 2(1 + α), i.e. (n, α) ∈ {(3, 1
+2), (4, 1)}. Then there exist constants c, t0 > 0 such
+that
+sup
+ν∈P
+Eν[W2(µB
+t , µ)2] ≤ ct−1 log t,
+t ≥ t0.
+If ∂M is convex or (n, α) = (3, 1
+2), then there exists a constant c′ > 0 such that
+inf
+ν∈P Eν[W2(µB
+t , µ)2] ≥ c′t−1 log t,
+t ≥ t0.
+(5) If n > 2(1 + α), then there exist constants c, t0 > 1 such that
+c−1t−
+2
+n−2α ≤
+�
+Eν[W1(µB
+t , µ)]
+�2 ≤ Eν[W2(µB
+t , µ)2] ≤ ct−
+2
+n−2α ,
+t ≥ t0.
+(6) If p, q ∈ [1, ∞) with γα,p,q < 0, then there exist constants c, t0 > 1 such that
+c−1t−1 ≤
+�
+Eν[W1(µB
+t , µ)]
+�2 ≤
+�
+Eν[W2p(µB
+t , µ)2q]
+� 1
+q ≤ ct−1,
+t ≥ t0, ν ∈ P.
+(7) Let p, q ∈ [1, ∞) with γα,p,q ≥ 0. Then for any γ > γα,p,q, there exists a constant c > 0
+such that
+sup
+ν∈P
+�
+Eν[W2p(µB
+t , µ)2q]
+� 1
+q ≤ ct−
+1
+1+γ ,
+t ≥ 1.
+If (2.17) holds, then there exists a constant c > 0 such that
+sup
+ν∈P
+�
+Eν[W2p(µB
+t , µ)2q]
+� 1
+q ≤ ct
+−
+1
+1+γα,p,q + ct−1 log(1 + t)1{γα,p,q=0}.
+5.3
+Subordinated Wright-Fisher diffusion process
+Let a, b > 1
+4 be two constants, and let
+µ := 1[0,1](x)Γ(2a + 2b2)
+Γ(2a)Γ(2b) x2a−1(1 − x)2b−1dx
+be the beta distribution on M = [0, 1]. The Fisher-Wright diffusion process ˆXt is generated by
+ˆL := 1
+2x(1 − x) d2
+dx + {a − (a + b)x} d
+dx.
+Under the Riemannian metric ⟨∂x, ∂x⟩ = 2{x(1 − x)}−1, we have
+Γ(f, g) = ⟨∇f, ∇g⟩ := 1
+2x(1 − x)f ′(x)g′(x),
+x ∈ M,
+ρ(x, y) =
+√
+2
+� y
+x
+{s(1 − s)}− 1
+2ds,
+0 < x ≤ y < 1.
+Since divµZ = 0 implies Z = 0, we have L = ˆL.
+49
+
+Lemma 5.5. For the present model with L = ˆL, (A1) with d = 4(a ∨ b) and (A2) hold, (2.23)
+holds for d′ = 2, (B) holds with h(r) = eKr for some constant K > 0, and (2.17) holds.
+Proof. Firstly, the condition (2.6) with d = 4(a ∨ b) is implied by [15, Corollary 2.3]. By [27,
+(2.4)], the Bakry-Emery curvature of ˆL is bounded below by a constant −K < 0, so that
+|∇Ptf| ≤ eKtPt|∇f|,
+(2.8) and (B) hold for θ = m = 1, k(p) = eK and h(r) = erK.
+Next, for any p ≥ 1 there exists a constant c1 > 0 such that
+ˆLρ(X0, ·)2p(Xt) = 2pρ(X0, Xt)2(p−1)Lρ(X0, ·)(Xt)2 + p(p − 1)ρ(X0, Xt)2(p−1)
+≤ c1ρ(X0, Xt)2(p−1),
+so that
+(5.11)
+Eµ[ρ(X0, Xt)2p] ≤ c1
+� t
+0
+Eµ[ρ(X0, Xs)2(p−1)]ds.
+In particular, for p = 1 we obtain (2.9), and for general p ∈ N it follows from (5.11) by the
+induction argument.
+Moreover, we have λi = (a + b)i so that (2.23) holds for d′ = 2. Indeed, according to the
+proof of [14, Theorem 1.1], all eigenfunctions are polynomials. The trivial eigenvalue is λ0 = 0
+with φ0 = 1. For any i ∈ N, let
+φi(x) :=
+i
+�
+j=0
+αjxj
+with αi > 0 be the unitary eigenfunction for λi. We have
+−λiφi(x) = ˆLφi(x).
+Since the coefficients of xi in left hand and right hand sides are −λiαi and −i(a + b)αi respec-
+tively, these two constants have to be equal each other, so that λi = i(a + b).
+Finally, as explained in step (3) in the proof of Lemma 5.1, for (2.17) it suffices to verify
+(5.3) and (5.4). Since the curvature is bounded from below as indicated in the beginning of
+the proof, and since a one-dimensional ball is convex, (5.4) follows from [29, Theorem 1.4].
+So, it remains to (5.3). With the transform x → 1 − x, we only need to prove this condition
+for x ∈ [0, 1
+2]. Let x ∈ [0, 1
+2] and B(x, r) := {y ∈ [0, 1] : ρ(x, y) ≤ r}. Take, for instance,
+r0 = 1
+8ρ( 1
+2, 1) such that
+x0 := sup B(1/2, 2r0) ∈ (1/2, 1).
+We have
+c0 :=
+inf
+x∈[0, 1
+2 ] µ(B(x, r0)) > 0,
+so that
+µ(B(x, 2r)) ≤ 1 ≤ c−1
+0 µ(B(x, r)),
+r ≥ r0.
+50
+
+Hence, we only need to consider r ∈ (0, r0). On the other hand, we find constants c1, c2 > 0
+such that
+c1
+��√x − √y
+��ρ(x, y) ≤ c2
+��√x − √y
+��,
+x ∈ [0, 1/2], r ∈ (0, 2r0),
+so that for some constants c3, c4 > 0,
+��
+(√x−c1r)+�2,
+�√x+c1r}2∧x0
+�
+⊂ B(x, r) ⊂
+��
+(√x−c2r)+�2,
+�√x+c2r}2∧x0
+�
+, r ∈ (0, 2r0).
+Noting that 0 < 1 − x0 ≤ 1 − s ≤ 1 for s ∈ B(x, 2r0), we find constants c4 > c3 > 0 such that
+c3
+�
+(x + c3r)2a − x2a�
+≤ µ(B(x, r))
+≤ µ(B(x, 2r)) ≤ c4
+�
+(x + c4r)2a − [(x − c4r)+]2a�
+,
+x ∈ [0, 1/2], r ∈ (0, r0).
+since (x+c4r)2a−{(x−c4r)+}2a
+(x+c3r)2a−x2a
+is a continuous function of (x, r) ∈ [0, 1
+2] × [0, r0], where when r = 0
+the function is understood as the limit c4
+c3 as r → 0, we obtain
+sup
+x∈[0,1/2],r∈(0,r0)
+µ(B(x, 2r))
+µ(B(x, r)) ≤
+sup
+x∈[0, 1
+2],r∈[0,r0]
+(x + c4r)2a − {(x − c4r)+}2a
+(x + c3r)2a − x2a
+< ∞.
+Therefore, (5.3) holds.
+In conclusion, all assertions in Theorems 2.1-2.6(1) hold for d = 4(a ∨ b) and d′ = 2 so that
+qα :=
+4(a ∨ b)
+4(a ∨ b) − α,
+α(d, d′) = ˜α := (a ∨ b)
+�√
+5 − 1
+�
+,
+γα,p,q := 2(a ∨ b)(2 − p−1 − q−1) − α.
+(5.12)
+So, we have the following result.
+Theorem 5.6. For the above L = ˆL and ηB = ηB
+Z for Z = 0, the following assertions hold for
+qα, ˜a and, γα,p,q in (5.12).
+(1) If α > 4
+3(a ∨ b) which implies qα > 4(a∨b)
+2α
+and α > ˜α, then
+lim
+t→∞ sup
+ν∈P
+Eν���tW2(µB
+t , µ)2 − ΞB(t)
+��q�
+= 0,
+q ∈ [1, qα).
+(2) For any q ∈ [1, qα) and k ∈ (
+d
+2αi(q), ∞] ∩ [1, ∞], where we set (
+d
+2αi(q), ∞] = {∞} if α = 0,
+lim
+t→∞ sup
+ν∈Pk,R
+Eν����tW2(µB
+t , µ)2 − ΞB(t)
+���
+q�
+= 0,
+R ∈ (0, ∞).
+(3) ηB < ∞ and lim supt→∞ supν∈P
+��tEν[W2(µB
+t , µ)2] − ηB�� = 0.
+(4) Let p, q ∈ [1, ∞) with γα,p,q < 0. There exist constants c, t0 > 1 such that
+c−1t−1 ≤
+�
+Eν[W1(µB
+t , µ)]
+�2 ≤
+�
+Eν[W2p(µB
+t , µ)2q]
+� 1
+q ≤ ct−1,
+t ≥ t0, ν ∈ P.
+(5) Let p, q ∈ [1, ∞) with γα,p,q ≥ 0. Then there exists a constant c > 0 such that
+sup
+ν∈P
+�
+Eν[W2p(µB
+t , µ)2q]
+� 1
+q ≤ ct
+−
+1
+1+γα,p,q + c1{γα,p,q=0}t−1 log(1 + t),
+t ≥ 1.
+51
+
+5.4
+Subordinated subelliptic diffusions on SU(2)
+Let M = SU(2) be the space of 2 × 2, complex, unitary matrices with determinant 1, which is
+a 3-dimensional compact Lie group with Lie algebra and Riemannian metric given by
+su(2) := span{U1, U2, U3},
+⟨Ui, Uj⟩ = 1{i=j},
+1 ≤ i, j ≤ 3,
+where for i = √−1,
+U1 :=
+� 0
+1
+−1
+0
+�
+,
+U2 :=
+�0
+i
+i
+0
+�
+,
+U3 :=
+�i
+0
+0
+−i
+�
+.
+For each 1 ≤ i ≤ 3, Ui is understood as a left-invariant vector field defined as
+Uif(x) := lim
+ε↓0
+f(eεUix) − f(x)
+ε
+,
+f ∈ C1(SU(2)).
+Then [U1, U2] = 2U3, so that
+ˆL := U2
+1 + U2
+2
+satisfies H¨ormader’s condition. Moreover, ˆL is symmetric in L2(µ) where µ is the normalized
+Haar measure on SU(2), and the intrinsic distance ρ induced by
+Γ(f, g) := (U1f)(U1g) + (U2f)(U2g)
+is the Carnot-Carath´eodory distance. By Chow’s theorem, (M, ρ) is a compact geodesic space.
+To formulate the diffusion process ˆXt generated by ˆL, we use the cylindrical coordinate
+introduced in [10]:
+�
+0, π
+2
+�
+× [0, 2π] × [−π, π] ∋ (r, θ, z) �→ er(cos θ)U1+r(cos θ)U2ezU3 ∈ M := SU(2).
+Under these coordinates, ˆXt := (rt, θt, zt) is constructed by solutions of the SDEs
+drt = 2 cot(2rt)dt + dBt,
+d(θt, zt) =
+�
+2
+sin θt
+, tan rt
+�
+d ˜Bt,
+(5.13)
+where (Bt, ˜Bt) is a two-dimensional Brownian motion, see [5, Remark 2.2]. The following lemma
+shows that conditions (A1), (A2), (2.17) and (2.23) hold. However, due to the degeneracy of the
+diffusion, assumption (B) may be invalid.
+Lemma 5.7. Conditions (A1), (A2), (2.17) and (2.23) hold for d = 4 and d′ = 3.
+Proof. By [5, Theorem 4.10], for any p > 1 there exists a constant c(p) > 0 such that
+|∇ ˆPtf| ≤ c(p)e−2t(Pt|∇f|p)
+1
+p,
+t ≥ 0.
+So, (2.8) holds.
+52
+
+According to [7], the generalized curvature-dimension condition CD(ρ1, 1
+2, 1, 2) holds, so
+that (2.17) is implied by [6, Theorem 1.2].
+Let ˆpt be the heat kernel of ˆPt with respect to µ. By [5, Proposition 3.1] and the spectral
+representation of heat kernel, see also [8], all eigenvalues with multiplicities of −ˆL are given by
+{λi}i≥0 =
+�
+4k(k + |n| + 1) + 2|n| :
+n ∈ Z, k ∈ Z+
+�
+.
+In particular, λ1 = 2. It is easy to see that for large i ∈ N,
+#
+�
+4k(k + |n| + 1) + 2|n| ≤ i :
+n ∈ Z, k ∈ Z+
+�
+has order i
+3
+2, so that (2.23) holds for d′ = 3.
+To verify (2.6), we use the cylindrical coordinates (r, θ, z), for which the identity matrix
+becomes 0 := (0, 0, 0). Let ˆpt be the heat kernel of ˆPt with respect to µ. By [5, Proposition
+3.9], there exists a constant c > 0 such that
+ˆpt(0, 0) ≤ ct−2,
+t ∈ (0, 1].
+By the left invariant of the heat kernel which follows from the same property of the generator
+ˆL, we obtain
+∥ ˆPt∥1→∞ = sup
+x∈M
+ˆpt(x, x) ≤ ct−2,
+t ∈ (0, 1].
+This together with λ1 = 2 implies (2.6) for λ = 2.
+It remains to verify (A2). For any (r, z) ∈ [0, π
+2)×[−π, π], let θ(r, z) ∈ [−π, π] be the unique
+solution to the equation
+θ(r, z) − z = (cos r)(sin θ(r, z)) arccos[(cos θ(r, z)) cos r]
+�
+1 − (cos2 r) cos2 θ(r, z)
+.
+By [5, Remark 3.12], the distance of x := (r, θ, z) to 0 depends only on (r, z), and there exists
+a constant c1 > 0 such that
+(5.14)
+ρ(0, x)2 = (θ(r, z) − z)2 tan2 r
+sin2 θ(r, z)
+≤ c1 sin2 r ≤ c1r2.
+On the other hand, letting ˆX0 = 0, by (5.13) and Itˆo’s formula, for any p ≥ 1 we find a constant
+c1(p) > 0 such that
+dr2p
+t ≤ c(p)r2(p−1)
+t
+dt + dMt
+for some martingale Mt. So, we find a constant c2(p) > 0 such that
+E[r2p
+t ] ≤ k(p)tp,
+t ≥ 0.
+Combining this with (5.14) and using the left invariance of the heat kernel, we obtain (A2).
+53
+
+By the above lemma and that M = SU(2) is a Polish space, we conclude that all assertions
+in Theorems 2.1-Theorem 2.3 and Theorem 2.6(1) hold for d = 4, d′ = 3 and
+qα :=
+8
+9 − 2α,
+γα,p,q := 1
+2 − α + 2(2 − p−1 − q−1).
+(5.15)
+So, we have the following result.
+Theorem 5.8. Let ˆL := U2
+1 + U2
+2, and L = ˆL + Z for some C1
+b -vector field Z satisfying (2.2).
+The following assertions hold for qα, α′ and γα,p,q in (5.15).
+(1) If α ∈ ( 1
+2, 1], then for any q ∈ [1, qα) and k ∈ (
+d
+2αi(q), ∞] ∩ [1, ∞],
+lim
+t→∞ sup
+ν∈Pk,R
+Eν����{tW2(µB
+t , µ)2 − ΞB(t)}+���
+q�
+= 0,
+R ∈ (0, ∞).
+(2) If α ∈ ( 1
+2, 1], then
+lim sup
+t→∞
+sup
+ν∈P
+tEν[W2(µB
+t , µ)2] ≤ ηB
+Z < ∞.
+(3) If α ∈ (0, 1
+2], then there exist constants c, t0 > 0 such that
+sup
+ν∈P
+Eν[W2(µB
+t , µ)2] ≤ ct−
+2
+3−2α + c1{α= 1
+2}t−1 log t,
+t ≥ t0.
+(4) If p, q ∈ [1, ∞) with γα,p,q < 0, then there exist constants c, t0 > 1 such that
+c−1t−1 ≤
+�
+Eν[W1(µB
+t , µ)]
+�2 ≤
+�
+Eν[W2p(µB
+t , µ)2q]
+� 1
+q ≤ ct−1,
+t ≥ t0, ν ∈ P.
+(5) Let p, q ∈ [1, ∞) with γα,p,q ≥ 0. Then for any γ > γα,p,q, there exists a constant c > 0
+such that for any t ≥ 1,
+sup
+ν∈P
+�
+Eν[W2p(µB
+t , µ)2q]
+� 1
+q ≤ ct
+−
+1
+1+γα,p,q + c1{γα,p,q=0}t−1 log(1 + t).
+References
+[1] L. Ambrosio, N. Gigli, G. Savar´e, Gradient Flows in Metric Spaces and in the Spaces of
+Probability Measures, Lectures in Mathematics ETH Z¨urich, Birkh¨auser Verlag, Basel,
+2005.
+[2] L. Ambrosio, F. Stra, D. Trevisan, A PDE approach to a 2-dimensional matching problem,
+Probab. Theory Relat. Fields 173(2019), 433–477.
+[3] P. Auscher, T. Coulhon, X. T. Duong, S. Hofmann, Riesz transform on manifolds and
+heat kernel regularity, Ann. Sci. ´Ecole Norm. Sup. (4)37(2004), 911–957.
+54
+
+[4] D. Bakry, On Sobolev and logarithmic Sobolev inequalities for Markov semigroups, in:
+New Trends in Stochastic Analysis, World Scientific, 1997, pp. 43–75.
+[5] F. Baudoin, M. Bonnefont, The subelliptic heat kernel on SU(2): Representations, asymp-
+totics and gradient bounds, Math. Zeit. 263(2009), 647–672.
+[6] F. Baudoin, N. Garofal, A note on the boundedness of Riesz transform for some subelliptic
+operators, 2013, pp. 398–421.
+[7] F. Baudoin, N. Garofal,
+Curvature-dimension inequalities and Ricci lower bounds for
+sub-Riemannian manifolds with transverse symmetries, J. Euro. Math. Soc. 19(2017),
+151–219.
+[8] R.O. Bauer, Analysis of the horizontal Laplacian for the Hopf fibration, Forum Math.
+17(2005), 903–920.
+[9] I. Chavel, Eigenvalues in Riemannian Geometry, Academic Press, 1984.
+[10] M. Cowling, A. Sikora, A spectral multiplier theorem for a sublaplacian on SU(2), Math.
+Zeit. 238(2001), 1–36.
+[11] E. B. Davies, Heat Kernels and Spectral Theory, Cambridge Univ. Press, 1989.
+[12] M. D. Donsker, S. R. S. Varadhan, Asymptotic evaluation of certain Markov process
+expectations for large time, I-IV, Comm. Pure Appl. Math. 28(1975), 1–47, 279–301;
+29(1976), 389–461; 36(1983), 183–212.
+[13] K. Du, Empirical approximation to invariant measures for McKean-Vlasov processes:
+mea-field interaction vs self-interaction, to appear in Bernoulli.
+[14] S. Feng, L. Miclo, F.-Y. Wang, Poincar´e inequality for Dirichlet distributions and infinite-
+dimensional generalizations, Lat. Am. J. Probab. Math. Stat. 14(2017), 361–380.
+[15] S. Feng, F.-Y. Wang, Harnack inequality and applications for infinite-dimensional GEM
+processes, Pot. Anal. 44(2016), 137–153.
+[16] S. German, D. German,
+Stochastic relaxation, Gibbs distribution, and the Baysian
+restoration of images, IEEE Trans. Pattern Analysis Machine Intelligence 6(1984), 721–
+741.
+[17] L.-J. Huang, Y.-H. Mao, On some mixing times for nonreversible finite Markov chains,
+J. Appl. Probab. 54(2017), 627–637.
+[18] M. Huesmann, F. Mattesini, D. Trevisan, Wasserstein asympototics for the empirical
+measure of fractional Brownian motion on a flat torus, arXiv:2205.01025.
+[19] C.-R. Hwang, S.-Y. Hwang-Ma, S.-J. Sheu, Accelerating Gaussian diffusions, Ann. Appl.
+Probab. 3(1993), 897–913.
+55
+
+[20] C.-R. Hwang, S.-Y. Hwang-Ma, S.-J. Sheu, Accelerating diffusions, Ann. Appl. Probab.
+15(2005), 1433–1444.
+[21] M. Ledoux, On optimal matching of Gaussian samples, in:
+Zap. Nauchn. Sem. S.-
+Peterburg. Otdel. Mat. Inst. Steklov. (POMI) vol. 457, in: Veroyatnost Statistika., vol.
+25, 2017, pp. 226–264.
+[22] H. Li, B. Wu, Wasserstein convergence for empirical measures of subordinated Dirichlet
+diffusions on Riemannian manifolds, arXiv:2206.03901.
+[23] H. Li, B. Wu, Wasserstein convergence for conditional empirical measures of subordinated
+Dirichlet diffusions on Riemannian manifolds, arXiv:2204.13559.
+[24] H. Li, B. Wu, Wasserstein convergence rates for empirical measures of subordinated pro-
+cesses on noncompact manifolds, arXiv:2201.12797.
+[25] E.M. Ouhabaz, F.-Y. Wang, Sharp estimates for instrinsic ultracontractivity on C1,α-
+domains, Manu. Math. 112(2007), 229–244.
+[26] M.-K. von Renesse, K.-T. Sturm, Transport inequalities, gradient estimates, entropy, and
+Ricci curvature, Comm. Pure Appl. Math. 58(2005), 923–940.
+[27] W. Stannat, On validity of the log-Sobolev inequality for symmetric Fleming-Viot opera-
+tors, Ann. Probab. 28(2000), 667–684.
+[28] C. Villani, Optimal transport: Old and New Part 1,2, Springer, 2009.
+[29] F.-Y. Wang, Application of coupling method to the Neumann eigenvalue problem, Probab.
+Theory Rel. Fields 98(1994), 299–306.
+[30] F.-Y. Wang, Functional Inequalities, Markov Semigroups and Spectral Theory, Science
+Press, Beijing, 2005.
+[31] F.-Y. Wang, Gradient estimates and the first Neumann eigenvalue on non-convex mani-
+folds, Stoch. Proc. Appl. 115(2005), 1475–1486.
+[32] F.-Y. Wang, Analysis of Diffusion Processes on Riemannian Manifolds, World Scientific,
+2014.
+[33] F.-Y. Wang, Precise limit in Wasserstein distance for conditional empirical measures of
+Dirichlet diffusion processes, J. Funct. Anal. 280(2021), 108998, 23pp.
+[34] F.-Y. Wang, Convergence in Wasserstein distance for empirical measures of Dirichlet
+diffusion processes on manifolds, to appear in J. Eur. Math. Soc.
+[35] F.-Y. Wang, Wasserstein convergence rate for empirical measures on noncompact mani-
+folds, Stoch. Proc. Appl. 144(2022), 271–287.
+[36] F.-Y. Wang, Convergence in Wasserstein distance for empirical Measures of semilinear
+SPDEs, to appear in Ann. Appl. Probab.
+56
+
+[37] F.-Y. Wang, B. Wu, Wasserstein convergence for empirical measures of subordinated
+diffusion processes on Riemannian manifolds, to appear in Pot. Anal.
+[38] F.-Y. Wang, J.-X. Zhu, Limit Theorems in Wasserstein Distance for Empirical Measures
+of Diffusion Processes on Riemannian Manifolds, Ann. Inst. Henri Poincar. Probab. Stat.
+59(2023), 437–475.
+[39] L. Wu, Moderate deviations of dependent random variables related to CLT, Ann. Probab.
+23(1995), 420–445.
+[40] L. Wu, Uniformly integrable operators and large deviations for Markov processes, J. Funct.
+Anal. 172(2000), 301–376.
+[41] H. Ye, J. Gao, Y. Ding, A generalized Gronwall inequality and its application to a frac-
+tional differential equation, J. Math. Anal. Appl. 328(2007), 1075–1081.
+57
+
diff --git a/3dFAT4oBgHgl3EQfERwH/content/tmp_files/load_file.txt b/3dFAT4oBgHgl3EQfERwH/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..be543875d1a4ebade0457fd71d8c53f7767d8ceb
--- /dev/null
+++ b/3dFAT4oBgHgl3EQfERwH/content/tmp_files/load_file.txt
@@ -0,0 +1,1684 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf,len=1683
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='08420v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='PR]  20 Jan 2023  Convergence in Wasserstein Distance for  Empirical Measures of Non-Symmetric  Subordinated Diffusion Processes∗  Feng-Yu Wang  Center for Applied Mathematics, Tianjin University, Tianjin 300072, China  Department of Mathematics, Swansea University, Bay Campus, SA1 8EN, United Kingdom  wangfy@tju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='cn  January 23, 2023  Abstract  By using the spectrum of the underlying symmetric diffusion operator, the convergence  in Wasserstein distance is characterized for the empirical measure of non-symmetric sub-  ordinated diffusion processes in an abstract framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  In particular, let µ(dx) := eV (x)dx be a probability measure on an n-dimensional com-  pact connected Riemannian manifold M, let Wp be the Lp-Wasserstein distance induced  by the Riemannian distance, and let µt be the empirical measure of the diffusion process  Xt generated by L := ∆ + ∇V + Z, where Z is a µ-divergence free vector field on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  When t → ∞, we derive the sharp convergence rate of Wp(µt, µ) for p ≥ 1, as well as the  precise limit of tW2(µt, µ)2 for n ≤ 3:  lim  t→∞ E  ����tW2(µt, µ)2 −  ∞  �  i=1  1  tλi  � � t  0  φi(Xs)ds  �2����  q  = 0,  q ∈  �  1,  n  2(n − 2)+  �  ,  where {λi}i≥1 are all non-trivial eigenvalues of −ˆL := −(∆+∇V ) with unit eigenfunctions  {φi}i≥1 ⊂ L2(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Consequently, for n ≤ 3 we have  lim  t→∞ E[tW2(µt, µ)2] =  ∞  �  i=1  2  λ2  i  �  1 − 1  λi  µ  �  |∇L−1(Zφi)|2��  ∈ (0, ∞),  which provides a precise characterization on the physical observation that a divergence  free perturbation Z ̸= 0 accelerates the convergence of the symmetric system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Moreover,  for n ∈ {2, 3, 4} a critical phenomenon appears to the convergence of E[Wp(µt, µ)2q]  1  q with  the critical rate t−1 log t as t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  These results are included by our main results established in an abstract framework for  subordinated diffusion processes, which are further illustrated by some other examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  ∗Supported in part by NNSFC (11921001) and the National Key R&D Program of China (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  2022YFA1006000, 2020YFA0712900).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  1  AMS subject Classification: 60B05, 60B10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Keywords: Empirical measure, Wasserstein distance, non-symmetric diffusion process, subor-  dination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  1  Introduction  In statistical physics, the empirical measure is a fundamental object to simulate the stationary  distribution (Gibbs measure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Since the ground breaking series work [12] where the celebrated  Donsker-Varadhan larger deviation principle was developed, the long time behavior of empirical  measures has become a key research topic in the study of Markov processes, see [39, 40] for  criteria on the central limit theorem and large deviations for hyperbounded Markov processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  On the other hand, the Wasserstein distance is an intrinsic object in the theory of optimal  transport and calculus in Wasserstein space, see [1, 28] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' So, it is crucial  and interesting to study the convergence in Wasserstein distance for the empirical measure of  Markov processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Moreover, it was observed in [16] that a divergence-free perturbation to symmetric stochastic  systems may accelerate the algorithm of Gibbs measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' This has been confirmed in several  papers for the convergence of Markov semigroups to stationary distributions, see [17, 19, 20]  and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' It is interesting to provide a sharp characterization on the acceleration  for the convergence of empirical measures in Wasserstein distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  In recent years, the sharp convergence rate in the second moment of L2-Wasserstein distance  has been derived in [33, 34, 35, 36, 38] for empirical measures of symmetric diffusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  In  particular, in lower dimensions the precise limit is explicitly formulated by using eigenvalues  and eigenfunctions of the generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' These results have been extended to subordinated processes  in [37, 22, 23, 24] and the fractional Brownian motion on torus in [18], see also [13] for the  study of McKean-Vlasov SDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Based on the above backgrounds, this paper investigates the convergence of empirical mea-  sures for non-symmetric subordinated diffusion processes in an abstract framework, describes  the acceleration of the convergence for divergence-free perturbations to symmetric systems, and  illustrates the main results by typical examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  To figure out a clear picture of the our general results (see Section 2 for details), in the  following we only consider non-symmetric diffusion processes on a compact manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' See Sec-  tion 5 for applications of the general results to three more examples including the subordinated  conditional diffusion process, the subordinated Wright-Fisher process, and the subordinated  subelliptic diffusion process on SU(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Let M be an n-dimensional compact connected Riemannian manifold possibly with a bound-  ary ∂M, let P be the space of all probability measures on M, let ρ be the Riemannian distance,  and for any p ≥ 1, let Wp be the Lp-Wasserstein distance induced by ρ, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='1) below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Let µ(dx) := eV (x)dx ∈ P, where V ∈ C2(M) and dx is the volume measure on M, and let  2  Z be a C1-vector field with divµZ = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  µ(Zf) :=  �  M  ⟨Z, ∇f⟩dµ = 0,  f ∈ C1(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  It is well known that the spectrum of ˆL := ∆ + ∇V (with Neumann boundary if ∂M exists) is  discrete and all eigenvalues {λi}i≥0 of −ˆL listed in the increasing order counting multiplicities  satisfy  c1i  n  2 ≤ λi ≤ c2λ  n  2  i ,  i ≥ 0  for some constants c1, c2 > 0, see for instance [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Let {φi}i≥0 with φ0 ≡ 1 being the corre-  sponding unitary eigenfunctions in L2(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Let Xt be the diffusion process on M generated by  L := ∆ + ∇V + Z,  with reflecting boundary if ∂M exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' We consider the empirical measure  µt := 1  t  � t  0  δXsds,  t > 0,  where δXs is the Dirac measure at Xs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' By the central limit theorem (see [39]), for any  f ∈ L2  0(µ) :=  �  f ∈ L2(µ) : µ(f) = 0  �  ,  we have  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='1)  lim  t→∞  √  tµt(f) = lim  t→∞  1  √  t  � t  0  f(Xs)ds = N(0, V(f)) in law,  where N(0, V(f)) is the centered normal distribution with variance  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='2)  V(f) :=  � ∞  0  µ(fPsf)ds = µ(|∇L−1f|2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  For any k, R ≥ 1, let  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='3)  Pk,R  µ  :=  �  ν ∈ P : dν = hdµ, ∥h∥k ≤ R  �  ,  where ∥ · ∥k is the norm in Lk(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' For any ν ∈ P, let Eν be the expectation for the diffusion  process Xt with initial distribution ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  We first consider the long time behavior of W2(µt, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' The following result shows that when  n ≤ 3 and ∂M is either empty or convex, tW2(µt, µ)2 behaves as  Ξ(t) :=  ∞  �  i=1  1  λi  |ψi(t)|2,  where ψi(t) :=  1  √  t  � t  0 φi(Xs)ds (i ≥ 1, t > 0), so that uniformly in ν ∈ P, tEν[W2(µt, µ)2]  converges to  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='4)  ηZ :=  ∞  �  i=1  2V(φi)  λi  =  ∞  �  i=1  2  λ2  i  �  1 − 1  λi  V(Zφi)  �  ,  where the second equality follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='2 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  3  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' There exists a constant κ ≥ 1 with κ = 1 when ∂M is either empty or convex,  such that the following assertions hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (1) When n ≤ 2, for any q ∈ [1, ∞),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='5)  lim  t→∞ sup  ν∈P  Eν����  tW2(µt, µ)2 − Ξ(t)  �+ +  �  Ξ(t) − κtW2(µt, µ)2�+��q�  = 0,  so that when ∂M is either empty or convex,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='6)  lim  t→∞ sup  ν∈P  Eν���tW2(µt, µ)2 − Ξ(t)|q�  = 0,  q ∈ [1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (2) When n = 3, for any R ∈ [1, ∞), k ∈ ( 3  2, ∞] and q ∈ [1, 3  2),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='7)  lim  t→∞ sup  ν∈Pk,R  µ  Eν����  tW2(µt, µ)2 − Ξ(t)  �+ +  �  Ξ(t) − κtW2(µt, µ)2�+��q�  = 0,  so that when ∂M is either empty or convex,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='8)  lim  t→∞ sup  ν∈Pk,R  µ  Eν���tW2(µt, µ)2 − Ξ(t)|q�  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (3) For n ≤ 3 we have ηZ ∈ (0, ∞) and  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='9)  lim  t→∞ sup  ν∈P  ��  tEν[W2(µt, µ)2] − ηZ  �+ +  �  ηZ − κtEν[W2(µt, µ)2]  �+�  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  In particular, when ∂M is either empty or convex,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='10)  lim  t→∞ sup  ν∈P  ��tEν[W2(µt, µ)2] − ηZ  �� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (4) For n = 4, there exist constants c1, c2, t0 > 0 such that  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='11) c1  t log(1 + t) ≤ inf  ν∈P Eν[W2(µt, µ)2] ≤ sup  ν∈P  Eν[W2(µt, µ)2] ≤ c2  t log(1 + t),  t ≥ t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (5) For n ≥ 5, there exist constants c1, c2, t0 > 0 such that  c1t−  2  d−2 ≤ inf  ν∈P  �  Eν[W1(µt, µ)]  �2 ≤ sup  ν∈P  Eν[W2(µt, µ)2] ≤ c2t−  2  d−2,  t ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='12)  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (1) By (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='4) we have ηZ < η0 for Z ̸= 0, so (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='9) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='10) provide a precise  characterization on the observation of [16] that a divergence-free perturbation Z accelerates  the convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (2) When Z = 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' in the symmetric case), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='9) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='10) have been presented in  [38] with a central limit theorem on tW2(µt, µ)2, which are covered by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' The Lq-  convergence (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='5) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='6) appear here for the first time, which are new also in the symmetric  case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  4  (3) In the symmetric case (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Z = 0), estimates (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='9), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='12) and the upper bound in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='11)  have been presented in [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Other estimates are new also in the symmetric case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Moreover, it  is proved in [38] that for M being the 4-dimensional torus and L = ∆, there exists a constant  c > 0 such that  inf  ν∈P  �  Eν[W1(µt, µ)]  �2 ≥ ct−1 log(1 + t),  t ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  We hope that this estimate also holds for general non-symmetric diffusions on 4-dimensional  compact manifolds, such that the lower bound estimate in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='11) is strengthened with W1  replacing W2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  In the next result, we estimate  �  E[Wp(µt, µ)2q]  � 1  q for all p, q ∈ [1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Besides the critical  phenomenon for the convergence rate in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='1 with (p, q) = (2, 1) and the critical rate  t−1 log t in dimension n = 4, the critical rate also appears to dimensions n = 2, 3 with different  (p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' There exist c, t0 ∈ (0, ∞) and κ : [2, ∞) × [1, ∞) → (0, ∞), such that the  following assertions hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (1) When n = 1, for any (p, q) ∈ [2, ∞) × [1, ∞),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='13)  c  t ≤ inf  ν∈P  �  Eν[W1(µt, µ)]  �2 ≤ sup  ν∈P  �  Eν[Wp(µt, µ)2q]  � 1  q ≤ κp,q  t ,  t ≥ t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (2) Let n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Then (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='13) holds for any p ∈ [2, ∞) and q ∈ [1,  p  p−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Next, for any p ∈ (2, ∞)  and q =  p  p−2,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='14)  sup  ν∈P  �  Eν[Wp(µt, µ)2q]  � 1  q ≤ κp,q  t  log(1 + t),  t ≥ t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Finally, for any p ∈ (2, ∞) and q ∈ (  p  p−2, ∞),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='15)  sup  ν∈P  �  Eν[Wp(µt, µ)2q]  � 1  q ≤ κp,qt−  2pq  3npq−2pq−2nq−np ,  t ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (3) Let n = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Then (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='13) holds for any p ∈ [2, 3) and q ∈ [1,  3p  5p−6);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='14) holds for p ∈ [2, 3)  and q =  3p  5p−6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='15) holds for any p ∈ [2, ∞) and q ∈ (  3p  5p−6), ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (4) Let n = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Then (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='15) holds for (2, 1) ̸= (p, q) ∈ [2, ∞) × [1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (5) When n ≥ 5, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='15) holds for any (p, q) ∈ [2, ∞) × [1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  The remainder of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' In Section 2, we state our main results  for non-symmetric subordinated diffusion processes in an abstract framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' In Sections 3  and 4, we prove the main results on upper and lower bound estimates respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' In Section 5,  we apply the main results to some concrete models, where the result for the first model covers  Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='2 as direct consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  5  2  Main results  We first introduce the framework of the study, then state the main results on the Wasserstein  distance of the empirical measures for subordinated non-symmetric diffusion processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='1  The framework  We first introduce the sate space of diffusion processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Let (M, ρ) be a length space, let P  be the set of all probability measures on M, let Bb(M) be the class of bounded measurable  functions on M, and let Cb,L(M) be the set of all bounded Lipschitz continuous functions on  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' For any p ∈ [1, ∞), the Lp-Wasserstein distance is defined as  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='1)  Wp(ν1, ν2) :=  inf  π∈C (ν,γ)  � �  M×M  ρ(x, y)pπ(dx, dy)  � 1  p  ,  ν1, ν2 ∈ P,  where P is the space of all probability measures on M, and C (ν1, ν2) is the set of all couplings  for ν1 and ν2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Next, we introduce the reversible Makrov process ˆXt on M with unique invariant probability  measure µ ∈ P having full support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' The Markov semigroup ˆPt is formulated as  ˆPtf(x) = Ex[f( ˆXt)],  t ≥ 0, x ∈ M, f ∈ Bb(M),  where Ex stands for the expectation for the underlying Markov process starting at point x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' In  general, for any ν ∈ P, Eν is the expectation for the Markov process with initial distribution  ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Let ( ˆE , D( ˆE )) and (ˆL, D(ˆL)) be the associated symmetric Dirichlet form and self-adjoint  generator in L2(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' We assume that Cb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='L(M) is a dense subset of D( ˆE ) under the ˆE1-norm  ∥f∥ ˆE1 :=  �  µ(f 2) + ˆE (f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' f),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  where µ(f 2) :=  �  M f 2dµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' and that  ˆE (f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' g) =  �  M  Γ(f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' g)dµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' g ∈ Cb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='L(M)  holds for a symmetric local square field (champ de carr´e)  Γ : Cb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='L(M) × Cb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='L(M) → Bb(M),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  such that for any f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' h ∈ Cb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='L and φ ∈ C1  b (R),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' we have  �  Γ(f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' f)(x) = |∇f(x)| := lim sup  y→x  |f(y) − f(x)|  ρ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' y)  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  x ∈ M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  and the chain rule  Γ(fg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' h) = fΓ(g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' h) + gΓ(f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' h),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Γ(φ(f),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' h) = φ′(f)Γ(f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  6  We also assume that ˆL satisfies the chain rule  ˆLΦ(f) = Φ′(f)ˆLf + Φ′′(f)|∇f|2,  f ∈ D(ˆL) ∩ Cb,l, Φ ∈ C2(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  For any q ≥ p ∈ [1, ∞], let ∥ · ∥p be the norm in Lp(µ), and let ∥ · ∥p→q the operator norm from  Lp(µ) to Lq(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Throughout the paper, we simply denote µ(f) =  �  M fdµ for f ∈ L1(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Moreover, we make a non-symmetric perturbation to the symmetric diffusion process by  adding a µ-divergence free drift Z to the generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' More precisely, let  Z : Cb,L(M) → Bb(M)  be a bounded vector field with divµZ = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Z(fg) = fZg + gZf,  Z(φ(f)) = φ′(f)Zf  holds for f, g ∈ Cb,L(M) and φ ∈ C1(R), and  ∥Z∥∞ := inf  �  K ≥ 0 : |Zf| ≤ K|∇f|, f ∈ Cb,L(M)  �  < ∞,  µ(Zf) :=  �  M  Zfdµ = 0,  f ∈ Cb,L(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Consequently, Z uniquely extends to a bounded linear operator from D( ˆE ) to L2(µ) with  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='2)  µ(Zf) = 0,  f ∈ D( ˆE ),  and  E (f, g) := ˆE (f, g) + µ(fZg),  f, g ∈ D(E ) = D( ˆE )  is a (non-symmetric) conservative Dirichlet form with generator  L := ˆL + Z,  D(L) = D(ˆL),  which satisfies the chain rule  LΦ(f) = Φ′(f)Lf + Φ′′(f)|∇f|2,  f ∈ D(ˆL) ∩ Cb,l, Φ ∈ C2(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Assume that L generates a unique diffusion process Xt on M, such that the associated Markov  semigroup is given by  Ptf(x) = Ex[f(Xt)],  x ∈ M, t ≥ 0, f ∈ Bb(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  By Duhamel’s formula,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='3)  Ptf = ˆPtf +  � t  0  Ps{Z ˆPt−sf}ds,  f ∈ D( ˆE ), t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Finally, we make subordination of the diffusion process Xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Let B be the set of Bernstein  functions B satisfying B(0) = 0, B(r) > 0 for r > 0 and  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='4)  � ∞  0  e− 1  2 B(r)dr < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  7  For each B ∈ B, there exists a unique stable process SB  t on [0, ∞) with Laplace transform  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='5)  E[e−λSB  t ] = e−B(λ)t,  t, λ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Let SB  t be independent of Xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' We consider the subordinated diffusion process  XB  t := XSB  t ,  t ≥ 0,  and study the convergence to µ in Wp (p ≥ 1) for the empirical measure  µB  t := 1  t  � t  0  δXB  s ds,  t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  We will mainly consider α-stable type time change for α ∈ [0, 1], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' the Bernstein function  B is in the classes  Bα :=  �  B ∈ B : lim inf  λ→∞ B(λ)λ−α > 0  �  ,  Bα :=  �  B ∈ B : lim sup  λ→∞  B(λ)λ−α < ∞  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='2  Upper bound estimates  We make the following assumption, where (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='6) implies that the spectrum of −ˆL is discrete  and all eigenvalues {λi}i≥0 listed in the increasing order counting multiplicities satisfy  λi ≥ ci  2  d,  i ∈ Z+  for some constant c > 0, where λ1 ≥ λ, see for instance see [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' In general, λi may increase  faster than λ  2  d  i , see for instance Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='2 where d = n + 2 but λi ∼ i  2  n, we make the  additional assumption (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (A1) There exist constants c, λ > 0, d ≥ d′ ≥ 1 and a map k : (1, ∞) → (0, ∞) such that  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='6)  ∥ ˆPt − µ∥1→∞ ≤ c(1 ∧ t)− d  2 e−λt,  t > 0,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='7)  λi ≥ ci  2  d′ ,  i ∈ Z+,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='8)  |∇ ˆPtf| ≤ k(p)( ˆPt|∇f|p)  1  p,  t ∈ [0, 1], p ∈ (1, ∞), f ∈ Cb,L(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  We will also need the following condition on the continuity of ˆXt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (A2) For any p ∈ [1, ∞) there exists a constant c(p) > 0 such that  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='9)  Eµ�  ρ( ˆX0, ˆXt)p�  ≤ c(p)t  p  2,  t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  8  Let {φi}i≥0 with φ0 ≡ 1 be the unitary eigenfunctions for {λi}i≥0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='10)  ˆLφi = −λiφi,  ˆPtφi = e−λitφi,  µ(φiφj) = 1{i=j},  i, j ∈ Z+, t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Let  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='11)  ΞB(t) :=  ∞  �  i=1  ψB  i (t)2  λi  ,  ψB  i (t) := 1  √  t  � t  0  φi(XB  s )ds,  t > 0, i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Let Pk,R be in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Finally, let  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='12)  qα := 1 +  2(1 + α) − d′  (2d + d′ − 2 − 2α)+,  and denote the integer parts of q ∈ [1, ∞) by  i(q) := sup  �  i ∈ N : i ≤ q  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  The first main result of the paper is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Assume (A1), (A2) and let B ∈ Bα for some α ∈ [0, 1] with d′ < 2(1 + α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (1) If qα >  d  2α and  α > α(d, d′) := 1  4  ��  (2 + d − d′)2 + 4d(d + d′ − 2) + d′ − d − 2  �  ,  then  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='13)  lim  t→∞ sup  ν∈P  Eν����  �  tW2(µB  t , µ)2 − ΞB(t)  �+���  q�  = 0,  q ∈ [1, qα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (2) For any q ∈ [1, qα) and k ∈ (  d  2αi(q), ∞] ∩ [1, ∞], where we set (  d  2αi(q), ∞] = {∞} if α = 0,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='14)  lim  t→∞ sup  ν∈Pk,R  Eν����  �  tW2(µB  t , µ)2 − ΞB(t)  �+���  q�  = 0,  R ∈ (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  To estimate E[W2(µB  t , µ)2], we let  ηB  Z :=  ∞  �  i=1  2VB(φi)  λi  ,  VB(φi) :=  � ∞  0  µ(φiP B  s φi)ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Assume (A1) and (A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Let q ∈ [1, ∞) and B ∈ Bα for some α ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (1) If d′ < 2(1 + α), then ηB  Z < ∞ and  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='15)  lim sup  t→∞  sup  ν∈P  tEν[W2(µB  t , µ)2] ≤ ηB  Z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  9  (2) Let d′ ≥ 2(1 + α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Then there exists a constant c > 0 such that for any t ≥ 1,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='16)  sup  ν∈P  Eν[W2(µB  t , µ)2] ≤  �  ct−1 log(1 + t),  if d′ = 2(1 + α),  ct−  2  d′−2α ,  if d′ > 2(1 + α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  To estimate Wp(µB  t , µ), we will use the Lp-boundedness of the Riesz transform ∇(a0 − ˆL)− 1  2  for some a0 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' According to [3], together with the non-degeneracy condition, the volume  doubling condition and the scaled Poincar´e inequality, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='8) implies  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='17)  ∥∇(a0 − ˆL)− 1  2∥p < ∞ for some a0 ∈ [0, ∞) and all p ∈ (2, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Under assumption (A1), let  γα,p,q := d′  2 + d  2  �  2 − p−1 − q−1�  − α − 1,  p, q ∈ [1, ∞), α ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Assume (A1) and (A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Let B ∈ Bα for some α ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (1) If γα,p,q < 0, then there exists a constant c > 0 such that  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='18)  sup  ν∈P  �  Eν[W2p(µB  t , µ)2q� 1  q ≤ ct−1,  t ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (2) If γα,p,q ≥ 0, then for any γ > γα,p,q, there exists a constant c > 0 such that  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='19)  sup  ν∈P  �  Eν[W2p(µB  t , µ)2q]  � 1  q ≤ ct−  1  1+γ ,  t ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (3) Let (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='17) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' If γα,p,q ≥ 0, then there exists a constant c > 0 such that for any t ≥ 1,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='20)  sup  ν∈P  �  Eν[W2p(µB  t , µ)2q]  � 1  q ≤  �  ct−1 log(1 + t),  if γα,p,q = 0,  ct  −  1  1+γα,p,q ,  if γα,p,q > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='3  Lower bound estimate  To derive sharp lower bound for E[W2(µB  t , µ)2], we make the following assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (B) (M, ρ) is a geodesic space, there exist constants θ, K > 0 and m ≥ 1 such that  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='21)  |∇Ptef|2 ≤ (Ptef)Pt(|∇f|2ef) + Ktθ∥∇f∥2  ∞(Pte2mf)  1  m,  t ∈ [0, 1], f ∈ Cb,L(M),  and there exists a function h ∈ C([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' [1, ∞)) such that  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='22)  W2(ν ˆPr, µ)2 ≤ h(r)W2(ν, µ)2,  ν ∈ P, r ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  When M is a Riemannian manifold without boundary or with convex boundary, if the  Bakry-Emery curvature of ˆL is bounded below by a constant −K/2, then (B) holds for m = 1  and h(r) = eKr, see for instance [32, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='3(2′)(9)] or [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  10  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Assume (A1) and (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Let B ∈ Bα for some α ∈ [0, 1] with d′ < 2(1 + α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (1) If α > α(d, d′) and qα >  d  2α, then  lim  t→∞ sup  ν∈P  Eν����  �  th(0)W2(µB  t , µ)2 − ΞB(t)  �−���  q�  = 0,  q ∈ [1, qα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (2) For any q ∈ [1, qα) and k ∈ (  d  2αi(q), ∞] ∩ [1, ∞), where we set (  d  2αi(q), ∞] = {∞} if α = 0,  we have  lim  t→∞ sup  ν∈Pk,R  Eν����  �  th(0)W2(µB  t , µ)2 − ΞB(t)  �−���  q�  = 0,  q ∈ [1, qα), R ∈ [1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (3) We have  lim inf  t→∞ sup  ν∈P  tEν[W2(µB  t , µ)2] ≥ h(0)−1ηB  Z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  The next result manages the critical case where the convergence rate of E[W2(µB  t , µ)2] is at  most t−1 log t, correspondingly to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='16) on the upper bound estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Assume (A1), (A2), (B) and that  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='23)  k′i  2  d′ ≤ λi ≤ ki  2  d′ ,  i ∈ N  holds for some constants k, k′ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Let B ∈ Bα for some α ∈ [0, 1] such that d′ = 2(1 + α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Then there exist constants c, t0 > 0 such that  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='24)  inf  ν∈P Eν[W2(µB  t , µ)2] ≥ ct−1 log(1 + t),  t ≥ t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Finally, we consider the lower bound estimate on W1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Let B ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Then the following assertions hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (1) Assume (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='6), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='8) and that the completion ¯  M of M is a Polish space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Then there exist  constants c, t0 > 0 such that  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='25)  inf  ν∈P Eν[W1(µB  t , µ)] ≥ ct− 1  2,  t ≥ t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (2) Assume that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='9) holds for p = 1, and there exists constants k, d′′ > 0 such that  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='26)  sup  x∈M  µ(B(x, r)) ≤ krd′′,  r ≥ 0,  where B(x, r) := {y ∈ M : ρ(x, y) ≤ r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' If B ∈ Bα for some α ∈ [0, 1] with d′′ > 2(1+α),  then there exist constants c, t0 > 0 such that  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='27)  inf  ν∈P Eν[W1(µB  t , µ)] ≥ ct−  1  d′′−2α,  t ≥ t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  11  3  Proofs of Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='1-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='3  For a density function f with respect to µ, let (fµ)(A) :=  �  A fdµ for measurable A ⊂ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Recall that for any probability density functions f, f1, f2 ∈ L2(µ), we have  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='1)  W2(fµ, µ)2 ≤  �  M  |∇ˆL−1(f − 1)|2  M (f)  dµ,  M (f) := 1{f>0}  f − 1  log f ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='2)  Wp(f1µ, f2µ)p ≤ pp  �  M  |∇ˆL−1(f1 − f2)|p  f p−1  2  dµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  These estimates have been presented in [2] and [21] respectively by using the Kantorovich dual  formula and properties on Hamilton-Jacobi equations, which are available when (M, ρ) is a  length space as we assumed, see [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Since the empirical measure µB  t is singular with respect to µ, to apply these estimates we  make the following regularization of µB  t :  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='3)  µB  t,r := f B  t,rµ,  f B  t,r := 1 +  ∞  �  i=1  e−λirψB  i (t),  t, r > 0,  where ψB  i (t) :=  1  √  t  � t  0 φi(XB  s )ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Letting ν ˆPr being the distribution of ˆXr with initial distribu-  tion ν, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='10) and the spectral representation  ˆpr(x, y) = 1 +  ∞  �  i=1  e−λirφi(x)φi(y)  for the heat kernel of ˆPr with respect to µ, we have  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='4)  µB  t,r = µB  t ˆPr,  t, r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  So, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='1) implies  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='5)  W2(µB  t,r, µ)2 ≤  �  M  |∇ˆL−1(f B  t,r − 1)|2  M (f B  t,r)  dµ,  t, r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  According to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='6), we have f B  t,r → 1 as t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' When f B  t,r → 1 fast enough, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='10) and  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='3), for large enough t we may bound tW2(µB  t,r, µ)2 by  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='6)  ΞB  r (t) := tµ(|∇ˆL−1(f B  t,r − 1)|2) =  ∞  �  i=1  e−2λir  λi  ψB  i (t)2,  t, r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  On the other hand, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='2) we have  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='7)  Wp(µB  t,r, µ)p ≤ ppµ  �  |∇ˆL−1(f B  t,r − 1)|p�  ,  t, r > 0, p ∈ [1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  With the above observations, in the following we present some lemmas to estimate ΞB  r (t),  µ  �  |∇ˆL−1(f B  t,r − 1)|p�  and Wp(µB  t,r, µB  t ), which will be used to prove Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='1-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  12  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='1  Some lemmas  To apply (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='7), we need estimate ∥∇ˆL−1(f B  t,r − 1)∥p, see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='12) below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' To this and also for  later use, we first estimate ∥P B  t − µ∥p→q and ∥PtZ∥2p for q ≥ p ≥ 1, where P B  t  is the Markov  semigroup for the subordinated diffusion process XB  t given by  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='8)  P B  t f(x) := Ex[f(XB  t )] = E[PSB  t f(x)],  t ≥ 0, x ∈ M, f ∈ Bb(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Assume (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Then the following assertions hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (1) Let B ∈ Bα for some α ∈ (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' There exists a constant k > 0 such that  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='9)  ∥P B  t − µ∥p→q ≤ k(1 ∧ t)− d(q−p)  2pqα e−λt,  t > 0, q ≥ p ∈ [1, ∞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (2) If (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='8) holds, then for any p ∈ [1, ∞) there exists a constant c(p) > 0 such that  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='10)  ∥∇Ptf∥p ≤ c(p)(1 ∧ t)− 1  2e−λt∥f∥p,  t > 0, f ∈ Cb,L(M),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='11)  ∥Pt(Zf)∥2p ≤ c(p)∥Z∥∞t− 1  2e−λt∥f∥2p,  f ∈ D( ˆE ) ∩ L2p(µ), t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Moreover, for any κ ∈ (0, 1  2) there exists a constant c(p, κ) > 0 such that  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='12)  ∥∇ˆL−1f∥2p ≤ c(p, κ)∥(−ˆL)  d(p−1)  4p  −κf∥2,  µ(f) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' (a) We will use some known results on functional inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Firstly, λ1 > 0 is equiva-  lent to the Poincar´e inequality  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='13)  µ(f 2) ≤ 1  λ1  ˆE (f, f),  f ∈ D( ˆE ), µ(f) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='2) we have E (f, f) = ˆE (f, f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' So, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='13) implies  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='14)  ∥Pt − µ∥2 ≤ e−λ1t,  t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Next, according to [30, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='14 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='15], (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='6) implies the super Poincar´e inequality  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='15)  µ(f 2) ≤ r ˆE (f, f) + c1(1 + r− d  2)µ(|f|)2,  r > 0, f ∈ D( ˆE )  for some constant c1 > 0, which further yields  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='16)  ∥Pt∥1→∞ ≤ c2(1 ∧ t)− d  2,  t > 0  for some constant c2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Noting that B ∈ Bα implies  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='17)  B(λ) ≥ k1{λ≥λ1}λα  13  for some constant k > 0, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='5) we find a constant c3 > 0 such that  E[(SB  t )−d] = E  � ∞  0 λd−1e−λSB  t dλ  � ∞  0 λd−1e−λdλ  ≤ c3(1 ∧ t)− d  α,  t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Combining this with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='5), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='8), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='14) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='16), we find constants c4, c5 ≥ 1 such that  ∥P B  t − µ∥1→∞ ≤ E[∥PSB  t − µ∥1→∞] ≤ c4E  ��  1 + (SB  t )− d  2�  e−λ1SB  t  �  ≤ 2c4  �  E[1 + (SB  t )−d]  � 1  2�  E[e−2λ1SB  t ]  � 1  2 ≤ c5(1 ∧ t)− d  2α e−B(2λ1)t/2,  t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  By the interpolation theorem, this and ∥P B  t − µ∥p ≤ 2 for p ∈ [1, ∞] imply (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='9) for some  constants c, λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' In particular, for B(r) = r and Z = 0 or Z ̸= 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='9) implies to  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='18)  ∥ ˆPt − µ∥p→q ∨ ∥Pt − µ∥p→q ≤ kt− d(q−p)  2pq e−λt,  t > 0, q ≥ p ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (b) To prove (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='10), we first prove that for some decreasing c : (1, ∞) → (0, ∞),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='19)  |∇ ˆPtf| ≤ c1(p)  √  t ( ˆPt|f|p)  1  p,  t ∈ (0, 1], f ∈ Cb,L(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  By H¨older’s inequality and using f = f + − f −, it suffices to prove for p ∈ (1, 2] and f ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Moreover, by first using f + ε replacing f for ε > 0 then letting ε ↓ 0, we may and do assume  that inf f > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='8) we have ˆPt−sf ∈ D(ˆL) ∩ Cb,L(M) for s ∈ [0, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Then by the chain rule we obtain  d  ds  ˆPs( ˆPt−sf)pds = ˆPs ˆL( ˆPt−sf)p − ˆPs  �  p( ˆPt−sf)p−1ˆL ˆPt−sf  �  = p(p − 1) ˆPs  �  ( ˆPt−sf)p−2|∇ ˆPt−sf|2�  ,  s ∈ [0, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  So, for p ∈ (1, 2], we have  I := ˆPtf p − ( ˆPtf)p =  � t  0  d  ds  ˆPs( ˆPt−sf)pds  = p(p − 1)  � t  0  ˆPs  �  ( ˆPt−sf)p−2|∇ ˆPt−sf|2�  ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='20)  By the H¨older and Jensen inequalities, we obtain  � ˆPs|∇ ˆPt−sf|p� 2  p ≤  � ˆPs{(Pt−sf)p−2|∇ ˆPt−sf|2}  �� ˆPs( ˆPt−sf)p� 2−p  p  ≤  � ˆPs{(Pt−sf)p−2|∇ ˆPt−sf|2}  �� ˆPtf p� 2−p  p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Combining this with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='20) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='8) where we may assume that k(p) is decreasing in p due to  Jensen’s inequality, we find increasing C : (1, ∞) → (0, ∞) such that  I ≥ p(p − 1)  � t  0  � ˆPs|∇ ˆPt−sf|p� 2  p( ˆPtf p)  p−2  p  14  ≥ C(p)  � t  0  |∇ ˆPtf|2( ˆPtf p)  p−2  p ds = C(p)t|∇ ˆPtf|2( ˆPtf p)  p−2  p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  This implies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='19) for some decreasing c : (1, ∞) → (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Next, we intend to prove that for some constant c > 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='21)  ∥∇Ptf∥∞ ≤ c∥f∥∞t− 1  2,  t ∈ (0, 1], f ∈ Bb(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  For any x ̸= y ∈ M, let  ht(x, y) :=  sup  ∥g∥∞≤1  |Ptg(x) − Ptg(y)|  ρ(x, y)  ,  t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='3) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='19), we find a constant c1 > 0 such that  ht(x, y) ≤ c1t− 1  2 +  � t  0  c1(t − s)− 1  2hs(x, y)ds,  t ∈ (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  By the generalized Gronwall inequality, see [41], this implies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Moreover, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='19), the Lp-contraction of Pt and ˆPt, and the Duhamel’s formula  Ptf = ˆPtf +  � t  0  ˆPs(ZPt−sf)ds,  we obtain  ∥∇Ptf∥p ≤ ∥∇ ˆPtf∥p +  � t  0  ∥∇ ˆPs(ZPt−sf)∥pds  ≤ c(p)t− 1  2∥f∥p +  � t  0  c(p)∥Z||∞s− 1  2∥∇Pt−sf∥pds,  t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  When f ∈ Bb(M), by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='21) and the generalized Gronwall inequality, this imply (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='10) for  t ∈ (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Finally, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='18) for p = q such that  ∥Pt − µ∥p ≤ ke−λt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Combining this with the semigroup property and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='10) for t ∈ (0, 1], for any t > 1 we have  ∥∇Ptf∥p = ∥∇P1(Pt−1f − 1)∥p ≤ c(p)∥Pt−1(f − 1)∥p ≤ c(p)ke−λ(t−1)∥f∥p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  So, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='10) also holds for t > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (c) Let P ∗  t be the L2(µ)-adjoint operator of Pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='2), P ∗  t is the diffusion semigroup  generated by L∗ := ˆL − Z, and satisfies  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='22)  P ∗  t g = ˆPtg −  � t  0  ˆPs(ZP ∗  t−sg)ds,  t > 0,  g ∈ L2(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  15  Let g ∈ D( ˆE ) with ∥g∥  2p  2p−1 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' We have  ∥∇P ∗  t g∥2  2p  2p−1 ≤ ∥∇P ∗  t g∥2  2 = ˆE (P ∗  t g, P ∗  t g) ≤ ˆE (g, g) < ∞,  t ≥ 0,  so that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='2), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='10) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='22) yield that for some constant c1 > 0,  ∥∇P ∗  t g∥  2p  2p−1 ≤ c1t− 1  2 + c1  � t  0  s− 1  2∥∇P ∗  t−sg∥  2p  2p−1ds < ∞,  t ∈ (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  By the generalized Gronwall inequality, see [41], we find a constant c2 > 0 such that  sup  g∈D( ˆ  E ),∥g∥  2p  2p−1  ≤1  ∥∇P ∗  t g∥  2p  2p−1 ≤ c2t− 1  2,  t ∈ (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Combining this with the semigroup property and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='18), we find a constant c2 > 0 such that  sup  g∈D( ˆ  E ),∥g∥  2p  2p−1  ≤1  ∥∇P ∗  t g∥  2p  2p−1 ≤ c3t− 1  2e−λt,  t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Therefore, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='2), for any f ∈ D( ˆE ) ∩ L2p(µ) we have  ∥Pt(Zf)∥2p =  sup  g∈D( ˆ  E ),∥g∥  2p  2p−1  ≤1  ��µ((P ∗  t g)(Zf))  ��  =  sup  g∈D( ˆE ),∥g∥  2p  2p−1  ≤1  ��µ(f(ZP ∗  t g))  �� ≤ c3∥Z∥∞t− 1  2e−λt∥f∥2p,  t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Therefore, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='11) holds for some constants c(p), λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (d) Noting that  (−ˆL)−(1+ d(p−1)  4p  −κ) =  1  Γ(1 + d(p−1)  4p  − κ)  � ∞  0  s  d(p−1)  4p  −κ ˆPsds,  by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='6) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='10), we find constants c1, c2, c3 > 0 such that  ∥∇ˆL−1f∥2p = ∥∇(−ˆL)−(1+ d(p−1)  4p  −κ)(−ˆL)  d(p−1)  4p  −κf∥2p  ≤  1  Γ(1 + d(p−1)  4p  − κ)  � ∞  0  (1 ∧ s)  d(p−1)  4p  −κ∥∇ ˆPs/2{ ˆPs/2(−ˆL)  d(p−1)  4p  −κf}∥2pds  ≤ c1  � ∞  0  (1 ∧ s)  d(p−1)  4p  −κ− 1  2e−λs/2∥ ˆPs/2(−ˆL)  d(p−1)  4p  −κf∥2pds  ≤ c2  � ∞  0  (1 ∧ s)−(κ+ 1  2 )e−λs∥(−ˆL)  d(p−1)  4p  −κf∥2ds ≤ c3∥(−ˆL)  d(p−1)  4p  −κf∥2,  where the last step is due to 1  2 + κ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Thus, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='12) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  16  Next, we present some consequence of (A1) and (A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' We have the following assertions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (1) If (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='6) holds, then (M, ρ) is bounded, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='23)  D := sup  x,y∈M  ρ(x, y) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (2) If (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='9) holds, then for any p, q ∈ [1, ∞), there exists a constant c > 0 such that  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='24)  �  Eµ[W2p(µB  t , µB  t,r)2q]  � 1  q ≤ cr,  r ∈ (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (3) If (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='8) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='21) hold, then there exist constants κ0, κ1 > 0 such that  |∇Ptef|2 ≤ (Ptef)Pt(|∇f|2ef) + κ1tθ∥∇f∥2  ∞(Ptef)2,  for t ∈ [0, 1], f ∈ Cb,L(M) with t∥∇f∥2  ∞ ≤ κ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='25)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' (1) According to [30, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='15(2)], (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='6) implies the super Poincar´e inequality  µ(f 2) ≤ r ˆE (f, f) + (1 + r− d  2 )µ(|f|)2,  r > 0, f ∈ D( ˆE ),  which further implies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='23) due to [30, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (2) By Jensen’s inequality, we only need to prove (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='24) for q ≥ p ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Recall that δXB  s ˆPr is  the distribution of ˆXr with initial value XB  s , we have  π := 1  t  � t  0  �  δXB  s × (δXB  s ˆPr)  �  ds ∈ C (µB  t , µB  t,r),  so that  W2p(µB  t , µB  t,r)2p ≤ 1  t  � t  0  Ex[ρ(x, ˆXr)2p]  ��  x=XB  s ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Noting that Eµ =  �  M Exµ(dx) and ˆXt is stationary with initial distribution µ, by combining  this with Jensen’s inequality and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='9), we obtain  Eµ[W2p(µB  t , µB  t,r)2q] ≤ Eµ  �1  t  � t  0  Ex[ρ(x, ˆXr)2q]  ��  x=XB  s ds  �  = 1  t  � t  0  Eµ[ρ( ˆX0, ˆXr)2q]ds ≤ c(2q)rq,  t > 0, r ∈ (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  So, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='24) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (3) Let f ∈ Cb,L(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='21), we have (Pt−sef)2m ∈ D(L) ∩ Cb,L(M) for s ∈ [0, t), so that  the chain rule implies  d  dsPs(Pt−sef)2m = PsL(Pt−sef)2m − Ps  �  2m(Pt−sef)2m−1LPt−sef�  17  = 2m(2m − 1)(Pt−sef)2m−2|∇Pt−sef|2,  s ∈ [0, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  By combining this with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='8) for p = 2 and Jensen’s inequality, we find a constant c1 > 0 such  that  Pte2mf − (Ptef)2m =  � t  0  d  dsPs(Pt−sef)2mds  =  � t  0  Ps  �  2m(2m − 1)(Pt−sef)2m−2|∇Pt−sef|2�  ds  ≤ c1∥∇f∥2  ∞  � t  0  Ps  �  (Pt−sef)2m−2Pt−se2f�  ds ≤ c1t∥∇f∥2  ∞Pte2mf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Taking κ0 =  1  2c2 such that t∥∇f∥2  ∞ ≤ κ0 implies c1t∥∇f∥2  ∞ ≤ 1  2, we derive  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='26)  Pte2mf ≤ 2(Ptef)2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Combining this with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='21) we obtain (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='25) for some constant κ1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Noting that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='10) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='3) imply  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='27)  ∥(−ˆL)β(f B  t,r − 1)∥2  2 = 1  t  ∞  �  i=1  λ2β  i e−2λirψB  i (t)2,  r, t > 0, β ∈ R,  to bound Eν[Wp(µB  t,r, µ)2q] from above using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='7) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='12), we estimate Eν[|ψB  i |2q] as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Assume (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Then the following assertions hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (1) For any q ∈ [1, ∞) there exists a constant c(q) > 0 such that  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='28)  sup  t>0 Eν[|ψB  i (t)|2q] ≤ c(q)∥h∥∞λ  d(q−1)  2  i  B(λi)−q,  i ∈ N, ν = hµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (2) Let B ∈ Bα for some α ∈ (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' For any q ∈ [1, ∞) and k ∈ (  d  2αi(q), ∞] ∩ [1, ∞], there  exists a constant c(q, k) > 0 such that  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='29)  Eν[|ψB  i (t)|2q] ≤ c(q, k)∥h∥k(1 ∧ t)−  d  2αk λ  d(q−1)  2  −qα  i  ,  i ∈ N, ν = hµ, t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Moreover, if i(q) >  d  2α, then there exists a constant c(q) > 0 such that  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='30)  sup  ν∈P  Eν[|ψB  i (t)|2q] ≤ c(q)(1 ∧ t)− d  2α λ  d(q−1)  2  −qα  i  ,  i ∈ N, t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' (1) We first prove the following estimate for some constants k1, k2 > 0:  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='31)  ∥P B  t φi − e−B(λi)tφi∥2q ≤ k1∥Z∥∞λ  d(q−1)  4q  −1  i  e−k2t,  t > 0, i ∈ N, q ∈ [1, ∞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  18  By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='10) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='18), we find constants c1, c2 > 0 such that  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='32)  ∥φi∥2q = inf  s>0 ∥ ˆPsφi∥2qe−λis ≤ c1 inf  s∈(0,1] s− d(2q−2)  8q  e−λis ≤ c2λ  d(q−1)  4q  i  ,  i ∈ N, q ∈ [1, ∞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='3), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='8) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='10), we obtain  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='33)  P B  t φi = E  �  e−λiSB  t φi +  � SB  t  0  e−λi(SB  t −s)Ps(Zφi) ds  �  ,  t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Combining this with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='5), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='11) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='32), we derive  ∥P B  t φi − e−B(λi)tφi∥2q ≤ c(q)∥Z∥∞∥φi∥2qλ  d(q−1)  4q  −1  i  E  � SB  t  0  e−λi(SB  t −s)(1 ∧ s)− 1  2e−λsds  ≤ c2c(q)∥Z∥∞λ  d(q−1)  4q  i  E  � SB  t  0  e−λi(SB  t −s)(1 ∧ s)− 1  2e−λsds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='34)  Noting that λi ≥ λ1 ≥ λ implies  −λi(SB  t − s) − λs ≤ −λi  2 (SB  t − s) − λ  2(SB  t − s) − λs = −λi  2 (SB  t − s) − λ  2SB  t − λ  2s,  by the FKG inequality, we find a constant c3 > 0 such that  � SB  t  0  e−λi(SB  t −s)(1 ∧ s)− 1  2e−λsds ≤ e−λSB  t /2  � SB  t  0  e−λi(SB  t −s)/2(1 ∧ s)− 1  2e−λs/2ds  ≤ e−λSB  t /2  � � SB  t  0  e−λi(SB  t −s)/2ds  � 1  SB  t  � SB  t  0  (1 ∧ s)− 1  2e−λsds  ≤ c3  λi  e−λSB  t /2(1 ∧ SB  t )− 1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='35)  Moreover, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='5), the increasing property of B and  � ∞  0 e− 1  2 B(λ)dλ < ∞ in the definition of B,  we find constants c4, c5, c6 > 0 such that  E  �  (1 ∧ SB  t )− 1  2e−λSB  t /2�  ≤ E  ��  1 + c2  � ∞  0  u− 1  2e−uSB  t du  �  e−λSB  t /2  �  = e−B(λ/2)t + c4  � ∞  0  u− 1  2e−B(u+λ/2)tdu  ≤ e−B(λ/2)t + c4  � ∞  0  u− 1  2e− 1  2 B(u)− 1  2B(λ/2)tdu ≤ c5e−c6t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='36)  Combining this with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='34) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='35), we obtain (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Next, we prove (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='28) for q ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' By [37, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='14)] for f = φi, we find a constant k0 > 0 such  that  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='37)  Eν�  |ψB  i (t)|2q�  ≤ k0  �1  t  � t  0  ds1  � s1  0  �  Eν[|φiP B  s1−sφi|q](XB  s )  � 1  q ds  �q  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  19  By ν = hµ and the Markov property, we obtain  Eν�  |φiP B  s1−sφi|q(XB  s )  �  = µ  �  hP B  s |φiP B  s1−sφi|q�  ≤ ∥h∥k  ��P B  s |φiP B  s1−sφi|q��  k  k−1  ≤ ∥h∥k∥P B  s ∥1→  k  k−1∥φiP B  s1−sφi∥q  q ≤ ∥h∥k∥P B  s ∥1→  k  k−1∥φi∥q  2q∥P B  s1−sφi∥q  2q,  k ∈ [1, ∞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='38)  Taking k = ∞ and combining with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='31), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='32) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='37), we find constants k1, k2 > 0 such  that  Eν�  |ψB  i (t)|2q�  ≤ k1∥h∥∞λ  d(q−1)  2  i  �1  t  � t  0  ds1  � s1  0  �  e−B(λi)(s1−s) + λ−1  i e−k2(s1−s)�  ds  �q  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Noting that B(λi) ≤ cλi for some constant c > 0 and all i ∈ N, this implies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='28) for q ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Finally, for any q ∈ (1, ∞), let i(q) be the integer part of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='28) for i(q) and 1 + i(q)  replacing q which have just been proved, and using H¨older’s inequality, we find a constant  c(q) > 0 such that  Eν�  |ψB  i (t)|2q�  ≤  �  Eν�  |ψB  i (t)|2i(q)��i(q)+1−q�  Eν�  |ψB  i (t)|2+2i(q)��q−i(q)  ≤ c(q)∥h∥∞λ  1  2{d(i(q)−1)(i(q)+1−q)+di(q)(q−i(q))}  i  B(λi)−i(q)(i(q)+1−q)−(1+i(q))(q−i(q))  = c(q)∥h∥∞λ  d(q−1)  2  i  B(λi)−q,  t > 0, i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='39)  Then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='28) is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (2) By the same reason leading to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='39), we only need to prove (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='29) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='30) for q ∈ N  so that i(q) = q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Let k ∈ ( d  2αq, ∞] ∩ [1, ∞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='17), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='31), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='37) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='38), we find  constants c1, c2 > 0 such that  Eν�  |ψB  i (t)|2q�  ≤ c1∥h∥kλ  d(q−1)  2  i  �1  t  � t  0  ds1  � s1  0  (1 ∧ s)−  d  2kqα�  e−c2λα  i (s1−s) + λ−1  i e−c2(s1−s)�  ds  �q  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='40)  By the FKG inequality, we find a constant c3 > 0 such that  � s1  0  (1 ∧ s)−  d  2kqα �  e−c2λα  i (s1−s) + λ−1  i e−c2(s1−s)�  ds  ≤  � 1  s1  � s1  0  (1 ∧ s)−  d  2kqα ds  � � s1  0  �  e−c2λα  i (s1−s) + λ−1  i e−c2(s1−s)�  ds  ≤ c3(1 ∧ s1)−  d  2kqαλ−α  i  ,  t > 0, i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='41)  This together with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='40) yields  Eν�  |ψB  i (t)|2q�  ≤ c∥h∥k(1 ∧ t)−  d  2αk λ  d(q−1)  2  −qα  i  ,  t > 0, i ∈ N  for some constant c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Therefore, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='29) holds for q ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  20  It remains to prove (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='30) for  d  2α < q ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='9), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='31), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='32) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='17), we find  constants c1, c2 > 0 such that  sup  ν∈P  �  Eν�  |φiP B  s1−sφi|q(XB  s )  �� 1  q = sup  ν∈P  �  ν  �  P B  s |φiP B  s1−sφi|q�� 1  q  ≤ ∥P B  s ∥  1  q  1→∞∥φiP B  s1−sφi∥q ≤ c1(1 ∧ s)−  d  2qα ∥φi∥2q∥P B  s1−sφi∥2q  ≤ c1(1 ∧ s)−  d  2qα λ  d(q−1)  2q  i  �  e−c2λα  i (s1−s) + λ−1  i e−c2(s1−s)�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Combining this with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='37) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='41) for k = 1, we get (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='30) for md,α ≤ q ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  We are now ready to estimate Eν[∥(−ˆL)β(f B  t,r − 1)∥2q  2 ] for β ∈ R and q ∈ [1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' More  generally, to apply (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='2) we estimate Eν[∥(a0 − ˆL)  1  2(−ˆL)β− 1  2(f B  t,r − 1)∥2q  2 ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Assume (A1), (A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Let a0 ∈ [0, ∞), q ∈ [1, ∞), β ∈ R, and B ∈ Bα for some  α ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (1) For any k ∈ (  d  2αi(q), ∞] ∩ [1, ∞] where k = ∞ if α = 0, for any R ∈ [1, ∞), there exists a  constant c > 0 such that  sup  t≥1,ν∈Pk,R  tqEν���(a0 − ˆL)  1  2(−ˆL)β− 1  2(f B  t,r − 1)  ��2q  2  �  ≤ c  �  r−(2β+ d′  2 + d(q−1)  2q  −α)+ + 1{α=2β+ d′  2 + d(q−1)  2q  } log(1 + r−1)  �q,  r ∈ (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='42)  (2) If i(q) >  d  2α, then there exists a constant c > 0 such that  sup  t≥1,ν∈P  tqEν���(a0 − ˆL)  1  2(−ˆL)β− 1  2(f B  t,r − 1)  ��2q  2  �  ≤ c  �  r−(2β+ d′  2 + d(q−1)  2q  −α)+ + 1{α=2β+ d′  2 + d(q−1)  2q  } log(1 + r−1)  �q,  r ∈ (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='43)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='17), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='27), H¨older’s inequality and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='29), we find a constant c1 > 0 such that  I :=  sup  t≥1,ν∈Pk,R  tqEν���(a0 − ˆL)  1  2(−ˆL)β− 1  2(f B  t,r − 1)  ��2q  2  �  =  sup  t≥1,ν∈Pk,R  Eν  �����  ∞  �  i=1  �λi + a0  λi  �  λ2β  i e−2λirψB  i (t)2  ����  q�  ≤  �λ1 + a0  λ1  �q  sup  t≥1,ν∈Pk,R  � ∞  �  i=1  λθ  i e−2λir�q−1  ∞  �  i=1  λ2qβ−θ(q−1)  i  e−2λirEν[|ψB  i (t)|2q]  ≤ c1  �  ∞  �  i=1  λθ  ie−2λir�q−1  ∞  �  i=1  λ  2qβ−θ(q−1)+ d(q−1)  2  −qα  i  e−2λir,  θ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Taking  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='44)  θ := 2β + d(q − 1)  2q  − α,  21  so that θ = 2qβ − θ(q − 1) + d(q−1)  2  − qα, and noting that  λθ+  i e−λir ≤ sup  s>0  sθ+e−sr ≤ cr−θ+,  r ∈ (0, 1]  holds for some constant c > 0 depending on θ+, we find a constant c2 > 0 such that  I ≤ c1  �  ∞  �  i=1  λθ  i e−2λir�q  ≤ c2r−qθ+�  ∞  �  i=1  λ−θ−  i  e−λir�q  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  On the other hand, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='7) and the integral transform t = rs  2  d′ , we find constants c3, c4, c5 > 0  such that  ∞  �  i=1  λ−θ−  i  e−λir ≤ c3  � ∞  1  s− 2θ−  d′ e−c3rs  2  d′ ds  = c3d′  2  � ∞  r  rθ−− d′  2 t  d′  2 −θ−−1e−c3tdt  ≤ c5  �  r−( d′  2 −θ−)+ + 1{ d′  2 =θ−} log(1 + r−1)  �  ,  r ∈ (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='45)  Thus, we find a constant c6 > 0 such that  I ≤ c6r−qθ+�  r−( d′  2 −θ−)+ + 1{ d′  2 =θ−} log(1 + r−1)  �q  = c6  �  r−( d′  2 +θ)+ + 1{ d′  2 +θ=0} log(1 + r−1)  �q,  r ∈ (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  This together with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='44) implies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  When i(q) >  d  2α, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='43) can be proved in the same way by using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='30) replacing (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  The next lemmas shows that the limit of E[ΞB  r (t)] as t → ∞ is  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='46)  ηB  Z,r :=  ∞  �  i=1  2e−2λir  λi  VB(φi),  r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Assume (A1), and let R ∈ [1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (1) There exists a constant c > 0 such that  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='47)  ηB  Z,r ≤  ∞  �  i=1  ce−2rλi  λiB(λi),  r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Consequently, ηB  Z < ∞ provided �∞  i=1  1  λiB(λi) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  22  (2) There exists a constant c > 0 such that  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='48)  sup  ν∈P∞,R  ��Eν[ΞB  r (t)] − ηB  Z,r  �� ≤ c  t  ∞  �  i=1  e−2λir  λiB(λi),  t, r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Consequently, when �∞  i=1  1  λiB(λi) < ∞, there exists a constant c′ > 0 such that  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='49)  sup  ν∈P2,R  Eν[ΞB(t)] ≤ ηB  Z + c′  t < ∞,  t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (3) Let B ∈ Bα for some α ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' For any k >  d  2α, there exists a constant c > 0 such that  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='50)  sup  t≥1,ν∈Pk,R  Eν[ΞB  r (t)] ≤ c2  �  r−( d′  2 −1−α)+ + 1{d′=2(1+α)} log(1 + r−1)  �  ,  r ∈ (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' (1) By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='31) for q = 1, and noting that B(λi) ≤ c0λi for some constant c0 > 0, we find  constants c1, c2, c3 > 0 such that  VB(φi) :=  � ∞  0  µ(φiPtφi)dt ≤ c1  � ∞  0  (e−B(λi)t + λ−1  i e−c2t)dt ≤  c3  B(λi),  i ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  This together with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='46) implies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' By the dominated convergence theorem with r → 0,  the claimed consequence follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (2) By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='3) and the Markov property, we obtain  Eν[ψB  i (t)2] = 2  t  � t  0  ds1  � s1  0  Eν[φi(XB  s1)φi(XB  s )]ds  = 2  t  � t  0  ds1  � s1  0  ν  �  P B  s {φiP B  s1−sφi}  �  ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='51)  Since ν ∈ P∞,R implies ν = hµ with ∥h − 1∥∞ ≤ R + 1, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='3), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='40) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='9), we find  constants c1, c2 > 0 such that  sup  ν∈P∞,R  ��ν  �  P B  s {φiP B  s1−sφi}  �  − µ(φiP B  s1−sφ)  �� =  sup  ν∈P∞,R  ��µ  �  {P B  s )∗h − 1}φiP B  s1−sφ)  ��  ≤ c1e−c2s∥φiP B  s1−sφi∥1 ≤ c1e−c2s∥P B  s1−sφi∥2 ≤ c1e−c2s�  e−B(λi)(s1−s) + λ−1  i e−c2(s1−s)�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Combining this with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='51) and that B(λi) ≤ c0λi for some constant c0 > 0, we find a constant  c3 > 0 such that  sup  ν∈P∞,R  ����tEν[ΞB  r (t)] −  ∞  �  i=1  2e−2λir  λit  � t  0  ds1  � s1  0  µ(φiP B  s1−sφi)ds  ����  ≤  ∞  �  i=1  2e−2λir  λit  � t  0  ds1  � s1  0  c1e−c2s�  e−B(λi)(s1−s) + λ−1  i e−c2(s1−s)�  ds  ≤ c3  t  ∞  �  i=1  e−2λir  λiB(λi),  t, r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='52)  23  Similarly, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='31) for q = 1, ∥φi∥2 = 1 and B(λi) ≤ c0λi, we find constants c4, c5, c6 > 0 such  that  ����  � s1  0  µ  �  φiP B  s1−sφi  �  ds − VB(φi)  ���� ≤  � ∞  s1  ��µ  �  φiP B  s φi  ��� ≤  � ∞  s1  ∥P B  s φi∥2ds  ≤ c4  � ∞  s1  �  e−B(λi)s + λ−1  i e−c5s�  ds ≤ c6e−c5s1  B(λi) ,  s1 > 0, i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  This together with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='52) implies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='48) for some constant c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (3) Let k >  d  2α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='29) for q = 1, and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='45) for θ− = 1 + α, we find constants c1, c2 > 0  such that  sup  t≥1,ν∈Pk,R  Eν[ΞB  r (t)] =  sup  t≥1,ν∈Pk,R  ∞  �  i=1  e−2λir  λi  Eν[ψB  i (t)2]  ≤ c1  ∞  �  i=1  e−2λir  λ1+α  i  ≤ c2  �  r−( d′  2 −1−α)+ + 1{d′=2(1+α)} log(1 + r−1)  �  ,  r ∈ (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Then the proof is finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Finally, to get rid of the term M (f B  t,r) from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='5), we present one more lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Assume (A1) and (A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Let B ∈ Bα for some α ∈ [0, 1] such that d′ < 2(1 + α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Then the following assertions hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (1) There exists a constant c > 0 and σ ∈ (0, 1) such that  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='53)  Eµ[µ(|f B  t,r − 1|2)] ≤ ct−1r−σ,  t ≥ 1, r ∈ (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (2) There exists a constant γ > 1 such that  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='54)  lim  t→∞ Eµ�  µ  �  |M (ft,t−γ)−1 − 1|q��  = 0,  q ∈ [1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (3) We have qα > 1 and  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='55)  sup  t,r>0 tqEµ�  µ  �  |∇ˆL−1(f B  t,r − 1)|2q��  < ∞,  q ∈ [1, qα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' (1) By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='3), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='28) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='17), we find constants c1, c2 > 0 such that  tEµ[µ(|f B  t,r − 1|2)] ≤ c1  ∞  �  i=1  e−2λirEµ[ψB  i (t)2] ≤ c2  ∞  �  i=1  e−2λirλ−α  i  ,  t, r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Since d′ < 2(1 + α) implies d′  2 − α < 1, combining this with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='45) we derive (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='53) for any  σ ∈ ( d′  2 − α, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  24  (2) Let σ ∈ (0, 1) be in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='53) and take θ ∈ (0, q−1(1 − σ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' According to [37, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='2],  for any η ∈ (0, 1) and δ(η) := |(1 − η)− 1  2 −  2  η+2|, we have  Eµ�  |M (ft,r(y))−1 − 1|q�  ≤ δ(η) + (1 + θ−1r− θ  2)qEµ[1{|fB  t,r(y)−1|>η}],  t ≥ 1, r ∈ (0, 1], y ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Next, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='53) and Chebyshev’s inequality, we obtain  �  M  Eµ[1{|fB  t,r(y)−1|>η}]µ(dy) ≤ η−2Eµ[µ(|f B  t,r − 1|2)] ≤ cη−2t−1r−σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Putting these two estimates together, we find a function C : (0, 1) → (0, ∞) such that  Eµ�  µ  �  |M (ft,r)−1 − 1|q��  ≤ δ(η) + C(η)t−1r−(σ+ θ  2 ),  t ≥ 1, r, η ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Noting that θ ∈ (0, q−1(1 − σ)) implies θ′ := σ + θ  2 ∈ (0, 1), by taking r = t−γ for γ ∈ (1, 1  θ′),  and letting first t → ∞ then η → 0, we derive (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (3) By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='12), d′ < 2(1 + α) implies qα > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' For any q ∈ [1, qα), we have  d(q − 1)  2q  − 1 < α < −d′  2 − d(q − 1)  2q  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  So, there exists κ ∈ (0, 1  2) such that β := d(q−1)  4q  − κ satisfies  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='56)  2β < α − d′  2 − d(q − 1)  2q  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='12), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='3), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='17) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='42), we find constants c1, c2 > 0 such that  tqEµ�  µ  �  |∇ˆL−1(f B  t,r − 1)|2q��  ≤ c1tqEµ�  ∥(−ˆL)  d(q−1)  4q  −κ(f B  t,r − 1)∥2q  2  �  ≤ c2r−(2β+ d′  2 + d(q−1)  2q  −α)+ = c2,  where the last step follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='56).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='55) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='2  Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='1  We first consider the stationary case where the initial distribution is the invariant measure µ,  then extend to more general setting by using an approximation argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' To this end, we need  the following further modification of the empirical measure:  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='57)  d˜µB  t,r = ˜f B  t,rdµ,  ˜f B  t,r := (1 − r)f B  t,r + r,  r ∈ (0, 1], t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='2), we have  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='58)  W2(µB  t,r, ˜µB  t,r)2 ≤ 4rΞB  r (t),  t > 0, r ∈ (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  25  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Assume (A1) and (A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Let B ∈ Bα for some α ∈ [0, 1] such that d′ <  2(1 + α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Then  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='59)  lim  t→∞ Eµ���{tW2(µB  t , µ)2 − ΞB(t)}+��q�  = 0,  q ∈ [1, qα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Let t ≥ 1 and take rt = t−γ for γ > 1 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='6) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='57), we have  tµ(|∇ˆL−1( ˜ft,rt − 1)|2) = (1 − rt)2ΞB  rt(t),  so that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='5), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='55) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='54) yield  lim  t→∞ Eµ���{tW2(˜µB  t,rt, µ)2 − (1 − rt)2ΞB  rt(t)}+��q�  = lim  t→∞ Eµ���{tW2(˜µB  t,rt, µ)2 − tµ(|∇ˆL−1( ˜ft,rt − 1)|2)}+��q�  ≤ lim  t→∞ tqEµ��  µ  �  |∇ˆL−1( ˜f B  t,rt − 1)|2|M ( ˜f B  t,rt)−1 − 1|  ��q�  ≤ lim  t→∞ tqEµ�  µ  �  |∇ˆL−1( ˜ft,rt − 1)|2q|M ( ˜f B  t,rt)−1 − 1|q��  ≤ lim  t→∞  �  tq′Eµ�  µ  �  |∇ˆL−1( ˜ft,rt − 1)|2q′��� q  q′ �  Eµ�  µ  �  |M ( ˜f B  t,rt)−1 − 1|  qq′  q′−q ��� q′−q  q′  = 0,  q′ ∈ (q, qα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Noting that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='11) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='6) imply  ΞB(t) ≥ ΞB  rt(t) ≥ (1 − rt)2ΞB  rt(t),  we derive  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='60)  lim  t→∞ Eµ���{tW2(˜µB  t,rt, µ)2 − ΞB(t)}+��q�  = 0,  q ∈ [1, qα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  On the other hand, noting that q ∈ [1,  d  (d+d′−2−2α)+ ) implies 1 + α − d′  2 − d(q−1)  2q  < 0, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='6),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='7) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='42), we obtain  sup  r>0,t≥1  4−qtqEµ[W2(µB  t,r, µ)2q] ≤  sup  r>0,t≥1  tqEµ�  ∥(−ˆL)− 1  2(f B  t,r − 1)∥2q  2  �  =  sup  r>0,t≥1  Eµ�  ΞB  r (t)q�  < ∞,  1 ≤ q <  d  (d + d′ − 2 − 2α)+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='61)  Since q ∈ [1, qα) implies q ∈ [1,  d  (d+d′−2−2α)+ ), by combining this with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='24), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='58), the triangle  inequality and rt = t−γ with γ > 1, we find a constant c > 0 such that  lim  t→∞ tqEµ[W2(µB  t , ˜µB  t,rt)2q]  ≤ 2q lim  t→∞ tqEµ[W2(µB  t,rt, ˜µB  t,rt)2q + W2(µB  t,rt, µB  t )2q]  ≤ c lim  t→∞(rtt)q�  1 + Eµ[ΞB  rt(t)q]  �  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='62)  This together with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='60) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='61) implies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='59).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  26  Next, we consider arbitrary initial distribution ν ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Let  νε := ν ˆPε,  ε ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='6), there exists a constant c > 0 such that  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='63)  νε ≤ cε− d  2αµ,  Eνε ≤ cε− d  2αEµ,  ε ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Let  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='64)  µB,ε  t  := 1  t  � t+ε  ε  δXB  s ds,  t, ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  By the Markov property, µB,ε  t  is the empirical measure with initial distribution νε, so that for  any nonnegative measurable function F on P,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='65)  Eν[F(µB,ε  t  )] = Eνε[F(µB  t )],  t, ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  To estimate W2(µB,ε  t  , µ), we take  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='66)  ΞB,ε(t) :=  ∞  �  i=1  1  λi  ψB,ε  i  (t)2,  ψB,ε  i  (t) := 1  √  t  � t+ε  ε  φi(XB  s )ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  We have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Assume (A1) and (A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Let B ∈ Bα for some α ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (1) If α > 0 such that d + d′ < 2 + 4α, then for any q ∈ [1,  d  (d+d′−2−2α)+ ),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='67)  lim  ε↓0  sup  t≥1,ν∈P  Eν�  |tW2(µB  t , µ)2 − tW2(µB,ε  t  , µ)2|2q�  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (2) If α > 0 and d + d′ ≥ 2 + 4α, then for any k > d+d′−2−2α  2α  and q ∈ [1,  d  (d+d′−2−2α)+ ),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='68)  lim  ε↓0  sup  t≥1,ν∈Pk,R  Eν�  |tW2(µB  t , µ)2 − tW2(µB,ε  t  , µ)2|2q�  = 0,  R ∈ [1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (3) For any q ∈ [1, ∞), k = ∞ or k ∈ (  d  2αi(q), ∞] ∩ [1, ∞], there exists a constant c > 0 such  that  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='69)  sup  ν∈Pk,R  Eν�  |ψB,ε  i  (t)2 − ψB  i (t)2|q�  ≤ cRε  q  2t− q  2λ  d(q−1)  2  −qα  i  ,  i ∈ N, t ≥ 1, ε ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Moreover, if i(q) >  d  2α, then there exists a constant c > 0 such that  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='70)  sup  ν∈P  Eν�  |ψB,ε  i  (t)2 − ψB  i (t)2|q�  ≤ cε  q  2t− q  2λ  d(q−1)  2  −qα  i  ,  i ∈ N, t ≥ 1, ε ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  27  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' (1) By d + d′ < 2 + 4α, we have  d  2α <  d  (d+d′−2−2α)+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' So, it suffices to prove for 1 ≤ q ∈  ( d  2α,  d  (d+d′−2−2α)+ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' For t > ε ≥ 0, we have  π := 1  t  � t  ε  δ(XB  s ,XB  s )ds + 1  t  � ε  0  δ(XB  s ,XB  t+s)ds ∈ C (µB  t , µB,ε  t  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  So, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='23) implies  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='71)  Wp(µB  t , µB,ε  t  )p ≤ 1  t  � ε  0  ρ(XB  s , XB  s+t)pds ≤ εDp  t ,  t > ε ≥ 0, p ∈ [1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  On the other hand, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='2), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='61), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='63) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='65), we find a map  c :  �  1,  d  (d + d′ − 2 − 2α)+  �  → (0, ∞)  such that  sup  t≥1,ν∈P  tqEν[W2(µB,ε  t  , µ)2q] =  sup  t≥1,r∈(0,1],ν∈P  tqEνε[W2(µB  t,r, µ)2q]  ≤ 4q  sup  t≥1,r∈(0,1],ν∈P  Eνε[ΞB  r (t)q] ≤ c(q)ε− d  2α ,  ε ∈ (0, 1), q ∈  �  1,  d  (d + d′ − 2 − 2α)+  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='72)  Combining this with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='71), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='63) and q >  d  2α, we find constants c1, c2 > 0 such that  lim  ε↓0  sup  t≥1,ν∈P  Eν�  |tW2(µB  t , µ)2 − tW2(µB,ε  t  , µ)2|2q�  ≤ lim  ε↓0  sup  t≥1,ν∈P  Eν�  |tW2(µB  t , µB,ε  t  )2 + 2tW2(µB  t , µB,ε  t  )W2(µB,ε  t  , µ)|2q�  ≤ lim  ε↓0  sup  t≥1,ν∈P  c1  �  ε2q + εq sup  ν∈P  Eνε[W2(µB  t , µ)2q]  �  ≤ lim  ε↓0 c2εq− d  2α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  So, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='67) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (2) Let α > 0, d + d′ ≥ 2 + 4α and k > d+d′−2−2α  2α  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' It suffices to prove (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='68) for 1 ≤ q ∈  ( d  2αk,  d  (d+d′−2−2α)+ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' By the same reason leading to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='63), we find a constant c > 0 such that  sup  ν∈Pk,R  Eνε ≤ cε−  d  2αk Eµ,  ε ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Hence, as shown above, we derive  lim  ε↓0  sup  t≥1,ν∈Pk,R  Eν�  |tW2(µB  t , µ)2 − tW2(µB,ε  t  , µ)2|2q�  ≤ lim  ε↓0 c2εq−  d  2αk = 0  since q >  d  2αk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  28  (3) Let ψB  i and ψB,ε  i  be in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='11) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='66).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' We have  ��ψB  i (t)2 − ψB,ε  i  (t)2��  = 1  t  ����  � ε  0  �  φi(XB  t+s) − φi(XB  s )  �  ds  ����  ����  � t  0  �  φi(XB  s+ε) + φi(XB  s )  �  ds  ����  ≤  √ε  √  t  �  |ψB,t  i  (ε) + |ψB  i (ε)|  ��  |ψB,ε  i  (t) + |ψB  i (t)|  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Since (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='65) implies {νt : ν ∈ P, t ≥ 1} ⊂ P∞,R for some R > 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='28) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='17) yield  sup  ε>0,t≥1,ν∈P  Eν[ψB,t  i  (ε)2q] ≤ c1λ  d(q−1)  2  −qα  i  Eµ  for some constant c1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Moreover, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='28) for k = ∞, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='29) for α > 0 and k ∈ (  d  2αi(q), ∞] ∩  [1, ∞], and the fact that ν ∈ Pk,R implies νε ∈ Pk,R for ε > 0, we find a constant c2 > 0 such  that  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='73)  sup  ε>0,t≥1,ν∈Pk,R  Eν[ψB,ε  i  (t)2q + ψB  i (ε)2q] ≤ c2λ  d(q−1)  2  −qα  i  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Combining these estimate we derive (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='69).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Finally, when i(q) >  d  2α, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='70) can be proved in the same way by using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='30) in place of  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  We are now ready to prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' (1) It suffices to prove for q ∈ ( d  2α, qα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' By α > α(d, d′), we have  d  2α <  2(1 + α) − d′  (d + d′ − 2 − 2α)+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  So, either i(q) >  d  2α or i(q) <  2+2α−d′  (d+d′−2−2α)+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Below we consider these two situations respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (1a) Let i(q) >  d  2α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='65), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='63) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='60), we obtain  lim  t→∞ sup  ν∈P  Eν����  �  tW2(µB,ε  t  , µ)2 − ΞB,ε(t)  �+���  q�  = lim  t→∞ sup  ν∈P  Eνε����  �  tW2(µB  t , µ)2 − ΞB(t)  �+���  q�  ≤ lim  t→∞ cε− d  2α Eµ����  �  tW2(µB  t , µ)2 − ΞB(t)  �+���  q�  = 0,  ε ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='74)  Next, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='11) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='66),  |ΞB(t) − ΞB,ε(t)| ≤  ∞  �  i=1  1  λi  |ψB,ε  i  (t)2 − ψB  i (t)2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  29  Combining this with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='70), when i(q) >  d  2α we find a constant k1 > 0 such that  sup  t≥1,ν∈P  t  q  2Eν�  |ΞB(t) − ΞB,ε(t)|q�  ≤  � ∞  �  i=1  λ−θ  i  �q−1  ∞  �  i=1  λθ(q−1)−q  i  sup  t≥1,ν∈P  t  q  2Eν�  |ψB,ε  i  (t)2 − ψB  i (t)2|q�  ≤ k1ε  q  2  �  ∞  �  i=1  λ−θ  i  �q−1  ∞  �  i=1  λ  θ(q−1)+ d(q−1)  2  −q−αq  i  ,  θ ∈ R, ε ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Taking  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='75)  θ = 1 + α − d(q − 1)  2q  such that −θ = θ(q − 1) + d(q−1)  2  − q − αq, we arrived at  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='76)  sup  t≥1,ε∈(0,1),ν∈P  ε− q  2t  q  2Eν�  |ΞB(t) − ΞB,ε(t)|q�  ≤ k1  �  ∞  �  i=1  λ−θ  i  �q  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Noting that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='12), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='75) and q < qα imply 2θ  d′ > 1, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='7) we find a constant k2 > 0 such  that  ∞  �  i=1  λ−θ  i  ≤ k2  ∞  �  i=1  i− 2θ  d′ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Thus,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='77)  sup  t≥1,ε∈(0,1),ν∈P  ε− q  2t  q  2Eν�  |ΞB(t) − ΞB,ε(t)|q�  < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Consequently,  lim  t→∞  sup  ε∈(0,1),ν∈P  Eν�  |ΞB(t) − ΞB,ε(t)|q�  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Combining this with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='67) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='74), we derive (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (1b) Let i(q) <  2+2α−d′  (d+d′−2−2α)+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Since q >  d  2α implies i(q) + 1 >  d  2α, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='30) and H¨older’s  inequality, we find a constant c1 > 0 such that  sup  t≥1,ν∈P  Eν�  |ψB  i (t)|2q�  ≤  sup  t≥1,ν∈P  �  Eν�  |ψB  i (t)|2{i(q)+1}��  q  i(q)+1  ≤ c1λ  dqi(q)  2(i(q)+1) −qα  i  ,  i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  By the calculations leading to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='76), we derive the same estimate for  θ := 1 + α −  di(q)  2(i(q) + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  We have θ > d′  2 due to i(q) <  2+2α−d′  (d+d′−2−2α)+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Hence, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='77) holds, which together with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='67) and  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='74) imply (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  30  (2) Let α ∈ (0, 1], q ∈ [1, qα) and k ∈ (  d  2αi(q), ∞] ∩ [1, ∞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' By using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='69) in place of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='70),  the proof of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='77) implies  sup  t≥1,ε∈(0,1),ν∈Pk,R  ε− q  2t  q  2Eν�  |ΞB(t) − ΞB,ε(t)|q�  < ∞,  R ∈ [1, ∞),  so that  lim  t→∞  sup  ε∈(0,1),ν∈Pk,R  Eν�  |ΞB(t) − ΞB,ε(t)|q�  = 0,  R ∈ [1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  This together with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='67) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='74) implies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  When k = ∞, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='14) follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='59) and that Eν ≤ ∥h∥∞Eµ for ν = hµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='3  Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='2  (1) Let d′ < 2(1 + α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='5, we have ηB  Z < ∞ and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='49) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Combining this with  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='59) and Eν ≤ ∥h∥∞Eµ for ν = hµ, we obtain  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='78)  lim sup  t→∞  sup  ν∈P∞,R  tEν�  W2(µB  t , µ)2�  ≤ lim sup  t→∞  sup  ν∈P∞,R  Eν�  ΞB(t)  �  ≤ ηB  Z ,  R ∈ [1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  On the other hand, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='63), for any ε ∈ (0, 1) there exists R ∈ [1, ∞) such that νε ∈ P∞,R  holds for all ν ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' So, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='78) together with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='65) implies  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='79)  lim sup  t→∞  sup  ν∈P  Eν�  W2(µB,ε  t  , µ)2�  = lim sup  t→∞  sup  ν∈P  Eνε�  W2(µB  t , µ)2�  ≤ ηB  Z ,  ε ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Combining this with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='71) for p = 2 and using the triangle inequality, we find a constant c > 0  such that  lim sup  t→∞  sup  ν∈P  Eν�  W2(µB  t , µ)2�  ≤ lim sup  t→∞  sup  ν∈P  Eν�  (1 + ε  1  2)W2(µB,ε  t  , µ)2 + (1 + ε− 1  2)W2(µB,ε  t  , µB  t )2�  ≤ (1 + ε  1  2)ηB  Z + c(ε + ε  1  2),  ε ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Letting ε ↓ 0 we obtain (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (2) Let d′ ≥ 2(1 + α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='6), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='7) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='42), we find a constant c1 > 0 such that  Eµ[W2(µB  t,r, µ)2] ≤ 4  t Eµ[µ(|∇ˆL−1(f B  t,r − 1)|2)] = 4  t Eµ[µ(|(−ˆL)− 1  2(f B  t,r − 1)|2)]  ≤ c1  t  �  r1+α− d′  2 + 1{d′=2+2α} log(1 + r−1)  �  ,  t ≥ 1, r ∈ (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='80)  Combining this with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='24) and the triangle inequality, we find a constant c2 > 0 such that  Eµ[W2(µB  t , µ)2] ≤ 2Eµ[W2(µB  t,r, µ)2] + 2Eµ[W2(µB  t,r, µB  t )2]  ≤ c1  t  �  r1+α− d′  2 + 1{d′=2+2α} log(1 + r−1)  �  + c2r,  t ≥ 1, r ∈ (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  31  Taking r = t−  2  d′−2α when d′ > 2 + 2α, and r = t−1 when d′ = 2 + 2α, we find a constant c3 > 0  such that  Eµ[W2(µB  t , µ)2] ≤  �  c3t−1 log(1 + t),  if d′ = 2(1 + α),  c3t−  2  d′−2α,  if d′ > 2(1 + α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  By combining this with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='63) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='65), we find a constant c4 > 0 such that  sup  ν∈P  Eν[W2(µB,1  t  , µ)2] = sup  ν∈P  Eν1[W2(µB  t , µ)2] ≤ c4Eµ[W2(µB  t , µ)2]  ≤ c3c4  �  1{d′=2(1+α)}t−1 log(1 + t) + t−  2  d′−2α  �  ,  t ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='81)  Noting that the triangle inequality implies  Eν[W2(µB  t , µ)2] ≤ 2W2(µB,1  t  , µ)2 + 2W2(µB,1  t  , µB  t )2,  we deduce (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='16) from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='71) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='81).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='4  Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='3  By approximating µB  t using µB,1  t  as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='71) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='81), we only need to prove for ν = µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='7), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='12) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='42), for any κ ∈ (0, 1  2) we find constants c1, c2 > 0 such that for any  t ≥ 1 and r ∈ (0, 1],  �  Eµ[W2p(µB  t,r, µ)2q]  � 1  q ≤ c1  �  Eµ[∥(−ˆL)  d(p−1)  4p  −κ(f B  t,r − 1)∥2q  2  � 1  q  ≤ c2  t  �  r−  �  d(p−1)  2p  −2κ+ d′  2 + d(q−1)  2q  −α  �+  + 1�  d(p−1)  2p  −2κ+ d′  2 + d(q−1)  2q  −α=0  � log(1 + r−1)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='82)  Below we prove assertions (1)-(3) in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='3 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (1) Let γα,p,q < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' We may take κ ∈ (0, 1  2) such that  d(p − 1)  2p  − 2κ + d′  2 + d(q − 1)  2q  − α < 0,  so that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='82) implies  �  Eµ[W2p(µB  t,r, µ)2q]  � 1  q ≤ c2  t ,  r ∈ (0, 1], t ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  By Fatou’s lemma for r → 0, we obtain (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='19) for ν = µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (2) Let γα,p,q ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' For any γ > γα,p,q, we find κ ∈ (0, 1  2) such that  d(p − 1)  2p  − 2κ + d′  2 + d(q − 1)  2q  − α ≤ γ,  so that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='82), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='24) and the triangle inequality imply  �  Eµ[W2p(µB  t , µ)2q]  � 1  q ≤ c1  �  t−1r−γ + r  �  ,  r ∈ (0, 1], t ≥ 1  32  for some constant c1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Taking r = t−  1  1+γ we obtain (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='20) for ν = µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (3) Let (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='17) hold and γα,p,q ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='7) we find constants c1, c2 > 0 such that  W2p(µB  t,r, µ)2p ≤ c1∥∇(−ˆL)−1(f B  t,r − 1)∥2p  2p ≤ c2∥(a0 − ˆL)  1  2(−ˆL)−1(f B  t,r − 1)∥2p  2p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='83)  On the other hand, by the Sobolev embedding theorem, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='6) implies that for any constants  k2 ≥ k1 > −∞ and q1 ≥ q2 ≥ 1 with  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='84)  1  q1  = 1  q2  + k1 − k2  d  ,  there exists a constant C > 0 such that  ∥(−ˆL)  k1  2 f∥q1 ≤ C∥(−ˆL)  k2  2 f∥q2,  µ(f) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Taking  k1 = −2,  k2 = d(p − 1)  2p  − 2,  q1 = 2p,  q2 = 2  such that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='84) holds, we find a constant c2 > 0 such that  ∥(a0 − ˆL)  1  2(−ˆL)−1(f B  t,r − 1)∥2p ≤ c2∥(a0 − ˆL)  1  2(−ˆL)  d(p−1)  4p  −1f∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Combining this with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='42) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='83), we find a constant c3 > 0 such that  �  Eµ[W2p(µB  t,r, µ)2q]  � 1  q ≤ c3  t  �  r−γα,p,q + 1{γα,p,q=0} log(1 + r−1)  �  ,  t ≥ 1, r ∈ (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  By this together with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='24) and the triangle inequality, we find a constant c4 > 0 such that  �  Eµ[W2p(µB  t , µ)2q]  � 1  q ≤ c4  �  t−1r−γα,p,q + t−11{γα,p,q=0} log(1 + r−1) + r  �  ,  t ≥ 1, r ∈ (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Taking r = t  −  1  1+γα,p,q we obtain (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  4  Proofs of Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='4 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='5  We will follow the line of [38] to estimate the lower bound of W2(µB  t , µ) by using an idea of [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  For any f ∈ D(L) with ∥f∥∞ + ∥∇f∥∞ + ∥Lf∥∞ < ∞, let  T σ  t f := −σ log ˆP σ  2 eσ−1f,  σ > 0, t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Assume that (M, ρ) is a geodesic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' If (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='8) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='21) hold, then there exists  constants k1, k2 > 0 such that ∥∇f∥2  ∞ ≤ k1σ implies  T σ  1 f(y) − f(x) ≤ 1  2ρ(x, y)2 + σ∥ˆLf∥2  ∞ + k2σθ∥∇f∥2  ∞,  µ(T σ  1 f − f) ≤ 1  2µ(|∇f|2) + k2σ−1∥∇f∥4  ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='1)  33  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Without loss of generality, we assume θ ≤ 1  4 such that in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='25) we have θ ∧ 1  2 = θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Let  k1 = 2κ0, then ∥∇f∥2  ∞ ≤ k1σ implies  tσ  2 ∥∇σ−1f∥2  ∞ ≤ 1  2σ−1∥∇f∥2  ∞ ≤ κ0,  t ∈ [0, 1],  so that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='26) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='26) for m = 1 with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='8) for σ−1f replacing f, we obtain  |∇T σ  t f|2 =  σ2|∇ ˆP tσ  2 eσ−1f|2  ( ˆP tσ  2 eσ−1f)2  ≤  (1 + k(2))σ2 ˆP tσ  2 (|∇σ−1f|2e2σ−1f)  ( ˆP tσ  2 eσ−1f)2  ≤  (1 + k(2))∥∇f∥2  ∞ ˆP tσ  2 (e2σ−1f)  ( ˆP tσ  2 eσ−1f)2  ≤ 2(1 + k(2))∥∇f∥2  ∞ =: c1∥∇f∥2  ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='2)  Next, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='25) with θ ≤ 1  4, we find a constant c2 > 0 such that  ˆLT σ  t f = −  σˆL ˆP tσ  2 e−σ−1f  ˆP tσ  2 eσ−1f  +  σ|∇ ˆP tσ  2 eσ−1f|2  ( ˆP tσ  2 eσ−1f)2  =  ˆP tσ  2 {eσ−1f ˆLf}  ˆP tσ  2 eσ−1f  +  σ|∇ ˆP tσ  2 eσ−1f|2 − σ( ˆP tσ  2 eσ−1f) ˆP tσ  2 (|∇σ−1f|2eσ−1f)  ( ˆP tσ  2 eσ−1f)2  ≤ ∥ˆLf∥∞ + c2σθ−1∥∇f∥2  ∞,  σ, t ∈ (0, 1], ∥∇f∥2  ∞ ≤ k1σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='3)  Moreover, for any two points x, y ∈ M, let γ : [0, 1] → M be the minimal geodesic from x to y  with  |˙γt| := lim sup  s→t  ρ(γt, γs)  |t − s| = ρ(x, y),  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  So,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='4)  lim sup  s→t  |f(γt) − f(γs)|  |t − s|  ≤ |∇f(γt)|ρ(x, y),  t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  By the Kolmogorov equation and the chain rule, we have  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='5)  ∂tT σ  t f = −  σ2 ˆL ˆP tσ  2 eσ−1f  2 ˆP tσ  2 eeσ−1f  = σ  2  ˆLT σ  t f − 1  2|∇T σ  t f|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  This together with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='3) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='4) yields  d  dtT σ  t f(γt) = (∂tT σ  t f)(γt) + d  dtT σ  s f(γt)  ���  s=t  ≤ σ  2  ˆLT σ  t f(γt) − 1  2|∇T σ  t f(γt)|2 + |∇T σ  t f(γt)|ρ(x, y)  ≤ 1  2  �  ρ(x, y)2 + σ  2 ∥ˆLf∥∞ + c2σθ∥∇f∥2  ∞  �  ,  t ∈ [0, 1], ∥∇f∥2  ∞ ≤ k1σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  34  Integrating over t ∈ [0, 1] and noting that T σ  0 f = f, we derive the first inequality in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  On the other hand, by T σ  0 f = f and µ(ˆLT σ  t f) = 0, we obtain  µ(f − T σ  1 f) = −  �  M  dµ  � 1  0  (∂tT σ  t f)dt  =  � t  0  dt  �  M  �1  2|∇T σ  t f|2 − σ  2  ˆLT σ  t f  �  dµ = 1  2  � 1  0  µ(|∇T σ  t f|2)dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='6)  Moreover, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='5) and the integration by parts formula, we obtain  d  dsµ(|∇T σ  s f|2) = − d  ds  �  M  (T σ  s f)ˆLT σ  s fdµ  = −  �  M  (ˆLT σ  s f)∂sT σ  s fdµ −  �  M  (T σ  s f)ˆL(∂sT σ  s f)dµ  = −2  �  M  (ˆLT σ  s f)∂sT σ  s f dµ = −2  �  M  (ˆLT σ  s f)  �σ  2  ˆLT σ  s f − 1  2|∇T σ  s f|2�  dµ  ≤ 1  4σ∥∇T σ  s f∥4  ∞,  s ∈ (0, 1], t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  This together with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='2) implies  µ(|∇T σ  t f|2) − µ(|∇f|2) ≤ c2  1  4σ∥∇f∥4  ∞,  t ∈ [0, 1], σ ∈ (0, 1], ∥∇f∥2  ∞ ≤ k1σ  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='7)  Substituting this into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='6), we derive the second estimate in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Similarly to the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='1 using Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='7 and the  approximation argument with Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='8, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='4(1)(2) follow from  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='8)  lim  t→∞ Eµ���{th(0)W2(µB  t , µ)2 − ΞB(t)}−��q�  = 0,  q ∈ [1, qα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Moreover, according to the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='2(1), Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='4(3) is implied by Theorem  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='4(1)(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' So, it remains to verify (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' The main idea for the proof of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='8) goes back to  [1, 38], but we have to make suitable modifications for the present more general situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Let  ˆft,r = (−ˆL)−1(f B  t,r − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Firstly, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='18), we find a constant c1 > 0 such that  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='9)  ∥ ˆft,r∥∞ ≤  � ∞  0  ∥ ˆPs(f B  t,r − 1)∥∞ds ≤ c1∥f B  t,r − 1∥∞  � ∞  0  e−λ1sds = c1  λ1  ∥f B  t,r − 1∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='10) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='9), we find a constant c2 > 0 such that  ∥∇ ˆPsg∥∞ ≤ c2(s ∧ 1)− 1  2e−λ1s∥g∥∞,  s > 0, g ∈ Bb(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Noting that ˆft,r := (−ˆL)−1(f B  t,r − 1) =  � ∞  0  ˆPs(f B  t,r − 1)ds, we obtain  ∥∇ ˆft,r∥∞ ≤  � ∞  0  ∥∇ ˆPs(f B  t,r − 1)∥∞ds ≤ c2∥f B  t,r − 1∥∞  � ∞  0  (1 ∧ s)− 1  2e−λ1sds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  35  Combining this with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='9) and |ˆL ˆft,r| = |f B  t,r − 1|, we find a constant c > 0 such that  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='10)  ∥ˆL ˆft,r∥∞ + ∥ ˆft,r∥∞ + ∥∇ ˆft,r∥∞ ≤ c∥f B  t,r − 1∥∞,  t, r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Next, let  C1( ˆft,r, σ) := σ∥ˆL ˆft,r∥2  ∞ + k2σθ∥∇ ˆft,r∥2  ∞,  C2( ˆft,r, σ) := k2σ−1∥∇ ˆft,r∥4  ∞,  Bt,r(σ) := {∥f B  t,r − 1∥2  ∞ ≤ k1c−1σ1+θ},  σ ∈ (0, 1], t, r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='11)  By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='1), the integration by parts formula, and the Kantorovich dual formula, we obtain  C1( ˆft,r, σ) + 1  2W2(µB  t,r, µ)2 ≥ µB  t,r( ˆft,r) − µ(T σ  1 ˆft,r)  = µ  �  (f B  t,r − 1)(−ˆL)−1(f B  t,r − 1)  �  + µ( ˆft,r) − µ(T σ  1 ˆft,r)  ≥ µ(|∇ ˆft,r|2) − 1  2µ(|∇ ˆft,r|2) − C2( ˆft,r, σ) = 1  2tΞB  r (t) − C2( ˆft,r, σ),  σ ∈ (0, 1], t, r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  This together with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='10) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='11) yields  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='12)  1Bt,r(σ)  �  tW2(µB  t,r, µ)2 − ΞB  r (t)  �  ≥ −c3tσ1+2θ,  σ ∈ (0, 1], t, r > 0  for some constant c3 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' On the other hand, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='6) we find a constant c4 > 0 such that  ∥f B  t,r − 1∥2  ∞ = ∥ ˆP r  2(f B  t, r  2 − 1)∥2  ∞ ≤ c4r− d  2∥f B  t, r  2 − 1∥2  2,  r ∈ (0, 1],  so that by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='53) with 1 replacing σ ∈ (0, 1), there exists a constant c5 > 0 such that  Eµ[∥f B  t,r − 1∥2  ∞] ≤ c4r− d  2 t−1  ∞  �  i=1  e−λirEµ[ψB  i (t)2]  ≤ c5r− d  2 −1t−1,  t ≥ 1, r ∈ (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Hence, we find a constant c6 > 0 such that  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='13)  Pµ(Bt,r(σ)c) = Pµ(∥f B  t,r − 1∥2  ∞ > k1c−1σ1+θ) ≤ c6r− d  2 −1t−1σ−(1+θ),  t ≥ 1, σ, r ∈ (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Taking σ = σt := t−  1  1+3θ/2 in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='12) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='13), we arrive at  lim sup  t→∞  Pµ�  {tW2(µB  t,r, µ)2 − ΞB  r (t)}− ≥ ε  �  ≤ lim sup  t→∞  Pµ(Bt,r(σt)c) = 0,  ε > 0, r ∈ (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  On the other hand, by B ∈ Bα, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='28) and �∞  i=1 λ−1−α  i  < ∞ due to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='7) and d′ < 2(1 + α),  we obtain  lim  r→0 sup  t≥1  Eµ�  |ΞB  r (t) − ΞB(t)|  �  ≤ lim  r→0  ∞  �  i=1  1 − e−λir  λi  sup  t≥1  Eµ[ψB  i (t)2]  36  ≤ c(1) lim  r→0  ∞  �  i=1  �  1 − e−λir�  λ−1−α  i  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Moreover, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='22) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='61) imply  lim  r→0 sup  t≥1  Eµ[{th(0)W2(µB  t , µ)2 − tW2(µB  t,r, µ)2}−] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Therefore, for any ε > 0,  lim sup  t→∞  Pµ�  {th(0)W2(µB  t , µ)2 − ΞB(t)}− ≥ 3ε  �  ≤ lim  r→0 lim sup  t→∞  Pµ�  {tW2(µB  t,r, µ)2 − ΞB  r (t)}− ≥ ε  �  + lim  r→0 sup  t≥1  �  Pµ�  |ΞB(t) − ΞB  r (t)| ≥ ε  �  + Pµ�  {th(0)W2(µB  t , µ)2 − tW2(µB  t,r, µ)2}− ≥ ε  ��  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Combining this with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='61) and applying the dominated convergence theorem, we prove (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  To prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='5, we need the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Assume (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (1) Let V = VB for B(λ) = λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' We have  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='14)  V(φi) = λ−1  i  − λ−2  i V(Zφi),  i ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (2) If (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='10) holds, then there exists a constant c > 0 such that  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='15)  VB(φi) ≥  1  B(λi) − cλ  − 3  2  i  ,  i ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' (1) By ˆLφi = −λiφi and using the Kolmogorov equation, we obtain  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='16)  Psφi = − 1  λi  Ps ˆLφi = − 1  λi  d  dsPsφi + 1  λi  Ps(Zφi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  This together with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='2) and µ(φ2  i ) = 1 implies  � t  0  µ(φiPsφi)ds = − 1  λi  � t  0  � d  dsµ(φiPsφi) − µ  �  φi(ZPsφi)  ��  ds  = 1  λi  �  1 − µ(φiPtφi)  �  − 1  λi  � t  0  µ  �  {Zφi}Psφi  �  ds  = 1  λi  �  1 − µ(φiPtφi)  �  + 1  λ2  i  � t  0  � d  dsµ  �  {Zφi}Psφi  �  − µ  �  {Zφi}Ps{Zφi}  ��  ds  = 1  λi  �  1 − µ(φiPtφi)  �  + 1  λ2  i  µ({Zφi}Ptφi) − 1  λ2  i  � t  0  µ  �  {Zφi}Ps{Zφi}  �  ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='17)  37  By (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='2) and ∥Pt − µ∥2 ≤ e−λ1t, we may let t → ∞ to derive  V(φi) = 1  λi  − 1  λ2  i  V(Zφi),  i ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (2) Let (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='10) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='10), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='5) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='33) we obtain  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='18)  VB(φi) :=  � ∞  0  µ(φiP B  t φi)dt =  1  B(λi) +  � ∞  0  �  E  � SB  t  0  e−λi(SB  t −s)µ  �  φiPs(Zφi)  �  ds  �  dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='10) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='3) for (P ∗  t , −Z) replacing (Pt, Z), we derive  P ∗  s φi = ˆPsφi −  � s  0  P ∗  r {Z ˆPs−rφi}dr = e−λisφi −  � s  0  e−λi(s−r)P ∗  r (Zφi)dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  This together with by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='2), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='11) and  ∥Zφi∥2  2 ≤ ∥Z∥∞∥∇φi∥2  2 = ∥Z∥2  ∞λi,  due to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='10), implies  µ(φiPs(Zφi)) = µ((P ∗  s φi)Zφi) = −  � s  0  e−λi(s−r)µ((Zφi)Pr(Zφi))dr  ≤ c(1)λ  1  2  i  � s  0  e−λi(s−r)r− 1  2e−λrdr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='19)  By λ ≤ λ1 ≤ λi, we have  −λi(s − r) − λr ≤ −(s − r)λi/2 − λs/2 − λr/2,  so that by the FGK inequality, we find a constant a1 > 0 such that  � s  0  e−λi(s−r)r− 1  2e−λrdr ≤ e−λs/2  � s  0  e−λi(s−r)/2r− 1  2e−λr/2dr  ≤ e−λs/2  � � s  0  e−λi(s−r)/2dr  �1  s  � s  0  r− 1  2e−λr/2dr  ≤ a1λ−1  i e−λs/2(1 ∧ s)− 1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='20)  So, by the same reason leading to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='20), this and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='19) imply  � SB  t  0  e−λi(SB  t −s)µ(φiPs(Zφi))ds ≤ c(1)a1λ  − 1  2  i  � SB  t  0  e−λi(SB  t −s)e−λs/2(1 ∧ s)− 1  2ds  ≤ a2λ  − 3  2  i  (1 ∧ SB  t )− 1  2e−λSB  t /4  for some constant a2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Combining this with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='36), we find a constant a3 > 0 such that  � ∞  0  �  E  � SB  t  0  e−λi(SB  t −s)µ(φiPs(Zφi))ds  �  dt ≤ a3λ  − 3  2  i  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Therefore, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='15) follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  38  Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='23), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='15), B ∈ Bα, d′ = 2(1 + α) and  5  2(1+α) ≥ 5  4 > 1, we find  constants a1, a2, a3, a4 > 0 such that  ηB  Z,r =  ∞  �  i=1  e−2λir VB(φi)  λi  ≥  ∞  �  i=1  e−2λir� a1  λ1+α  i  − a2  λ  5  2  i  �  ≥  ∞  �  i=1  e−2rc2i  1  1+α �  a1c−1−α  2  i−1 − a2c2  1i−  5  2(1+α)  �  ≥ a3 log(1 + r−1) − a4,  r ∈ (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='21)  Next, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='12), we obtain  Eµ[W2(µB  t,r, µ)2] ≥ Eµ[1BσW2(µB  t,r, µ)2] ≥ t−1Eµ[1Bt,r(σ)ΞB  r (t)] − c3σ1+2θ  ≥ t−1Eµ[ΞB  r (t)] − t−1Eµ[1BcσΞB  r (t)] − c3σ1+2θ,  t ≥ 1, r, σ ∈ (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='22)  By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='48), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='23) and d′ = 2(1 + α), we find constants k1, k2 > 0 such that  Eµ[ΞB  r (t)] =  ∞  �  i=1  e−2λir  λi  Eµ[ψB  i (t)2] ≥ ηB  Z,r − k1t−1  ∞  �  i=1  e−2rλiλ−(1+α)  i  ≥ ηB  Z,r − k1t−1  ∞  �  i=1  e−2rc2i  1  1+α c−1−α  2  i−1 ≥ ηB  Z,r − k2t−1 log(1 + r−1),  t ≥ 0, r ∈ (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Combining this with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='21) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='22), we derive  Eµ[W2(µB  t,r, µ)2] ≥ a3t−1 log(1 + r−1) − a4t−1 − k2t−2 log(1 + r−1)  − t−1Eµ[1Bt,r(σ)cΞB  r (t)] − c3σ1+2θ,  t ≥ 1, r, σ ∈ (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='23)  On the other hand, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='42) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='13), we find constants k3, k4 > 0 such that  Eµ[ΞB  r (t)2] = Ex[∥(−ˆL)− 1  2(f B  t,r − 1)∥4  2] ≤ k3r−k4,  Pµ(Bt,r(σ)c) ≤ k3r−k4t−1σ−(1+θ) t ≥ 1, r ∈ (0, 1], σ ∈ (0, 1],  where θ > 0 is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Thus,  Eµ[1Bt,r(σ)cΞB  r (t)] ≤  �  Pµ(Bt,r(σ)c)Eµ[ΞB  r (t)2]] ≤ k3r−k4t− 1  2σ− 1+θ  2 ,  t ≥ 1, r, σ ∈ (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Combining this with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='23), we arrive at  Eµ[W2(µB  t,r, µ)2] ≥ a3t−1 log(1 + r−1) − a4t−1 − k2t−2 log(1 + r−1)  − k3r−k4t− 3  2σ− 1+θ  2 − c3σ1+2θ,  t ≥ 1, r, σ ∈ (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Taking σ = t−  1  1+2θ and r = t−  θ  2k4(1+2θ) , obtain  Eµ[W2(µB  t,r, µ)2] ≥  a3  1 + 2θt−1 log t − a4t−1 − k2t−2 log(1 + t) − (k3 + c3)t−1,  t ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Therefore, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='24) holds for some constants c, t0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  39  Finally, to prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='6, we present one more lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Let (E, ρ) be a Polish space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Let Xt be a continuous time Markov process on E  such that the associated semigroup Pt satisfies  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='24)  ∥Pt − µ∥2 ≤ c1e−λ1t,  t ≥ 0  for some constants c1, λ1 > 0 and a probability measure µ on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' If there exists φ ∈ Cb,L(E)  such that µ(φ) = 0 and  V(φ) :=  � ∞  0  µ(φPtφ)dt > 0,  then there exist constants c, t0 > 0 such that µt := 1  t  � t  0 δXsds satisfies  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='25)  Eµ[W1(µt, µ)] ≥ ct− 1  2,  t ≥ t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  If M ⊂ E such that µ(M) = 1 and ∥Pt∥1→2 < ∞ for t > 0, then (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='25) holds for infν∈P Eν  replacing Eµ, where P is the set of all probability measures on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' By [39, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='1(c)], we have  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='26)  lim  t→∞  √  tEµ[|µt(φ)|] =  �  2πV(φ)  �− 1  2  � ∞  −∞  re−  r2  2V(φ) dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  So, by the Kantorovich dual formula, there exist constants c, t0 > 0 such that  √  tEµ[W1(µt, µ)] ≥  √  tEµ[|µt(φ) − µ(φ)|] ≥ c,  t ≥ t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Next, let M ⊂ E such that µ(M) = 1 and ∥Pt∥1→2 < ∞ for t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Then  {νP1 : ν ∈ P} ⊂  �  ν ∈ P : ν = hµ, ∥h∥2 ≤ ∥P1∥1→2  �  ,  so that [39, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='1(c)], ˆµt := 1  t  � t+1  1  δXsds satisfies  lim  t→∞ inf  ν∈P  √  tEν[|ˆµt(φ)|] > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Noting that  |ˆµt(φ) − µt(φ)| ≤ ∥φ∥∞t−1,  t > 1,  we obtain  lim  t→∞ inf  ν∈P  √  tEν[|µt(φ)|] > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  By the Kantorovich dual formula and µ(φ) = 0, this implies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='25) for infν∈P replacing Eµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  40  Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' (1) By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='6) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='8), for any i ≥ 1 we have ∥φi∥∞ < ∞ and there  exist constants c1, c2 > 0 such that  ∥∇φi∥∞ = eλi∥∇ ˆP1φi∥∞ ≤ c1eλi��(P1|∇φi|2)  1  2��  ∞  ≤ c2eλi∥∇φi∥2 = c2eλi�  λi < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  So, φi ∈ Cb,L(M), and hence is uniquely extended to φi ∈ Cb,L(E) for E := ¯  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' On the other  hand, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='15) we have VB(φi) > 0 for large enough i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Moreover, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='9) implies ∥P B  t ∥1→2 < ∞  for t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' So, the first assertion follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='3 with E = ¯  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (2) Let d′′ > 2(1 + α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='9) for p = 1 we find a constant c1 > 0 such that  Eµ[ρ( ˆXt, ˆX0)] ≤ c1t  1  2,  t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Combining this with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='3) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='8), we find a constant c2 > 0 such that  Eµ[ρ(Xt, X0)] =  �  M  ˆPtρ(x, ·)(x)µ(dx)  = Eµ[ρ( ˆXt, ˆX0)] +  �  M  µ(dx)  � t  0  Ps{Z ˆPt−sρ(x, ·)}(x)ds  ≤ c1t− 1  2 + c2  � t  0  (t − s)− 1  2ds ≤ c3t  1  2,  t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  According to the proof of [37, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='1(2)], this and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='26) imply  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='27)  Eµ[W1(µB  t , µ)] ≥ c4t−  1  d′′−2α ,  t ≥ t1  for some constants c4, t1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Finally, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='9), we find a constant t2 > 0 such that ∥Pt2 − µ∥1→∞ ≤ 1  2, so that  νt2 := νP B  t2 ≥ 1  2µ,  ν ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Let µB,t2  t  be in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='64) for ε = t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Then the Markov property and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='27) yield  inf  ν∈P Eν[W1(µB,t2  t  , µ)] = inf  ν∈P Eνt2[W1(µB  t , µ)] ≥ 1  2c4t−  1  d′′−2α ,  t ≥ t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Combining this with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='71), the triangle inequality and d′′−2α > 2, we find constants c5, c, t0 >  t1 such that  inf  ν∈P Eν[W1(µB  t , µ)] ≥ 1  2c4t−  1  d′′−2α − c5t−1 ≥ ct−  1  d′′−2α ,  t ≥ t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  41  5  Some concrete models  In this part, we apply our general results to the (reflecting) subordinated diffusion process on a  compact manifold, the subordinated conditional diffusion process on a bounded open domain,  the subordinated Wright-Fisher diffusion process, and subordinated subelliptic diffusion process  on SU(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' It is possible to consider more general hypoelliptic diffusion processes studied in  [6, 7] by using the generalized curvature-dimension condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  For simplicity, throughout this section, we take  B ∈ Bα ∩ Bα for some α ∈ (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='1  Subordinated (Reflecting) diffusion process  In this part, we consider the model stated in Introduction, for which all conditions in Theorems  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='1-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='6 are satisfied for d = d′ = d′′ = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Indeed, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='6) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='23) are well known (see [9, 11]), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='8) follows from [31, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='1],  and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='22) with h(r) = κeKr is implied by [38, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='36), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='37)], where κ ≥ 1 and K ≥ 0 are  constants with κ = 1 when ∂M is empty or convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Moreover, the following lemma confirms  other conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='21), (A2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='17) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' (1) Let lt be the local time of ˆXt on ∂M if ∂M exists, and let lt = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' By [31,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='1)] and the proof of [31, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='1], there exist constants c1, K, δ > 0 such that  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='1)  |∇ ˆPtf(x)| ≤ Ex[|∇f( ˆXt)|eKt+δlt],  t ≥ 0, x ∈ M, f ∈ C1  b (M),  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='2)  sup  x∈M  Ex[l2  t ] ≤ c1t,  sup  x∈M  Ex[eλlt] < ∞,  λ, t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  By the Schwarz inequality, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='1) implies  |∇ ˆPtef(x)| ≤ ˆPt|∇ef|(x) + Ex�  |∇ef( ˆXt)|(eKt+δlt − 1)  �  ≤  � ˆPtef(x)  � 1  2� ˆPt(|∇f|2ef)(x)  � 1  2 + ∥∇f∥∞( ˆPte2f)  1  2�  Ex[e2Kt+2δlt − 1]  � 1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  On the other hand, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='2) we find a constant c2 > 0 such that  Ex[e2Kt+2δlt − 1] ≤ Ex[(2K + 2δlt)e2Kt+2δlt]  ≤  �  Ex[(2K + 2δlt)2]  � 1  2�  Ex[e4Kt+4δlt]  � 1  2 ≤ c2t  1  2,  t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Therefore, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='21) holds for θ = 1  4 and m = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (2) When ∂M is empty or convex, we have  ⟨∇ρ( ˆX0, ·), N⟩( ˆXt)dlt ≤ 0,  42  where N is the inward unit normal vector on ∂M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  On the other hand, by the Laplacian  comparison theorem, there exits a constant c0 > 0 such that  ˆLρ( ˆX0, ·)2 ≤ c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  So, by Itˆo’s formula, we find a constant c1 > 0 such that  dρ( ˆX0, ˆXt)2 ≤ c1dt + 2  √  2ρ( ˆX0, ˆXt)dBt,  where Bt is the one-dimensional Brownian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Thus, for any p ≥ 1, there exists a constant  c(p) > 0 such that  dρ( ˆX0, ˆXt)2p ≤ c(p)ρ( ˆX0, ˆXt)2(p−1)dt + dMt  holds for some martingale Mt, so that  E[ρ( ˆX0, ˆXt)2p] ≤ c(p)  � t  0  E[ρ( ˆX0, ˆXs)2(p−1)]ds,  t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Consequently, for p = 1 we get  Ex[ρ( ˆX0, ˆXt)2] ≤ c(1)t,  and by inducting in p ∈ N, we derive (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='9) for p ∈ 2N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Therefore, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='9) holds for all p ≥ 1 due  to Jensen’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  When ∂M is non-convex, as explained in the proof of [32, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='7], there exists a  function 1 ≤ φ ∈ C∞  b (M) such that ∂M is convex under the metric  ⟨·, ·⟩′ := φ−1⟨·, ·⟩,  and ˆL = φ−2∆′ +Z′, where ∆′ is the Laplacian induced by the new metric and Z′ is a C1  b vector  field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Let ρ′ be the Riemannian distance induced by the new metric, we have ρ ≤ ∥φ∥∞ρ′, so  that the above argument for convex ∂M leads to  E[ρ( ˆX0, ˆXt)2p] ≤ ∥φ∥2p  ∞E[ρ′( ˆX0, ˆXt)2p] ≤ k(p)tp,  t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (3) To verify (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='17), we follow the line of [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' As explained in the end of page 12 in [3],  see also the proof of [3, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='5], under (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='10), it remains to verify the volume doubling  condition and scaled Poincar´e inequalities on balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' More precisely, we only need to find a  distance ˜ρ and constants c1, c2, c3 > 0 such that  c1˜ρ ≤ ρ ≤ c2˜ρ,  and the balls ˜B(x, r) := {y ∈ M : ˜ρ(x, y) ≤ r} for all x ∈ M and r > 0 satisfy  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='3)  µ( ˜B(x, 2r)) ≤ c3µ( ˜B(x, r)),  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='4)  µ(1 ˜B(x,r)f 2) ≤ c3r2µ(1 ˜B(x,r)|∇f|2) + µ(1 ˜B(x,r)f)2,  f ∈ C1  b (M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Since for the present model we have ˜D := ∥˜ρ∥∞ < ∞ and krd ≤ µ( ˜B(x, r)) ≤ Krd for some  constants K > k > 0 and all r ∈ [0, ˜D], (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='3) holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' To verify (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='4), by the conformal  43  change of metric used in step (2), we may and do assume that ∂M is either empty or convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  In this case, there exists a constant r0 > 0 such that B(x, r) := {y ∈ M : ρ(x, y) ≤ r} is convex  for all x ∈ M and r ∈ (0, r0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Then we take ˜ρ := ρ ∧ r0, so that ˜B(x, r) is convex for all r > 0,  since ˜B(x, r) = B(x, r) for r < r0 and ˜B(x, r) = M for r ≥ r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Thus, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='4) follows from [29,  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  According to the above observations, we conclude that all assertions in Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='1-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='6  hold for d = d′ = d′′ = n, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  qα =  n  2(n − 1 − α)+,  α(d, d′) = α(n) := 1  2  �  1 + 2n(n − 1) − 2,  γα,p,q = n  2(3 − p−1 − q−1) − 1 − α,  p, q ∈ [1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='5)  Below we summarize these results, which in particular imply Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='2 stated in  Introduction, by B(λ) = λ, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='14) and  V(Zφi) := µ((Zφi)(−L)−1(Zφi)) = µ(|∇L−1(Zφi)|2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Let qα, α(n), γα,p,q for p, q ∈ [1, ∞) be in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Then the following assertions  hold for some constant κ ∈ [1, ∞), where κ = 1 when ∂M is either empty or convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (1) If n < 2(1 + α), qα >  n  2α and α > α(n), then  lim  t→∞ sup  ν∈P  Eν���{tW2(µB  t , µ)2 − ΞB(t)}+ + {tκW2(µB  t , µ)2 − ΞB(t)}−��q�  = 0,  q ∈ [1, qα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (2) If n < 2(1 + α), then for any q ∈ [1, qα) and k ∈ (  d  2αi(q), ∞] ∩ [1, ∞],  lim  t→∞ sup  ν∈Pk,R  Eν����{tW2(µB  t , µ)2 − ΞB(t)}+ + {tκW2(µB  t , µ)2 − ΞB(t)}−���  q�  = 0,  R ∈ (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (3) If n < 2(1 + α), then  κ−1ηB  Z ≤ lim inf  t→∞  inf  ν∈P tEν[W2(µB  t , µ)2] ≤ lim sup  t→∞  sup  ν∈P  tEν[W2(µB  t , µ)2] ≤ ηB  Z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (4) If n = 2(1 + α), then there exist constants c, t0 > 1 such that  c−1t−1 log t ≤ Eν[W2(µB  t , µ)2] ≤ ct−1 log t,  t ≥ t0, ν ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (5) If n > 2(1 + α), then there exist constants c, t0 > 1 such that  c−1t−  2  n−2α ≤  �  Eν[W1(µB  t , µ)]  �2 ≤ Eν[W2(µB  t , µ)2] ≤ ct−  2  n−2α ,  t ≥ t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (6) If p, q ∈ [1, ∞) with n(3 − p−1 − q−1) < 2(1 + α), then there exist constants c, t0 > 1 such  that  c−1t−1 ≤  �  Eν[W1(µB  t , µ)]  �2 ≤  �  Eν[W2p(µB  t , µ)2q]  � 1  q ≤ ct−1,  t ≥ t0, ν ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  44  (7) If p, q ∈ [1, ∞) such that n(3 − p−1 − q−1) ≥ 2(1 + α), then there exists a constant c > 0  such that for any t ≥ 1,  sup  ν∈P  �  Eν[W2p(µB  t , µ)2q]  � 1  q ≤  �  ct−1 log(1 + t),  if n(3 − p−1 − q−1) = 2(1 + α),  ct  −  1  1+γα,p,q ,  if n(3 − p−1 − q−1) > 2(1 + α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='2  Subordinated conditional diffusion process  Let M be a bounded connected C2 open domain in an n-dimensional complete Riemannian  manifold, and let V ∈ C2  b (M) be such that µ0(dx) := eV (x)dx is a probability measure on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Consider the Dirichlet eigenproblem for ˆL0 := ∆ + ∇V in M:  ˆL0hi = −θihi,  hi|∂M = 0,  i ≥ 0,  where {θi}i≥0 are listed in the increasing order counting multiplicities, and {hi}i≥0 are the  associated unitary eigenfunctions in L2(µ0) with h0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Let  ˆL := ˆL0 + 2∇ log h0 = ∆ + ∇(V + 2 log h0),  µ(dx) := h0(x)2µ0(dx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Then the diffusion process ˆXt generated by ˆL is non-explosive in M, whose distribution coincides  with the conditional distribution of the ˆL0-diffusion process ˆX0  t under the condition that  τ := inf{t ≥ 0 : ˆX0  t ∈ ∂M} = ∞,  in the sense that for any T > 0 and any F ∈ Cb(C[0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' M)),  E[F( ˆX[0,T])] = lim  m→∞ E[F( ˆX[0,T]0)|τ > m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Let Z be a C1  b -vector field on M satisfying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  It is well known that {λi := θi −θ0}i≥0 are all eigenvalues of −ˆL with unitary eigenfunctions  {φi := hih−1  0 }i≥0, and that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='6), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='23) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='26) hold for  d = n + 1,  d′ = d′′ = n,  see for instance [9, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Next, by [34, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='6], (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='22) holds for h(r) = κeKr for some  constants κ ≥ 1 and K ≥ 0, where κ = 1 when ∂M is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' The following lemma confirms  other conditions in Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='1-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='6, except (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='17) which is not yet verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' For the present model, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='8) and (A2) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' When ∂M is convex or n ≤ 3, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='21)  is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' As explained in the proof of [34, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='6], if ∂M is convex then the Bakry-Emery  curvature ˆL is bounded from below by a constant −K, so that  |∇Ptf| ≤ eKtPt|∇f|,  which implies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='8) for k(p) = eK as well as (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='21) for θ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' So, in the following we only prove  these conditions for non-convex ∂M, and to verify (A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  45  (1) When ∂M is non-convex, let ρ′ be the Riemannian distance induced by ⟨·, ·⟩′ introduced  in the proof of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' According to the proof of [34, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='6], for any x, y ∈ M, there exists  a coupling ( ˆXt, ˆYt) of the diffusion process generated by ˆL starting from (x, y), such that for  some constant c1 > 0 we have  dρ′( ˆXt, ˆYt) ≤ c1ρ′( ˆXt, ˆYt)dt + dMt,  where Mt is a martingale with d⟨M⟩t ≤ c1ρ′( ˆXt, ˆYt)dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Thus, for any q ∈ (1, ∞), there exists a  constant K(q) > 0 such that  �  E[ρ′( ˆXt, ˆYt)q]  � 1  q ≤ K(q)ρ′(x, y),  t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Therefore, by ρ′ ≤ ρ ≤ ∥φ∥∞ρ′,  |∇ ˆPtf(x)| := lim sup  y→x  | ˆPtf(x) − ˆPtf(y)|  ρ(x, y)  ≤ lim sup  y→x  E  �|f( ˆXt) − f( ˆYt)|  ρ′( ˆXt, ˆYt)  ρ′( ˆXt, ˆYt)  ρ(x, y)  �  ≤ lim sup  y→x  �  E  �|f( ˆXt) − f( ˆYt)|p  ρ′( ˆXt, ˆYt)p  �� 1  p (E[ρ′( ˆXt, ˆYt)  p  p−1])  p−1  p  ρ(x, y)  ≤ K  �  p  p − 1  �  ∥φ∥∞(Pt|∇f|p)  1  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  So, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='8) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (2) Let ∂M be non-convex and n ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Since ∇V ∈ C1  b (M) and  −Hesslog h0 = −Hessh0  h0  + ∇h0 ⊗ h0  h2  0  ≥ −∥∇2h0∥∞  h0  ,  there exists a constant c1 > 0 such that the Bakry-Emery curvature of ˆL is bounded below by  − c1  2h0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Γ2(g) := 1  2L|∇g|2 − ⟨∇g, ∇ˆLg⟩ ≥ −c1  2 h−1  0 |∇g|2,  g ∈ C(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  So, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='18) for d = n + 2, and applying Jensen’s inequality, we find a constant c2 > 0 such  that  |∇ ˆPtef|2 − ˆPt|∇ef|2 = −  � t  0  d  ds  ˆPs|∇ ˆPt−sef|2ds  ≤ c1  � t  0  ˆPs  �  h−1  0 |∇ˆˆPt−sef|2�  ds ≤ c1  � t  0  ( ˆPsh  −  m  m−1  0  )  m−1  m ( ˆPs|∇ ˆPt−sef|2m)  1  mds  ≤ c2  � t  0  s− (n+2)(m−1)  2m  µ(h  −  m  m−1  0  )  m−1  m ( ˆPs|∇ ˆPt−sef|2m)  1  mds  ≤ c2∥∇f∥2  ∞( ˆPte2mf)  1  m  � t  0  s− (n+2)(m−1)  2m  µ(h  −  m  m−1  0  )  m−1  m ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Noting that n + 2 ≤ 5 and µ(h−r  0 ) = µ0(h2−r  0  ) < ∞ for r < 3, for any m ∈ ( 3  2, 5  3), we have  θ1 := 1 − (n + 2)(m − 1)  2m  ∈ (0, 1),  µ  �  h  −  m  m−1  0  �  < ∞,  46  so that for some constant c3 > 0 we have  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='6)  |∇ ˆPtef|2 − ˆPt|∇ef|2 ≤ c3tθ1∥∇f∥2  ∞( ˆPte2m)  1  m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Similarly, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='8) and its consequence  |∇ ˆPtg| ≤ c  √  t( ˆPt|∇g|2)  1  2,  we find a constant c4 > 0 such that  ˆPt|∇ef|2 − ( ˆPtef) ˆPt(|∇f|2ef) =  � t  0  d  ds  ˆPt−s  �  ( ˆPsef) ˆPs(|∇f|2ef)  �  ds  = −  � t  0  ˆPt−s⟨∇ ˆPsef, ∇ ˆPs(|∇f|2ef)⟩ds  ≤ c4∥∇f∥2  ∞  � t  0  s− 1  2 ˆPt−s( ˆPse2f)ds = 2c4t  1  2∥∇f∥2  ∞ ˆPte2f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  This together with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='6) implies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='21) for θ = θ1 ∧ 1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (3) It remains to verify (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' By the conformal change of metric as in the end of the proof  of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='1, we only consider the case where ∂M is convex, so that  ⟨N, ∇ρ(x, ·)⟩|∂M ≤ 0,  x ∈ M,  where N is the inward unit normal vector field of ∂M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Let ρ∂ be the distance to ∂M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' It  is well known (see for instance [25]) that ∇h0 is inward normal on the boundary and c1 :=  ∥h−1  0 ρ∂∥∞ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' So,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='7)  ρ∂ ≤ c1h0,  ⟨∇h0, ∇ρ(x, ·)⟩|∂M ≤ 0,  x ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Moreover, by the Hessian comparison theorem, there exists a constant c2 > 0 such that  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='8)  Hessρ(x,·)2(v, v) ≤ c2|v|2,  x ∈ M, v ∈ TM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  We intend to show that these two estimates imply  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='9)  sup  x,y∈M  ⟨∇ log h0, ∇ρ(x, ·)2⟩(y) ≤ c  for some constant c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' To see this, for any x, y ∈ M, let z ∈ ∂M such that ρ(y, z) = ρ∂(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Let  γ : [0, 1] → M,  γ0 = z,  γ1 = y,  |˙γ| = ρ∂(y)  be the minimal geodesic from z to y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Let vs = ∇h0(γs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' We have  v0 = a0 ˙γ0,  ∥0→s v0 = a0 ˙γs,  s ∈ [0, 1],  where a0 := ⟨N, ∇h0⟩(z) and ∥0,s is the parallel displacement along the geodesic γs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Since  h0 ∈ C2  b (M), we find a constant c3 > 0 such that  |v1 − a0 ˙γ1| = |v1− ∥0→1 v0| ≤ c3ρ∂(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  47  Combining this with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='7) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='8), and noting that |∇ρ2| ≤ 2∥ρ∥∞ < ∞, we find a constant  c4 > 0 such that  ⟨∇h0, ∇ρ(x, ·)2⟩(y) = ⟨v1, ∇ρ(x, ·)2(γ1)⟩ ≤ a0⟨˙γ1, ∇ρ(x, ·)2(γ1)⟩ + c3ρ∂(y)  = a0⟨˙γ0, ∇ρ(x, ·)2(γ0)⟩ + a0  � 1  0  d  ds⟨˙γs, ∇ρ(x, ·)2(γs)⟩ds + c3ρ∂(y)  ≤ a0  � 1  0  Hessρ(x,·)2(˙γs, ˙γs)ds + c3ρ∂(y) ≤ a0c2ρ∂(y)2 + c3ρ∂(y) ≤ c4h0(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Therefore, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='9) holds for c = c4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  By (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='9) and Itˆo’s formula, we obtain  dρ( ˆX0, ·)2( ˆXt) ≤ cdt + 2  √  2ρ( ˆX0, ˆXt)dBt,  where Bt is the one-dimensional Brownian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' This implies (A2) as explained in the proof  of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='1(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  We now conclude that all assertions in Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='1-Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='5 hold, except Theorem  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='2(4) where the condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='17) is to be verified for this model, for d = n + 2, d′ = d′′ = n and  qα := 1 + 2(1 + α) − n  4n + 2 − 2α ,  α(d, d′) = ˜α(n) := 1  2  ��  4 + 2n(n + 2) − 2  �  ,  γα,p,q := n  2 + n + 2  2  (2 − p−1 − q−1) − α − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='10)  So, we have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Let ˆL := L0 + 2∇ log h0, and L = ˆL + Z for some C1  b -vector field Z satisfying  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' The following assertions hold for qα, ˜a(n) and, γα,p,q in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='10), and a constant κ ≥ 1 with  κ = 1 when ∂M is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (1) If n < 2(1 + α), qα >  n  2α and α > ˜α(n), then  lim  t→∞ sup  ν∈P  Eν���{tW2(µB  t , µ)2 − ΞB(t)}+ + {tκW2(µB  t , µ)2 − ΞB(t)}−��q�  = 0,  q ∈ [1, qα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (2) If n < 2(1 + α), then for any q ∈ [1, qα) and k ∈ (  d  2αi(q), ∞] ∩ [1, ∞],  lim  t→∞ sup  ν∈Pk,R  Eν����{tW2(µB  t , µ)2 − ΞB(t)}+ + {tκW2(µB  t , µ)2 − ΞB(t)}−���  q�  = 0,  R ∈ (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (3) If n < 2(1 + α), then  κ−1ηB  Z ≤ lim inf  t→∞  inf  ν∈P tEν[W2(µB  t , µ)2] ≤ lim sup  t→∞  sup  ν∈P  tEν[W2(µB  t , µ)2] ≤ ηB  Z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  48  (4) Let n = 2(1 + α), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' (n, α) ∈ {(3, 1  2), (4, 1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Then there exist constants c, t0 > 0 such  that  sup  ν∈P  Eν[W2(µB  t , µ)2] ≤ ct−1 log t,  t ≥ t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  If ∂M is convex or (n, α) = (3, 1  2), then there exists a constant c′ > 0 such that  inf  ν∈P Eν[W2(µB  t , µ)2] ≥ c′t−1 log t,  t ≥ t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (5) If n > 2(1 + α), then there exist constants c, t0 > 1 such that  c−1t−  2  n−2α ≤  �  Eν[W1(µB  t , µ)]  �2 ≤ Eν[W2(µB  t , µ)2] ≤ ct−  2  n−2α ,  t ≥ t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (6) If p, q ∈ [1, ∞) with γα,p,q < 0, then there exist constants c, t0 > 1 such that  c−1t−1 ≤  �  Eν[W1(µB  t , µ)]  �2 ≤  �  Eν[W2p(µB  t , µ)2q]  � 1  q ≤ ct−1,  t ≥ t0, ν ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (7) Let p, q ∈ [1, ∞) with γα,p,q ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Then for any γ > γα,p,q, there exists a constant c > 0  such that  sup  ν∈P  �  Eν[W2p(µB  t , µ)2q]  � 1  q ≤ ct−  1  1+γ ,  t ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  If (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='17) holds, then there exists a constant c > 0 such that  sup  ν∈P  �  Eν[W2p(µB  t , µ)2q]  � 1  q ≤ ct  −  1  1+γα,p,q + ct−1 log(1 + t)1{γα,p,q=0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='3  Subordinated Wright-Fisher diffusion process  Let a, b > 1  4 be two constants, and let  µ := 1[0,1](x)Γ(2a + 2b2)  Γ(2a)Γ(2b) x2a−1(1 − x)2b−1dx  be the beta distribution on M = [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' The Fisher-Wright diffusion process ˆXt is generated by  ˆL := 1  2x(1 − x) d2  dx + {a − (a + b)x} d  dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Under the Riemannian metric ⟨∂x, ∂x⟩ = 2{x(1 − x)}−1, we have  Γ(f, g) = ⟨∇f, ∇g⟩ := 1  2x(1 − x)f ′(x)g′(x),  x ∈ M,  ρ(x, y) =  √  2  � y  x  {s(1 − s)}− 1  2ds,  0 < x ≤ y < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Since divµZ = 0 implies Z = 0, we have L = ˆL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  49  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' For the present model with L = ˆL, (A1) with d = 4(a ∨ b) and (A2) hold, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='23)  holds for d′ = 2, (B) holds with h(r) = eKr for some constant K > 0, and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='17) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Firstly, the condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='6) with d = 4(a ∨ b) is implied by [15, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' By [27,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='4)], the Bakry-Emery curvature of ˆL is bounded below by a constant −K < 0, so that  |∇Ptf| ≤ eKtPt|∇f|,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='8) and (B) hold for θ = m = 1, k(p) = eK and h(r) = erK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Next, for any p ≥ 1 there exists a constant c1 > 0 such that  ˆLρ(X0, ·)2p(Xt) = 2pρ(X0, Xt)2(p−1)Lρ(X0, ·)(Xt)2 + p(p − 1)ρ(X0, Xt)2(p−1)  ≤ c1ρ(X0, Xt)2(p−1),  so that  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='11)  Eµ[ρ(X0, Xt)2p] ≤ c1  � t  0  Eµ[ρ(X0, Xs)2(p−1)]ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  In particular, for p = 1 we obtain (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='9), and for general p ∈ N it follows from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='11) by the  induction argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Moreover, we have λi = (a + b)i so that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='23) holds for d′ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Indeed, according to the  proof of [14, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='1], all eigenfunctions are polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' The trivial eigenvalue is λ0 = 0  with φ0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' For any i ∈ N, let  φi(x) :=  i  �  j=0  αjxj  with αi > 0 be the unitary eigenfunction for λi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' We have  −λiφi(x) = ˆLφi(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Since the coefficients of xi in left hand and right hand sides are −λiαi and −i(a + b)αi respec-  tively, these two constants have to be equal each other, so that λi = i(a + b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Finally, as explained in step (3) in the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='1, for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='17) it suffices to verify  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='3) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Since the curvature is bounded from below as indicated in the beginning of  the proof, and since a one-dimensional ball is convex, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='4) follows from [29, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  So, it remains to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' With the transform x → 1 − x, we only need to prove this condition  for x ∈ [0, 1  2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Let x ∈ [0, 1  2] and B(x, r) := {y ∈ [0, 1] : ρ(x, y) ≤ r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Take, for instance,  r0 = 1  8ρ( 1  2, 1) such that  x0 := sup B(1/2, 2r0) ∈ (1/2, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  We have  c0 :=  inf  x∈[0, 1  2 ] µ(B(x, r0)) > 0,  so that  µ(B(x, 2r)) ≤ 1 ≤ c−1  0 µ(B(x, r)),  r ≥ r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  50  Hence, we only need to consider r ∈ (0, r0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' On the other hand, we find constants c1, c2 > 0  such that  c1  ��√x − √y  ��ρ(x, y) ≤ c2  ��√x − √y  ��,  x ∈ [0, 1/2], r ∈ (0, 2r0),  so that for some constants c3, c4 > 0,  ��  (√x−c1r)+�2,  �√x+c1r}2∧x0  �  ⊂ B(x, r) ⊂  ��  (√x−c2r)+�2,  �√x+c2r}2∧x0  �  , r ∈ (0, 2r0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Noting that 0 < 1 − x0 ≤ 1 − s ≤ 1 for s ∈ B(x, 2r0), we find constants c4 > c3 > 0 such that  c3  �  (x + c3r)2a − x2a�  ≤ µ(B(x, r))  ≤ µ(B(x, 2r)) ≤ c4  �  (x + c4r)2a − [(x − c4r)+]2a�  ,  x ∈ [0, 1/2], r ∈ (0, r0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  since (x+c4r)2a−{(x−c4r)+}2a  (x+c3r)2a−x2a  is a continuous function of (x, r) ∈ [0, 1  2] × [0, r0], where when r = 0  the function is understood as the limit c4  c3 as r → 0, we obtain  sup  x∈[0,1/2],r∈(0,r0)  µ(B(x, 2r))  µ(B(x, r)) ≤  sup  x∈[0, 1  2],r∈[0,r0]  (x + c4r)2a − {(x − c4r)+}2a  (x + c3r)2a − x2a  < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Therefore, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='3) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  In conclusion, all assertions in Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='1-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='6(1) hold for d = 4(a ∨ b) and d′ = 2 so that  qα :=  4(a ∨ b)  4(a ∨ b) − α,  α(d, d′) = ˜α := (a ∨ b)  �√  5 − 1  �  ,  γα,p,q := 2(a ∨ b)(2 − p−1 − q−1) − α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='12)  So, we have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' For the above L = ˆL and ηB = ηB  Z for Z = 0, the following assertions hold for  qα, ˜a and, γα,p,q in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (1) If α > 4  3(a ∨ b) which implies qα > 4(a∨b)  2α  and α > ˜α, then  lim  t→∞ sup  ν∈P  Eν���tW2(µB  t , µ)2 − ΞB(t)  ��q�  = 0,  q ∈ [1, qα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (2) For any q ∈ [1, qα) and k ∈ (  d  2αi(q), ∞] ∩ [1, ∞], where we set (  d  2αi(q), ∞] = {∞} if α = 0,  lim  t→∞ sup  ν∈Pk,R  Eν����tW2(µB  t , µ)2 − ΞB(t)  ���  q�  = 0,  R ∈ (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (3) ηB < ∞ and lim supt→∞ supν∈P  ��tEν[W2(µB  t , µ)2] − ηB�� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (4) Let p, q ∈ [1, ∞) with γα,p,q < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' There exist constants c, t0 > 1 such that  c−1t−1 ≤  �  Eν[W1(µB  t , µ)]  �2 ≤  �  Eν[W2p(µB  t , µ)2q]  � 1  q ≤ ct−1,  t ≥ t0, ν ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (5) Let p, q ∈ [1, ∞) with γα,p,q ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Then there exists a constant c > 0 such that  sup  ν∈P  �  Eν[W2p(µB  t , µ)2q]  � 1  q ≤ ct  −  1  1+γα,p,q + c1{γα,p,q=0}t−1 log(1 + t),  t ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  51  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='4  Subordinated subelliptic diffusions on SU(2)  Let M = SU(2) be the space of 2 × 2, complex, unitary matrices with determinant 1, which is  a 3-dimensional compact Lie group with Lie algebra and Riemannian metric given by  su(2) := span{U1, U2, U3},  ⟨Ui, Uj⟩ = 1{i=j},  1 ≤ i, j ≤ 3,  where for i = √−1,  U1 :=  � 0  1  −1  0  �  ,  U2 :=  �0  i  i  0  �  ,  U3 :=  �i  0  0  −i  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  For each 1 ≤ i ≤ 3, Ui is understood as a left-invariant vector field defined as  Uif(x) := lim  ε↓0  f(eεUix) − f(x)  ε  ,  f ∈ C1(SU(2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Then [U1, U2] = 2U3, so that  ˆL := U2  1 + U2  2  satisfies H¨ormader’s condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Moreover, ˆL is symmetric in L2(µ) where µ is the normalized  Haar measure on SU(2), and the intrinsic distance ρ induced by  Γ(f, g) := (U1f)(U1g) + (U2f)(U2g)  is the Carnot-Carath´eodory distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' By Chow’s theorem, (M, ρ) is a compact geodesic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  To formulate the diffusion process ˆXt generated by ˆL, we use the cylindrical coordinate  introduced in [10]:  �  0, π  2  �  × [0, 2π] × [−π, π] ∋ (r, θ, z) �→ er(cos θ)U1+r(cos θ)U2ezU3 ∈ M := SU(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Under these coordinates, ˆXt := (rt, θt, zt) is constructed by solutions of the SDEs  drt = 2 cot(2rt)dt + dBt,  d(θt, zt) =  �  2  sin θt  , tan rt  �  d ˜Bt,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='13)  where (Bt, ˜Bt) is a two-dimensional Brownian motion, see [5, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' The following lemma  shows that conditions (A1), (A2), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='17) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='23) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' However, due to the degeneracy of the  diffusion, assumption (B) may be invalid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Conditions (A1), (A2), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='17) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='23) hold for d = 4 and d′ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' By [5, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='10], for any p > 1 there exists a constant c(p) > 0 such that  |∇ ˆPtf| ≤ c(p)e−2t(Pt|∇f|p)  1  p,  t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  So, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='8) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  52  According to [7], the generalized curvature-dimension condition CD(ρ1, 1  2, 1, 2) holds, so  that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='17) is implied by [6, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Let ˆpt be the heat kernel of ˆPt with respect to µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' By [5, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='1] and the spectral  representation of heat kernel, see also [8], all eigenvalues with multiplicities of −ˆL are given by  {λi}i≥0 =  �  4k(k + |n| + 1) + 2|n| :  n ∈ Z, k ∈ Z+  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  In particular, λ1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' It is easy to see that for large i ∈ N,  #  �  4k(k + |n| + 1) + 2|n| ≤ i :  n ∈ Z, k ∈ Z+  �  has order i  3  2, so that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='23) holds for d′ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  To verify (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='6), we use the cylindrical coordinates (r, θ, z), for which the identity matrix  becomes 0 := (0, 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Let ˆpt be the heat kernel of ˆPt with respect to µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' By [5, Proposition  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='9], there exists a constant c > 0 such that  ˆpt(0, 0) ≤ ct−2,  t ∈ (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  By the left invariant of the heat kernel which follows from the same property of the generator  ˆL, we obtain  ∥ ˆPt∥1→∞ = sup  x∈M  ˆpt(x, x) ≤ ct−2,  t ∈ (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  This together with λ1 = 2 implies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='6) for λ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  It remains to verify (A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' For any (r, z) ∈ [0, π  2)×[−π, π], let θ(r, z) ∈ [−π, π] be the unique  solution to the equation  θ(r, z) − z = (cos r)(sin θ(r, z)) arccos[(cos θ(r, z)) cos r]  �  1 − (cos2 r) cos2 θ(r, z)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  By [5, Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='12], the distance of x := (r, θ, z) to 0 depends only on (r, z), and there exists  a constant c1 > 0 such that  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='14)  ρ(0, x)2 = (θ(r, z) − z)2 tan2 r  sin2 θ(r, z)  ≤ c1 sin2 r ≤ c1r2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  On the other hand, letting ˆX0 = 0, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='13) and Itˆo’s formula, for any p ≥ 1 we find a constant  c1(p) > 0 such that  dr2p  t ≤ c(p)r2(p−1)  t  dt + dMt  for some martingale Mt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' So, we find a constant c2(p) > 0 such that  E[r2p  t ] ≤ k(p)tp,  t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Combining this with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='14) and using the left invariance of the heat kernel, we obtain (A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  53  By the above lemma and that M = SU(2) is a Polish space, we conclude that all assertions  in Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='1-Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='3 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='6(1) hold for d = 4, d′ = 3 and  qα :=  8  9 − 2α,  γα,p,q := 1  2 − α + 2(2 − p−1 − q−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='15)  So, we have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Let ˆL := U2  1 + U2  2, and L = ˆL + Z for some C1  b -vector field Z satisfying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  The following assertions hold for qα, α′ and γα,p,q in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (1) If α ∈ ( 1  2, 1], then for any q ∈ [1, qα) and k ∈ (  d  2αi(q), ∞] ∩ [1, ∞],  lim  t→∞ sup  ν∈Pk,R  Eν����{tW2(µB  t , µ)2 − ΞB(t)}+���  q�  = 0,  R ∈ (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (2) If α ∈ ( 1  2, 1], then  lim sup  t→∞  sup  ν∈P  tEν[W2(µB  t , µ)2] ≤ ηB  Z < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (3) If α ∈ (0, 1  2], then there exist constants c, t0 > 0 such that  sup  ν∈P  Eν[W2(µB  t , µ)2] ≤ ct−  2  3−2α + c1{α= 1  2}t−1 log t,  t ≥ t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (4) If p, q ∈ [1, ∞) with γα,p,q < 0, then there exist constants c, t0 > 1 such that  c−1t−1 ≤  �  Eν[W1(µB  t , µ)]  �2 ≤  �  Eν[W2p(µB  t , µ)2q]  � 1  q ≤ ct−1,  t ≥ t0, ν ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  (5) Let p, q ∈ [1, ∞) with γα,p,q ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Then for any γ > γα,p,q, there exists a constant c > 0  such that for any t ≥ 1,  sup  ν∈P  �  Eν[W2p(µB  t , µ)2q]  � 1  q ≤ ct  −  1  1+γα,p,q + c1{γα,p,q=0}t−1 log(1 + t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  References  [1] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Ambrosio, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Gigli, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Savar´e, Gradient Flows in Metric Spaces and in the Spaces of  Probability Measures, Lectures in Mathematics ETH Z¨urich, Birkh¨auser Verlag, Basel,  2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  [2] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Ambrosio, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Stra, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Trevisan, A PDE approach to a 2-dimensional matching problem,  Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Theory Relat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Fields 173(2019), 433–477.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  [3] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Auscher, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Coulhon, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Duong, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Hofmann, Riesz transform on manifolds and  heat kernel regularity, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' ´Ecole Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Sup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' (4)37(2004), 911–957.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  54  [4] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Bakry, On Sobolev and logarithmic Sobolev inequalities for Markov semigroups, in:  New Trends in Stochastic Analysis, World Scientific, 1997, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' 43–75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  [5] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Baudoin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Bonnefont, The subelliptic heat kernel on SU(2): Representations, asymp-  totics and gradient bounds, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Zeit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' 263(2009), 647–672.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  [6] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Baudoin, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Garofal, A note on the boundedness of Riesz transform for some subelliptic  operators, 2013, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' 398–421.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  [7] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Baudoin, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Garofal,  Curvature-dimension inequalities and Ricci lower bounds for  sub-Riemannian manifolds with transverse symmetries, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Euro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' 19(2017),  151–219.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  [8] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Bauer, Analysis of the horizontal Laplacian for the Hopf fibration, Forum Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  17(2005), 903–920.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  [9] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Chavel, Eigenvalues in Riemannian Geometry, Academic Press, 1984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  [10] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Cowling, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Sikora, A spectral multiplier theorem for a sublaplacian on SU(2), Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Zeit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' 238(2001), 1–36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  [11] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Davies, Heat Kernels and Spectral Theory, Cambridge Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Press, 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  [12] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Donsker, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Varadhan, Asymptotic evaluation of certain Markov process  expectations for large time, I-IV, Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Pure Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' 28(1975), 1–47, 279–301;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  29(1976), 389–461;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' 36(1983), 183–212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  [13] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Du, Empirical approximation to invariant measures for McKean-Vlasov processes:  mea-field interaction vs self-interaction, to appear in Bernoulli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  [14] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Feng, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Miclo, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Wang, Poincar´e inequality for Dirichlet distributions and infinite-  dimensional generalizations, Lat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' 14(2017), 361–380.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  [15] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Feng, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Wang, Harnack inequality and applications for infinite-dimensional GEM  processes, Pot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' 44(2016), 137–153.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  [16] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' German, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' German,  Stochastic relaxation, Gibbs distribution, and the Baysian  restoration of images, IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Pattern Analysis Machine Intelligence 6(1984), 721–  741.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  [17] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Huang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Mao, On some mixing times for nonreversible finite Markov chains,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' 54(2017), 627–637.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  [18] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Huesmann, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Mattesini, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Trevisan, Wasserstein asympototics for the empirical  measure of fractional Brownian motion on a flat torus, arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='01025.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  [19] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Hwang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Hwang-Ma, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Sheu, Accelerating Gaussian diffusions, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' 3(1993), 897–913.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  55  [20] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Hwang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Hwang-Ma, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Sheu, Accelerating diffusions, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  15(2005), 1433–1444.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  [21] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Ledoux, On optimal matching of Gaussian samples, in:  Zap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Nauchn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Sem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='-  Peterburg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Otdel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Steklov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' (POMI) vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' 457, in: Veroyatnost Statistika.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  25, 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' 226–264.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  [22] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Li, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Wu, Wasserstein convergence for empirical measures of subordinated Dirichlet  diffusions on Riemannian manifolds, arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='03901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  [23] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Li, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Wu, Wasserstein convergence for conditional empirical measures of subordinated  Dirichlet diffusions on Riemannian manifolds, arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='13559.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  [24] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Li, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Wu, Wasserstein convergence rates for empirical measures of subordinated pro-  cesses on noncompact manifolds, arXiv:2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='12797.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  [25] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Ouhabaz, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Wang, Sharp estimates for instrinsic ultracontractivity on C1,α-  domains, Manu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' 112(2007), 229–244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  [26] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' von Renesse, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Sturm, Transport inequalities, gradient estimates, entropy, and  Ricci curvature, Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Pure Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' 58(2005), 923–940.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  [27] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Stannat, On validity of the log-Sobolev inequality for symmetric Fleming-Viot opera-  tors, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' 28(2000), 667–684.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  [28] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Villani, Optimal transport: Old and New Part 1,2, Springer, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  [29] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Wang, Application of coupling method to the Neumann eigenvalue problem, Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Theory Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Fields 98(1994), 299–306.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  [30] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Wang, Functional Inequalities, Markov Semigroups and Spectral Theory, Science  Press, Beijing, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  [31] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Wang, Gradient estimates and the first Neumann eigenvalue on non-convex mani-  folds, Stoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' 115(2005), 1475–1486.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  [32] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Wang, Analysis of Diffusion Processes on Riemannian Manifolds, World Scientific,  2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  [33] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Wang, Precise limit in Wasserstein distance for conditional empirical measures of  Dirichlet diffusion processes, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' 280(2021), 108998, 23pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  [34] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Wang, Convergence in Wasserstein distance for empirical measures of Dirichlet  diffusion processes on manifolds, to appear in J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  [35] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Wang, Wasserstein convergence rate for empirical measures on noncompact mani-  folds, Stoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' 144(2022), 271–287.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  [36] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Wang, Convergence in Wasserstein distance for empirical Measures of semilinear  SPDEs, to appear in Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  56  [37] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Wang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Wu, Wasserstein convergence for empirical measures of subordinated  diffusion processes on Riemannian manifolds, to appear in Pot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  [38] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Zhu, Limit Theorems in Wasserstein Distance for Empirical Measures  of Diffusion Processes on Riemannian Manifolds, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Henri Poincar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  59(2023), 437–475.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  [39] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Wu, Moderate deviations of dependent random variables related to CLT, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  23(1995), 420–445.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  [40] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Wu, Uniformly integrable operators and large deviations for Markov processes, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' 172(2000), 301–376.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  [41] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Ye, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Gao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Ding, A generalized Gronwall inequality and its application to a frac-  tional differential equation, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content=' 328(2007), 1075–1081.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
+page_content='  57' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFAT4oBgHgl3EQfERwH/content/2301.08420v1.pdf'}
diff --git a/4NFST4oBgHgl3EQfZjjS/content/tmp_files/2301.13792v1.pdf.txt b/4NFST4oBgHgl3EQfZjjS/content/tmp_files/2301.13792v1.pdf.txt
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@@ -0,0 +1,1732 @@
+arXiv:2301.13792v1  [math.FA]  31 Jan 2023
+Linear extension operators for Sobolev spaces on
+uniform trees
+Charles Fefferman1 and Bo’az Klartag2
+Dedicated in friendship to David Jerison
+Abstract
+Let 1 < p < ∞ and suppose that we are given a function f defined on the leaves of a
+weighted tree. We would like to extend f to a function F defined on the entire tree, so as
+to minimize the weighted W 1,p-Sobolev norm of the extension. An easy situation is when
+p = 2, where the harmonic extension operator provides such a function F. In this note
+we record our analysis of the particular case of a uniform binary tree, which is a complete,
+finite, binary tree with weights that depend only on the distance from the root. Neither the
+averaging operator nor the harmonic extension operator work here in general. Nevertheless,
+we prove the existence of a linear extension operator whose norm is bounded by a constant
+depending solely on p. This operator is a variant of the standard harmonic extension operator,
+and in fact it is harmonic extension with respect to a certain Markov kernel determined by p
+and by the weights.
+1
+Introduction
+Consider a full binary tree of height N, whose set of vertices is denoted by
+V =
+N
+�
+k=0
+{0, 1}k,
+i.e., the vertices are strings of zeroes and ones of length at most N. For x ∈ {0, 1}k and ℓ ≤ k
+we write πℓ(x) ∈ {0, 1}ℓ for the prefix of x of length ℓ. Thus for k ≥ 1, the parent of a vertex
+x ∈ {0, 1}k ⊆ V is the vertex πk−1(x). The set {0, 1}0 is a singleton whose unique element
+is denoted by ∅, the empty string, which is the root of the tree. The set of leaves of the tree is
+1Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, New Jersey 08544,
+USA. Email: cf@math.princeton.edu. Supported by the Air Force Office of Scientific Research, grant number
+FA9950-18-1-0069 and the National Science Foundation (NSF), grant number DMS-1700180.
+2Department of Mathematics,
+Weizmann Institute of Science,
+Rehovot 7610001,
+Israel.
+Email:
+boaz.klartag@weizmann.ac.il. Supported by a grant from the Israel Science Foundation (ISF).
+1
+
+{0, 1}N, and all other vertices in V are internal vertices. A vertex x ∈ {0, 1}k is said to have
+depth
+d(x) = k,
+thus leaves have depth N and the root has depth 0. The collections of bijections from V to
+V that preserve depth and parenthood relations form a group. This group is referred to as the
+symmetry group of the tree. It has 22N−1 elements, and it is a 2-Sylow subgroup of the group of
+all permutations of the leaves. The set
+E =
+N
+�
+k=1
+{0, 1}k = V \ {∅}
+is referred to as the set of edges of the tree, and the depth of an edge e ∈ {0, 1}k is d(e) = k.
+That is, we think of e ∈ E as an undirected edge connecting the vertex whose string corresponds
+to e to its unique parent. Each internal vertex other than the root is connected to three vertices,
+which are its parent and its two children. Assume that we are given edge weights
+W1, W1, . . . , WN > 0,
+where we view Wk as the weight of all edges of depth k. For 1 < p < ∞, the associated
+˙W 1,p-seminorm is defined, for F : V → R, via
+∥F∥ ˙W 1,p(V ) =
+
+
+N
+�
+k=1
+Wk
+�
+x∈{0,1}k
+|F(x) − F(πk−1(x))|p
+
+
+1/p
+.
+(1)
+We write ∂V = {0, 1}N ⊆ V for the set of leaves of the tree. The trace of the ∥·∥ ˙W 1,p-seminorm
+is defined, for f : ∂V → R, via
+∥f∥ ˙W 1,p(∂V ) = inf
+�
+∥F∥ ˙W 1,p(V ) ; F|∂V = f
+�
+,
+(2)
+i.e., the infimum of the ˙W 1,p-seminorm over all extensions of f from the leaves to the entire tree.
+We write RV for the collection of all functions f : V → R, and similarly R∂V is the collection
+of all functions f : ∂V → R. Our main result is the following:
+Theorem 1.1. Let 1 < p < ∞ and let W1, . . . , WN > 0. Then there exists a linear operator
+H : R∂V → RV with the following properties:
+1. It is a linear extension operator, i.e., (Hf)(x) = f(x) for any x ∈ ∂V and any function
+f : ∂V → R.
+2. Its norm is bounded by a constant ¯Cp depending only on p, i.e., for any f : ∂V → R,
+∥Hf∥ ˙W 1,p(V ) ≤ ¯Cp∥f∥ ˙W 1,p(∂V ).
+2
+
+In fact, we have the bound
+¯Cp ≤ 4p1/pq1/q ·
+�
+1 + max{(p − 1)−1/p, (q − 1)−1/q}
+�
+≤ C · max
+�
+1
+p − 1,
+1
+q − 1
+�
+,
+(3)
+where q = p/(p − 1) and where C > 0 is a universal constant.
+The proof of Theorem 1.2 is constructive, and the extension operator H that we construct is
+in fact a harmonic extension operator with respect to a certain random walk defined on the tree.
+At each step the random walk jumps from a vertex to one of its neighbors, where of course the
+neighbors of a vertex are its parent and its children. The Markov kernel corresponding to the
+random walk is invariant under the symmetries of the tree, thus the probability to move from a
+vertex to its neighbor depends only on the weights of the vertex and of its neighbor. The Markov
+kernel of our random walk is determined by the following requirement: For any s = 1, . . . , N,
+the probability that a random walk starting at some vertex of depth s will reach a leaf before
+reaching a vertex of depth s − 1 equals
+qs =
+(2sWs)−1/(p−1)
+�N
+k=s(2kWk)−1/(p−1).
+(4)
+Thus the weights of our random walk typically depend on p ∈ (1, ∞), except for the case where
+Ws is proportional to 2−s. This seems inevitable. Indeed, in some examples such as the 3rd
+example in Section 2 below, the linear extension operator H : R∂V → RV that corresponds to
+the parameter value p = p0, is not uniformly bounded for any p ∈ (1, ∞) \ {p0}. When p = 2,
+our random walk coincides with the usual random walk corresponding to the given weights on
+the edges of the binary tree, and thus in this case H is the standard harmonic extension operator
+(hence ¯Cp = 1 for p = 2).
+There is a certain range of weights where the averaging operator yields a uniformly bounded
+linear extension operator, as proven by Bj¨orn ×2, Gill and Shanmugalingam [1]. The averaging
+operator is the extension operator that assigns to each internal vertex v the average of the function
+values on the leaves of the subtree whose root is v. This averaging operator seems natural also
+from the point of view of Whitney’s extension theory, see the work by Shvartsman [5] on Sobolev
+extension in W 1,p(Rn). However, there are examples of uniform binary trees where the averaging
+operator does not provide a uniformly bounded operator, such as the case where Wk = 2−k for
+all k.
+What about non-uniform trees? Suppose that the edge weights are arbitrary positive numbers
+(We)e∈E that are not necessarily determined by the depth of the edge. When p = 2, there is still
+a harmonic extension operator of norm one from ˙W 1,p(∂V ) to ˙W 1,p(V ). However, for p ̸= 2 the
+situation seems subtle. We conjecture that in the general case of non-uniform tree weights there
+is no linear extension operator whose norm is bounded by a function of p alone. This conjecture
+is closely related to questions about well-complemented subspaces of ℓn
+p that are beyond the
+scope of this note.
+3
+
+In order to prove Theorem 1.1 we reformulate the problem in a way that brings us closer to
+analysis in ℓp-spaces. For 1 < p < ∞, the associated Lp(E)-norm is defined, for f : E → R,
+via
+∥f∥p = ∥f∥Lp(E) =
+
+
+N
+�
+k=1
+Wk
+�
+x∈{0,1}k
+|f(x)|p
+
+
+1/p
+.
+(5)
+We think of a function f : E → R as the gradient of a function ˜f : V → R, uniquely determined
+up to an additive constant. Given f : E → R we thus define a function ˜f : V → R as follows:
+For any x ∈ V other then the root,
+˜f(x) =
+d(x)
+�
+i=1
+f(πix),
+(6)
+while ˜f(x) = 0 if x = ∅ is the root. The only property of ˜f that matters is that for all x ∈ E,
+f(x) = ˜f(x) − ˜f(πd(x)−1(x)).
+We are interested in finding a linear operator T : Lp(E) → Lp(E), with a uniform bound on its
+operator norm, that has the following properties:
+1. The operator T takes the form
+Tf(x) = ˜T ˜f(x) − ˜T ˜f(πd(x)−1x)
+(7)
+for some linear operator ˜T : RV → RV . That is, ˜T takes functions on V to functions on
+V , and T is induces from ˜T via formula (7).
+2. The operator ˜T is equivariant with respect to the tree symmetries and it satisfies ˜T(1) ≡ 1,
+i.e., it maps the constant function 1 to itself.
+3. The function ˜Tg coincides with the function g on the leaves of the tree, i.e. ( ˜Tg)|∂V = g|∂V
+for any g : V → R.
+4. The function ˜Tg is determined by the values of the function g on the leaves of the tree.
+The operator norm of ˜T with respect to the ˙W 1,p-seminorm equals to the operator norm
+of T with respect to the Lp(E)-norm. Defining H(f|∂V ) = ˜Tf, Theorem 1.1 may thus be
+reformulated as follows:
+Theorem 1.2. Let 1 < p < ∞ and let W1, . . . , WN > 0. Then there exists a linear operator
+T : Lp(E) → Lp(E) with the above properties, whose operator norm is at most a certain
+constant ¯Cp depending only on p. In fact, we have the bound (3) for the constant ¯Cp.
+4
+
+The proof of Theorem 1.2 occupies the next three sections. In Section 2 we discuss invariant
+random walks on a full binary tree and describe the corresponding harmonic extension operator.
+In Section 3 we deal with the problem of bounding the norm of this operator, and use the sym-
+metries of the problem in order to reduce it to a one-dimensional question. This one-dimensional
+question is then answered in Section 4 using the Muckenhoupt criterion [4].
+When analyzing the binary tree we use the following notation: We write dlca(x, y) for the
+length of the maximal prefix shared by the strings x ∈ {0, 1}k and y ∈ {0, 1}ℓ, while lca(x, y)
+is the maximal prefix itself. Thus for two vertices x, y ∈ V , their least common ancestor is
+lca(x, y) ∈ V and its depth is dlca(x, y) ∈ {0, . . . , N}. Note that for any x ∈ E and ω ∈ ∂V ,
+dlca(πd(x)−1x, ω) = min{d(x) − 1, dlca(x, ω)}.
+(8)
+Acknowledgements. We would like to thank Jacob Carruth, Arie Israel, Anna Skorobogatova
+and Ignacio Uriarte-Tuero for helpful conversations. This research was conducted while BK
+was visiting Princeton University’s Department of Mathematics; he is grateful for their gracious
+hospitality.
+2
+Invariant random walks
+A Markov chain on V is a sequence of random variables R1, R2, . . . ∈ V such that the distribution
+of Ri+1 conditioned on R1, . . . , Ri is the same as its distribution conditioned on Ri. A Markov
+chain is time-homogeneous if for any x, y ∈ V , the probability that Ri+1 = x conditioned on the
+event Ri = y does not depend on i. A random walk on V is a time-homogeneous Markov chain
+R1, R2, . . . ∈ V such that Ri+1 is a neighbor of Ri with probability one.
+We say that the random walk is invariant if the probability to jump from a vertex x to a vertex
+y depends only on the depths d(x) and d(y). Our random walk will be invariant, and it will stop
+when it reaches a leaf, i.e., we have the stopping time
+τ = min{i ≥ 1 ; Ri is a leaf}.
+For s ≥ 1 we define qs to be the probability of the following event: Assuming that R1 is a vertex
+of depth s, the event is that Ri will remain at the subtree whose root is R1 for all 1 ≤ i ≤ τ.
+Equivalently, define
+Xi = d(Ri) ∈ {0, . . . , N}.
+Then X1, X2, . . . ∈ {0, . . . , N} is a random walk, since |Xi+1 − Xi| = 1 for all i. Furthermore,
+qs = P(∀1 ≤ i ≤ τ, Xi ≥ s | X1 = s).
+(9)
+Clearly
+q0 = qN = 1.
+(10)
+For r, s ∈ {0, . . . , N} with r ≤ s we set
+ps,r = P(min{Xi ; i ≤ τ} = r | X1 = s).
+5
+
+That is, the number ps,r is the probability that r is the minimal node that the walker visits when
+starting from node s, before reaching the terminal node N. Clearly �s
+r=0 ps,r = 1.
+Lemma 2.1. For 0 ≤ r ≤ s ≤ N − 1,
+ps,r = qr ·
+s
+�
+k=r+1
+(1 − qk)
+(11)
+where an empty product equals one. Moreover, pN,r = δrN, where δrN is the Kronecker delta.
+Proof. The expression on the right-hand side of (11) is the probability to ever reach s − 1 when
+starting from X1 = s, and from s − 1 to ever reach s − 2, etc. until we finally reach r, yet from
+r we require to never reach r − 1. Alternatively, when 0 ≤ r ≤ s, s ≥ 1 we have the recurrence
+relation
+ps,r = qsδs,r + (1 − qs) · ps−1,r.
+(12)
+This recurrence relation leads to another proof of (11).
+Suppose that our random walk R1, R2, . . . ∈ V begins at a vertex R1 = x with d(x) = s.
+Consider a leaf y ∈ ∂V with dlca(x, y) = r. What is the probability that our random walk will
+reach the leaf y? We claim that this probability is
+bs,r := P(Rτ = y) =
+r
+�
+k=0
+2k−Nps,k.
+(13)
+Indeed, conditioning on the value of k = mini≤τ d(Ri), by symmetry we know that Rτ is dis-
+tributed uniformly among the 2N−k leaf-descendants of the vertex πk(x). When k ≤ r, exactly
+one of these leaf-descendants is the leaf y, since the vertex of minimal depth that (Ri) visits must
+be the vertex πk(x), which is a prefix of y as k ≤ r = dlca(x, y). Hence the probability that
+Rτ = y, conditioning on the value of k, equals to 1/2N−k when k ≤ r and it vanishes other-
+wise. By using the definition of ps,k and the complete probability formula, we obtain (13). The
+harmonic extension operator associated with our invariant random walk is given by
+˜Tg(x) =
+�
+ω∈{0,1}N
+bd(x),dlca(x,ω) · g(ω)
+(x ∈ V ).
+(14)
+The operator T is induced from ˜T via formula (7) above. Requirements 1,...,4 from Section 1 are
+clearly satisfied.
+We stipulate that the collection of descendants of a vertex x ∈ V , denoted by D(x) ⊆ V ,
+includes the vertex x itself. Abbreviate a ∧ b = min{a, b} and a ∨ b = max{a, b}. The operator
+T takes the form
+Tf(x) =
+�
+y∈E
+K(x, y)f(y),
+(15)
+where the kernel K is described next.
+6
+
+Proposition 2.2. Let x, y ∈ E and denote s = d(x), t = d(y), r = dlca(x, y). Then the following
+hold: If x ̸∈ D(y) and y ̸∈ D(x), then r ≤ s ∧ t − 1 and
+K(x, y) = −qs · 2−t ·
+r
+�
+k=0
+2kps−1,k ≤ 0.
+(16)
+Otherwise, i.e., if y ∈ D(x) or if x ∈ D(y) then r = s ∧ t and
+K(x, y) = qs · 2−t ·
+r−1
+�
+k=0
+(2r − 2k)ps−1,k ≥ 0.
+(17)
+Proof. By (7) and (14) we have, for any x ∈ E,
+Tf(x) = ˜T ˜f(x) − ˜T ˜f(πd(x)−1x)
+=
+�
+ω∈{0,1}N
+bd(x),dlca(x,ω) ˜f(ω) −
+�
+ω∈{0,1}N
+bd(x)−1,dlca(πd(x)−1x,ω) ˜f(ω)
+=
+�
+ω∈{0,1}N
+ad(x),dlca(x,w) ˜f(ω)
+(18)
+where for any 0 ≤ r ≤ s, by (8) and (13),
+as,r = bs,r − bs−1,min{s−1,r} =
+r
+�
+k=0
+2k−Nps,k −
+min{s−1,r}
+�
+k=0
+2k−Nps−1,k.
+(19)
+Hence, by (6) and (18),
+Tf(x) =
+�
+ω∈{0,1}N
+ad(x),dlca(x,ω)
+N
+�
+i=1
+f(πiω)
+=
+�
+y∈E
+
+
+�
+ω∈{0,1}N;πd(y)ω=y
+ad(x),dlca(x,ω)
+
+ f(y) =
+�
+y∈E
+K(x, y)f(y),
+where the kernel K of the operator T satisfies, for any x, y ∈ E,
+K(x, y) =
+�
+ω∈{0,1}N ;πd(y)ω=y
+ad(x),dlca(x,ω).
+(20)
+Fix x, y ∈ E with s = d(x), t = d(y) and r = dlca(x, y). Let us consider first the case where x
+is not a descendant of y. This means that the prefix of y that is shared by x, is not the entire string
+y. Hence for any ω ∈ {0, 1}N with πd(y)ω = y we have dlca(x, ω) = dlca(x, y) ≤ d(y) − 1.
+Therefore, from (20) and (19),
+K(x, y) = 2N−d(y)ad(x),dlca(x,y)
+= 2N−t �
+bs,r − bs−1,(s−1)∧r
+�
+=
+r
+�
+k=0
+2k−tps,k −
+(s−1)∧r
+�
+k=0
+2k−tps−1,k,
+7
+
+as bs,r = �r
+k=0 2k−Nps,k. Since ps,s = qs, we have
+K(x, y) = δrs2s−tqs +
+(s−1)∧r
+�
+k=0
+2k−t [ps,k − ps−1,k] = qs ·
+
+δrs2s−t −
+(s−1)∧r
+�
+k=0
+2k−tps−1,k
+
+ ,
+(21)
+where we used the relation (12), which implies that when k ≤ s − 1,
+ps,k − ps−1,k = −qs · ps−1,k.
+(22)
+We may now prove the conclusion of the proposition in the case where x ̸∈ D(y) and y ̸∈ D(x).
+Indeed, in this case r ≤ s ∧ t − 1 and formula (21) applies. Since δrs = 0 in this case, we deduce
+formula (16) from (21).
+The next case we consider is the case where r = s ≤ t − 1, or equivalently, where y ∈
+D(x) \ {x}. Thus x ̸∈ D(y) and formula (21) applies. Recalling that �s−1
+k=0 ps−1,k = 1 we obtain
+from (21) that
+K(x, y) = qs ·
+s−1
+�
+k=0
+(2s−t − 2k−t)ps−1,k = qs · 2−t ·
+s−1
+�
+k=0
+(2r − 2k)ps−1,k,
+proving formula (17) in the case y ∈ D(x) \ {x}.
+We move on to the case where x ∈ D(y) \ {y}, thus dlca(x, y) = t ≤ s − 1. In this case, by
+applying (20), (19), (13) and then (22),
+K(x, y) =
+�
+ω∈{0,1}N ;πd(y)ω=y
+ad(x),dlca(x,ω) = 2N−d(x)ad(x),d(x) +
+d(x)−1
+�
+k=d(y)
+2N−k−1ad(x),k
+= 2N−sas,s +
+s−1
+�
+k=t
+2N−k−1as,k = 2N−s(bs,s − bs−1,s−1) +
+s−1
+�
+k=t
+2N−k−1(bs,k − bs−1,k)
+= 2N−s
+�
+s
+�
+k=0
+2k−Nps,k −
+s−1
+�
+k=0
+2k−Nps−1,k
+�
++
+s−1
+�
+k=t
+2N−k−1
+k
+�
+ℓ=0
+2ℓ−N(ps,ℓ − ps−1,ℓ)
+= qs − qs
+s−1
+�
+k=0
+2k−sps−1,k − qs
+s−1
+�
+ℓ=0
+s−1
+�
+k=ℓ∨t
+2ℓ−k−1ps−1,ℓ
+= qs
+�
+1 −
+s−1
+�
+k=0
+2k−sps−1,k −
+s−1
+�
+ℓ=0
+[2ℓ−ℓ∨t − 2ℓ−s]ps−1,ℓ
+�
+= qs
+�
+1 −
+s−1
+�
+k=0
+2k−k∨tps−1,k
+�
+= qs ·
+�
+1 −
+t−1
+�
+k=0
+2k−tps−1,k −
+s−1
+�
+k=t
+ps−1,k
+�
+= qs ·
+t−1
+�
+k=0
+(1 − 2k−t)ps−1,k = qs · 2−t ·
+t−1
+�
+k=0
+(2t − 2k)ps−1,k.
+8
+
+Since r = t in this case, we have proved formula (17) in the case where y ∈ D(x) \ {x}. Finally,
+the last case that remains is when x = y. In this case r = s = t and
+K(x, y) =
+�
+ω∈{0,1}N;πd(y)ω=y
+ad(x),dlca(x,ω) = 2N−d(x)ad(x),d(x) = 2N−sas,s = 2N−s(bs,s − bs−1,s−1)
+= 2N−s
+�
+s
+�
+k=0
+2k−Nps,k −
+s−1
+�
+k=0
+2k−Nps−1,k
+�
+= qs − qs
+s−1
+�
+k=0
+2k−sps−1,k
+= qs ·
+s−1
+�
+k=0
+(1 − 2k−s)ps−1,k,
+completing the proof of formula (17).
+Some examples.
+1. The simplest example is when qs = 1 for all s ≥ 1. In this case the operator ˜T is the
+familiar averaging operator. That is, the extension operator ˜T is the operator that assigns
+to each vertex the average of the values at the leaves of its subtree. In this case
+ps,r = δs,r.
+2. Consider the case where the invariant random walk is such that Xi := d(Ri) is a symmetric
+random walk on {0, . . . , N}, i.e., the probability to jump from i to i + 1 is exactly 1/2 for
+i = 1, . . . , N − 1. Recall that qs is the probability to never leave the subtree when starting
+at a vertex of depth s. We claim that in this example, for s = 1, . . . , N,
+qs =
+1
+N − s + 1.
+(23)
+Indeed, the function f(i) = i is harmonic on {0, 1, . . . , N} and hence f(Xi) is a mar-
+tingale. Thus for any stopping time ˜τ we have f(X1) = Ef(X˜τ). We pick the stopping
+time
+˜τ = min{ i; Xi ∈ {s − 1, N} }
+and obtain (23) since
+N · qs + (s − 1) · (1 − qs) = s.
+Next we use Lemma 2.1 and find a formula for ps,r. Since formula (23) is valid for any
+s ≥ 1, we conclude that for any r ≥ 0 and s ≥ r + 1,
+s�
+k=r+1
+(1 − qk) =
+s�
+k=r+1
+N − k
+N − k + 1 = N − s
+N − r.
+(24)
+9
+
+Formula (24) is actually valid for any 0 ≤ r ≤ s, since an empty product equals one.
+Recall that q0 = 1. We thus conclude from Lemma 2.1 that for s = 0, . . . , N − 1,
+ps,r = N − s
+N − r ·
+�
+1
+r = 0
+1
+N−r+1
+1 ≤ r ≤ s
+while pN,r = δN,r.
+3. Let 0 < δ < 1, and consider the case where (Xi) is a random walk on {0, . . . , N} such
+that the probability to jump from k to k + 1 equals 1/2 if k < N − 1, and it equals δ if
+k = N − 1. A harmonic function here is
+f(k) =
+�
+k
+k ≤ N − 1
+N − 2 + 1/δ
+k = N
+Therefore, for s = 1, . . . , N − 1,
+(N − 2 + 1/δ) · qs + (s − 1)(1 − qs) = s.
+Thus q0 = qN = 1 while for s = 1, . . . , N − 1,
+qs =
+1
+N − s + 1/δ − 1.
+Hence for any s ≤ N − 1 and r ≤ s − 1,
+s
+�
+k=r+1
+(1 − qk) =
+s
+�
+k=r+1
+N − k + δ−1 − 2
+N − k + δ−1 − 1 = N − s + δ−1 − 2
+N − r + δ−1 − 2.
+We conclude from Lemma 2.1 that for s = 0, . . . , N − 1 and 0 ≤ r ≤ s,
+ps,r = qr ·
+s�
+k=r+1
+(1 − qk) = N − s + δ−1 − 2
+N − r + δ−1 − 2 ·
+�
+1
+r = 0
+1
+N−r+1/δ−1
+1 ≤ r ≤ s
+while pN,r = δN,r.
+We conclude this section with the following:
+Lemma 2.3. For any numbers q1, . . . , qN−1 ∈ (0, 1) there exists a random walk
+X1, X2, . . . ∈ {0, . . . , N}
+satisfying (9) with τ = min{i ≥ 1 ; Xi = N} for s = 1, . . . , N − 1.
+Proof. Write xk for the probability that the random walk jumps from k to k + 1. Then for
+s = 0, . . . , N − 1,
+qs = xs(qs+1 + (1 − qs+1)xs),
+(25)
+where we set q0 = qN = 1. We claim that xs ∈ [0, 1] is determined by equation (25). Indeed, the
+right-hand side of (25) is a continuous, increasing function of xs in the interval [0, 1], that maps
+this interval to itself.
+10
+
+3
+The ancestral and non-ancestral parts of the kernel
+We need to bound the operator norm in Lp(E) of the operator T whose kernel is described in
+Proposition 2.2. Let us consider first the non-ancestral part of the operator, given by the kernel
+K0(x, y) = K(x, y) · 1{x̸∈D(y),y̸∈D(x)} = −1{r≤s∧t−1} · qs · 2−t ·
+r
+�
+k=0
+2kps−1,k.
+(26)
+Here as usual s = d(x), t = d(y) and r = dlca(x, y). Write T0 : Lp(E) → Lp(E) for the
+operator whose kernel is K0. A function f : E → R is invariant under the symmetries of the
+tree, or invariant in short, if it takes the form
+f(x) = F(d(x))
+for some function F : {1, . . . , N} → R. The operator T0 is equivariant under the symmetries
+of the tree. Therefore, if f(x) = F(d(x)) is an invariant function, then so is T0f. In fact, in the
+case where f(x) = F(d(x)) we can write
+T0f(x) =
+N
+�
+t=1
+L0(d(x), t)F(t)
+(27)
+for a certain kernel L0(s, t) defined for s, t = 1, . . . , N.
+Lemma 3.1. For s, t = 1, . . . , N,
+L0(s, t) = −qs ·
+m−1
+�
+k=0
+(1 − 2k−m)ps−1,k ≤ 0.
+Proof. Let r ≤ min{t, s} − 1. A moment of reflection reveals that for x ∈ E with d(x) = s,
+n(t; s, r) := #{y ∈ E ; d(y) = t, dlca(x, y) = r} = 2t−r−1.
+By (26), (27) and the definition of T0, for any x ∈ E with d(x) = s,
+L0(s, t) =
+�
+y∈E;d(y)=t
+K0(x, y) = −
+t∧s−1
+�
+r=0
+n(t; s, r) · qs · 2−t ·
+r
+�
+k=0
+2kps−1,k.
+Denote m = s ∧ t. Then for s, t = 1, . . . , N,
+L0(s, t) = −qs ·
+t∧s−1
+�
+r=0
+r
+�
+k=0
+2k−r−1ps−1,k = −qs ·
+m−1
+�
+k=0
+m−1
+�
+r=k
+2k−r−1ps−1,k
+= −qs ·
+m−1
+�
+k=0
+(1 − 2k−m)ps−1,k.
+11
+
+Write ΩN = {1, . . . , N} and for F : ΩN → R define
+∥F∥p = ∥F∥Lp(ΩN) =
+� N
+�
+k=1
+2k · Wk · |F(k)|p
+�1/p
+.
+(28)
+Observe that ∥f∥Lp(E) = ∥F∥p if f(x) = F(d(x)). Let
+S0F(s) =
+N
+�
+t=1
+L0(s, t)F(t)
+so that by (27),
+T0(F ◦ d) = (S0F) ◦ d.
+(29)
+For f, g : E → R we consider the scalar product
+⟨f, g⟩ =
+�
+x∈E
+Wd(x)f(x)g(x)
+while for F, G : ΩN → R we set
+⟨F, G⟩ := ⟨F ◦ d, G ◦ d⟩ =
+N
+�
+k=1
+2kWkF(k)G(k).
+The adjoint operators T ∗
+0 and S∗
+0 are defined with respect to these scalar products. The follow-
+ing lemma is probably well-known to experts (see, e.g., Howard and Schep [2] for a related
+argument), and its proof is provided for completeness.
+Lemma 3.2. Let 1 < p < ∞. Then the norm of the operator T0 : Lp(E) → Lp(E) is attained at
+an invariant, non-negative function f, and it equals to the norm of the operator S0 : Lp(Ωn) →
+Lp(Ωn).
+Proof. Denote momentarily T = −T0 and S = −S0. By Lemma 3.1 the kernel −L0 of the
+operator S is non-negative, and by (26) the kernel of the operator T is non-negative as well.
+By approximation, we may assume that these two kernels are strictly positive, while keeping
+condition (29), thus
+T (F ◦ d) = (SF) ◦ d.
+(30)
+We deduce that T ∗(F ◦ d) = (S∗F) ◦ d. By compactness,
+sup
+0̸≡F ∈Lp(ΩN )
+∥SF∥p
+∥F∥p
+is attained at some function F. Since the kernel of S is non-negative, we may assume that
+the extremal function F is non-negative. By the Lagrange multipliers theorem, the function F
+satisfies a certain eigenvalue equation, and in fact there exists λ ∈ R such that
+S∗(SF)p−1 = λF p−1.
+(31)
+12
+
+Since the kernel of S is positive and F is non-negative and not identically zero, it follows from
+(31) that F is actually positive. The norm of the operator S : Lp(Ωn) → Lp(Ωn) equals λ1/p > 0,
+since
+∥SF∥p
+p = ⟨(SF)p−1, SF⟩ = ⟨S∗(SF)p−1, F⟩ = λ⟨F p−1, F⟩ = λ∥F∥p
+p.
+Denoting f = F ◦ d, we find from (30) that f is a positive invariant function satisfying
+T ∗(T f)p−1 = λf p−1.
+Since the kernel of T is non-negative, we have the pointwise H¨older inequality
+T (uv) ≤ T (up)1/p · T (vq)1/q,
+valid for any non-negative functions u, v ∈ Lp(E), where q = p/(p − 1). The operator T has
+a non-negative kernel, and hence its norm is attained at a non-negative function u ∈ Lp(E). By
+the pointwise H¨older inequality,
+(T u)p ≤ T (upf −p/q) · (T f)p/q = T (upf 1−p) · (T f)p−1.
+Therefore,
+∥T u∥p
+Lp(E) ≤ ⟨T (upf 1−p), (T f)p−1⟩ = ⟨upf 1−p, T ∗(T f)p−1⟩ = λ⟨upf 1−p, f p−1⟩ = λ∥u∥p
+Lp(E).
+Thus the norm of T : Lp(E) → Lp(E) is at most λ1/p, which is the norm of S : Lp(ΩN) →
+Lp(ΩN). The two norms must therefore be equal, since the operator S is equivalent to the
+restriction of T to the space of invariant functions.
+We move on to the ancestral part of the operator, which according to Proposition 2.2 is given
+by
+K1(x, y) = K(x, y) − K0(x, y) = 1{r=s∧t} · qs · 2−t ·
+r−1
+�
+k=0
+(2r − 2k)ps−1,k ≥ 0.
+(32)
+Write T1 for the operator whose kernel is K1. As before, for an invariant function f(x) =
+F(d(x)) we may write
+T1f(x) =
+N
+�
+t=1
+L1(d(x), t)F(t)
+(33)
+for a certain kernel L1(s, t) defined for s, t = 1, . . . , N. We also write
+S1F(s) =
+N
+�
+t=1
+L1(s, t)F(t).
+It is possible to use formula (7) for the operator T and the definition (14) of the harmonic exten-
+sion operator ˜T and deduce that
+S0 + S1 ≡ 0,
+(34)
+essentially because the only invariant, harmonic function on the vertices of the tree is the constant
+function. An alternative, more direct proof of (34) is provided in the following:
+13
+
+Lemma 3.3. Let 1 < p < ∞. Then the norm of the operator T1 : Lp(E) → Lp(E) is equal to
+the norm of S1 : Lp(Ωn) → Lp(Ωn). Additionally, for s, t = 1, . . . , N with m = s ∧ t we have
+L1(s, t) = qs ·
+m−1
+�
+k=0
+(1 − 2k−m)ps−1,k = −L0(s, t).
+Proof. The first assertion of the lemma follows from fact that the kernel of T1 is non-negative
+and invariant under the symmetries of the tree, as in Lemma 3.2. For the second part, let s, t =
+1, . . . , N and denote m = s ∧ t. We claim that for x ∈ E with d(x) = s,
+n(t; s, m) := #{y ∈ E ; d(y) = t, dlca(x, y) = m} = max{1, 2t−s}.
+(35)
+Indeed, assume first that t ≥ s. How many y’s are there with d(y) = t and dlca(x, y) = m?
+Since m = s, the answer is 2t−s. Next, if s ≥ t, then the number of such y’s is one. This proves
+(35). Therefore,
+L1(s, t) =
+�
+y∈E;d(y)=t
+K1(x, y) = n(t; s, m) · qs · 2−t ·
+m−1
+�
+k=0
+(2m − 2k)ps−1,k
+= max{2−t, 2−s} · qs ·
+m−1
+�
+k=0
+(2m − 2k)ps−1,k.
+= 2−m · qs ·
+m−1
+�
+k=0
+(2m − 2k)ps−1,k.
+Corollary 3.4. We have
+∥T∥Lp(E)→Lp(E) ≤ 2∥S0∥Lp(ΩN)→Lp(ΩN).
+Proof. This follows from the fact that T = T0 + T1 together with the facts that ∥T0∥ = ∥S0∥ and
+∥T1∥ = ∥S1∥ while S1 = −S0.
+In view of Corollary 3.4, we are interested in bounds for the norm of the operator S = −S0 =
+S1 : Lp(ΩN) → Lp(ΩN) whose non-negative kernel is
+L(s, t) = qs ·
+s∧t−1
+�
+k=0
+(1 − 2k−s∧t)ps−1,k ≤ qs ·
+s∧t−1
+�
+k=0
+ps−1,k.
+(36)
+14
+
+From Lemma 2.1 we know that ps,r = qr · �s
+k=r+1(1 − qk) for s ≤ N − 1. A little exercise in
+probability shows that for any 1 ≤ m ≤ s ≤ N,
+m−1
+�
+k=0
+ps−1,k =
+s−1
+�
+k=m
+(1 − qk).
+(37)
+Alternatively, (37) holds true for m = 0 as q0 = 1, and it may be proven by induction on m since
+ps−1,m +
+s−1
+�
+k=m
+(1 − qk) = qm ·
+s−1
+�
+k=m+1
+(1 − qk) +
+s−1
+�
+k=m
+(1 − qk) =
+s�
+k=m+1
+(1 − qk).
+From (36) and (37) we thus obtain
+Corollary 3.5. For s, t = 1, . . . , N, with m = min{s, t},
+0 ≤ L(s, t) ≤ qs
+s−1
+�
+k=m
+(1 − qk),
+where an empty product equals one.
+Some examples (parallel to the ones discussed in Section 2).
+1. For the averaging operator, where qs = 1 and ps,r = δs,r, we have
+L(s, t) = −1
+2 · 1{s≤t},
+i.e., this is the matrix whose entries equal 0 below the diagonal and −1/2 on and above
+the diagonal. This is a rather simple matrix, and it is bounded with respect to the weighted
+Lp-norm for quite a few sequences of weights.
+2. For the symmetric random walk matrix, we have q0 = 1 while for 1 ≤ s ≤ N,
+qs =
+1
+N − s + 1.
+Hence in view of Corollary 3.5, with m = min{s, t},
+0 ≤ L(s, t) ≤
+1
+N − s + 1
+s−1
+�
+k=m
+N − k
+N − k + 1 =
+1
+N − m + 1.
+3. In the case where
+qs =
+1
+N − s + 1/δ − 1
+for some 0 < δ < 1, we have
+0 ≤ L(s, t) ≤
+1
+N − s + 1/δ − 1
+s−1
+�
+k=m
+N − k + 1/δ − 2
+N − k + 1/δ − 1 =
+1
+N − m + 1/δ − 1.
+15
+
+All that remains is to bound the Lp(ΩN)-norm of the operator whose kernel is discussed
+in Corollary 3.5. Recall from Lemma 2.3 that we have the freedom to choose the parameters
+q1, . . . , qN−1 ∈ (0, 1) as we please.
+How should we choose these parameters? Since L(N, t) ≤ �N−1
+k=t (1−qk) and we are looking
+for upper bounds for the norm, the qs should not be too tiny. On the other hand, L(s, t) ≤ qs for
+s ≤ N − 1 and hence it is beneficial to choose qs rather small. We would therefore need some
+balance for the qs, which is the subject of the next section.
+4
+One-dimensional analysis
+Let 1 < p < ∞. It will be slightly more convenient to denote
+K(s, t) = L(N + 1 − s, N + 1 − t)
+and
+Qs = qN+1−s.
+Recalling from (10) that qN = 1, we see that
+Q1 = 1.
+From Corollary 3.5 we know that for s, t = 1, . . . , N,
+K(s, t) = L(N + 1 − s, N + 1 − t) ≤ qN+1−s
+N−s
+�
+k=N+1−max{s,t}
+(1 − qk) = Qs
+max{s,t}
+�
+k=s+1
+(1 − Qk).
+From Corollary 3.5 we know that L ≥ 0. Consequently, for s, t = 1, . . . , N,
+0 ≤ K(s, t) ≤
+�
+Qs
+t ≤ s
+Qs · �t
+k=s+1(1 − Qk)
+t ≥ s + 1
+(38)
+Recall that we are given edge weights W1, . . . , WN > 0, and that the associated Lp(E)-norm is
+given by (5). Denote
+ws := 2N+1−s · WN+1−s > 0.
+Consider the weighted ℓp-norm
+∥f∥p,w =
+� N
+�
+k=1
+wk|f(k)|p
+�1/p
+(39)
+and the operator T whose kernel is K(s, t). We are allowed to choose the weights q1, . . . , qN−1 ∈
+(0, 1) as we please, or equivalently, we have the freedom to determine Q2, . . . , QN ∈ (0, 1). We
+must keep Q1 = 1. Based on considerations related to the Muckenhoupt criterion discussed
+below, we set
+Qs =
+w−1/(p−1)
+s
+�s
+k=1 w−1/(p−1)
+k
+.
+(40)
+It is clear that Q1 = 1 and that Qs ∈ (0, 1) for all s ≥ 2. Recall that q = p/(p − 1).
+16
+
+Lemma 4.1. In order to prove Theorem 1.2, it suffices to show that the operator norm of T with
+respect to the ∥ · ∥p,w-norm is bounded by a constant ˆCp > 0 depending only on p, where in fact
+ˆCp ≤ 2p1/pq1/q ·
+�
+1 + max{(p − 1)−1/p, (q − 1)−1/q}
+�
+.
+(41)
+Proof. In view of Corollary 3.4, it suffices to bound the operator norm of −S0, whose kernel is
+L, with respect to the Lp(ΩN)-norm defined in (28). Under the transformation
+s �→ N + 1 − s
+the operator −S0 whose kernel is L transforms to the operator T whose kernel is K. The Lp(ΩN)-
+norm from (28) transforms to the ∥·∥p,w-norm defined in (39). Hence Theorem 1.2 would follow
+once we obtain the bound (41), where ¯Cp ≤ 2 ˆCp by Corollary 3.4.
+The remainder of this section is devoted to the proof of the following:
+Proposition 4.2. The operator norm of T with respect to the norm (39) is bounded by a number
+ˆCp depending only on p ∈ (1, ∞). In fact, we have the bound (41) for the constant ˆCp.
+Our main tool in the proof of Proposition 4.2 is the Muckenhoupt criterion [4], which is an
+indispensable tool for proving one-dimensional inequalities of Poincar´e-Sobolev type. For the
+reader’s convenience, we include here a statement and a proof of a straightforward modification
+of the Muckenhoupt criterion, with sums in place of integrals:
+Theorem 4.3. (Muckenhoupt) Let 1 < p < ∞ and write ΩN = {1, . . . , N}. Let U, V : ΩN →
+(0, ∞) and let A > 0 be such that for all r = 1, . . . , N,
+� N
+�
+k=r
+|U(k)|p
+�1/p
+≤ A
+�
+r
+�
+k=1
+|V (k)|−q
+�−1/q
+.
+(42)
+Then for any function f : ΩN → R,
+� N
+�
+k=1
+�����U(k)
+k
+�
+ℓ=1
+f(ℓ)
+�����
+p�1/p
+≤ CpA
+� N
+�
+k=1
+|V (k)f(k)|p
+�1/p
+,
+(43)
+with Cp = p1/pq1/q.
+By continuity, the analog of Theorem 4.3 for p = 1, ∞ holds true with C1 = C∞ = 1. We
+remark that as in [4], this criterion is tight, in the sense that the infimum over all A > 0 satisfying
+(42) is equivalent to the best constant in inequality (43). For the proof of Theorem 4.3 we require
+the following:
+17
+
+Lemma 4.4. For α1, . . . , αN > 0 and r = 1, . . . , N,
+r
+�
+k=1
+αk
+�
+k
+�
+ℓ=1
+αℓ
+�−1/p
+≤ q ·
+�
+r
+�
+k=1
+αk
+�1/q
+.
+(44)
+Proof. We will use the simple inequality
+(a + b)1/q − a1/q ≥ b ·
+min
+ξ∈(a,a+b)
+ξ1/q−1
+q
+= 1
+q(a + b)1/q−1 · b,
+(45)
+valid for any a, b ≥ 0 with a + b > 0. From (45), for k = 1, . . . , N,
+� k
+�
+ℓ=1
+αℓ
+�1/q
+−
+�k−1
+�
+ℓ=1
+αℓ
+�1/q
+≥ 1
+q · αk ·
+�
+k
+�
+ℓ=1
+αℓ
+�1/q−1
+,
+where an empty sum equals zero. By summing this for k = 1, . . . , r we obtain (44).
+Proof of Theorem 4.3 (Muckenhoupt). By the H¨older inequality, for any function f : ΩN → R
+and weights h : ΩN → (0, ∞),
+N
+�
+k=1
+�����U(k)
+k
+�
+ℓ=1
+f(ℓ)
+�����
+p
+≤
+N
+�
+k=1
+Up(k) ·
+k
+�
+ℓ=1
+|f(ℓ)V (ℓ)h(ℓ)|p ·
+�
+k
+�
+j=1
+|V (j)h(j)|−q
+�p/q
+=
+N
+�
+ℓ=1
+|f(ℓ)V (ℓ)h(ℓ)|p
+N
+�
+k=ℓ
+Up(k)
+� k
+�
+j=1
+|V (j)h(j)|−q
+�p/q
+.
+(46)
+Set h(k) =
+��k
+ℓ=1 V (ℓ)−q�1/(pq)
+. By applying Lemma 4.4 with αk = V (k)−q we obtain
+r
+�
+k=1
+|V (k)h(k)|−q =
+r
+�
+k=1
+αk
+� k
+�
+ℓ=1
+αℓ
+�−1/p
+≤ q ·
+�
+r
+�
+k=1
+αk
+�1/q
+= q ·
+�
+r
+�
+k=1
+V (k)−q
+�1/q
+.
+Hence for any f : ΩN → R, the expression in (46) is at most
+qp/q ·
+N
+�
+ℓ=1
+|f(ℓ)V (ℓ)h(ℓ)|p
+N
+�
+k=ℓ
+Up(k)
+�
+k
+�
+j=1
+|V (j)|−q
+�p/q2
+.
+(47)
+By applying (42) and then Lemma 4.4 with αk = Up(N + 1 − k) and with p ∈ (1, ∞) playing
+the rˆole of q ∈ (1, ∞), we see that
+N
+�
+k=ℓ
+|U(k)|p
+�
+k
+�
+j=1
+|V (j)|−q
+�p/q2
+≤ Ap/q
+N
+�
+k=ℓ
+|U(k)|p
+� N
+�
+j=k
+|U(j)|p
+�−1/q
+= Ap/q
+N+1−ℓ
+�
+k=1
+αk
+� k
+�
+j=1
+αj
+�−1/q
+≤ p · Ap/q
+�N+1−ℓ
+�
+k=1
+αk
+�1/p
+= p · Ap/q
+� N
+�
+k=ℓ
+Up(k)
+�1/p
+.
+18
+
+Hence the expression in (47) is at most
+p · (qA)p/q ·
+N
+�
+ℓ=1
+|f(ℓ)V (ℓ)h(ℓ)|p ·
+� N
+�
+k=ℓ
+Up(k)
+�1/p
+.
+Applying (42) again we bound the last expression from above by
+p · (qA)p/q · A ·
+N
+�
+ℓ=1
+|f(ℓ)V (ℓ)h(ℓ)|p ·
+�
+ℓ
+�
+k=1
+V −q(k)
+�−1/q
+= pqp/qAp
+N
+�
+ℓ=1
+|f(ℓ)V (ℓ)|p,
+completing the proof.
+Corollary 4.5. Let 1 < p < ∞ and ΩN = {1, . . . , N}. Let U, V : ΩN → (0, ∞) and let A > 0
+be such that for all r = 1, . . . , N,
+�
+r
+�
+k=1
+|U(k)|p
+�1/p
+≤ A
+� N
+�
+k=r
+|V (k)|−q
+�−1/q
+.
+(48)
+Then for any f : ΩN → R,
+� N
+�
+k=1
+�����U(k)
+N
+�
+ℓ=k
+f(ℓ)
+�����
+p�1/p
+≤ CpA
+� N
+�
+k=1
+|V (k)f(k)|p
+�1/p
+,
+(49)
+with Cp = p1/pq1/q.
+Proof. Denote ˜U(k) = U(N + 1 − k) and ˜V (k) = V (N + 1 − k). Then from (48), for all
+r = 1, . . . , N,
+� N
+�
+k=r
+| ˜U(k)|p
+�1/p
+≤ A
+�
+r
+�
+k=1
+| ˜V (k)|−q
+�−1/q
+.
+By Theorem 4.3, this implies that for any g : ΩN → R, denoting f(r) = g(N + 1 − r),
+� N
+�
+k=1
+�����
+˜U(k)
+k
+�
+ℓ=1
+f(N + 1 − ℓ)
+�����
+p�1/p
+≤ CpA
+� N
+�
+k=1
+| ˜V (k)f(N + 1 − k)|p
+�1/p
+,
+or equivalently,
+� N
+�
+k=1
+�����
+˜U(N + 1 − k)
+N
+�
+ℓ=k
+f(ℓ)
+�����
+p�1/p
+≤ CpA
+� N
+�
+k=1
+| ˜V (N + 1 − k)f(k)|p
+�1/p
+.
+This implies (49).
+19
+
+Proposition 4.6. For f : Ωn → R and s = 1, . . . , N denote
+T0f(s) = Qs
+s
+�
+t=1
+f(t).
+Then the operator norm of T0 with respect to the norm ∥ · ∥p,w defined in (39) is bounded by a
+number ˜Cp depending only on p ∈ (1, ∞). In fact, ˜Cp ≤ 21/pp1/pq1/q · max{1, (p − 1)−1/p}.
+The proof of Proposition 4.6 requires the following:
+Lemma 4.7. For any α1, . . . , αN > 0 and r = 1, . . . , N,
+N
+�
+k=r
+αk
+� k
+�
+ℓ=1
+αℓ
+�−p
+≤
+2
+min{1, p − 1} ·
+�
+r
+�
+k=1
+αk
+�−p+1
+.
+(50)
+Proof. We use the inequality
+(p − 1) · b(a + b)−p ≤ a−p+1 − (a + b)−p+1,
+which is valid for any a, b > 0. Then for k = 2, . . . , N,
+(p − 1) · αk
+� k
+�
+ℓ=1
+αℓ
+�−p
+≤
+�k−1
+�
+ℓ=1
+αℓ
+�−p+1
+−
+�
+k
+�
+ℓ=1
+αℓ
+�−p+1
+.
+By summing this for k = r + 1, . . . , N we obtain
+(p − 1) ·
+N
+�
+k=r+1
+αk
+� k
+�
+ℓ=1
+αℓ
+�−p
+≤
+�
+r
+�
+k=1
+αk
+�−p+1
+−
+� N
+�
+ℓ=1
+αℓ
+�−p+1
+≤
+�
+r
+�
+k=1
+αk
+�−p+1
+,
+where an empty sum equals zero. We conclude (50) by summing this with the trivial inequality
+min{1, p − 1} · αr
+�
+r
+�
+ℓ=1
+αℓ
+�−p
+≤
+�
+r
+�
+k=1
+αk
+�−p+1
+.
+Proof of Proposition 4.6. Define
+U(k) = w1/p
+k
+· Qk
+and
+V (k) = w1/p
+k .
+Let us verify the condition of the Muckenhoupt criterion. We need to find A > 0 such that for
+all r = 1, . . . , N inequality (42) holds true, that is,
+N
+�
+k=r
+wkQp
+k ≤ Ap
+�
+r
+�
+k=1
+w−q/p
+k
+�−p/q
+.
+(51)
+20
+
+Recall that p/q = p − 1. From the definition (40) of Qk, we need
+N
+�
+k=r
+w−1/(p−1)
+k
+�
+k
+�
+ℓ=1
+w−1/(p−1)
+ℓ
+�−p
+≤ Ap
+�
+r
+�
+k=1
+w−1/(p−1)
+k
+�−(p−1)
+.
+Setting αk = w−1/(p−1)
+k
+and using Lemma 4.7, we see that (51) holds true with
+A = 21/p · max{1, (p − 1)−1/p}.
+From Theorem 4.3 we thus conclude that for any f : ΩN → R,
+� N
+�
+k=1
+wk
+�����Qk
+k
+�
+ℓ=1
+f(ℓ)
+�����
+p�1/p
+≤ p1/pq1/qA ·
+� N
+�
+k=1
+wk|f(k)|p
+�1/p
+.
+This implies the required bound for the operator norm of f.
+Proposition 4.8. For f : Ωn → R and s = 1, . . . , N denote
+T1f(s) =
+N
+�
+t=s+1
+�
+Qs
+t�
+k=s+1
+(1 − Qk)
+�
+f(t).
+Then the operator norm of T1 with respect to the norm ∥ · ∥p,w defined in (39) is bounded by
+21/qp1/pq1/q · max{1, (q − 1)−1/q}.
+Proof. Denote αk = w−1/(p−1)
+k
+and recall from (40) that Qk = αk/ �k
+ℓ=1 αℓ. We claim that for
+any t ≥ s + 1,
+Qs
+t�
+k=s+1
+(1 − Qk) =
+αs
+�t
+k=1 αk
+.
+(52)
+Indeed, (52) holds true for t = s, since an empty product equals one, and for t ≥ s + 1 it is
+proven by an easy induction on t. Consequently,
+T1f(s) = αs
+N
+�
+t=s+1
+1
+�t
+j=1 αj
+f(t).
+Since p, q ∈ (1, ∞), the elementary inequality of Lemma 4.7 is valid also when p is replaced by
+q. It implies that for r = 1, . . . , N,
+�
+r
+�
+k=1
+αk
+�−q+1
+≥ min{1, q − 1}
+2
+·
+N
+�
+k=r
+αk
+�
+k
+�
+j=1
+αj
+�−q
+.
+(53)
+21
+
+Set A = 21/q min{1, q − 1}−1/q. Since αk = w−1/(p−1)
+k
+and q/p = q − 1 = 1/(p − 1), it follows
+from (53) that for r = 1, . . . , N,
+�
+r
+�
+k=1
+wkαp
+k
+�1/p
+≤ A
+
+
+N
+�
+k=r
+w−q/p
+k
+� k
+�
+j=1
+αj
+�−q
+
+−1/q
+.
+This is precisely the Muckenhoupt criterion from Corollary 4.5, with
+U(k) = w1/p
+k αk
+and
+V (k) = w1/p
+k
+k
+�
+j=1
+αj.
+Thus, by Corollary 4.5, for any g : ΩN → R,
+N
+�
+k=1
+wk
+�����αk
+N
+�
+ℓ=k
+g(ℓ)
+�����
+p
+≤ (CpA)p
+N
+�
+k=1
+wk
+�����
+�
+k
+�
+j=1
+αj
+�
+g(k)
+�����
+p
+,
+(54)
+with Cp = p1/pq1/q. By restricting attention to non-negative functions g in (54), we may alter
+(54) and replace �N
+ℓ=k by the shorter sum �N
+ℓ=k+1. Inequality (54) remains correct, for non-
+negative g, also after this modification. Denoting g(k) = f(k)/ �k
+j=1 αj, we conclude that for
+any non-negative function f : ΩN → R,
+N
+�
+k=1
+wk
+�����αk
+N
+�
+ℓ=k+1
+1
+�ℓ
+j=1 αj
+f(ℓ)
+�����
+p
+≤ (CpA)p
+N
+�
+k=1
+wk |f(k)|p .
+(55)
+Since the kernel of T1 is non-negative, its operator norm is attained at a non-negative function
+f : ΩN → R. Therefore (55) implies the required bound for the operator norm of T1.
+Proof of Proposition 4.2. The kernel of the operator T is given in (38). It is a non-negative
+kernel, and therefore the operator norm of T is at most the operator norm of the operator whose
+kernel is the expression on the right-hand side of (38). The latter operator equals
+T0 + T1
+with T0 from Proposition 4.6 and T1 from Proposition 4.8. From these two propositions it follows
+that the operator norm of T is at most
+21/pp1/pq1/q· max{1, (p − 1)−1/p} + 21/qp1/pq1/q · max{1, (q − 1)−1/q}.
+Theorem 1.2 follows from Lemma 4.1 and Proposition 4.2.
+22
+
+Remarks.
+1. In this paper we have left open several natural questions, including the existence of linear
+extension operators for ˙W 1,p(T) for weighted trees T in the extreme cases p = 1, p = ∞,
+as well as the analog of our result for the inhomogeneous Sobolev space W 1,p(T) in place
+of ˙W 1,p(T).
+2. The problem of existence of linear Sobolev extension operators for weighted trees arose in
+connection with an extension problem for W 1,p(R2). More precisely, given E ⊆ R2, let
+˙W 1,p(E) denote the space of restrictions to E of functions in ˙W 1,p(R2), endowed with the
+natural seminorm. Does there exist a linear extension operator from ˙W 1,p(E) to ˙W 1,p(R2)?
+The answer is affirmative for p ≥ 2; see A. Israel [3]. For 1 < p < 2, the answer
+is unknown. For a particular class of examples E, the problem reduces to the question
+answered by Theorem 1.1.
+References
+[1] Bj¨orn, A., Bj¨orn, J., Gill, J. T., Shanmugalingam, N., Geometric analysis on Cantor sets
+and trees. J. Reine Angew. Math., Vol. 725, (2017), 63–114.
+[2] Howard, R., Schep, A. R., Norms of positive operators on Lp-spaces. Proc. Amer. Math.
+Soc., Vol. 109, no. 1, (1990), 135–146.
+[3] Israel, A., A bounded linear extension operator for L2,p(R2). Ann. of Math. (2), Vol. 178,
+no. 1, (2013), 183–230.
+[4] Muckenhoupt, B., Hardy’s inequality with weights. Studia Math., Vol. 44, (1972), 31–38.
+[5] Shvartsman, P., Sobolev W 1
+p -spaces on closed subsets of Rn. Adv. Math., Vol. 220, no. 6,
+(2009), 1842–1922.
+23
+
diff --git a/4NFST4oBgHgl3EQfZjjS/content/tmp_files/load_file.txt b/4NFST4oBgHgl3EQfZjjS/content/tmp_files/load_file.txt
new file mode 100644
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+++ b/4NFST4oBgHgl3EQfZjjS/content/tmp_files/load_file.txt
@@ -0,0 +1,727 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf,len=726
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='13792v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='FA]  31 Jan 2023  Linear extension operators for Sobolev spaces on  uniform trees  Charles Fefferman1 and Bo’az Klartag2  Dedicated in friendship to David Jerison  Abstract  Let 1 < p < ∞ and suppose that we are given a function f defined on the leaves of a  weighted tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' We would like to extend f to a function F defined on the entire tree, so as  to minimize the weighted W 1,p-Sobolev norm of the extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' An easy situation is when  p = 2, where the harmonic extension operator provides such a function F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' In this note  we record our analysis of the particular case of a uniform binary tree, which is a complete,  finite, binary tree with weights that depend only on the distance from the root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Neither the  averaging operator nor the harmonic extension operator work here in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Nevertheless,  we prove the existence of a linear extension operator whose norm is bounded by a constant  depending solely on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' This operator is a variant of the standard harmonic extension operator,  and in fact it is harmonic extension with respect to a certain Markov kernel determined by p  and by the weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  1  Introduction  Consider a full binary tree of height N, whose set of vertices is denoted by  V =  N  �  k=0  {0, 1}k,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=', the vertices are strings of zeroes and ones of length at most N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' For x ∈ {0, 1}k and ℓ ≤ k  we write πℓ(x) ∈ {0, 1}ℓ for the prefix of x of length ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Thus for k ≥ 1, the parent of a vertex  x ∈ {0, 1}k ⊆ V is the vertex πk−1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' The set {0, 1}0 is a singleton whose unique element  is denoted by ∅, the empty string, which is the root of the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' The set of leaves of the tree is  1Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, New Jersey 08544,  USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Email: cf@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='princeton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Supported by the Air Force Office of Scientific Research, grant number  FA9950-18-1-0069 and the National Science Foundation (NSF), grant number DMS-1700180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  2Department of Mathematics,  Weizmann Institute of Science,  Rehovot 7610001,  Israel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Email:  boaz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='klartag@weizmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='il.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Supported by a grant from the Israel Science Foundation (ISF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  1  {0, 1}N, and all other vertices in V are internal vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' A vertex x ∈ {0, 1}k is said to have  depth  d(x) = k,  thus leaves have depth N and the root has depth 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' The collections of bijections from V to  V that preserve depth and parenthood relations form a group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' This group is referred to as the  symmetry group of the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' It has 22N−1 elements, and it is a 2-Sylow subgroup of the group of  all permutations of the leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' The set  E =  N  �  k=1  {0, 1}k = V \\ {∅}  is referred to as the set of edges of the tree, and the depth of an edge e ∈ {0, 1}k is d(e) = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  That is, we think of e ∈ E as an undirected edge connecting the vertex whose string corresponds  to e to its unique parent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Each internal vertex other than the root is connected to three vertices,  which are its parent and its two children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Assume that we are given edge weights  W1, W1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , WN > 0,  where we view Wk as the weight of all edges of depth k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' For 1 < p < ∞, the associated  ˙W 1,p-seminorm is defined, for F : V → R, via  ∥F∥ ˙W 1,p(V ) =  \uf8eb  \uf8ed  N  �  k=1  Wk  �  x∈{0,1}k  |F(x) − F(πk−1(x))|p  \uf8f6  \uf8f8  1/p  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (1)  We write ∂V = {0, 1}N ⊆ V for the set of leaves of the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' The trace of the ∥·∥ ˙W 1,p-seminorm  is defined, for f : ∂V → R, via  ∥f∥ ˙W 1,p(∂V ) = inf  �  ∥F∥ ˙W 1,p(V ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' F|∂V = f  �  ,  (2)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=', the infimum of the ˙W 1,p-seminorm over all extensions of f from the leaves to the entire tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  We write RV for the collection of all functions f : V → R, and similarly R∂V is the collection  of all functions f : ∂V → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Our main result is the following:  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Let 1 < p < ∞ and let W1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , WN > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Then there exists a linear operator  H : R∂V → RV with the following properties:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' It is a linear extension operator, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=', (Hf)(x) = f(x) for any x ∈ ∂V and any function  f : ∂V → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Its norm is bounded by a constant ¯Cp depending only on p, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=', for any f : ∂V → R,  ∥Hf∥ ˙W 1,p(V ) ≤ ¯Cp∥f∥ ˙W 1,p(∂V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  2  In fact, we have the bound  ¯Cp ≤ 4p1/pq1/q ·  �  1 + max{(p − 1)−1/p, (q − 1)−1/q}  �  ≤ C · max  �  1  p − 1,  1  q − 1  �  ,  (3)  where q = p/(p − 1) and where C > 0 is a universal constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='2 is constructive, and the extension operator H that we construct is  in fact a harmonic extension operator with respect to a certain random walk defined on the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  At each step the random walk jumps from a vertex to one of its neighbors, where of course the  neighbors of a vertex are its parent and its children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' The Markov kernel corresponding to the  random walk is invariant under the symmetries of the tree, thus the probability to move from a  vertex to its neighbor depends only on the weights of the vertex and of its neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' The Markov  kernel of our random walk is determined by the following requirement: For any s = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N,  the probability that a random walk starting at some vertex of depth s will reach a leaf before  reaching a vertex of depth s − 1 equals  qs =  (2sWs)−1/(p−1)  �N  k=s(2kWk)−1/(p−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (4)  Thus the weights of our random walk typically depend on p ∈ (1, ∞), except for the case where  Ws is proportional to 2−s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' This seems inevitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Indeed, in some examples such as the 3rd  example in Section 2 below, the linear extension operator H : R∂V → RV that corresponds to  the parameter value p = p0, is not uniformly bounded for any p ∈ (1, ∞) \\ {p0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' When p = 2,  our random walk coincides with the usual random walk corresponding to the given weights on  the edges of the binary tree, and thus in this case H is the standard harmonic extension operator  (hence ¯Cp = 1 for p = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  There is a certain range of weights where the averaging operator yields a uniformly bounded  linear extension operator, as proven by Bj¨orn ×2, Gill and Shanmugalingam [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' The averaging  operator is the extension operator that assigns to each internal vertex v the average of the function  values on the leaves of the subtree whose root is v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' This averaging operator seems natural also  from the point of view of Whitney’s extension theory, see the work by Shvartsman [5] on Sobolev  extension in W 1,p(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' However, there are examples of uniform binary trees where the averaging  operator does not provide a uniformly bounded operator, such as the case where Wk = 2−k for  all k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  What about non-uniform trees?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Suppose that the edge weights are arbitrary positive numbers  (We)e∈E that are not necessarily determined by the depth of the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' When p = 2, there is still  a harmonic extension operator of norm one from ˙W 1,p(∂V ) to ˙W 1,p(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' However, for p ̸= 2 the  situation seems subtle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' We conjecture that in the general case of non-uniform tree weights there  is no linear extension operator whose norm is bounded by a function of p alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' This conjecture  is closely related to questions about well-complemented subspaces of ℓn  p that are beyond the  scope of this note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  3  In order to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='1 we reformulate the problem in a way that brings us closer to  analysis in ℓp-spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' For 1 < p < ∞, the associated Lp(E)-norm is defined, for f : E → R,  via  ∥f∥p = ∥f∥Lp(E) =  \uf8eb  \uf8ed  N  �  k=1  Wk  �  x∈{0,1}k  |f(x)|p  \uf8f6  \uf8f8  1/p  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (5)  We think of a function f : E → R as the gradient of a function ˜f : V → R, uniquely determined  up to an additive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Given f : E → R we thus define a function ˜f : V → R as follows:  For any x ∈ V other then the root,  ˜f(x) =  d(x)  �  i=1  f(πix),  (6)  while ˜f(x) = 0 if x = ∅ is the root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' The only property of ˜f that matters is that for all x ∈ E,  f(x) = ˜f(x) − ˜f(πd(x)−1(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  We are interested in finding a linear operator T : Lp(E) → Lp(E), with a uniform bound on its  operator norm, that has the following properties:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' The operator T takes the form  Tf(x) = ˜T ˜f(x) − ˜T ˜f(πd(x)−1x)  (7)  for some linear operator ˜T : RV → RV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' That is, ˜T takes functions on V to functions on  V , and T is induces from ˜T via formula (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' The operator ˜T is equivariant with respect to the tree symmetries and it satisfies ˜T(1) ≡ 1,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=', it maps the constant function 1 to itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' The function ˜Tg coincides with the function g on the leaves of the tree, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' ( ˜Tg)|∂V = g|∂V  for any g : V → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' The function ˜Tg is determined by the values of the function g on the leaves of the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  The operator norm of ˜T with respect to the ˙W 1,p-seminorm equals to the operator norm  of T with respect to the Lp(E)-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Defining H(f|∂V ) = ˜Tf, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='1 may thus be  reformulated as follows:  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Let 1 < p < ∞ and let W1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , WN > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Then there exists a linear operator  T : Lp(E) → Lp(E) with the above properties, whose operator norm is at most a certain  constant ¯Cp depending only on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' In fact, we have the bound (3) for the constant ¯Cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  4  The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='2 occupies the next three sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' In Section 2 we discuss invariant  random walks on a full binary tree and describe the corresponding harmonic extension operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  In Section 3 we deal with the problem of bounding the norm of this operator, and use the sym-  metries of the problem in order to reduce it to a one-dimensional question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' This one-dimensional  question is then answered in Section 4 using the Muckenhoupt criterion [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  When analyzing the binary tree we use the following notation: We write dlca(x, y) for the  length of the maximal prefix shared by the strings x ∈ {0, 1}k and y ∈ {0, 1}ℓ, while lca(x, y)  is the maximal prefix itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Thus for two vertices x, y ∈ V , their least common ancestor is  lca(x, y) ∈ V and its depth is dlca(x, y) ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Note that for any x ∈ E and ω ∈ ∂V ,  dlca(πd(x)−1x, ω) = min{d(x) − 1, dlca(x, ω)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (8)  Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' We would like to thank Jacob Carruth, Arie Israel, Anna Skorobogatova  and Ignacio Uriarte-Tuero for helpful conversations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' This research was conducted while BK  was visiting Princeton University’s Department of Mathematics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' he is grateful for their gracious  hospitality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  2  Invariant random walks  A Markov chain on V is a sequence of random variables R1, R2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' ∈ V such that the distribution  of Ri+1 conditioned on R1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , Ri is the same as its distribution conditioned on Ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' A Markov  chain is time-homogeneous if for any x, y ∈ V , the probability that Ri+1 = x conditioned on the  event Ri = y does not depend on i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' A random walk on V is a time-homogeneous Markov chain  R1, R2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' ∈ V such that Ri+1 is a neighbor of Ri with probability one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  We say that the random walk is invariant if the probability to jump from a vertex x to a vertex  y depends only on the depths d(x) and d(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Our random walk will be invariant, and it will stop  when it reaches a leaf, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=', we have the stopping time  τ = min{i ≥ 1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Ri is a leaf}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  For s ≥ 1 we define qs to be the probability of the following event: Assuming that R1 is a vertex  of depth s, the event is that Ri will remain at the subtree whose root is R1 for all 1 ≤ i ≤ τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Equivalently, define  Xi = d(Ri) ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Then X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N} is a random walk, since |Xi+1 − Xi| = 1 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Furthermore,  qs = P(∀1 ≤ i ≤ τ, Xi ≥ s | X1 = s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (9)  Clearly  q0 = qN = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (10)  For r, s ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N} with r ≤ s we set  ps,r = P(min{Xi ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' i ≤ τ} = r | X1 = s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  5  That is, the number ps,r is the probability that r is the minimal node that the walker visits when  starting from node s, before reaching the terminal node N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Clearly �s  r=0 ps,r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' For 0 ≤ r ≤ s ≤ N − 1,  ps,r = qr ·  s  �  k=r+1  (1 − qk)  (11)  where an empty product equals one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Moreover, pN,r = δrN, where δrN is the Kronecker delta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' The expression on the right-hand side of (11) is the probability to ever reach s − 1 when  starting from X1 = s, and from s − 1 to ever reach s − 2, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' until we finally reach r, yet from  r we require to never reach r − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Alternatively, when 0 ≤ r ≤ s, s ≥ 1 we have the recurrence  relation  ps,r = qsδs,r + (1 − qs) · ps−1,r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (12)  This recurrence relation leads to another proof of (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Suppose that our random walk R1, R2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' ∈ V begins at a vertex R1 = x with d(x) = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Consider a leaf y ∈ ∂V with dlca(x, y) = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' What is the probability that our random walk will  reach the leaf y?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' We claim that this probability is  bs,r := P(Rτ = y) =  r  �  k=0  2k−Nps,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (13)  Indeed, conditioning on the value of k = mini≤τ d(Ri), by symmetry we know that Rτ is dis-  tributed uniformly among the 2N−k leaf-descendants of the vertex πk(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' When k ≤ r, exactly  one of these leaf-descendants is the leaf y, since the vertex of minimal depth that (Ri) visits must  be the vertex πk(x), which is a prefix of y as k ≤ r = dlca(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Hence the probability that  Rτ = y, conditioning on the value of k, equals to 1/2N−k when k ≤ r and it vanishes other-  wise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' By using the definition of ps,k and the complete probability formula, we obtain (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' The  harmonic extension operator associated with our invariant random walk is given by  ˜Tg(x) =  �  ω∈{0,1}N  bd(x),dlca(x,ω) · g(ω)  (x ∈ V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (14)  The operator T is induced from ˜T via formula (7) above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Requirements 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=',4 from Section 1 are  clearly satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  We stipulate that the collection of descendants of a vertex x ∈ V , denoted by D(x) ⊆ V ,  includes the vertex x itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Abbreviate a ∧ b = min{a, b} and a ∨ b = max{a, b}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' The operator  T takes the form  Tf(x) =  �  y∈E  K(x, y)f(y),  (15)  where the kernel K is described next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  6  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Let x, y ∈ E and denote s = d(x), t = d(y), r = dlca(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Then the following  hold: If x ̸∈ D(y) and y ̸∈ D(x), then r ≤ s ∧ t − 1 and  K(x, y) = −qs · 2−t ·  r  �  k=0  2kps−1,k ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (16)  Otherwise, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=', if y ∈ D(x) or if x ∈ D(y) then r = s ∧ t and  K(x, y) = qs · 2−t ·  r−1  �  k=0  (2r − 2k)ps−1,k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (17)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' By (7) and (14) we have, for any x ∈ E,  Tf(x) = ˜T ˜f(x) − ˜T ˜f(πd(x)−1x)  =  �  ω∈{0,1}N  bd(x),dlca(x,ω) ˜f(ω) −  �  ω∈{0,1}N  bd(x)−1,dlca(πd(x)−1x,ω) ˜f(ω)  =  �  ω∈{0,1}N  ad(x),dlca(x,w) ˜f(ω)  (18)  where for any 0 ≤ r ≤ s, by (8) and (13),  as,r = bs,r − bs−1,min{s−1,r} =  r  �  k=0  2k−Nps,k −  min{s−1,r}  �  k=0  2k−Nps−1,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (19)  Hence, by (6) and (18),  Tf(x) =  �  ω∈{0,1}N  ad(x),dlca(x,ω)  N  �  i=1  f(πiω)  =  �  y∈E  \uf8ee  \uf8f0  �  ω∈{0,1}N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='πd(y)ω=y  ad(x),dlca(x,ω)  \uf8f9  \uf8fb f(y) =  �  y∈E  K(x, y)f(y),  where the kernel K of the operator T satisfies, for any x, y ∈ E,  K(x, y) =  �  ω∈{0,1}N ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='πd(y)ω=y  ad(x),dlca(x,ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (20)  Fix x, y ∈ E with s = d(x), t = d(y) and r = dlca(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Let us consider first the case where x  is not a descendant of y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' This means that the prefix of y that is shared by x, is not the entire string  y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Hence for any ω ∈ {0, 1}N with πd(y)ω = y we have dlca(x, ω) = dlca(x, y) ≤ d(y) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Therefore, from (20) and (19),  K(x, y) = 2N−d(y)ad(x),dlca(x,y)  = 2N−t �  bs,r − bs−1,(s−1)∧r  �  =  r  �  k=0  2k−tps,k −  (s−1)∧r  �  k=0  2k−tps−1,k,  7  as bs,r = �r  k=0 2k−Nps,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Since ps,s = qs, we have  K(x, y) = δrs2s−tqs +  (s−1)∧r  �  k=0  2k−t [ps,k − ps−1,k] = qs ·  \uf8ee  \uf8f0δrs2s−t −  (s−1)∧r  �  k=0  2k−tps−1,k  \uf8f9  \uf8fb ,  (21)  where we used the relation (12), which implies that when k ≤ s − 1,  ps,k − ps−1,k = −qs · ps−1,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (22)  We may now prove the conclusion of the proposition in the case where x ̸∈ D(y) and y ̸∈ D(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Indeed, in this case r ≤ s ∧ t − 1 and formula (21) applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Since δrs = 0 in this case, we deduce  formula (16) from (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  The next case we consider is the case where r = s ≤ t − 1, or equivalently, where y ∈  D(x) \\ {x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Thus x ̸∈ D(y) and formula (21) applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Recalling that �s−1  k=0 ps−1,k = 1 we obtain  from (21) that  K(x, y) = qs ·  s−1  �  k=0  (2s−t − 2k−t)ps−1,k = qs · 2−t ·  s−1  �  k=0  (2r − 2k)ps−1,k,  proving formula (17) in the case y ∈ D(x) \\ {x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  We move on to the case where x ∈ D(y) \\ {y}, thus dlca(x, y) = t ≤ s − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' In this case, by  applying (20), (19), (13) and then (22),  K(x, y) =  �  ω∈{0,1}N ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='πd(y)ω=y  ad(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='dlca(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='ω) = 2N−d(x)ad(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='d(x) +  d(x)−1  �  k=d(y)  2N−k−1ad(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='k  = 2N−sas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='s +  s−1  �  k=t  2N−k−1as,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='k = 2N−s(bs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='s − bs−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='s−1) +  s−1  �  k=t  2N−k−1(bs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='k − bs−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='k)  = 2N−s  �  s  �  k=0  2k−Nps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='k −  s−1  �  k=0  2k−Nps−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='k  �  +  s−1  �  k=t  2N−k−1  k  �  ℓ=0  2ℓ−N(ps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='ℓ − ps−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='ℓ)  = qs − qs  s−1  �  k=0  2k−sps−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='k − qs  s−1  �  ℓ=0  s−1  �  k=ℓ∨t  2ℓ−k−1ps−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='ℓ  = qs  �  1 −  s−1  �  k=0  2k−sps−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='k −  s−1  �  ℓ=0  [2ℓ−ℓ∨t − 2ℓ−s]ps−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='ℓ  �  = qs  �  1 −  s−1  �  k=0  2k−k∨tps−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='k  �  = qs ·  �  1 −  t−1  �  k=0  2k−tps−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='k −  s−1  �  k=t  ps−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='k  �  = qs ·  t−1  �  k=0  (1 − 2k−t)ps−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='k = qs · 2−t ·  t−1  �  k=0  (2t − 2k)ps−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  8  Since r = t in this case, we have proved formula (17) in the case where y ∈ D(x) \\ {x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Finally,  the last case that remains is when x = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' In this case r = s = t and  K(x, y) =  �  ω∈{0,1}N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='πd(y)ω=y  ad(x),dlca(x,ω) = 2N−d(x)ad(x),d(x) = 2N−sas,s = 2N−s(bs,s − bs−1,s−1)  = 2N−s  �  s  �  k=0  2k−Nps,k −  s−1  �  k=0  2k−Nps−1,k  �  = qs − qs  s−1  �  k=0  2k−sps−1,k  = qs ·  s−1  �  k=0  (1 − 2k−s)ps−1,k,  completing the proof of formula (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Some examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' The simplest example is when qs = 1 for all s ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' In this case the operator ˜T is the  familiar averaging operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' That is, the extension operator ˜T is the operator that assigns  to each vertex the average of the values at the leaves of its subtree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' In this case  ps,r = δs,r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Consider the case where the invariant random walk is such that Xi := d(Ri) is a symmetric  random walk on {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=', the probability to jump from i to i + 1 is exactly 1/2 for  i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Recall that qs is the probability to never leave the subtree when starting  at a vertex of depth s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' We claim that in this example, for s = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N,  qs =  1  N − s + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (23)  Indeed, the function f(i) = i is harmonic on {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N} and hence f(Xi) is a mar-  tingale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Thus for any stopping time ˜τ we have f(X1) = Ef(X˜τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' We pick the stopping  time  ˜τ = min{ i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Xi ∈ {s − 1, N} }  and obtain (23) since  N · qs + (s − 1) · (1 − qs) = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Next we use Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='1 and find a formula for ps,r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Since formula (23) is valid for any  s ≥ 1, we conclude that for any r ≥ 0 and s ≥ r + 1,  s�  k=r+1  (1 − qk) =  s�  k=r+1  N − k  N − k + 1 = N − s  N − r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (24)  9  Formula (24) is actually valid for any 0 ≤ r ≤ s, since an empty product equals one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Recall that q0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' We thus conclude from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='1 that for s = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N − 1,  ps,r = N − s  N − r ·  �  1  r = 0  1  N−r+1  1 ≤ r ≤ s  while pN,r = δN,r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Let 0 < δ < 1, and consider the case where (Xi) is a random walk on {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N} such  that the probability to jump from k to k + 1 equals 1/2 if k < N − 1, and it equals δ if  k = N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' A harmonic function here is  f(k) =  �  k  k ≤ N − 1  N − 2 + 1/δ  k = N  Therefore, for s = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N − 1,  (N − 2 + 1/δ) · qs + (s − 1)(1 − qs) = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Thus q0 = qN = 1 while for s = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N − 1,  qs =  1  N − s + 1/δ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Hence for any s ≤ N − 1 and r ≤ s − 1,  s  �  k=r+1  (1 − qk) =  s  �  k=r+1  N − k + δ−1 − 2  N − k + δ−1 − 1 = N − s + δ−1 − 2  N − r + δ−1 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  We conclude from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='1 that for s = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N − 1 and 0 ≤ r ≤ s,  ps,r = qr ·  s�  k=r+1  (1 − qk) = N − s + δ−1 − 2  N − r + δ−1 − 2 ·  �  1  r = 0  1  N−r+1/δ−1  1 ≤ r ≤ s  while pN,r = δN,r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  We conclude this section with the following:  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' For any numbers q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , qN−1 ∈ (0, 1) there exists a random walk  X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N}  satisfying (9) with τ = min{i ≥ 1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Xi = N} for s = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Write xk for the probability that the random walk jumps from k to k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Then for  s = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N − 1,  qs = xs(qs+1 + (1 − qs+1)xs),  (25)  where we set q0 = qN = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' We claim that xs ∈ [0, 1] is determined by equation (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Indeed, the  right-hand side of (25) is a continuous, increasing function of xs in the interval [0, 1], that maps  this interval to itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  10  3  The ancestral and non-ancestral parts of the kernel  We need to bound the operator norm in Lp(E) of the operator T whose kernel is described in  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Let us consider first the non-ancestral part of the operator, given by the kernel  K0(x, y) = K(x, y) · 1{x̸∈D(y),y̸∈D(x)} = −1{r≤s∧t−1} · qs · 2−t ·  r  �  k=0  2kps−1,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (26)  Here as usual s = d(x), t = d(y) and r = dlca(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Write T0 : Lp(E) → Lp(E) for the  operator whose kernel is K0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' A function f : E → R is invariant under the symmetries of the  tree, or invariant in short, if it takes the form  f(x) = F(d(x))  for some function F : {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N} → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' The operator T0 is equivariant under the symmetries  of the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Therefore, if f(x) = F(d(x)) is an invariant function, then so is T0f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' In fact, in the  case where f(x) = F(d(x)) we can write  T0f(x) =  N  �  t=1  L0(d(x), t)F(t)  (27)  for a certain kernel L0(s, t) defined for s, t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' For s, t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N,  L0(s, t) = −qs ·  m−1  �  k=0  (1 − 2k−m)ps−1,k ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Let r ≤ min{t, s} − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' A moment of reflection reveals that for x ∈ E with d(x) = s,  n(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' s, r) := #{y ∈ E ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' d(y) = t, dlca(x, y) = r} = 2t−r−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  By (26), (27) and the definition of T0, for any x ∈ E with d(x) = s,  L0(s, t) =  �  y∈E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='d(y)=t  K0(x, y) = −  t∧s−1  �  r=0  n(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' s, r) · qs · 2−t ·  r  �  k=0  2kps−1,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Denote m = s ∧ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Then for s, t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N,  L0(s, t) = −qs ·  t∧s−1  �  r=0  r  �  k=0  2k−r−1ps−1,k = −qs ·  m−1  �  k=0  m−1  �  r=k  2k−r−1ps−1,k  = −qs ·  m−1  �  k=0  (1 − 2k−m)ps−1,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  11  Write ΩN = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N} and for F : ΩN → R define  ∥F∥p = ∥F∥Lp(ΩN) =  � N  �  k=1  2k · Wk · |F(k)|p  �1/p  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (28)  Observe that ∥f∥Lp(E) = ∥F∥p if f(x) = F(d(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Let  S0F(s) =  N  �  t=1  L0(s, t)F(t)  so that by (27),  T0(F ◦ d) = (S0F) ◦ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (29)  For f, g : E → R we consider the scalar product  ⟨f, g⟩ =  �  x∈E  Wd(x)f(x)g(x)  while for F, G : ΩN → R we set  ⟨F, G⟩ := ⟨F ◦ d, G ◦ d⟩ =  N  �  k=1  2kWkF(k)G(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  The adjoint operators T ∗  0 and S∗  0 are defined with respect to these scalar products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' The follow-  ing lemma is probably well-known to experts (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=', Howard and Schep [2] for a related  argument), and its proof is provided for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Let 1 < p < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Then the norm of the operator T0 : Lp(E) → Lp(E) is attained at  an invariant, non-negative function f, and it equals to the norm of the operator S0 : Lp(Ωn) →  Lp(Ωn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Denote momentarily T = −T0 and S = −S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='1 the kernel −L0 of the  operator S is non-negative, and by (26) the kernel of the operator T is non-negative as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  By approximation, we may assume that these two kernels are strictly positive, while keeping  condition (29), thus  T (F ◦ d) = (SF) ◦ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (30)  We deduce that T ∗(F ◦ d) = (S∗F) ◦ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' By compactness,  sup  0̸≡F ∈Lp(ΩN )  ∥SF∥p  ∥F∥p  is attained at some function F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Since the kernel of S is non-negative, we may assume that  the extremal function F is non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' By the Lagrange multipliers theorem, the function F  satisfies a certain eigenvalue equation, and in fact there exists λ ∈ R such that  S∗(SF)p−1 = λF p−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (31)  12  Since the kernel of S is positive and F is non-negative and not identically zero, it follows from  (31) that F is actually positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' The norm of the operator S : Lp(Ωn) → Lp(Ωn) equals λ1/p > 0,  since  ∥SF∥p  p = ⟨(SF)p−1, SF⟩ = ⟨S∗(SF)p−1, F⟩ = λ⟨F p−1, F⟩ = λ∥F∥p  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Denoting f = F ◦ d, we find from (30) that f is a positive invariant function satisfying  T ∗(T f)p−1 = λf p−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Since the kernel of T is non-negative, we have the pointwise H¨older inequality  T (uv) ≤ T (up)1/p · T (vq)1/q,  valid for any non-negative functions u, v ∈ Lp(E), where q = p/(p − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' The operator T has  a non-negative kernel, and hence its norm is attained at a non-negative function u ∈ Lp(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' By  the pointwise H¨older inequality,  (T u)p ≤ T (upf −p/q) · (T f)p/q = T (upf 1−p) · (T f)p−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Therefore,  ∥T u∥p  Lp(E) ≤ ⟨T (upf 1−p), (T f)p−1⟩ = ⟨upf 1−p, T ∗(T f)p−1⟩ = λ⟨upf 1−p, f p−1⟩ = λ∥u∥p  Lp(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Thus the norm of T : Lp(E) → Lp(E) is at most λ1/p, which is the norm of S : Lp(ΩN) →  Lp(ΩN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' The two norms must therefore be equal, since the operator S is equivalent to the  restriction of T to the space of invariant functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  We move on to the ancestral part of the operator, which according to Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='2 is given  by  K1(x, y) = K(x, y) − K0(x, y) = 1{r=s∧t} · qs · 2−t ·  r−1  �  k=0  (2r − 2k)ps−1,k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (32)  Write T1 for the operator whose kernel is K1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' As before, for an invariant function f(x) =  F(d(x)) we may write  T1f(x) =  N  �  t=1  L1(d(x), t)F(t)  (33)  for a certain kernel L1(s, t) defined for s, t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' We also write  S1F(s) =  N  �  t=1  L1(s, t)F(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  It is possible to use formula (7) for the operator T and the definition (14) of the harmonic exten-  sion operator ˜T and deduce that  S0 + S1 ≡ 0,  (34)  essentially because the only invariant, harmonic function on the vertices of the tree is the constant  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' An alternative, more direct proof of (34) is provided in the following:  13  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Let 1 < p < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Then the norm of the operator T1 : Lp(E) → Lp(E) is equal to  the norm of S1 : Lp(Ωn) → Lp(Ωn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Additionally, for s, t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N with m = s ∧ t we have  L1(s, t) = qs ·  m−1  �  k=0  (1 − 2k−m)ps−1,k = −L0(s, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' The first assertion of the lemma follows from fact that the kernel of T1 is non-negative  and invariant under the symmetries of the tree, as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' For the second part, let s, t =  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N and denote m = s ∧ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' We claim that for x ∈ E with d(x) = s,  n(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' s, m) := #{y ∈ E ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' d(y) = t, dlca(x, y) = m} = max{1, 2t−s}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (35)  Indeed, assume first that t ≥ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' How many y’s are there with d(y) = t and dlca(x, y) = m?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Since m = s, the answer is 2t−s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Next, if s ≥ t, then the number of such y’s is one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' This proves  (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Therefore,  L1(s, t) =  �  y∈E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='d(y)=t  K1(x, y) = n(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' s, m) · qs · 2−t ·  m−1  �  k=0  (2m − 2k)ps−1,k  = max{2−t, 2−s} · qs ·  m−1  �  k=0  (2m − 2k)ps−1,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  = 2−m · qs ·  m−1  �  k=0  (2m − 2k)ps−1,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' We have  ∥T∥Lp(E)→Lp(E) ≤ 2∥S0∥Lp(ΩN)→Lp(ΩN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' This follows from the fact that T = T0 + T1 together with the facts that ∥T0∥ = ∥S0∥ and  ∥T1∥ = ∥S1∥ while S1 = −S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  In view of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='4, we are interested in bounds for the norm of the operator S = −S0 =  S1 : Lp(ΩN) → Lp(ΩN) whose non-negative kernel is  L(s, t) = qs ·  s∧t−1  �  k=0  (1 − 2k−s∧t)ps−1,k ≤ qs ·  s∧t−1  �  k=0  ps−1,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (36)  14  From Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='1 we know that ps,r = qr · �s  k=r+1(1 − qk) for s ≤ N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' A little exercise in  probability shows that for any 1 ≤ m ≤ s ≤ N,  m−1  �  k=0  ps−1,k =  s−1  �  k=m  (1 − qk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (37)  Alternatively, (37) holds true for m = 0 as q0 = 1, and it may be proven by induction on m since  ps−1,m +  s−1  �  k=m  (1 − qk) = qm ·  s−1  �  k=m+1  (1 − qk) +  s−1  �  k=m  (1 − qk) =  s�  k=m+1  (1 − qk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  From (36) and (37) we thus obtain  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' For s, t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N, with m = min{s, t},  0 ≤ L(s, t) ≤ qs  s−1  �  k=m  (1 − qk),  where an empty product equals one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Some examples (parallel to the ones discussed in Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' For the averaging operator, where qs = 1 and ps,r = δs,r, we have  L(s, t) = −1  2 · 1{s≤t},  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=', this is the matrix whose entries equal 0 below the diagonal and −1/2 on and above  the diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' This is a rather simple matrix, and it is bounded with respect to the weighted  Lp-norm for quite a few sequences of weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' For the symmetric random walk matrix, we have q0 = 1 while for 1 ≤ s ≤ N,  qs =  1  N − s + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Hence in view of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='5, with m = min{s, t},  0 ≤ L(s, t) ≤  1  N − s + 1  s−1  �  k=m  N − k  N − k + 1 =  1  N − m + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' In the case where  qs =  1  N − s + 1/δ − 1  for some 0 < δ < 1, we have  0 ≤ L(s, t) ≤  1  N − s + 1/δ − 1  s−1  �  k=m  N − k + 1/δ − 2  N − k + 1/δ − 1 =  1  N − m + 1/δ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  15  All that remains is to bound the Lp(ΩN)-norm of the operator whose kernel is discussed  in Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Recall from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='3 that we have the freedom to choose the parameters  q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , qN−1 ∈ (0, 1) as we please.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  How should we choose these parameters?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Since L(N, t) ≤ �N−1  k=t (1−qk) and we are looking  for upper bounds for the norm, the qs should not be too tiny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' On the other hand, L(s, t) ≤ qs for  s ≤ N − 1 and hence it is beneficial to choose qs rather small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' We would therefore need some  balance for the qs, which is the subject of the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  4  One-dimensional analysis  Let 1 < p < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' It will be slightly more convenient to denote  K(s, t) = L(N + 1 − s, N + 1 − t)  and  Qs = qN+1−s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Recalling from (10) that qN = 1, we see that  Q1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  From Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='5 we know that for s, t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N,  K(s, t) = L(N + 1 − s, N + 1 − t) ≤ qN+1−s  N−s  �  k=N+1−max{s,t}  (1 − qk) = Qs  max{s,t}  �  k=s+1  (1 − Qk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  From Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='5 we know that L ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Consequently, for s, t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N,  0 ≤ K(s, t) ≤  �  Qs  t ≤ s  Qs · �t  k=s+1(1 − Qk)  t ≥ s + 1  (38)  Recall that we are given edge weights W1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , WN > 0, and that the associated Lp(E)-norm is  given by (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Denote  ws := 2N+1−s · WN+1−s > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Consider the weighted ℓp-norm  ∥f∥p,w =  � N  �  k=1  wk|f(k)|p  �1/p  (39)  and the operator T whose kernel is K(s, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' We are allowed to choose the weights q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , qN−1 ∈  (0, 1) as we please, or equivalently, we have the freedom to determine Q2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , QN ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' We  must keep Q1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Based on considerations related to the Muckenhoupt criterion discussed  below, we set  Qs =  w−1/(p−1)  s  �s  k=1 w−1/(p−1)  k  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (40)  It is clear that Q1 = 1 and that Qs ∈ (0, 1) for all s ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Recall that q = p/(p − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  16  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' In order to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='2, it suffices to show that the operator norm of T with  respect to the ∥ · ∥p,w-norm is bounded by a constant ˆCp > 0 depending only on p, where in fact  ˆCp ≤ 2p1/pq1/q ·  �  1 + max{(p − 1)−1/p, (q − 1)−1/q}  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (41)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' In view of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='4, it suffices to bound the operator norm of −S0, whose kernel is  L, with respect to the Lp(ΩN)-norm defined in (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Under the transformation  s �→ N + 1 − s  the operator −S0 whose kernel is L transforms to the operator T whose kernel is K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' The Lp(ΩN)-  norm from (28) transforms to the ∥·∥p,w-norm defined in (39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Hence Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='2 would follow  once we obtain the bound (41), where ¯Cp ≤ 2 ˆCp by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  The remainder of this section is devoted to the proof of the following:  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' The operator norm of T with respect to the norm (39) is bounded by a number  ˆCp depending only on p ∈ (1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' In fact, we have the bound (41) for the constant ˆCp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Our main tool in the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='2 is the Muckenhoupt criterion [4], which is an  indispensable tool for proving one-dimensional inequalities of Poincar´e-Sobolev type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' For the  reader’s convenience, we include here a statement and a proof of a straightforward modification  of the Muckenhoupt criterion, with sums in place of integrals:  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' (Muckenhoupt) Let 1 < p < ∞ and write ΩN = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Let U, V : ΩN →  (0, ∞) and let A > 0 be such that for all r = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N,  � N  �  k=r  |U(k)|p  �1/p  ≤ A  �  r  �  k=1  |V (k)|−q  �−1/q  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (42)  Then for any function f : ΩN → R,  � N  �  k=1  �����U(k)  k  �  ℓ=1  f(ℓ)  �����  p�1/p  ≤ CpA  � N  �  k=1  |V (k)f(k)|p  �1/p  ,  (43)  with Cp = p1/pq1/q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  By continuity, the analog of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='3 for p = 1, ∞ holds true with C1 = C∞ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' We  remark that as in [4], this criterion is tight, in the sense that the infimum over all A > 0 satisfying  (42) is equivalent to the best constant in inequality (43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' For the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='3 we require  the following:  17  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' For α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , αN > 0 and r = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N,  r  �  k=1  αk  �  k  �  ℓ=1  αℓ  �−1/p  ≤ q ·  �  r  �  k=1  αk  �1/q  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (44)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' We will use the simple inequality  (a + b)1/q − a1/q ≥ b ·  min  ξ∈(a,a+b)  ξ1/q−1  q  = 1  q(a + b)1/q−1 · b,  (45)  valid for any a, b ≥ 0 with a + b > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' From (45), for k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N,  � k  �  ℓ=1  αℓ  �1/q  −  �k−1  �  ℓ=1  αℓ  �1/q  ≥ 1  q · αk ·  �  k  �  ℓ=1  αℓ  �1/q−1  ,  where an empty sum equals zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' By summing this for k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , r we obtain (44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='3 (Muckenhoupt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' By the H¨older inequality, for any function f : ΩN → R  and weights h : ΩN → (0, ∞),  N  �  k=1  �����U(k)  k  �  ℓ=1  f(ℓ)  �����  p  ≤  N  �  k=1  Up(k) ·  k  �  ℓ=1  |f(ℓ)V (ℓ)h(ℓ)|p ·  �  k  �  j=1  |V (j)h(j)|−q  �p/q  =  N  �  ℓ=1  |f(ℓ)V (ℓ)h(ℓ)|p  N  �  k=ℓ  Up(k)  � k  �  j=1  |V (j)h(j)|−q  �p/q  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (46)  Set h(k) =  ��k  ℓ=1 V (ℓ)−q�1/(pq)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' By applying Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='4 with αk = V (k)−q we obtain  r  �  k=1  |V (k)h(k)|−q =  r  �  k=1  αk  � k  �  ℓ=1  αℓ  �−1/p  ≤ q ·  �  r  �  k=1  αk  �1/q  = q ·  �  r  �  k=1  V (k)−q  �1/q  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Hence for any f : ΩN → R, the expression in (46) is at most  qp/q ·  N  �  ℓ=1  |f(ℓ)V (ℓ)h(ℓ)|p  N  �  k=ℓ  Up(k)  �  k  �  j=1  |V (j)|−q  �p/q2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (47)  By applying (42) and then Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='4 with αk = Up(N + 1 − k) and with p ∈ (1, ∞) playing  the rˆole of q ∈ (1, ∞), we see that  N  �  k=ℓ  |U(k)|p  �  k  �  j=1  |V (j)|−q  �p/q2  ≤ Ap/q  N  �  k=ℓ  |U(k)|p  � N  �  j=k  |U(j)|p  �−1/q  = Ap/q  N+1−ℓ  �  k=1  αk  � k  �  j=1  αj  �−1/q  ≤ p · Ap/q  �N+1−ℓ  �  k=1  αk  �1/p  = p · Ap/q  � N  �  k=ℓ  Up(k)  �1/p  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  18  Hence the expression in (47) is at most  p · (qA)p/q ·  N  �  ℓ=1  |f(ℓ)V (ℓ)h(ℓ)|p ·  � N  �  k=ℓ  Up(k)  �1/p  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Applying (42) again we bound the last expression from above by  p · (qA)p/q · A ·  N  �  ℓ=1  |f(ℓ)V (ℓ)h(ℓ)|p ·  �  ℓ  �  k=1  V −q(k)  �−1/q  = pqp/qAp  N  �  ℓ=1  |f(ℓ)V (ℓ)|p,  completing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Let 1 < p < ∞ and ΩN = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Let U, V : ΩN → (0, ∞) and let A > 0  be such that for all r = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N,  �  r  �  k=1  |U(k)|p  �1/p  ≤ A  � N  �  k=r  |V (k)|−q  �−1/q  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (48)  Then for any f : ΩN → R,  � N  �  k=1  �����U(k)  N  �  ℓ=k  f(ℓ)  �����  p�1/p  ≤ CpA  � N  �  k=1  |V (k)f(k)|p  �1/p  ,  (49)  with Cp = p1/pq1/q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Denote ˜U(k) = U(N + 1 − k) and ˜V (k) = V (N + 1 − k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Then from (48), for all  r = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N,  � N  �  k=r  | ˜U(k)|p  �1/p  ≤ A  �  r  �  k=1  | ˜V (k)|−q  �−1/q  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='3, this implies that for any g : ΩN → R, denoting f(r) = g(N + 1 − r),  � N  �  k=1  �����  ˜U(k)  k  �  ℓ=1  f(N + 1 − ℓ)  �����  p�1/p  ≤ CpA  � N  �  k=1  | ˜V (k)f(N + 1 − k)|p  �1/p  ,  or equivalently,  � N  �  k=1  �����  ˜U(N + 1 − k)  N  �  ℓ=k  f(ℓ)  �����  p�1/p  ≤ CpA  � N  �  k=1  | ˜V (N + 1 − k)f(k)|p  �1/p  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  This implies (49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  19  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' For f : Ωn → R and s = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N denote  T0f(s) = Qs  s  �  t=1  f(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Then the operator norm of T0 with respect to the norm ∥ · ∥p,w defined in (39) is bounded by a  number ˜Cp depending only on p ∈ (1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' In fact, ˜Cp ≤ 21/pp1/pq1/q · max{1, (p − 1)−1/p}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  The proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='6 requires the following:  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' For any α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , αN > 0 and r = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N,  N  �  k=r  αk  � k  �  ℓ=1  αℓ  �−p  ≤  2  min{1, p − 1} ·  �  r  �  k=1  αk  �−p+1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (50)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' We use the inequality  (p − 1) · b(a + b)−p ≤ a−p+1 − (a + b)−p+1,  which is valid for any a, b > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Then for k = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N,  (p − 1) · αk  � k  �  ℓ=1  αℓ  �−p  ≤  �k−1  �  ℓ=1  αℓ  �−p+1  −  �  k  �  ℓ=1  αℓ  �−p+1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  By summing this for k = r + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N we obtain  (p − 1) ·  N  �  k=r+1  αk  � k  �  ℓ=1  αℓ  �−p  ≤  �  r  �  k=1  αk  �−p+1  −  � N  �  ℓ=1  αℓ  �−p+1  ≤  �  r  �  k=1  αk  �−p+1  ,  where an empty sum equals zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' We conclude (50) by summing this with the trivial inequality  min{1, p − 1} · αr  �  r  �  ℓ=1  αℓ  �−p  ≤  �  r  �  k=1  αk  �−p+1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Define  U(k) = w1/p  k  Qk  and  V (k) = w1/p  k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Let us verify the condition of the Muckenhoupt criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' We need to find A > 0 such that for  all r = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N inequality (42) holds true, that is,  N  �  k=r  wkQp  k ≤ Ap  �  r  �  k=1  w−q/p  k  �−p/q  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (51)  20  Recall that p/q = p − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' From the definition (40) of Qk, we need  N  �  k=r  w−1/(p−1)  k  �  k  �  ℓ=1  w−1/(p−1)  ℓ  �−p  ≤ Ap  �  r  �  k=1  w−1/(p−1)  k  �−(p−1)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Setting αk = w−1/(p−1)  k  and using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='7, we see that (51) holds true with  A = 21/p · max{1, (p − 1)−1/p}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  From Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='3 we thus conclude that for any f : ΩN → R,  � N  �  k=1  wk  �����Qk  k  �  ℓ=1  f(ℓ)  �����  p�1/p  ≤ p1/pq1/qA ·  � N  �  k=1  wk|f(k)|p  �1/p  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  This implies the required bound for the operator norm of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' For f : Ωn → R and s = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N denote  T1f(s) =  N  �  t=s+1  �  Qs  t�  k=s+1  (1 − Qk)  �  f(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Then the operator norm of T1 with respect to the norm ∥ · ∥p,w defined in (39) is bounded by  21/qp1/pq1/q · max{1, (q − 1)−1/q}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Denote αk = w−1/(p−1)  k  and recall from (40) that Qk = αk/ �k  ℓ=1 αℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' We claim that for  any t ≥ s + 1,  Qs  t�  k=s+1  (1 − Qk) =  αs  �t  k=1 αk  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (52)  Indeed, (52) holds true for t = s, since an empty product equals one, and for t ≥ s + 1 it is  proven by an easy induction on t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Consequently,  T1f(s) = αs  N  �  t=s+1  1  �t  j=1 αj  f(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Since p, q ∈ (1, ∞), the elementary inequality of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='7 is valid also when p is replaced by  q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' It implies that for r = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N,  �  r  �  k=1  αk  �−q+1  ≥ min{1, q − 1}  2 N  �  k=r  αk  �  k  �  j=1  αj  �−q  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (53)  21  Set A = 21/q min{1, q − 1}−1/q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Since αk = w−1/(p−1)  k  and q/p = q − 1 = 1/(p − 1), it follows  from (53) that for r = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N,  �  r  �  k=1  wkαp  k  �1/p  ≤ A  \uf8eb  \uf8ed  N  �  k=r  w−q/p  k  � k  �  j=1  αj  �−q\uf8f6  \uf8f8  −1/q  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  This is precisely the Muckenhoupt criterion from Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='5, with  U(k) = w1/p  k αk  and  V (k) = w1/p  k  k  �  j=1  αj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Thus, by Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='5, for any g : ΩN → R,  N  �  k=1  wk  �����αk  N  �  ℓ=k  g(ℓ)  �����  p  ≤ (CpA)p  N  �  k=1  wk  �����  �  k  �  j=1  αj  �  g(k)  �����  p  ,  (54)  with Cp = p1/pq1/q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' By restricting attention to non-negative functions g in (54), we may alter  (54) and replace �N  ℓ=k by the shorter sum �N  ℓ=k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Inequality (54) remains correct, for non-  negative g, also after this modification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Denoting g(k) = f(k)/ �k  j=1 αj, we conclude that for  any non-negative function f : ΩN → R,  N  �  k=1  wk  �����αk  N  �  ℓ=k+1  1  �ℓ  j=1 αj  f(ℓ)  �����  p  ≤ (CpA)p  N  �  k=1  wk |f(k)|p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (55)  Since the kernel of T1 is non-negative, its operator norm is attained at a non-negative function  f : ΩN → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Therefore (55) implies the required bound for the operator norm of T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' The kernel of the operator T is given in (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' It is a non-negative  kernel, and therefore the operator norm of T is at most the operator norm of the operator whose  kernel is the expression on the right-hand side of (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' The latter operator equals  T0 + T1  with T0 from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='6 and T1 from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' From these two propositions it follows  that the operator norm of T is at most  21/pp1/pq1/q· max{1, (p − 1)−1/p} + 21/qp1/pq1/q · max{1, (q − 1)−1/q}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='2 follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='1 and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  22  Remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' In this paper we have left open several natural questions, including the existence of linear  extension operators for ˙W 1,p(T) for weighted trees T in the extreme cases p = 1, p = ∞,  as well as the analog of our result for the inhomogeneous Sobolev space W 1,p(T) in place  of ˙W 1,p(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' The problem of existence of linear Sobolev extension operators for weighted trees arose in  connection with an extension problem for W 1,p(R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' More precisely, given E ⊆ R2, let  ˙W 1,p(E) denote the space of restrictions to E of functions in ˙W 1,p(R2), endowed with the  natural seminorm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Does there exist a linear extension operator from ˙W 1,p(E) to ˙W 1,p(R2)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  The answer is affirmative for p ≥ 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' see A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Israel [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' For 1 < p < 2, the answer  is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' For a particular class of examples E, the problem reduces to the question  answered by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  References  [1] Bj¨orn, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=', Bj¨orn, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=', Gill, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=', Shanmugalingam, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=', Geometric analysis on Cantor sets  and trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Reine Angew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' 725, (2017), 63–114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  [2] Howard, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=', Schep, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=', Norms of positive operators on Lp-spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' 109, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' 1, (1990), 135–146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  [3] Israel, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=', A bounded linear extension operator for L2,p(R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' (2), Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' 178,  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' 1, (2013), 183–230.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  [4] Muckenhoupt, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=', Hardy’s inequality with weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Studia Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' 44, (1972), 31–38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  [5] Shvartsman, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=', Sobolev W 1  p -spaces on closed subsets of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' 220, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' 6,  (2009), 1842–1922.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  23' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
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+arXiv:2301.08639v1  [math.AC]  20 Jan 2023
+NOTES ON VALUATION THEORY FOR KRASNER HYPERFIELDS
+ALESSANDRO LINZI ID
+Abstract. The main aim of this article is to study and develop valuation theory for Krasner
+hyperfields. In analogy with classical valuation theory for fields, we generalise the formalism of
+valuation rings to describe equivalence of valuations on hyperfields. After proving basic results
+and discussing several examples, we focus on the valued hyperfields that Krasner originally defined
+in 1957. We find that these must have a particular additive structure which in turns implies the
+existence of a valuation a’la Krasner. We note that given such a valued hyperfield (F, v), the
+valuation induced by its additive structure does not have to be equivalent to v. We discuss the
+cases in which it does.
+1. Introduction
+In 1957, M. Krasner in [24] (the article is included in Krasner’s collected works [4, pages 413–
+490]) formulated for the first time an axiomatisation of structures that generalise fields by allowing
+the additive operation to be multivalued (see [4, pages 407–490] for more details). He called these
+hyperfields (in french hypercorps).
+On the one hand, he found his inspiration in the work of Marty [35, 36, 37] which is considered
+as the starting point of hypergroup theory. In fact, the additive part of a hyperfield is a hypergroup
+(see also [48, 38] for a more detailed historical overview).
+On the other hand, Krasner was motivated by his interest in valuation theory (in connection with
+p-adic numbers) and he sensed the importance of some hyperfields which are canonically associated
+to any valued field (cf. Example 4.11). Let us briefly recall some basic notions of classical valuation
+theory.
+Let K be a field and Γ a linearly ordered abelian group (written additively). A surjective map
+v : K → Γ ∪ {∞}
+is called a (Krull) valuation on K if it satisfies for all x, y ∈ K:
+– v(x) = ∞ if and only if x = 0,
+– v(xy) = v(x) + v(y),
+– v(x + y) ≥ min{v(x), v(y)}.
+Here, ∞ is a symbol such that γ + ∞ = ∞ + γ = ∞ > γ for all γ ∈ Γ. If a valuation v on a field
+K is given, then (K, v) is called a valued field. One usually denotes Γ by vK and call it the value
+group of (K, v). The value v(x) of x ∈ K will often be written as vx, if there is no risk of confusion.
+2020 Mathematics Subject Classification. Primary: 12J20, 20N20 Secondary: 13A18.
+Key words and phrases. Hyperfield, multifield, hyperring, valuation, ordered abelian group, tropical hyperfield.
+The author would like to spend a few words to thank H. Stojałowska, P. Touchard, Franz-Viktor and Katarzyna
+Kuhlmann, I. Cristea as well as Ch. Massouros, who all, in one way or another, contributed to the realisation of the
+final version of this manuscript.
+1
+
+2
+LINZI, A.
+If (K, v) is a valued field, then
+Ov := {x ∈ K | vx ≥ 0}
+is a subring of K, called the valuation ring of (K, v). It determines the valuation map v up to
+equivalence, i.e., up to composition with an order preserving isomorphism of the value group. Any
+valuation ring has a unique maximal ideal
+Mv := {x ∈ K | vx > 0}
+and the field Kv := Ov/Mv is called the residue field of (K, v). For further details, let us mention
+[10, 44] as general references on classical valuation theory.
+While it is quite natural to generalise the concept of valuation to hyperfields (see Definition 3.1),
+Krasner noticed that the structures which attracted his attention come equipped with a map similar
+to a valuation which satisfies two additional properties (see Definition 4.7). These properties would
+be vacuous if postulated for valued fields. Thus, Krasner included them in his axiomatisation of
+valued hyperfields (hypercorps valué).
+Krasner’s motivation was not model theoretical. Nevertheless, it turns out that the structures he
+studied play an important role in the model theory of valued fields; specifically for the problem of
+quantifier elimination for henselian valued fields (of characteristic 0). In this setting, these objects
+are known as RV-structures or leading-term structures and have been considered, independently of
+Krasner, by Flenner in [11] (see also [3, 27, 33, 34] for more details). In addition, an interesting
+application of hyperfields in the model theory of valued fields recently appeared in [32].
+The interest for valued hyperfields may also be motivated by the following observations. Clas-
+sically, real algebra, which studies real fields (i.e., linearly ordered fields, with an order compatible
+with the operations) and was developed by E. Artin in his solution of Hilbert’s seventeenth problem,
+is in relation with valuation theory. After the works of Marshall and Gładki [15, 16] on real hyper-
+fields, it is to expect that a development of valuation theory for hyperfields would be beneficial for
+this line of research. Significant developments have already been achieved with the generalisation
+of classical results, such as the Baer–Krull Theorem, to the multivalued setting (see [26, 31]). The
+reader interested in these aspects may look also at [12, 13, 14]. Furthermore, the unpublished work
+of Viro [47], followed up by Jun and Jell et al. [20, 18], indicate applications in the realm of trop-
+icalization maps and analytification, topics that are as well, classically, in relation with valuation
+theory.
+We believe that developing valuation theory for hyperfields will eventually lead to further appli-
+cations back to the classical theory of valuations for fields.
+The switch from singlevalued operations to multivalued ones may be not difficult to describe
+and understand. Nevertheless, the consequences of this switch have to be handled very carefully.
+For instance, valuation theorists are accostumed to the fact that v(x − y) always defines a metric
+(in fact, an ultrametric, cf. Section 4.1) on a valued field (K, v), but this clearly ceases to be true
+(in general) if the operation of addition (and hence of difference) is multivalued. Indeed, in the
+latter case, the distance map may depend on the choice of some element of the set x − y and this
+choice is not canonical, in general. This obstacle is overtaken using one of Krasner’s postulates
+which forces all the elements of x − y, for x ̸= y, to have the same value. However, one may expect
+the latter being a quite strong requirement. In fact, the restrictions due to Krasner’s axioms for
+valued hyperfields are so strong that, e.g., the trivial valuation on a hyperfield F (Example 3.2)
+satisfies them only if F is actually a field (Remark 4.10). For this and other reasons (which we
+will discuss later in the paper), it makes sense to consider also valuations on hyperfields which do
+not satisfy the two additional properties imposed by Krasner (cf. Example 4.17) and it turns out
+
+NOTES ON VALUATION THEORY FOR KRASNER HYPERFIELDS
+3
+that this choice is more beneficial than harmful. As a consequence of these observations, the term
+“valued hyperfield” will be used in this paper in a broader sense than the original one; the valued
+hyperfields satisfying in addition Krasner’s axioms will be called “Krasner valued hyperfields”and
+their valuations “Krasner valuations”.
+The manuscript is organised as follows. In Section 2 we introduce the necessary concepts and
+terminology from hypercompositional algebra, taking the opportunity to discuss several examples.
+In Section 3, we show that the formalism of valuation rings to describe valuations up to equivalence
+generalises without major modifications to the multivalued framework. We use this formalism to
+state and prove one of our main results Theorem 4.22 on Krasner valued hyperfields in Section 4.
+Our main theorem states that the existence of a Krasner valuation on a hyperfield F implies that
+the additive structure of F satisfies certain additional axioms (cf. Proposition 4.20). Conversely, if
+the additive structure of a hyperfield F satisfies those axioms, then F admits a Krasner valuation
+(cf. Proposition 4.21). Let v be a Krasner valuation on a hyperfield F. Then the additive structure
+of F induces a Krasner valuation w on F. We observe that, in general, w is not equivalent to
+v (Example 4.25). After Section 5, where we put into our context the notion of coarsening of a
+valuation, which turns out to be necessary for our discussion, we describe situations in which v and
+w are equivalent in Section 6. The final Section 7 of this manuscript contains possible further lines
+of investigation.
+Besides presenting some new results, the article is also thought to serve as a unified reference for
+basic hyperring and hyperfield theory and related terminology. On the one hand, these hypercom-
+positional structures have recently attracted increasing interest from the mathematical community
+(including top mathematicians such as A. Connes [7, 8]). On the other hand, we found the relevant
+literature on hyperrings and hyperfields to be quite fragmented. After the time and effort spent
+during the PhD studies and thanks to the fruitful connections with some of the mathematicians who
+are part of the early history of hypercompositional algebra, we felt that we could give a contribution
+from this point of view as well.
+2. Preliminaries
+Let H be a non-empty set and P(H) its power set. A multivalued operation + on H is a function
+which associates to every pair (x, y) ∈ H × H an element of P(H), denoted by x + y. If + is a
+multivalued operation on H ̸= ∅, then for x ∈ H and A, B ⊆ H we set
+A + B :=
+�
+a∈A,b∈B
+a + b,
+A + x := A + {x} and x + A := {x} + A. If A or B is empty, then so is A + B.
+A hypergroup can be defined as a non-empty set H with a multivalued operation + which is
+associative (see Definition 2.3 (CH1) below) and reproductive on H (i.e., x+H = H +x = H for all
+x ∈ H). This notion was first considered by F. Marty in [35, 36, 37]. The theory of hypergroups,
+with a detailed historical overview, is presented in [38], where an extensive bibliography is also
+provided.
+Definition 2.1. A hyperoperation + on H is a multivalued operation such that x + y ̸= ∅ for all
+x, y ∈ H.
+Lemma 2.2 (Theorem 12 in [38]). If (H, +) is a hypergroup, then + is a hyperoperation on H.
+Proof. Aiming for a contradiction, suppose that x + y = ∅ for some x, y ∈ H. Then
+H = x + H = x + (y + H) = (x + y) + H = ∅ + H = ∅,
+
+4
+LINZI, A.
+which is excluded.
+□
+The following special class of hypergroups (cf. Remark 2.4 below) will be of interest for us.
+Definition 2.3. A canonical hypergroup is a triple (H, +, 0), where H ̸= ∅, + is a multivalued
+operation on H and 0 is an element of H such that the following axioms hold:
+(CH1) + is associative, i.e., (x + y) + z = x + (y + z) for all x, y, z ∈ H,
+(CH2) x + y = y + x for all x, y ∈ H,
+(CH3) for every x ∈ H there exists a unique x′ ∈ H such that 0 ∈ x + x′ (the element x′ will be
+denoted by −x),
+(CH4) z ∈ x + y implies y ∈ z − x := z + (−x) for all x, y, z ∈ H.
+Lemma 2.4. Let (H, +, 0) be a canonical hypergroup. Then the multivalued operation + is repro-
+ductive on H. In particular, (H, +) is a hypergroup and + is a hyperoperation.
+Proof. Fix a ∈ H. For all x ∈ H + a there exists y ∈ H such that x ∈ y + a ⊆ H. Therefore,
+H + a ⊆ H. For the other inclusion, observe that for all x ∈ H we have that
+x ∈ x + 0 ⊆ x + (a − a) = (x − a) + a,
+so there exists y ∈ x − a ⊆ H such that x ∈ y + a ⊆ H + a. We have proved that H = H + a
+for an arbitrary a ∈ H. Now the conclusions of the lemma follow from (CH2), the definition of
+hypergroups and Lemma 2.2.
+□
+Let (H, +, 0) be a canonical hypergroup. Then 0 is called the neutral element for + in H because
+of the following observation.
+Lemma 2.5 (Section III, (b) in [39]). Let (H, +, 0) be a canonical hypergroup. Then x + 0 = {x}
+for all x ∈ H.
+Proof. Take x ∈ H. If y ∈ x + 0, then 0 ∈ y − x follows by (CH4). Now y = x follows from the
+uniqueness required in (CH3).
+□
+Remark 2.6. Note that an abelian group (G, ∗, 0) is not a priori a canonical hypergroup, because the
+operation on G is not a multivalued operation, as it takes values in G and not in P(G). However,
+it can be turned into a canonical hypergroup (G, +, 0) by setting x + y := {x ∗ y} for all x, y ∈ G.
+Conversely, if a canonical hypergroup (H, +, 0) satisfies that x + y is a singleton for all x, y ∈ H,
+then one can define on H a standard operation which makes it an abelian group with identity 0. In
+these cases, we sometimes abusively say that (G, ∗, 0) is a canonical hypergroup or that (H, +, 0)
+is a group.
+2.1. Hyperrings and hyperfields. Let us now define the structures of main interest in this paper.
+Definition 2.7. A (commutative) hyperring is a tuple (R, +, ·, 0) which satisfies the following
+axioms:
+(HR1) (R, +, 0) is a canonical hypergroup,
+(HR2) (R, ·) is a (commutative) semigroup and 0 is an absorbing element, i.e., 0 · x = x · 0 = 0, for
+all x ∈ R,
+(HR3) the operation · is distributive with respect to +. That is, for all x, y, z ∈ R,
+x(y + z) = xy + xz,
+
+NOTES ON VALUATION THEORY FOR KRASNER HYPERFIELDS
+5
+where, for x ∈ R and A ⊆ R, we have set
+xA := {xa | a ∈ A}.
+If the operation · has a neutral element 1 ̸= 0, then we say that (R, +, ·, 0, 1) is a hyperring with unity.
+If (R, +, ·, 0, 1) is a hyperring with unity and (R \ {0}, ·, 1) is an abelian group, then (R, +, ·, 0, 1) is
+called a hyperfield. If R is a hyperring with unity, then we denote by R× the multiplicative group
+of the units of R, i.e.,
+R× := {x ∈ R | ∃y ∈ R : xy = 1}.
+In particular, if R is a hyperfield, then R× = R \ {0}.
+Since we will only consider the commutative case, in the following sections we will call commu-
+tative hyperrings simply hyperrings.
+Remark 2.8. In the literature, the name “hyperring” can be found for structures (R, +, ·) where +
+is an operation and · is a multivalued operation or where both + and · are multivalued operations.
+Some authors refer to the structures (R, +, ·, 0) where only the addition is multivalued as Krasner
+hyperrings (some more historical remarks on this are given in in [17, Section 4]).
+Structures (R, +, ·, 0) where + is an operation and · is a multivalued operation (satisfying similar
+axioms) were introduced in [45] and called multiplicative hyperrings (in italian iperanelli moltiplica-
+tivi).
+Structures (R, +, ·, 0) with both + and · multivalued operations (satisfying similar axioms) are
+instead called general hyperrings (see e.g. [9, Section 2]).
+In this paper, the name “hyperring” will be used for Krasner hyperrings exclusively, as indicated
+in the above definition.
+In the literature, one may also find the term multiring referring to a structure (R, +, ·, 0), where
++ is a multivalued operation and · is an operation, satisfying (HR1), (HR2) and the following
+weaker version of (HR3):
+(MR) x · (y + z) ⊆ x · y + x · z, for all x, y, z ∈ M.
+Thus, in this case, the different name reflects a difference in the axioms rather than in the structure.
+Similarly, multifields are defined as hyperfields satisfying (MR) instead of (HR3). Multirings and
+multifields have been considered for instance in [12], where, among other facts, it is observed that
+all multifields are hyperfields, while there are several meaningful examples of multirings that are
+not hyperrings.
+In this paper, we preferred to use the name “hyperfield” solely.
+We say that a hyperring (resp. hyperfield) R is a ring (resp. field) if the additive canonical
+hypergroup of R is a group (cf. Remark 2.6). The next observation gives a necessary condition for
+a hyperring with unity to be a ring. This fact is an immediate corollary of a result already noted
+in [42, page 369] (see also Example 2.45 below). We wish to state it for later reference and we take
+the opportunity to write a quick proof .
+Lemma 2.9 ([42]). A hyperring with unity R is a ring if and only if 1 − 1 = {0}.
+Proof. If R is a field, then 1 − 1 = {0} holds trivially. Conversely, by axiom (HR3) 1 − 1 = {0}
+implies x − x = {0} for all x ∈ R. Take a, b ∈ R and x, y ∈ a + b. We have that
+x − y ⊆ (a + b) − (a + b) = (a − a) + (b − b) = {0}
+In particular, 0 ∈ x − y and x = y follows fro (CH3). We have proved that a + b is a singleton for
+all a, b ∈ R.
+□
+
+6
+LINZI, A.
+Example 2.10. The K-hyperfield K is the set {0, 1} with the hyperoperation + which has 0 as its
+neutral element and satisfies 1 + 1 = {0, 1}. The multiplication is the obvious one.
+Remark 2.11. It is customary among some authors to call the above hyperfield the Krasner hy-
+perfield. We prefer to avoid that terminology which, in view of Remark 2.8 above, might lead to
+unnecessary confusion.
+Example 2.12. The sign hyperfield S is the set {−1, 0, 1} with the hyperoperation + which has
+0 as its neutral element and satisfies x + x = {x} for x ∈ {−1, 1} and 1 − 1 = {−1, 0, 1}. The
+multiplication is the obvious one.
+Example 2.13. The weak sign hyperfield W is the set {−1, 0, 1} with the hyperoperation + which
+has 0 as its neutral element and satisfies x + x = {x, −x} for x ∈ {−1, 1} and 1 − 1 = {−1, 0, 1}.
+The multiplication is the obvious one.
+Other examples of finite hyperfields are described e.g. in [1]. Let us now consider some infinite
+examples.
+Example 2.14. Let (Γ, +, <, 0) be an ordered abelian group and ∞ be a symbol such that γ+∞ =
+∞+γ = ∞ > γ for all γ ∈ Γ. For x, y ∈ Γ∪{∞} such that x ≤ y (i.e., x < y or x = y), let us denote
+by [x, y] the set consisting of all z ∈ Γ ∪ {∞} such that x ≤ z ≤ y. Consider the hyperoperation ⊞
+defined on T (Γ) := Γ ∪ {∞} as x ⊞ ∞ = ∞ ⊞ x = {x} for all x ∈ T (Γ) and:
+x ⊞ y :=
+�
+{min{x, y}}
+if x ̸= y,
+[x, ∞]
+if x = y.
+(x, y ∈ T (Γ))
+It is not difficult to check that (T (Γ), ⊞, +, ∞, 0) is a hyperfield. The hyperfield T (R, +, 0, >),
+where > denotes the standard order of the real numbers, is known as the tropical hyperfield (see
+[47, Section 5.3]). Therefore, we call the hyperfields of the form T (Γ) generalised tropical hyperfields.
+Example 2.15. Let (Γ, +, <, 0) be an ordered abelian group and ∞ be a symbol such that γ+∞ =
+∞ + γ = ∞ > γ for all γ ∈ Γ. For x, y ∈ Γ ∪ {∞} such that x < y, let us denote by (x, y] the set
+consisting of all z ∈ Γ∪{∞} such that x < z ≤ y. Consider the hyperoperation ⊞′ defined on T (Γ)
+as x + ∞ = ∞ + x = {x} for all x ∈ T (Γ) and:
+x ⊞′ y :=
+�
+{min{x, y}}
+if x ̸= y,
+(x, ∞]
+if x = y.
+(x, y ∈ T (Γ))
+It is not difficult to check that (T (Γ), ⊞′, +, ∞, 0) is a hyperfield. We will denote it by T ′(Γ) and
+call it the strict generalised tropical hyperfield.
+The above examples are (up to isomorphism) all instances of a general construction which yields
+a hyperring from a ring and a subgroup of its multiplicative semigroup (see Proposition 2.17 below).
+This construction was first described by Krasner in [25]. We briefly recall it in the following example.
+Example 2.16. Let A be a commutative ring and T a subgroup of the commutative semigroup
+(A, ·). Let AT denote the set of all multiplicative cosets xT for x ∈ A, in particular 0T = {0}.
+Krasner showed that by setting
+xT + yT := {(x + yt)T | t ∈ T }
+and
+xT · yT := xyT
+
+NOTES ON VALUATION THEORY FOR KRASNER HYPERFIELDS
+7
+we have that (AT , +, ·, 0T ) is a hyperring where −(xT ) = (−x)T for all x ∈ A. In addition, if A
+is a field, then this construction always yields a hyperfield. This is called the factor (or quotient)
+hyperring (resp. hyperfield) of A modulo T .
+For the next proposition we employ the notion of isomorphism of hyperfields (see Definition 2.33
+below).
+Proposition 2.17. The following assertions hold:
+(i) For any field K with more than two elements, we have that K is isomorphic to KK×.
+(ii) For any ordered field K with positive cone P, we have that S is isomorphic to KP .
+(iii) If p > 3 is a prime such that p ≡ 3 mod 4, then W is isomorphic to (Fp)(F×
+p )2.
+(iv) Let Γ be an ordered abelian group. For any field k with more than two elements, let K :=
+k((tΓ)) denote the Hahn series field and let v be its canonical t-adic valuation. Then KO×
+v
+is isomorphic to T (Γ).
+(v) Let Γ be an ordered abelian group. Let K := F2((tΓ)) denote the Hahn series field and let
+v be its canonical t-adic valuation. Then KO×
+v is isomorphic to T ′(Γ).
+Proof. Let K be a field with more than two elements. Then KK× contains precisely two elements,
+the coset 0K× and the coset 1K×, the latter containing all non-zero elements of K. If x ∈ K×, then
+−x ∈ K× as well and thus 0K× belongs to 1K× + 1K×. Since by assumption there are x, y ∈ K×
+with x ̸= y, also (x − y)K× ̸= 0K× is an element of 1K× + 1K×. This shows (i).
+Let K be an ordered field with positive cone P. Here, this means that P is a subset of K× such
+that P + P, P · P ⊆ P and K× is the disjoint union of P and −P. It follows that KP contains
+precisely three1 elements: 0P, 1P (containing all the elements of P) and −1P (containing all the
+elements of −P). Since P + P ⊆ P, we have that 1P + 1P only contains 1P. By definition, if
+x ∈ P, then −x ∈ −P. Thus, 1P − 1P contains 0P. It does also contain 1P and −1P since e.g.
+1 + 1 ∈ P (because 1 ∈ P) and (1 + 1) − 1 = 1 ∈ P while 1 − (1 + 1) = −1 ∈ −P. This shows (ii).
+Let K be Fp (the finite field with p elements) for some prime number p > 3 such that p ≡ 3
+mod 4. The prime number p is certainly odd and thus the cardinality of (K×)2 is p−1
+2 . If −1 is
+a square in K, then K× contains an element of order 4 but this is excluded by the assumption
+p ≡ 3 mod 4. We have that x ∈ K× is a square if and only if −x is not a square. It follows that
+K(K×)2 contains precisely three elements: 0(K×)2, 1(K×)2 (containing all the non-zero squares)
+and −1(K×)2 (containing all the non-squares). Moreover, 1(K×)2 − 1(K×)2 contains 0(K×)2.
+This shows that −1(K×)2 is the hyperinverse of 1(K×)2 in K(K×)2. Since 1 + 1 ̸= 0 in K and
+1 = (1 + 1) − 1, we have that either 1(K×)2 or −1(K×)2 (depending on which among 1 + 1 and
+−(1 + 1) is a square) belongs to 1(K×)2 − 1(K×)2. By the symmetry of 1(K×)2 − 1(K×)2 under
+the application of − (i.e., multiplication by −1(K×)2), we obtain that both 1(K×)2 and −1(K×)2
+must be elements of 1(K×)2 −1(K×)2. In a finite field, any non-square is the sum of two (non-zero)
+squares. Thus, −1(K×)2 is an element of 1(K×)2 + 1(K×)2. In order to show that 1(K×)2 is an
+element of 1(K×)2+1(K×)2 as well, we distinguish two cases. If 1+1 is a square in K, then 1(K×)2
+is in 1(K×)2 + 1(K×)2 trivially. Otherwise, 1 + 1 is not a square. In this case, either 1 + 1 + 1 is a
+square, in which case 1 = (1+1+1)−(1+1) shows that 1(K×)2 is an element of 1(K×)2 +1(K×)2,
+or −(1 + 1 + 1) = −((1 + 1) + 1)) is a square, in which case −(1 + 1) = −(1 + 1 + 1) + 1 implies
+that 1(K×)2 is an element of 1(K×)2 + 1(K×)2. This proves (iii).
+1In the literature, sometimes positive cones are defined as “non-negative cones”, i.e., containing 0. Notice that for
+our statement to hold it is necessary to exclude 0 from positive cones.
+
+8
+LINZI, A.
+Let k be a field with more thatn 2 elements and consider the Hahn series field K = k((tΓ))
+equipped with the t-adic valuation v. A non-zero element x of K× is represented as a formal series
+x =
+�
+γ∈Γ
+aγtγ
+with well-ordered support, i.e., {γ ∈ Γ | aγ ̸= 0} is a well-ordered set with respect to the order of
+Γ. The t-adic value of x under v is defined to be the minimum of the support of x (cf. [10, Exercise
+3.5.6]).
+The value group of v is thus Γ and it follows from general valuation theory that K×/O×
+v is
+isomorphic to Γ as an ordered abelian group (cf. [10, Section 2.1] or Lemma 3.20 below).
+For
+x ∈ K×, we have that xO×
+v contains all the elements of K with the same value of x under v. If
+x, y ∈ K are such that vx ̸= vy (i.e., xO×
+v ̸= yO×
+v ), then
+v(x + yt) = v(x + y) = min{vx, vy}
+for any t ∈ O×
+v (i.e., any t with vt = 0). This follows from [10, Section 1.3 (1.3.4)] (or Corollary 3.5
+(iv) below). We now note that if x ∈ K has any positive value under v, then v(x − 1) = 0 and thus
+vx = v(1 + (x − 1)) is the value of some element of K generating a coset in KO×
+v which belongs to
+1O×
+v +1O×
+v . Moreover, by the assumption on the cardinality of k, there exists a ∈ k \{1}, so by the
+definition of the t-adic valuation, we have that va = 0 and v(1 − a) = 0. Since v(1) = v(−1) = 0,
+we conclude that the image of 1O×
+v − aO×
+v = 1O×
+v + 1O×
+v under v is [0, ∞]. By distributivity,
+this suffices to show that T (Γ) ≃ KO×
+v (see also Lemma 3.3 below). The proof of (iv) is now also
+complete.
+The proof for (iv) can easily be adapted to prove (v).
+□
+Remark 2.18. Let us mention at this point that not all hyperfields are factor hyperfields. This fact
+was proved by Massouros in [39] who then further improved his results in [40]. According to [21,
+Theorem 6] finite hyperfields such that 1 /∈ 1 + 1 and with the property that 0 does not belong to
+any finite sum of 1 with itself are also not quotient hyperfields.
+2.2. Subhyperrings. To choose a good notion of subhyperring is not as straightforward as it might
+seem. Two options are presented in the next definition.
+Definition 2.19. Let (R, +, ·, 0) be a hyperring. A subset S of R is a relational subhyperring of R
+if it is multiplicatively closed and with the induced multivalued operation:
+x +S y := (x + y) ∩ S
+(x, y ∈ S)
+we have that (S, +S, ·, 0) is a hyperring as well.
+A subset S of R is a (traditional) subhyperring of R if 0 ∈ S and for all x, y ∈ S we have that
+x − y ⊆ S and xy ∈ S.
+If R is a hyperring with unity 1, then we say that a relational subhyperring S of R is a relational
+subhyperfield of R if 1 ∈ S and (S, +S, ·, 0, 1) is a hyperfield. In addition, a subhyperring S of R is
+called a (traditional) subhyperfield if 1 ∈ S and (S \ {0}, ·, 1) is an abelian group.
+Remark 2.20. The adjective “relational” has been chosen for the following reason. A multivalued
+operation can be encoded in a first-order language via the ternary relation z ∈ x + y (as noticed
+e.g. in [32]). If we consider a first-order language L with a constant symbol for 0, a binary function
+symbol for · and a ternary relation symbol for +, then a hyperring naturally becomes a structure on
+L, which is a model of all the axioms of Definitions 2.7 and 2.3 (written as sentences in L). Under
+this interpretation, the relational subhyperrings of a hyperring R are precisely the submodels of R,
+
+NOTES ON VALUATION THEORY FOR KRASNER HYPERFIELDS
+9
+i.e., the substructures of R (see [43, Section 2.3]) which satisfy the axioms. For more details on the
+model theoretical point of view let us refer the reader to [33].
+Our choices regarding terminology are also motivated by historical reasons. In fact, subhyper-
+groups (and consequently subhyperfields) have been defined long time ago (see e.g. Definition 2 and
+the subsequent remark in [22]2).
+Remark 2.21. It is clear that a subhyperring S of a hyperring (R, +, ·, 0) is a relational subhyperring
+of R and that +S = + in this case.
+On the other hand, not all relational subhyperrings are
+subhyperrings, as the following example shows.
+Example 2.22. Consider an ordered abelian group (Γ, +, 0, <). Let R := T (Γ) be the hyperfield
+obtained as in Example 2.14 and consider the subset S := {∞, 0} of R. It is straightforward to check
+that S is a relational subhyperfield of R. However, S is not a subhyperfield of R since 0⊞0 = [0, ∞]
+is not a subset of S.
+Remark 2.23. Let Γ be an ordered abelian group. It is not difficult to see that the only subhyperfield
+of T (Γ) is T (Γ) itself. On the other hand, if ∆ is a subgroup of Γ, then T (∆) is a relational
+subhyperfield of T (Γ) (Example 2.22 above corresponds to the case ∆ = {0}).
+2.3. Homomorphisms. Next, let us consider a notion of homomorphism for hyperrings and hy-
+perfields.
+Definition 2.24. Let (R, +R, ·R, 0R) and (S, +S, ·S, 0S) be hyperrings. We say that a map σ :
+R → S is a homomorphism of hyperrings if
+(HH1) σ(0R) = 0S,
+(HH2) σ(x ·R y) = σ(x) ·S σ(y), for all x, y ∈ R,
+(HH3) σ(x +R y) ⊆ σ(x) +S σ(y), for all x, y ∈ R.
+If (R, +R, ·R, 0R, 1R) and (S, +S, ·S, 0S, 1S) are hyperfields, then we say that a homomorphism of
+hyperrings σ : R → S is a homomorphism of hyperfields when the following properties
+(HH4) σ(1R) = 1S,
+(HH5) σ(x−1) = σ(x)−1, for all x ∈ R \ {0},
+hold as well.
+Example 2.25. Let R be a hyperring. The map defined by σ(0) := 0 and σ(x) := 1 for x ∈ R\{0}
+is a homomorphism of hyperrings R → K.
+Example 2.26. Let (Γ, <, +, 0Γ) be an ordered abelian group. The map
+ι : K → T (Γ)
+0K �→ ∞
+1K �→ 0Γ
+is a homomorphism of hyperfields.
+Example 2.27. Let K be a field and T a subgroup of K×. The function K → KT mapping x ∈ K
+to [x]T ∈ KT is a homomorphism of hyperfields.
+Remark 2.28. Let σ : R → S be a homomorphism of hyperrings. Then the kernel of σ
+ker σ := {x ∈ R | σ(x) = 0S}
+2This article is included in Krasner’s collected works [4, pages 280–406]
+
+10
+LINZI, A.
+is a subhyperring of R. Indeed, for x, y ∈ ker σ, we have that
+σ(x − y) ⊆ σ(x) − σ(y) = 0S − 0S = {0S}.
+Thus, σ(z) = 0S for all z ∈ x − y, i.e., x − y ⊆ ker σ.
+Remark 2.29. Let (R, +, ·, 0, 1) be a hyperring and S ⊆ R. Consider the inclusion map ι : S ֒→ R.
+It is straightforward to verify that if S is a relational subhyperring of R, then ι is a homomoprhism
+of hyperrings.
+On the other hand, if (R, +R, ·R, 0R) and (S, +S, ·S, 0S) are hyperrings with S ⊆ R and the
+inclusion ι : S ֒→ R is a homomorphism of hyperrings, then one cannot conclude that S is a
+relational subhyperring of R, as the following example shows.
+Example 2.30. The identity S → W is a homomorphism of hyperrings. However, S is not a
+subhyperring of W as the hypersum of 1 with itself in S is {1}, while the hypersum of 1 with itself
+in W intersected with S is {−1, 1}.
+The above example motivates the following definition.
+Definition 2.31. Let (R, +R, ·R, 0R) and (S, +S, ·S, 0S) be hyperrings (resp. hyperfields with uni-
+ties 1R and 1S). An injective homomormphism of hyperrings (resp. hyperfields) σ : R → S is called
+an embedding of hyperrings (resp. hyperfields) if
+(EM1) σ(x +R y) = (σ(x) +S σ(y)) ∩ Im σ, for all x, y ∈ R.
+We leave the straightforward proof of following lemma to the reader.
+Lemma 2.32. If S is a relational subhyperring of a hyperring R, then the inclusion map S ֒→ R is
+an embedding of hyperrings. Conversely, If R and S are hyperrings with S ⊆ R and the inclusion
+map S ֒→ R is an embedding of hyperrings, then S is a relational subhyperring of R.
+Definition 2.33. A homomorphism of hyperrings (resp. hyperfields) σ : R → S is called an
+isomorphism of hyperrings (resp. hyperfields) if it is a bijective map and σ−1 : S → R is also a
+homomorphism of hyperrings (resp. hyperfields). We say that two hyperrings (resp. hyperfields) R
+and S are isomorphic and we write R ≃ S if there exists an isomorphism σ : R → S.
+Lemma 2.34. Let R and S be hyperrings (resp. hyperfields) and denote by +R and +S their
+hyperoperations, respectively. A map σ : R → S is an isomorphism of hyperrings (resp. hyperfields)
+if and only if σ is a surjective embedding of hyperrings (resp. hyperfields).
+Proof. If σ is an isomorphism, then it is bijective by definition, in particular Im σ = S. Since σ is
+a homomorphism, we have that σ(x +R y) ⊆ σ(x) +S σ(y) holds. On the other hand, since σ−1
+satisfies (HH3) as well, we obtain that
+σ−1(σ(x) +S σ(y)) ⊆ x +R y.
+Applying σ to both sides then yields σ(x +R y) ⊇ σ(x) +S σ(y).
+Assume now that σ is a surjective embedding and let us show that σ−1 is a homomorphism.
+Clearly, (HH1) and (HH2) hold for σ−1.
+Since by (EM1) and the surjectivity we have that
+σ(x +R y) = σ(x) +S σ(y), property (HH3) follows applying σ−1 to both sides and using again
+the surjectivity of σ.
+□
+Remark 2.35. Note that a bijective homomorphism is not necessarily an isomorphism as it can be
+deduced from Example 2.30.
+
+NOTES ON VALUATION THEORY FOR KRASNER HYPERFIELDS
+11
+Example 2.36. Let (Γ, <, +, 0Γ) be an ordered abelian group. Consider the relational subhyper-
+field S := {∞, 0} of T (Γ) as in Example 2.22. Then (S, +S, ·, ∞, 0Γ) is isomorphic to K.
+As usual we identify isomorphic structures: Examples 2.22 and 2.36 can be expressed by saying
+that K is a subhyperfield of T (Γ) for any ordered abelian group Γ.
+2.4. Hyperideals. We now briefly discuss how the classical ring theory notion of ideal generalises
+in the multivalued setting.
+Definition 2.37 (Definition 2.1 in [5] and Definition 2.9 in [31]). Let R be a hyperring. A subhy-
+perring I of R is a hyperideal of R if it satisfies
+(HID1) xy ∈ I, for all x ∈ R and y ∈ I.
+Lemma 2.38. Let R be a hyperring with no zero divisors, i.e., xy = 0R implies x = 0R or y = 0R.
+A relational subhyperring I of R satisfying (HID1) is a hyperideal of R.
+Proof. Pick x, y ∈ I and let z ∈ x − y. It suffices to prove that z ∈ I. If x = 0R or y = 0R,
+then there is nothing to show. Otherwise, by distributivity in R we have that zy ∈ xy − y2 and by
+(HID1) we obtain that zy ∈ I, since y ∈ I. Therefore, zy ∈ (xy − y2) ∩ I. Now, distributivity in I
+(with respect to the induced multivalued operation +I) yields zy ∈
+�
+(x − y) ∩ I
+�
+y, so there exists
+z′ ∈ (x − y) ∩ I such that zy = z′y. Hence, 0R ∈ zy − z′y = (z − z′)y. Since R has no zero divisors,
+this implies that 0R ∈ z − z′ and hence z = z′ ∈ I.
+□
+Remark 2.39. We wish to justify the choice of requiring hyperideals to be traditional subhyperrings.
+In [19, Section 3.1], Jun provides a generalisation of the classical quotient construction of a ring
+modulo an ideal in the multivalued setting. One property that is certainly desirable to preserve is for
+the canonical epimorphism from a hyperring R to the quotient hyperring R/I modulo a hyperideal
+I to be a homomorphism of hyperrings R → R/I having I as its kernel. We have no examples
+of hyperrings with a relational subhyperring satisfying (HID1), but at the same time we did not
+succeed in proving Lemma 2.38 without the assumption on zero divisors. If such an example would
+exist, then that relational subhyperring could not be the kernel of a homomorphism of hyperrings
+by Remark 2.28 and so we should not call it a hyperideal.
+Remark 2.40. Let σ : R → S be a homomorphism of hyperrings. Then it is easy to show that ker σ
+is a hyperideal of R. Conversely, for all hyperideals I of a hyperring R the map σ : R → K defined
+as
+σ(x) :=
+�
+0
+if x ∈ I,
+1
+otherwise.
+is a homomorphism of hyperrings such that ker σ = I.
+The following statements can be shown to hold as in the classical theory of rings.
+Lemma 2.41 (Lemma 2.12 in [31]). If a hyperideal I of a hyperring R contains a unit, then I = R.
+Corollary 2.42 (Corollary 2.13 in [31]). The only hyperideals of a hyperfield F are {0} and F.
+Definition 2.43 (Definition 2.14 (ii) in [31]). Let I be a hyperideal of a hyperring R. Then I is
+called maximal if I ⊊ R and for all hyperideals J of R we have that I ⊊ J implies J = R.
+We recall without proof the following result.
+Proposition 2.44 (Proposition 2.16 (ii) in [31]). Assume that R is a hyperring with unity. Let I
+be a hyperideal R. Then I is maximal if and only if R/I is a hyperfield.
+
+12
+LINZI, A.
+Example 2.45 ([42]). Let R be a hyperring with unity. An element s of R is called a scalar if
+x + s is a singleton for all x ∈ R. As in the proof of Lemma 2.9, it is not difficult to show that
+s ∈ R is a scalar if and only if s − s = {0}. Interestingly, the set of all scalar elements of R forms a
+hyperideal of R.
+Assume now that R is a hyperring with unity. If 1 is a scalar, then by Lemma 2.41 the hyperideal
+of scalars is R and thus R is a ring (as all of its elements are scalars). Moreover, we deduce from
+Corollary 2.42 that if F is a hyperfield, but not a field, then the only scalar of F is 0, i.e., x − x is
+not a singleton for all x ∈ F ×.
+3. Valued hyperfields
+Some of the results presented in this section can be found in [31].
+Some proofs have been
+improved, others we will repeat for the convenience of the reader.
+3.1. Valuations. The next definition is a straightforward generalisation of the definition of valua-
+tion for fields.
+Definition 3.1 (Definition 4.1 in [31]). Take a hyperfield F and an ordered abelian group Γ (written
+additively). A surjective map v : F → Γ ∪ {∞} is called a valuation on F if it has the following
+properties:
+(V1) vx = ∞ ⇐⇒ x = 0, for all x ∈ F,
+(V2) v(xy) = vx + vy, for all x, y ∈ F,
+(V3) z ∈ x + y =⇒ vz ≥ min{vx, vy}, for all x, y, z ∈ F.
+If v is a valuation on a hyperfield F we call (F, v) a valued hyperfield and we denote by vF the
+ordered abelian group v(F ×), i.e., the value group of (F, v).
+Example 3.2. Let F be a hyperfield. The trivial valuation vx = 0 for all x ∈ F × is always a
+valuation on F with value group {0}.
+The next result provides further examples of valued hyperfields.
+Lemma 3.3 (Corollary 4.2 in [31]). Let (K, v) be a valued field and take a subgroup T of K×.
+Denote by O×
+v the group of units of the valuation ring of K, i.e., O×
+v := {x ∈ K | vx = 0}. If
+T ⊆ O×
+v , then
+vT : KT → vK ∪ {∞}
+xT �→ vx
+is a valuation on KT.
+Proof. The assumption T ⊆ O×
+v guarantees that the map vT is well-defined. Moreover, surjectivity,
+(V1) and (V2) follow immediately from the corresponding properties of v and the definition of the
+factor hyperfield KT . It remains to verify (V3). Take x, y ∈ K and t ∈ T . We obtain that
+v(x + yt) ≥ min{vx, v(yt)} = min{vx, vy + vt} = min{vx, vy}.
+From this it easily follows that (V3) holds for (KT , vT ).
+□
+The next lemma gives an alternative definition of valuation on (hyper)fields (cf. [2, Example
+1.8]).
+Lemma 3.4. Let F be a hyperfield and Γ an ordered abelian group. Then (F, v) is a valued hyperfield
+with vF = Γ if and only if the map v : F → T (Γ) is a surjective homomorphism of hyperfields.
+
+NOTES ON VALUATION THEORY FOR KRASNER HYPERFIELDS
+13
+Proof. Assume that (F, v) is a valued hyperfield. Then v : F → T (vF) is a surjective map, (HH1)
+follows from (V1), (HH2) follows from (V2), (HH3) follows from (V3). Property (HH4) follows
+because vx = vx + v(1) for all x ∈ F and (HH5) follows since, for x ∈ F ×, we have that
+0 = v(1) = v(xx−1) = vx + v(x−1).
+Thus, v is a homomorphism of hyperfields.
+Conversely, let F be a hyperfield and v : F → T (Γ) a surjective homomoprhism of hyperfields.
+Property (V2) for v follows from (HH2) and (V3) follows from (HH3) together with the definition
+of the hyperoperation of T (Γ). Property (V1) follows from Corollary 2.42 since v(1) = 0 ̸= ∞ and
+thus ker v = {x ∈ F | vx = ∞} is a proper hyperideal of F. We proved that v is a valuation on F
+with vF = Γ.
+□
+On the basis of the previous lemma, the following result is almost immediate to verify. Thus,
+we omit its proof.
+Corollary 3.5 (Lemma 4.5 in [31]). Let v : F → Γ ∪ {∞} be a valuation on a hyperfield F. Then:
+(i) v(1) = v(−1) = 0,
+(ii) v(−x) = vx for all x ∈ F,
+(iii) vx−1 = −vx for all x ∈ F ×,
+(iv) if vx ̸= vy, then vz = min{vx, vy}, for every x, y ∈ F and z ∈ x + y.
+Lemma 3.6. Let (KT , w) be a valued hyperfield which is a factor hyperfield. Then there exists a
+valuation v on K such that T ⊆ O×
+v and w = vT .
+Proof. Let Γ be the value group of (KT , w). Since K → KT , x �→ [x]T and w : KT → T (Γ) are
+surjective homomorphisms of hyperfields, their composition v : K → T (Γ) is a valuation on K
+satisfying the conditions of the statement.
+□
+Example 3.7. By Proposition 2.17 (iv) we have that T (Γ) is isomorphic to all factor hyperfields of
+the form k((tΓ))O×
+vt for some field k. By the above lemmas, the isomorphism σ : k((tΓ))O×
+vt → T (Γ)
+is a valuation on k((tΓ))O×
+vt and the identity map id = σ ◦ σ−1 : T (Γ) → T (Γ) is a valuation on
+T (Γ).
+3.2. Valuation hyperrings. The next definition is inspired by classical valuation theory.
+Definition 3.8 (Definition 4.6 in [31]). Let F be a hyperfield. A relational subhyperring O of F
+is called a valuation hyperring in F if for all x ∈ F × we have that either x ∈ O or x−1 ∈ O.
+Observe that, by definition, it follows that 1 ∈ O for any valuation hyperring O in F. Let us
+now prove some more basic properties of valuation hyperrings.
+Lemma 3.9 (Proposition 4.7 in [31]). A valuation hyperring O in a hyperfield F is a subhyperring
+of F.
+Proof. It suffices to show that a − b ⊆ O for all a, b ∈ O. Take a, b ∈ O and x ∈ a − b. If x ∈ O,
+then there is nothing to show (note that this case also includes x = 0). Otherwise, we have that
+x−1 ∈ O and thus ax−1, bx−1 ∈ O. Since x ∈ a − b we obtain from (CH4) that a ∈ x + b, so, using
+axiom (HR3),
+ax−1 ∈ (x + b)x−1 = 1 + bx−1.
+
+14
+LINZI, A.
+We have obtained that ax−1 ∈ (1 + bx−1) ∩ O = 1 +O bx−1. By (CH4) and (HR3) applied to the
+hyperring (O, +O, ·, 0), it follows that
+xx−1 = 1 ∈ ax−1 +O (−bx−1) = (a +O (−b))x−1.
+Therefore, x ∈ a +O (−b) ⊆ O. This shows that a − b ⊆ O.
+□
+Lemma 3.10 (Lemma 4.8 in [31]). Let O be a valuation hyperring in a hyperfield F.
+Then
+M := O \ O× is the unique maximal hyperideal of O.
+Proof. Take a ∈ M and c ∈ O. If ca is invertible in O, then there exists x ∈ O such that x(ca) = 1.
+Hence (xc)a = 1 and a−1 = xc ∈ O contradicting a ∈ M. This proves that ca ∈ M and shows that
+M satisfies (HID1).
+Take a, b ∈ M. We may assume that ab−1 ∈ O (otherwise ba−1 ∈ O and we can interchange the
+roles of a and b). Since O is a subhyperring of F (cf. Lemma 3.9), we obtain that 1 − ab−1 ⊆ O
+and therefore, using what we have just proved, we conclude that
+b − a = b(1 − ab−1) ⊆ M.
+We have shown that M is a hyperideal of O.
+Since, by the definition of M, we have that O\M = O×, by Lemma 2.41, every proper hyperideal
+of O must be contained in M, showing that M is the unique maximal hyperideal of O.
+□
+3.3. Residue hyperfield. Any valuation on a hyperfield F induces a valuation hyperring in F.
+Proposition 3.11 (Proposition 4.11 in [31]). Let v : F → Γ ∪ {∞} be a valuation on a hyperfield
+F. Then
+Ov := {x ∈ F | vx ≥ 0}
+is a valuation hyperring in F and
+Mv := {x ∈ F | vx > 0}
+is its unique maximal hyperideal.
+Proof. We first prove that Ov is a subhyperring of F. Take a, b ∈ Ov. By (V3), for all c ∈ a − b we
+have vc ≥ min{va, v(−b)} = min{va, vb} ≥ 0, so a − b ⊆ Ov. Further, we have ab ∈ Ov by (V2).
+By Corollary 3.5 (iii) we conclude that if x /∈ Ov, then x−1 ∈ Ov so Ov is a valuation hyperring in
+F.
+Next we show that Mv is the unique maximal hyperideal of Ov. Observe that, by virtue of
+Corollary 3.5 (iii),
+O×
+v = {x ∈ Ov | vx = 0}.
+Hence, Mv = Ov \ O×
+v and then Mv is the unique maximal hyperideal of Ov by Lemma 3.10.
+□
+Remark 3.12. Let (F, v) be a valued hyperfield. Note that Ov can be described in terms of the
+hyperoperation ⊞ of T (vF) as the preimage in F of v(1) ⊞ v(1) under v:
+Ov = v−1(v(1) ⊞ v(1)).
+It follows from Proposition 2.44 that, for a valuation hyperring O with maximal hyperideal M,
+the quotient hyperring
+O/M = {x + M | x ∈ O},
+with the multivalued operation ⊕ defined as
+(x + M) ⊕ (y + M) := {z + M | z ∈ x + y},
+
+NOTES ON VALUATION THEORY FOR KRASNER HYPERFIELDS
+15
+is a hyperfield (see [19, Section 3] and Remark 2.39 above).
+Definition 3.13. If (F, v) is a valued hyperfield, then we call the hyperfield Ov/Mv the residue
+hyperfield of (F, v) and we denote it by Fv. For an element x ∈ Ov, we denote by xv its natural
+image in Fv.
+Proposition 3.14. Let (K, v) be a valued field and T ⊆ O×
+v a subgroup of K×. Then KT vT ≃
+(Kv)(T v), where T v := {tv | t ∈ T }.
+Proof. By definition vT [x]T = vx for all x ∈ K, thus OvT = (Ov)T and MvT = (Mv)T . It follows
+that
+KT vT = OvT /MvT = (Ov)T /(Mv)T .
+Let us define
+σ : (Ov)T /(Mv)T → (Kv)(T v)
+[x]T + (Mv)T �→ [xv]T v
+and show that σ is an isomorphism of hyperfields. By [19, Lemma 3.3] we have that [x]T +(Mv)T =
+[y]T + (Mv)T if and only if v(x − yt) > 0 for some t ∈ T . This implies that
+0 = (x − yt)v = xv − yv · tv ∈ [xv]T v − [yv]T v ,
+where we have used the assumption T ⊆ O×
+v . This shows that σ is well-defined and injective. The
+surjectivity of σ is clear. Moreover, σ is easily seen to be a homomorphism of the corresponding
+multiplicative groups. Finally, we observe that
+σ
+�
+[x]T + (Mv)T ⊕ ([y]T + (Mv)T )
+�
+= {σ
+�
+[z]T + (Mv)T
+�
+| [z]T ∈ [x]T + [y]T }
+= {[zv]T v | z = x + yt for some t ∈ T }
+= {[xv + yv · tv]T v | t ∈ T }
+= [xv]T v + [yv]T v
+hence σ is an isomorphism of hyperfields by Lemma 2.34.
+□
+Example 3.15. By Proposition 2.17, T (Γ) ≃ KO×
+v , where K = k((tΓ)) for some field k with more
+than two elements and v is its canonical t-adic valuation. Since Kv = k and O×
+v v = k×, by the
+above proposition and Proposition 2.17 (i) we have that the residue field of T (Γ) with respect to
+its valuation given by the identity map is isomorphic to kk× ≃ K which, moreover, is a relational
+subhyperfield of T (Γ) (cf. Examples 2.22 and 2.36).
+For a valued field (K, v) we denote by 1 + Mv the set of 1-units. That is, those x ∈ K such that
+v(x − 1) > 0 or equivalently, xv = 1v.
+Proposition 3.16. Let (K, v) be a valued field and T ⊆ O×
+v a subgroup of K×. Then the map
+ι : KT vT → KT
+[xv]T v �→ [x]T
+is an embedding of hyperfields if and only if 1 + Mv ⊆ T .
+Proof. If 1 + Mv ⊆ T and x, y ∈ O×
+v , then [xv]T v = [yv]T v if and only if xv = yv · tv = (yt)v for
+some t ∈ T if and only if v(1 − ytx−1) = v(x − yt) > 0 if and only if ytx−1 ∈ 1 + Mv ⊆ T . It
+follows that [y]T [x]−1
+T
+= [ytx−1]T = [1]T and ι is well-defined. On the other hand if [x]T = [y]T for
+
+16
+LINZI, A.
+some x, y ∈ O×
+v , then for some t ∈ T we have that x = yt and thus xv = (yt)v = yv · tv. It follows
+that ι is injective. Moreover, we have that
+ι
+�
+[xv]T v · [yv]−1
+T v
+�
+= ι
+�
+[xv · y−1v]T v
+�
+= [xy−1]T = [x]T [y]−1
+T
+and hence ι satisfies (HH2) and (HH5). It clearly satisfies (HH1) and (HH4). It remains to show
+that (EM1) holds. First observe that Im ι consists of classes [x]T where x = 0 or x ∈ R ⊆ O×
+v ,
+where R is a set of some selected representatives for Kv. Take again x, y ∈ O×
+v . We have that
+ι ([xv]T v + [yv]T v) = {[x + yt]T | x, y ∈ R, t ∈ T ∩ R} = ([x]T + [y]T ) ∩ Im ι.
+This completes the proof of one implication.
+If a ∈ 1 + Mv is not in T , then av = 1v and thus [av]T v = [1v]T v holds. On the other hand, we
+have that
+ι[av] = [a]T ̸= [1]T = ι[1v]T
+and thus ι is not well-defined and in particular not an embedding of hyperfields.
+□
+3.4. Equivalence of valuations. In analogy with classical valuation theory, our aim is now to
+show that valuation hyperrings can be used to describe valuations up to composition with an order
+preserving isomorphism of the value group.
+Proposition 3.17 (Proposition 4.12 in [31]). Let F be a hyperfield and O a valuation hyperring
+in F. Consider the multiplicative group Γ := F ×/O× and define a relation ≤ on Γ as follows:
+aO× ≤ bO× ⇐⇒ ba−1 ∈ O.
+Then (Γ, ·, ≤) is an ordered abelian group and the canonical projection
+π : F → Γ ∪ {∞},
+extended so that π(0F ) = ∞, is a valuation on F. Furthermore, Oπ = O.
+Proof. First we show that ≤ is an ordering for (Γ, ·). Since aa−1 = 1F ∈ O, reflexivity is clear. If
+ab−1, ba−1 ∈ O, then ab−1 ∈ O× so aO× = bO×. Hence ≤ is antisymmetric. If ab−1, bc−1 ∈ O,
+then ac−1 = ab−1bc−1 ∈ O, showing that ≤ is transitive. Take now a, b ∈ F × such that aO× ≤ bO×
+and c ∈ F ×. We have that bc(ac)−1 = bcc−1a−1 = ba−1 ∈ O, whence acO× ≤ bcO×. This shows
+that ≤ is compatible with the operation of Γ. Finally, ≤ is a total order since O is a valuation
+hyperring, so that ab−1 ∈ O or ba−1 ∈ O for all a, b ∈ F ×.
+We now show that π is a valuation on F. Clearly, π is a surjective map, onto the ordered abelian
+group Γ with ∞ and (V1) holds. Since π is a homomorphism of groups we obtain (V2). It remains
+to show that (V3) holds for π. Take x, y ∈ F. If one of them is 0F , then (V3) is straightforward.
+We may then assume that x, y ∈ F × and that xO× ≤ yO×. Take z ∈ x + y. We wish to show that
+zx−1 ∈ O. By assumption we have that yx−1 ∈ O, thus
+zx−1 ∈ (x + y)x−1 = 1 + yx−1 ⊆ O,
+where we used Lemma 3.9.
+Finally, we observe that, by definition
+Oπ = {x ∈ F | πx ≥ 1} = {x ∈ F | xO× ≥ 1O×} = {x ∈ F | x ∈ O} = O.
+□
+Definition 3.18. Let (Γi, <i) be (partially) ordered sets for i = 1, 2. A map σ : Γ1 → Γ2 is said
+to be order preserving if γ1 ≤1 γ2 implies that σ(γ1) ≤2 σ(γ2) for all γ1, γ2 ∈ Γ1.
+
+NOTES ON VALUATION THEORY FOR KRASNER HYPERFIELDS
+17
+Definition 3.19. For i = 1, 2 let vi : F → Γi ∪ {∞} be valuations on a hyperfield F. We say
+that v1 and v2 are equivalent if there exists an isomorphism of groups σ : Γ1 → Γ2 which is order
+preserving and such that v2 = σ ◦ v1.
+Lemma 3.20. Let v : F → Γ ∪ {∞} be a valuation on a hyperfield F. Then Γ ≃ F ×/O×
+v with an
+isomorphism of groups which is order preserving.
+Proof. We consider F ×/O×
+v as an ordered abelian group with the ordering defined in Proposition
+3.17. Using the surjectivity of v, we define a map
+σ : Γ → F ×/O×
+v
+by σ(va) = aO×
+v for all a ∈ F ×. This is well-defined since if va = vb, then va − vb = v(ab−1) = 0
+so that ab−1 ∈ O×
+v and then aO×
+v = bO×
+v . Using property (V2) of valuations, we obtain that σ is
+a homomorphism of groups. Further, if va ≤ vb, then ba−1 ∈ Ov which means that σ(va) ≤ σ(vb).
+Thus, σ is order preserving. It is clear that σ is surjective. It therefore remains to show that σ is
+injective. To this end, assume that aO×
+v = bO×
+v for some a, b ∈ F ×. Then there exists c ∈ O×
+v such
+that a = bc. Since vc = 0, by (V2) we obtain that va = vb. This completes the proof.
+□
+Remark 3.21. Observe that by construction of σ in the above proof, we have that σ ◦ v = π where
+π is the canonical epimorphism F × → F ×/O×
+v .
+Corollary 3.22. For i = 1, 2 let vi : F → Γi ∪ {∞} be valuations on a hyperfield F. Then v1 and
+v2 are equivalent if and only if Ov1 = Ov2.
+Proof. By the previous lemma we obtain for i = 1, 2 that Γi ≃ F ×/O×
+vi as ordered abelian groups
+with isomorphisms σi such that σi ◦ vi = πi where πi : F × → F ×/O×
+vi is the canonical projection
+for i = 1, 2. Thus, if Ov1 = Ov2, then π1 = π2 and σ := σ−1
+2
+◦ σ1 is an isomorphism of ordered
+abelian groups Γ1 → Γ2. Further we have that
+σ ◦ v1 = σ−1
+2
+◦ (σ1 ◦ v1) = σ−1
+2
+◦ π2 = v2.
+Hence, v1 and v2 are equivalent.
+On the other hand, if v1 and v2 are equivalent, then we obtain that F ×/O×
+v1 ≃ F ×/O×
+v2 as ordered
+abelian groups. In particular, for a ∈ F × we have that 1O×
+v1 ≤ aO×
+v1 if and only if 1O×
+v2 ≤ aO×
+v2.
+Using the definition of the ordering in F ×/O×
+vi we see that this means that a ∈ Ov1 if and only if
+a ∈ Ov2. Since 0 ∈ Ovi for i = 1, 2, we conclude that Ov1 = Ov2 as claimed.
+□
+In the following sections we will always consider valuations on hyperfields up to equivalence.
+4. Krasner valued hyperfields
+In this section, we focus our attention on those valued hyperfields which satisfy the original more
+restrictive axioms of Krasner and were called by him hypercorps valué. These valued hyperfields
+still attract the attention of mathematicians. For instance, they have recently been considered by
+Tolliver and Lee (see [46, 32]) who called them simply “valued hyperfields”.
+4.1. Ultrametric spaces. We begin by presenting some basic theory of ultrametric spaces, a no-
+tion as well studied by Krasner (cf. [23]). The axioms for an ultrametric distance can be formulated
+using just the linear order of non-negative real numbers where 0 is a bottom element. Since nothing
+but the order is used from the structure of real numbers, we will use the term ultrametric space in a
+broader sense allowing ultrametric distances to take values in any linearly ordered set. In addition,
+since the value group of a valuation has a top element, our definition of ultrametric space below
+
+18
+LINZI, A.
+is a modification which better fits into our context (the same approach can be found e.g. in [28]).
+In the case of real-valued distances, this modification corresponds to a replacement of the linearly
+ordered set (R≥0, <) with (R ∪ {∞}, >).
+Definition 4.1. An ultrametric distance (or simply an ultrametric) on a set X is a function
+d : X × X → Γ ∪ {∞}, where (Γ, <) is a linearly ordered set and ∞ satisfies γ < ∞ for all γ ∈ Γ,
+such that for all x, y, z ∈ X
+(U1) d(x, y) = ∞ if and only if x = y,
+(U2) d(x, y) = d(y, x),
+(U3) d(x, z) ≥ min{d(x, y), d(y, z)}.
+We call (X, d) an ultrametric space whenever d is an ultrametric on X. We call the set dX :=
+{d(x, y) | x, y ∈ X, x ̸= y} ⊆ Γ the value set of d.
+Example 4.2. Let (K, v) be a valued field. Then the function
+K × K → Γ ∪ {∞}
+(x, y) �→ v(x − y)
+is an ultrametric on K.
+Definition 4.3. Let (X, d) be an ultrametric space. A subset B ⊆ X is called a ball if for all
+y, z ∈ B and all x ∈ X we have that the following implication
+d(x, y) ≥ d(y, z)
+=⇒
+x ∈ B
+holds for all x, y, z ∈ X.
+Definition 4.4. A subset ρ of a linearly ordered set (Γ, <) is called an initial segment (resp. final
+segment) if for all δ ∈ ρ and all γ ∈ Γ if γ < δ (resp. γ > δ), then γ ∈ ρ.
+Lemma 4.5. For every element x of an ultrametric space (X, d) and every final segment ρ of
+dX ∪ {∞}, we have that
+Bρ(x) := {y ∈ X | d(x, y) ∈ ρ}.
+is a ball in X. Conversely, if B is a ball in X and ρ is the smallest (with respect to inclusion) final
+segment of dX ∪ {∞} containing d(y, z) for all y, z ∈ B, then for every x ∈ B,
+B = Bρ(x).
+In particular, Bρ(x) = Bρ(y) for every y ∈ Bρ(x).
+Proof. Assume that y, z ∈ Bρ(x), that is, d(x, y) ∈ ρ and d(x, z) ∈ ρ.
+If t ∈ X is such that
+d(y, t) ≥ d(y, z), then the inequalities
+d(x, t) ≥ min{d(x, y), d(y, t)} ≥ min{d(x, y), d(y, z)} ≥ min{d(x, y), d(y, x), d(x, z)} ∈ ρ
+follow from (U2) and (U3). Since ρ is a final segment of dX ∪ {∞}, we conclude that d(x, t) ∈ ρ.
+Hence, t ∈ Bρ(x)and Bρ(x) is a ball.
+For the converse, assume that B is a ball and let ρ be as in the assertion. Further, let x be any
+element in B. If y ∈ B, then d(x, y) ∈ ρ and thus y ∈ Bρ(x). On the other hand, if y ∈ Bρ(x),
+then d(x, y) ∈ ρ. So by definition of ρ, there is some z ∈ B such that d(x, z) ≤ d(x, y). Since B is
+a ball, it follows that y ∈ B. We have proved that B = Bρ(x).
+□
+Corollary 4.6. Let (X, d) be an ultrametric space. Every two balls with non-empty intersection
+are comparable by inclusion.
+
+NOTES ON VALUATION THEORY FOR KRASNER HYPERFIELDS
+19
+Proof. Take two balls B and B′ and suppose that z ∈ B ∩ B′. By Lemma 4.5 there are final
+segments ρ, ς of dX ∪ {∞} such that B = Bρ(z) and B′ = Bς(z). Since ρ and ς are final segments,
+we must have ρ ⊆ ς or ς ⊆ ρ. Hence, B ⊆ B′ or B′ ⊆ B.
+□
+4.2. Krasner valuations. Let (Γ, <, +, 0) be an ordered abelian group, ρ be an initial segment of
+Γ and γ an element of Γ. We will sometimes write γ > ρ to indicate that γ /∈ ρ.
+The following class of valued hyperfields is of special interest.
+Definition 4.7 (Section 3 of [24], Definition 1.4 in [46] and Definition 2.4 in [32]). We call a valued
+hyperfield (F, v) a Krasner valued hyperfield if
+(KVH1) For all x, y ∈ F, v(x + y) is a singleton unless 0 ∈ x + y.
+(KVH2) There exists an initial segment ρv of vF such that 0 ∈ ρv and for all x, y, z, t ∈ F we have
+that z ∈ x + y implies that t ∈ x + y if and only if vs > ρv + min{vx, vy} for all s ∈ z − t.
+The initial segment ρv is called the norm of v. We will also say that v is a Krasner valuation on F
+when (F, v) is a Krasner valued hyperfield.
+Example 4.8. Let (K, v) be a valued field and consider K as a hyperfield. Then v is a Krasner
+valuation on K with norm vK.
+Proposition 4.9. Let (F, v) be a Krasner valued hyperfield. For all x, y ∈ F such that x ̸= y, by
+axiom (KVH1), v(x−y) contains a unique element γx,y ∈ vF. Define a map dv : F ×F → Γ∪{∞}
+as
+dv(x, y) :=
+�
+γx,y
+if x ̸= y,
+∞
+otherwise.
+Then dv is an ultrametric on F. Moreover, for all x, y ∈ F and any z ∈ x + y, in the ultrametric
+space (F, dv) we have that
+x + y = Bρv+min{vx,vy}(z).
+Proof. Axiom (U1) follows from axiom (V1), axiom (U2) follows from Corollary 3.5 (ii) and axiom
+(U3) is a direct consequence of axiom (V3). The last statement is just a reformulation of axiom
+(KVH2).
+□
+We call the ultrametric dv on a Krasner valued hyperfield (F, v) the ultrametric induced by v on
+F.
+Remark 4.10. Let F be a hyperfield and let v be the trivial valuation on F. If v is a Krasner
+valuation of norm ρv, then by (KVH2) for all x, y ∈ F × we have that x ∈ 1 − 1 if and only if
+0 = vx > ρv. Since 0 ∈ ρv, it follows that x − x = {0} for all x ∈ F and then F is a field by Lemma
+2.9.
+Conversely, if K is a field, then the map v(0) := ∞ and vx := 0 for all x ∈ K× is a Krasner
+valuation on K with value group vK = {0} and norm vK.
+Krasner’s main motivation probably came from the following example.
+Example 4.11. For a valued field (K, v) and an initial segment ρ ⊆ vK containing 0, let us
+consider the (multiplicative) group of 1-units of level ρ:
+1 + Mρ
+v = {x ∈ K | v(x − 1) > ρ} ⊆ O×
+v .
+Then vρ := v1+Mρ
+v is a Krasner valuation on Kρ := K1+Mρ
+v and the norm of vρ is ρ.
+
+20
+LINZI, A.
+Actually, any Krasner valued hyperfield which is a factor hyperfield is of this form, as we show
+below.
+Proposition 4.12. Let F be a factor hyperfield admitting a Krasner valuation w. Then F = Kρ
+and w is vρ for some valued field (K, v) and some initial segment ρ of vK = wF containing 0.
+Proof. By Lemma 3.6 there is a valued field (K, v) and a subgroup T ⊆ O×
+w of K× such that
+vT = w. Suppose that t ∈ T is not a 1-unit in K. Then vt = 0 and v(t − 1) = 0 must hold.
+now, on the one hand, since w = vT is a Krasner valuation, for all [x]T ∈ [1]T − [1]T we have that
+vx = vT [x]T > 0 by (KVH2). On the other hand, since t ∈ T we have that [t − 1]T ∈ [1]T − [1]T .
+This contradiction proves that T ⊆ 1 + Mv must hold and the result follows.
+□
+Remark 4.13. We do not know if all Krasner valued hyperfields are factor hyperfields. Some results
+connected to this problem are provided in [34].
+Example 4.14. Consider the Hahn series field K := F2((tΓ)) for some non-trivial ordered abelian
+group Γ and let v denote its canonical t-adic valuation. In this case, since Kv ≃ F2 we have that
+O×
+v = 1 + Mv. We conclude that
+T ′(Γ) ≃ Kρ,
+where ρ = {γ ∈ Γ | γ ≤ 0}. We have that the identity map vρ is the identity map on T (Γ) = T ′(Γ)
+and it is a Krasner valuation on T ′(Γ) with norm ρ. Note that, the same map is not a Krasner
+valuation on T (Γ) as 0 ∈ [0, ∞] = 0 ⊞ 0 violates (KVH2).
+We now study the residue hyperfield of a Krasner valued hyperfield.
+Proposition 4.15. The residue hyperfield Fv of a Krasner valued hyperfield (F, v) is a field.
+Proof. Take xv ∈ 1v − 1v. This means that x ∈ 1 − 1 and by axiom (KVH2), we deduce that
+vx > ρv. Since 0 ∈ ρv, it follows that xv = 0v and therefore Fv is a field by Lemma 2.9.
+□
+Example 4.16. Let (K, v) be a valued field. It follows from Proposition 3.14 that, for any initial
+segment ρ of vK containing 0, the residue hyperfield of Kρ is a field isomorphic to Kv.
+Let us now consider a valued hyperfield which is not a Krasner valued hyperfield.
+Example 4.17. Consider the rational function field K := Q(X) over the rationals, and its mul-
+tiplicative subgroup T := Q×. Under the canonical X-adic valuation v, we have that T ⊆ O×
+v .
+Hence, Lemma 3.3 yields a valued hyperfield (KT , vT ). By Proposition 3.14 and Proposition 2.17
+(i), we have that
+KT vT ≃ QQ× ≃ K.
+Since K is not a field, the Proposition 4.15 implies that (KT , vT ) is not a Krasner valued hyperfield.
+4.3. Mittas and superiorly canonical hypergroups. J. Mittas was a student of Krasner. He
+introduced superiorly canonical hypergroups (defined below) and published (sometimes without
+proofs) some results which connect them to Krasner valuations (see e.g. [41]). Some of the contents
+of this subsection were inspired by his work.
+Definition 4.18 ([41] and page 81 of [38]). A canonical hypergroup H is called superiorly canonical
+if
+(SCH1) For all x, y ∈ H, if x ∈ x + y, then x + y = {x}.
+(SCH2) For all x, y, z, t ∈ H, if (x + y) ∩ (z + t) ̸= ∅, then x + y ⊆ z + t or z + t ⊆ x + y.
+(SCH3) For all x, y ∈ H such that x ̸= y we have that z, t ∈ x − y implies z − z = t − t.
+
+NOTES ON VALUATION THEORY FOR KRASNER HYPERFIELDS
+21
+(SCH4) For all x, y, z ∈ H, if x ∈ z − z and y /∈ z − z, then x − x ⊆ y − y.
+Example 4.19. Any abelian group is a superiorly canonical hypergroup.
+Other examples of superiorly canonical hypergroups are provided by the following result.
+Proposition 4.20. Let (F, v) be a Krasner valued hyperfield. Then the additive canonical hyper-
+group of F is superiorly canonical.
+Proof. We verify the axioms one by one. Assume that x ∈ x+ y for some x, y ∈ F. By reversibility,
+y ∈ x − x, so the inequalities vy > ρv + vx ≥ vx follow from axiom (KVH2) and 0 ∈ ρv. Therefore,
+if z ∈ x + y, then vx = vz by Corollary 3.5 (iv) and since by reversibility y ∈ z − x and dv(y, 0) =
+vy > ρv + min{vx, vz}, axiom (KVH2) implies that 0 ∈ z − x and so x = z must hold. This shows
+(SCH1).
+Axiom (SCH2) follows from Proposition 4.9 and Corollary 4.6.
+For (SCH3), take x, y ∈ F and assume that x ̸= y, i.e., 0 /∈ x − y. Take z, t ∈ x − y. We
+claim that vz = vt. Suppose that vz < vt, then by Corollary 3.5 (iv) we have that va = vz for all
+a ∈ z − t, thus
+dv(z, 0) = vz = dv(z, t) > ρv + min{vx, vy}.
+Axiom (KVH2) now implies that 0 ∈ x − y, a contradiction. Therefore, vz ≥ vt. Symmetrically,
+vt ≥ vz and so vz = vt as claimed.
+Now, by (KVH2) we have that a ∈ z − z if and only if
+va > ρv + vz and a ∈ t − t if and only if va > ρv + vt. Since vz = vt we conclude that z − z = t − t.
+This shows that (SCH3) holds.
+For (SCH4), take x ∈ z − z and y /∈ z − z. By axiom (KVH2) it follows that vx > ρv + vz and
+vy ∈ ρv +vz. In particular, vx > vy. Now, again by axiom (KVH2), if a ∈ x−x, then va > ρv +vx
+which implies va > ρv + vy and so a ∈ y − y. Hence, x − x ⊆ y − y. This completes the proof.
+□
+The theory of superiorly canonical hypergroups and the theory of Krasner valued hyperfields are
+even more deeply related.
+Proposition 4.21. Let F be a hyperfield with a superiorly canonical additive hypergroup. Then
+O := {x ∈ F | x − x ⊆ 1 − 1}
+is a valuation hyperring in F. Moreover, if v is the valuation such that Ov = O, then (F, v) is a
+Krasner valued hyperfield.
+Proof. If F is a field, then O = F and thus v is the trivial valuation on F. Since F is a field, (F, v)
+is a Krasner valued hyperfield. We can then assume that F is not a field.
+If x − x ⊆ 1 − 1 and y − y ⊆ 1 − 1, then
+xy − xy = (x − x)y ⊆ (1 − 1)y = y − y ⊆ 1 − 1.
+Hence, O is multiplicatively closed.
+If x − x ⊈ 1 − 1, then 1 − 1 ⊊ x − x by axiom (SCH2), since 0 ∈ (x − x) ∩ (1 − 1). Multiplying
+by x−1 we obtain that x−1 − x−1 ⊊ 1 − 1. Therefore, x−1 ∈ O.
+Assume that x − x ⊆ 1 − 1 and y − y ⊆ 1 − 1. We claim that if z ∈ x + y, then z − z ⊆ 1 − 1.
+Pick a ∈ z − z. Since z ∈ x + y we have that
+a ∈ z − z ⊆ (x − x) + (y − y) ⊆ (1 − 1) + (1 − 1),
+so there exists x′ ∈ 1 − 1 and y′ ∈ 1 − 1 such that a ∈ x′ + y′. Hence, 1 ∈ x′ + 1 by reversibility, and
+a ∈ x′ + y′ ⊆ (x′ + 1) − 1 = 1 − 1,
+
+22
+LINZI, A.
+where we have used axiom (SCH1). We proved that O is a valuation hyperring in F.
+Clearly, O× = {x ∈ F | x−x = 1−1}. Set vF := F ×/O× and let v : F × → vF be the canonical
+epimorphism. We have to show that (F, v) is a Krasner valued hyperfield. It is a valued hyperfield
+by Proposition 3.17. Let us now verify the validity of axioms (KVH1) and (KVH2).
+Take x, y ∈ F × and assume that 0 /∈ x + y. Pick z, t ∈ x + y and suppose that vz < vt. This
+means that tz−1 ∈ O \ O×, so
+tz−1 − tz−1 ⊊ 1 − 1.
+But then t − t ⊊ z − z which contradicts axiom (SCH3). We have proved that (KVH1) holds for
+(F, v).
+We now have to verify (KVH2). Our first claim is that v(1 − 1) is a final segment of vF which
+does not contain 0. Pick a ∈ 1 − 1 and γ > va. Let b ∈ F × be such that vb = γ. Since vb > va, we
+have that b − b ⊊ a − a. Now, if b /∈ 1 − 1, then axiom (SCH4) implies a − a ⊆ b − b. Therefore,
+b ∈ 1 − 1 must hold and thus γ = vb ∈ v(1 − 1) and v(1 − 1) is a final segment.
+For x ∈ F ×, if x ∈ x − x, then x − x = {x} by axiom (SCH1). However, since 0 ∈ x − x, this
+cannot be. An element x ∈ F × has value 0 if and only if x − x = 1 − 1. Hence it cannot belong to
+1 − 1 as x /∈ x − x for all x ∈ F ×. This shows that v(1 − 1) does not contain 0.
+We let ρv be the complement in vF of v(1 − 1). Then ρv is an initial segment of vF which
+contains 0.
+Observe that for all x ∈ F we have that a ∈ x − x = (1 − 1)x if and only if there exists y ∈ 1 − 1
+such that a = yx. This implies that va = vy + vx > ρv + vx. Conversely, if va > ρv + vx, then
+v(ax−1) ∈ v(1 − 1), so there exists b ∈ 1 − 1 such that vb = v(ax−1). Therefore,
+b ∈ 1 − 1 = bxa−1 − bxa−1 = b(xa−1 − xa−1),
+so 1 ∈ xa−1 − xa−1 implying that a ∈ x − x.
+Take now x, y ∈ F such that x ̸= y and vx ≤ vy. Fix z ∈ x − y. We have to show that t ∈ x − y
+if and only if dv(z, t) > ρv + vx.
+Assume first that t ∈ x − y. If t = z there is nothing to show. Otherwise, it suffices to show that
+z − t ⊆ x − x by what we have already shown above. Take a ∈ z − t. Since z ∈ x − y, we have that
+a ∈ z − t ⊆ x − (y + t).
+Hence, there exists b ∈ y+t such that a ∈ x−b. We obtain that b ∈ (x−a)∩(y+t), so y+t ⊆ x−a
+or x − a ⊆ y + t by axiom (SCH2). In the first case, since x ∈ y + t we have that x ∈ x − a and
+a ∈ x − x follows. It remains to deal with the case x − a ⊊ y + t. In this case, since t ∈ x − y and
+y − y ⊆ x − x, we obtain that
+x − a ⊊ y + t ⊆ y + x − y ⊆ x + (x − x).
+Take c ∈ x − a. There exists x′ ∈ x − x such that c ∈ x + x′. This implies that
+a ∈ x − c ⊆ x − (x + x′) = x − x
+where we have used axiom (CH4) and axiom (SCH1) to conclude that x + x′ = {x}.
+Now, assume that dv(z, t) > ρv + vx. We have to prove that t ∈ x − y. If z = t, then there is
+nothing to show. Hence, we can assume that z ̸= t and so z − t ⊊ x − x must hold. We have that
+t ∈ t + 0 ⊆ t + z − z ⊊ x − x + z.
+Hence, there is b ∈ z − x such that t ∈ x + b. We conclude that b ∈ (z − x) ∩ (t − x). By axiom
+(SCH2) we then obtain that z − x ⊆ t − x or t − x ⊆ z − x. In the first case, −y ∈ z − x ⊆ t − x
+
+NOTES ON VALUATION THEORY FOR KRASNER HYPERFIELDS
+23
+and t ∈ x − y follows. It remains to deal with the case t − x ⊊ z − x. In this case, since z ∈ x − y,
+we have that
+t − x ⊊ z − x ⊆ x − y − x = x − x − y.
+Take a ∈ t − x. There exists x′ ∈ x − x such that a ∈ x′ − y. Now, by reversibility and since
+a ∈ t − x, we have that
+−y ∈ a − x′ ⊆ t − (x + x′) = t − x,
+where for the last equality we have used axiom (SCH1). Now, t ∈ x−y follows by axiom (CH4).
+□
+Directly from Proposition 4.20 and Proposition 4.21 above, we deduce the following characteri-
+zation theorem for the hyperfields which admit a Krasner valuation.
+Theorem 4.22. Let F be a hyperfield.
+Then F admits a Krasner valuation if and only if the
+additive hypergroup of F is superiorly canonical.
+Example 4.23. The additive hypergroup of the hyperfield that we have considered in Example
+4.17 above is not superiorly canonical. Indeed,
+1T + 1T = {0T, 1T }
+and thus (SCH1) fails. It follows from Theorem 4.22 that this hyperfield does not admit Krasner
+valuations at all.
+Example 4.24. Consider a generalised tropical hyperfield T (Γ), where Γ is some non-trivial ordered
+abelian group (see Example 2.14). By Proposition 2.17 (iv) and Lemma 3.3 we conclude that T (Γ)
+is a valued hyperfield (see also the discussion after Proposition 5.9). Nevertheless, by Theorem 4.22,
+T (Γ) does not admit Krasner valuations as its additive hypergroup does not satisfy (SCH1).
+Let us now analyse another example.
+Example 4.25. Take K = Q(X) with the valuation v := vp ◦ vX where vX denotes the X-adic
+valuation on Q(X) and vp denotes the p-adic valuation on Q, for some prime number p. More
+explicitly, for a polynomial f(X) = �n
+i=0 aiXi in Q[X], we have
+vf(X) =
+�
+vXf(X), vp(avXf(X))
+�
+∈ Z × Z.
+The order relation in vK is the lexicographic order of Z × Z, that is, (n1, n2) < (m1, m2) if and
+only if n1 < m1 ∨ (n1 = m1 ∧ n2 < m2).
+Consider the Krasner valued hyperfield F := Kρ where ρ is the smallest initial segment of Z × Z
+which contains {0} × Z (cf. Example 4.11). Let us denote its Krasner valuation vρ by w.
+Note that, if (m1, m2) ∈ ρ and m1 > 0, then, since ρ is an initial segment, we have that
+{0} × Z ⊆ {(k1, k2) ∈ Z × Z | (k1, k2) ≤ (m1 − 1, m2)} ⊊ ρ,
+in contradiction with the minimality of ρ. Therefore, m1 ≤ 0 for all (m1, m2) ∈ ρ. Conversely, if
+m1 ≤ 0, then (m1, m2) ≤ (0, m2) for all m2 ∈ Z and therefore (m1, m2) ∈ ρ since ρ is an initial
+segment containing {0} × Z. It follows that
+(1)
+ρ = {(m1, m2) ∈ Z × Z | m1 ≤ 0} = {(m1, m2 + n) ∈ Z × Z | m1 ≤ 0} = ρ + (0, n)
+for any (0, n) ∈ {0} × Z.
+By Proposition 4.20, the additive hypergroup of F is superiorly canonical. Therefore, by Propo-
+sition 4.21, we have the valuation hyperring
+Ou = {x ∈ F | x − x ⊆ 1 − 1}
+
+24
+LINZI, A.
+for some Krasner valuation u on F. Let us now show that w and u are not equivalent valuations
+on F.
+In fact, we will show that Ow ⊊ Ou. Pick x ∈ Ow, i.e., wx ≥ (0, 0). By axiom (KVH2) we have
+that y ∈ x − x if and only if wy > ρ + wx ≥ ρ. Another application of axiom (KVH2) shows that
+x − x ⊆ 1 − 1 so that x ∈ Ou. Now, consider e.g. the element x of F corresponding to the rational
+number p−1 in K. Since vp(p−1) = −1 we have that wx = (0, −1) < (0, 0) so that x /∈ Ow. On
+the other hand, since wx = (0, −1) ∈ {0} × Z, we have that ρ + wx = ρ by (1) and thus using
+(KVH2), we obtain that y ∈ x − x if and only if wy > ρ if and only if y ∈ 1 − 1. This implies that
+x − x ⊆ 1 − 1 and thus x ∈ Ou.
+The above example constitutes the starting point for the discussion that we will have in Section
+6.
+5. More on ordered abelian groups
+In this section, we characterise generalised tropical hyperfields and discuss the concept of coars-
+ening of a valuation in the multivalued setting. We derive some conclusions on the relation between
+ordered abelian groups and generalised tropical hyperfields.
+5.1. Characterisation of generalised tropical hyperfields. To characterise generalised tropi-
+cal hyperfields, we will apply another remarkable characterisation result. A hyperfield is stringent
+if x + y is a singleton whenever 0 /∈ x + y. Bowler and Su in [6] characterised stringent hyperfields
+and we will now briefly recall their result.
+In [6, Section 4] a natural construction of a hyperfield arising from a short exact sequence
+(2)
+{1}
+F ×
+H×
+Γ
+{0}
+ϕ
+ψ
+where Γ is an ordered abelian group and F is any hyperfield is described.
+The thus obtained
+hyperfield has H× as multiplicative group and is called the Γ-layering of F along the short exact
+sequence (2). After that the following theorem is proved.
+Theorem 5.1 (Theorem 4.10 in [6]). A hyperfield H is stringent if and only if it is the Γ-layering
+of F along the short exact sequence (2) with F being (isomorphic to) either K, S or a field.
+From the details of the construction (which we omit for brevity) it is not difficult to verify that
+the extension of the map ϕ sending 0F to 0H is in any case an embedding of hyperfields.
+We say that a hyperfield has characteristic 2 if 0 ∈ 1 + 1 and that it has C-characteristic 1 if
+1 ∈ 1 + 1. For example, S satisfies the latter but not the former while K satisfies both. For more
+information on characteristic and C-characteristic of hyperfields the reader can see [21].
+We are now ready to state and prove our characterisation theorem.
+Theorem 5.2 (Characterisation of generalised tropical hyperfields). Generalised tropical hyper-
+fields are precisely stringent hyperfields of characteristic 2 and C-characteristic 1.
+Proof. The reader can easily verify from the relevant definitions that a generalised tropical hyperfield
+T (Γ) is a stringent hyperfield of characteristic 2 and C-characteristic 1.
+For the converse, let H be a stringent hyperfield of characteristic 2 and C-characteristic 1. By
+Theorem 5.1 and our observations on the extension of the map ϕ above, we have that either K, S or
+a field embed into H. Since H has characteristic 2 we have that 1 = −1, thus S cannot embed into
+H. On the other hand, any field cannot embed into H neither. Indeed, our assumption 0, 1 ∈ 1 + 1
+
+NOTES ON VALUATION THEORY FOR KRASNER HYPERFIELDS
+25
+implies that {0, 1} = (1 + 1) ∩ {0, 1} ⊆ (1 + 1) ∩ F = 1 +F 1 is not a singleton. It follows that H is
+the Γ-layering of K along a short exact sequence
+{1}
+K×
+H×
+Γ
+{0}
+ϕ
+ψ
+for some ordered abelian group Γ. Now, since K× = {1}, we obtain that the multiplicative group
+of H is isomorphic to Γ and the details of the construction given in [6, Section 4] show that the
+hyperoperation of H is the one of T (Γ). We conclude that H is a generalised tropical hyperfield.
+□
+An analogous reasoning yields to the following (but maybe less interesting) characterisation of
+strict generalised tropical hyperfields.
+Theorem 5.3. Generalised tropical hyperfields are precisely the Γ-layerings of F along the short
+exact sequence (2), with F = F2.
+5.2. Coarsenings.
+Definition 5.4. Let (Γ, <, +, 0) be an ordered abelian group. A subgroup ∆ of Γ is convex if for
+all γ ∈ Γ we have that if there exist δ1, δ2 ∈ ∆ such that δ1 < γ < δ2, then γ ∈ ∆.
+It is not difficult to see that the intersection of a family of convex subgroups of an ordered
+abelian group is again a convex subgroup and that the collection of all convex subgroups of an
+ordered abelian group is linearly ordered by inclusion. Let us recall another basic fact which makes
+convex subgroups important.
+Fact 5.5. Let (Γ, <, +, 0) be an ordered abelian group and ∆ a convex subgroup of Γ. Then (Γ/∆, ≺
+, +, 0) is an ordered abelian group, where
+x + ∆ ≺ y + ∆ ⇐⇒ x < y and y − x /∈ ∆.
+In particular, the canonical epimorphism Γ → Γ/∆ is order preserving.
+The following notion will play a fundamental role later.
+Definition 5.6. Let (Γ, <, +, 0) be an ordered abelian group and ρ an initial segment of Γ. We
+call the set
+ig(ρ) := {γ ∈ Γ | ρ + γ = ρ}
+the invariance group of ρ. If ig(ρ) = {0}, then we say that ρ has trivial invariance group.
+Remark 5.7. In [29, 30] F.-V. Kuhlmann proves several results about invariance groups associated
+to initial segments in ordered abelian groups. In particular, it follows from [30, Lemma 2.3] that
+the invariance group of an initial segment ρ of an ordered abelian group Γ such that 0 ∈ ρ is a
+convex subgroup of Γ contained in ρ.
+Definition 5.8. Let v, w be two valuations on a hyperfield F. We say that w is a coarsening of v
+if Ov ⊆ Ow.
+In classical valuation theory for fields, it is well-known that to any convex subgroup of the value
+group one can associate a coarsening. We note that this result generalises to valued hyperfields,
+actually with a much more conceptual proof.
+Proposition 5.9. Let (F, v) be a valued hyperfield and let ∆ be a convex subgroup of vF. Then
+v∆ : F → (vF/∆) ∪ {∞}
+x �→ vx + ∆
+is a valuation on F which is a coarsening of v.
+
+26
+LINZI, A.
+Proof. By Lemma 3.4, v : F → T (vF) is a surjective homomorphism of hyperfields. In addition,
+the canonical epimorphism vF → vF/∆ extends to a surjective map T (vF) → T (vF/∆). From the
+fact that the latter is order preserving, it easily follows that it is a homomorphism of hyperfields.
+We conclude that the composition v∆ : F → T (vF/∆) of v with the latter map is a surjective
+homomorphism of hyperfields. This proofs the first part of the statement. Now, since
+Ov∆ = {x ∈ F | v∆x ⪰ 0vF/∆}
+= {x ∈ F | vx + ∆ ⪰ ∆}
+= {x ∈ F | vx ≥ 0 or vx ∈ ∆}
+⊇ {x ∈ F | vx ≥ 0} = Ov,
+we conclude that v∆ is a coarsening of v.
+□
+In the above proof, we have seen that if ∆ is a convex subgroup of an ordered abelian group Γ,
+then the natural map
+π∆ : T (Γ) → T (Γ/∆)
+is a surjective homomorphism of hyperfields, which, by Lemma 3.4, is a valuation on T (Γ) with
+value group Γ/∆. The valuation hyperring of this valuation is
+O∆ = {γ ∈ Γ | γ + ∆ ⪰ 0 + ∆} = {γ ∈ Γ | γ ∈ ∆ or γ > ∆}
+and its unique maximal hypeideal is
+M∆ = {γ ∈ Γ | γ > ∆}
+It follows that the residue hyperfield T (Γ)π∆ = O∆/M∆ of T (Γ) with respect to the valuation π∆
+is isomorphic to T (∆). An isomorphism is given by the map
+T (∆) → T (Γ)π∆
+∞ �→ ∞
+δ �→ δπ∆
+In the next result we show that the valuations of T (Γ) are (up to equivalence) all of this form.
+Proposition 5.10. Let Γ be an ordered abelian group and assume that v : T (Γ) → T (Γ′) is a
+valuation. Then there exists a convex subgroup ∆ of Γ such that Γ/∆ ≃ Γ′ via an order preserving
+isomorphism.
+Proof. Let 0′ ∈ Γ′ denote the neutral element of the group Γ′, ⊞ the hyperoperation of T (Γ) and ⊞′
+the hyperoperation of T (Γ′). Set ∆ := v−1(0′). Since v is a homomorphism of groups Γ → Γ′, we
+have that ∆ is a subgroup of Γ. Moreover, since v is surjective, we have that Γ/∆ ≃ Γ′ as groups
+by the first homomorphism theorem for groups. Assume that δ1, δ2 ∈ ∆ and that γ ∈ Γ is such
+that δ1 ≤ γ ≤ δ2. Hence, δ2 ∈ γ ⊞ γ and so 0′ = v(δ2) ∈ v(γ) ⊞′ v(γ) since v is a homomorphism
+of hyperfields. On the other hand, γ ∈ δ1 ⊞ δ1 so that v(γ) ∈ 0′ ⊞′ 0′ = [0′, ∞]. If v(γ) > 0′, then
+0′ /∈ [v(γ), ∞] = v(γ) ⊞′ v(γ), a contradiction. We conclude that v(γ) = 0′ must hold and thus
+γ ∈ ∆. We have shown that ∆ is a convex subgroup of Γ. It remains to show that the isomorphism
+of groups σ : γ + ∆ �→ v(γ) is order preserving. Assume that γ1 + ∆ ≺ γ2 + ∆. This means that
+γ1 < γ2 and γ2 −γ1 /∈ ∆ hold in Γ. Thus, {γ1} = γ1 ⊞γ2 and then v(γ1) ∈ v(γ1)⊞′ v(γ2). Since σ is
+bijective and γ1 ̸= γ2, we have that v(γ1) ̸= v(γ2), so v(γ1) ∈ v(γ1) ⊞′ v(γ2) =
+�
+min{v(γ1), v(γ2)}
+�
+.
+This shows that v(γ1) < v(γ2) must hold in Γ′ and thus σ is order preserving and the proof is
+complete.
+□
+
+NOTES ON VALUATION THEORY FOR KRASNER HYPERFIELDS
+27
+The next result now follows from Corollary 3.22.
+Corollary 5.11. Let Γ be an ordered abelian group. There is a bijective correspondence between
+convex subgroups of Γ and valuation hyperrings of T (Γ).
+6. Krasner valuations induced by the additive structure
+In this section we apply some of the results obtained in the previous section to Krasner valued
+hyperfields. We begin by observing that coarsenings of Krasner valuations might not be Krasner
+valuations.
+Example 6.1. Let Γ be a non-trivial ordered abelian group with a non-trivial convex subgroup ∆
+and consider the identity map as a Krasner valuation v : T ′(Γ) → T (Γ). As a map, the coarsening
+v∆ : T ′(Γ) → T (Γ/∆)
+is just the canonical epimorphism Γ → Γ/∆ extended to send ∞ to ∞. Suppose that v∆ is a
+Krasner valuation of norm ρ for some initial segment ρ of Γ/∆ containing 0Γ/∆. Then by (KVH2)
+for all γ ∈ Γ we would have that
+γ > 0 ⇐⇒ γ ∈ 0 ⊞′ 0 ⇐⇒ γ + ∆ > ρ + (0 + ∆).
+On the other hand, since 0Γ/∆ ∈ ρ , any positive γ ∈ ∆ would violate this equivalence.
+Nevertheless, we have the following result.
+Theorem 6.2. Let (F, v) be a Krasner valued hyperfield and let w be the Krasner valuation on F
+determined by
+Ow = {x ∈ F | x − x ⊆ 1 − 1}.
+Then w is equivalent to the coarsening of v corresponding to ig(ρv). In particular, the coarsening
+of a Krasner valuation v corresponding to ig(ρv) is a Krasner valuation.
+Proof. We show that these two valuations have the same valuation hyperring. Indeed, for all x ∈ F
+we have that vx + ig(ρv) ⪰ 0vF/ ig(ρv) if and only if vx ∈ ig(ρv) or vx > ig(ρv). In both cases by
+(KVH2) applied to v we have that
+y ∈ x − x ⇐⇒ vy > ρv + vx ⊇ ρv =⇒ y ∈ 1 − 1,
+i.e., y ∈ Ow. Conversely, vx + ig(ρv) ≺ 0vF/ ig(ρv) if and only if vx /∈ ig(ρv) and vx < 0. Therefore,
+ρv + vx ⊊ ρv and there exists y ∈ F such that vy ∈ ρv and vy /∈ ρv + vx. By (KVH2) applied to v,
+the former implies y /∈ 1 − 1 while the latter is equivalent to y ∈ x − x. We conclude that x /∈ Ow.
+We have proved that vig(ρv) and w have the same valuation hyperring in F. The theorem follows
+from Corollary 3.22.
+□
+Corollary 6.3. Let F be a hyperfield admitting two Krasner valuations v1 and v2 of norm ρ1 and
+ρ2, respectively. Then the coarsening of v1 corresponding to ig(ρ1) is equivalent to the coarsening
+of v2 corresponding to ig(ρ2).
+Corollary 6.4. Let (F, v) be a Krasner valued hyperfield. If ρv has trivial invariance group, then
+Ov = {x ∈ F | x − x ⊆ 1 − 1}.
+Corollary 6.5. Let F be a hyperfield with a superiorly canonical additive hypergroup. Then the
+norm of the Krasner valuation v determined on F by the valuation hyperring
+Ov = {x ∈ F | x − x ⊆ 1 − 1}
+has trivial invariance group.
+
+28
+LINZI, A.
+Corollary 6.6. Let F be a hyperfield with a superiorly canonical additive hypergroup. There is a
+unique (up to equivalence) Krasner valuation v on F such that ig(ρv) = {0}.
+Remark 6.7. In the model theory of valued fields, the RV-structure (mentioned in the introduction)
+of level γ of a valued field (K, v), where γ is a non-negative element of vK, is essentially the Krasner
+valued hyperfield3 Kγ := Kργ, where ργ := {δ ∈ vK | δ ≤ γ}. Since ig(ργ) = {0}, it follows, as a
+consequence of Corollary 6.4, that the valuation hyperring of the Krasner valuation on Kγ induced
+by v is always definable in the language of hyperfields, e.g., using the ternary relation z ∈ x + y to
+encode the multivalued operation +.
+7. Further research
+There are at least three interesting points that have been touched but not developed further in
+this manuscript. Below we briefly describe them.
+(1) In the multivalued setting, (F, v), vF and Fv can all be described as (valued) hyperfields
+and they are not distinct structures. This applies in particular when F is a field.
+(2) In the literature there are not many examples of infinite hyperfields that are not factor
+hyperfields. Essentially, only the work of Massouros [39, 40] provides such examples. In
+all those examples M we have by definition x + x = M \ {x} for all x ∈ M ×. If v is a
+valuation on M and x ̸= 1, then x, x−1 ∈ 1 + 1, so by (V3) and Corollary 3.5 (iii) we have
+that vx = 0 must hold for all x ∈ M ×. It follows that M only admits the trivial valuation.
+Is it true that any hyperfield admitting a non-trivial valuation is a factor hyperfield?
+(3) In view of Theorem 5.2, a non-trivial valuation on a hyperfield F is a surjective homomor-
+phism onto a hyperfield H satisfying the following three conditions:
+• H has characteristic 2;
+• H has C-characteristic 1;
+• H is stringent.
+From this point of view, the image of a valuation is a quite special hyperfield. For example,
+one could think that responsible for the fact that finite (hyper)fields only admit the trivial
+valuation is the third property of H, since there are many examples of finite hyperfields
+satisfying 0, 1 ∈ 1+1. We speculate that investigating weakenings of the stringency property
+for H may lead to a notion of generalised valuation which have the potential to be applied
+also in the classical singlevalued setting e.g. to use (generalised) valuation-theoretic methods
+over finite fields.
+References
+[1] R. Ameri, M. Eyvazi, and S. Hoskova-Mayerova. Advanced results in enumeration of hyperfields. AIMS Math.,
+5(6):6552–6579, 2020.
+[2] M. Baker and O. Lorscheid. Descartes’ rule of signs, Newton polygons, and polynomials over hyperfields. J.
+Algebra, 569:416–441, 2021.
+[3] S. A. Basarab. Relative elimination of quantifiers for Henselian valued fields. Ann. Pure Appl. Logic, 53(1):51–74,
+1991.
+[4] L. Bélair and B. Poizat. Oeuvres Choisies de Marc Krasner. Editions L’Harmattan, 2019.
+[5] H. Bordbar and I. Cristea. Height of prime hyperideals in Krasner hyperrings. Filomat, 31(19):6153–6163, 2017.
+[6] N. Bowler and T. Su. Classification of doubly distributive skew hyperfields and stringent hypergroups. J. Algebra,
+574:669–698, 2021.
+[7] A. Connes and C. Consani. From monoids to hyperstructures: in search of an absolute arithmetic. In Casimir
+force, Casimir operators and the Riemann hypothesis, pages 147–198. Walter de Gruyter, Berlin, 2010.
+3This hyperfield is called the valued γ-hyperfield in [32]
+
+NOTES ON VALUATION THEORY FOR KRASNER HYPERFIELDS
+29
+[8] A. Connes and C. Consani. The hyperring of adèle classes. J. Number Theory, 131(2):159–194, 2011.
+[9] I. Cristea and S. Jančić-Rašović. Composition hyperrings. An. Ştiinţ. Univ. “Ovidius” Constanţa Ser. Mat.,
+21(2):81–94, 2013.
+[10] A. J. Engler and A. Prestel. Valued fields. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2005.
+[11] J. Flenner. Relative decidability and definability in Henselian valued fields. J. Symbolic Logic, 76(4):1240–1260,
+2011.
+[12] P. Gładki. Orderings of higher level in multifields and multirings. Ann. Math. Sil., (24):15–25, 2010.
+[13] P. Gładki. Witt equivalence of fields: a survey with a special emphasis on applications of hyperfields. In Ordered
+algebraic structures and related topics, volume 697 of Contemp. Math., pages 169–185. Amer. Math. Soc.,
+Providence, RI, 2017.
+[14] P.
+Gładki.
+A
+survey
+on
+multirings,
+hyperrings
+and
+hyperfields.
+Notes
+available
+at:
+http://www.math.us.edu.pl/ pgladki/inedita/hypsurvey.pdf, Unknown.
+[15] P. Gładki and M. Marshall. Orderings and signatures of higher level on multirings and hyperfields. J. K-Theory,
+10(3):489–518, 2012.
+[16] P. Gładki and M. Marshall. Witt equivalence of function fields over global fields. Trans. Amer. Math. Soc.,
+369(11):7861–7881, 2017.
+[17] A. C. Golzio. A brief historical survey on hyperstructures in algebra and logic. South Amer. J. Log., 4(1):91–119,
+2018.
+[18] P. Jell, C. Scheiderer, and J. Yu. Real tropicalization and analytification of semialgebraic sets. Int. Math. Res.
+Not. IMRN, (2):928–958, 2022. [Initially appeared as 2021, no. 24, 19178–19208].
+[19] J. Jun. Algebraic geometry over hyperrings. Adv. Math., 323:142–192, 2018.
+[20] J. Jun. Geometry of hyperfields. J. Algebra, 569:220–257, 2021.
+[21] D. E. Kędzierski, A. Linzi, and H. Stojałowska. Characteristic, C-characterisitic and positive cones in hyperfields.
+Submitted, 2022.
+[22] M. Krasner. Sur la primitivité des corps P-adiques. Mathematika (Cluj), 13(4):72–191, 1937.
+[23] M. Krasner. Nombres semi-réels et espaces ultramétriques. C. R. Acad. Sci. Paris, 219:433–435, 1944.
+[24] M. Krasner. Approximation des corps valués complets de caractéristique p ̸= 0 par ceux de caractéristique 0.
+In Colloque d’algèbre supérieure, tenu à Bruxelles du 19 au 22 décembre 1956, Centre Belge de Recherches
+Mathématiques, pages 129–206. Établissements Ceuterick, Louvain; Librairie Gauthier-Villars, Paris, 1957.
+[25] M. Krasner. A class of hyperrings and hyperfields. Internat. J. Math. Math. Sci., 6(2):307–311, 1983.
+[26] W. Krull. Allgemeine Bewertungstheorie. J. Reine Angew. Math., 167:160–196, 1932.
+[27] F.-V. Kuhlmann. Quantifier elimination for Henselian fields relative to additive and multiplicative congruences.
+Israel J. Math., 85(1-3):277–306, 1994.
+[28] F.-V. Kuhlmann. Maps on ultrametric spaces, Hensel’s lemma, and differential equations over valued fields.
+Comm. Algebra, 39(5):1730–1776, 2011.
+[29] F.-V. Kuhlmann. Selected methods for the classification of cuts, and their applications. In Algebra, logic and
+number theory, volume 121 of Banach Center Publ., pages 85–106. Polish Acad. Sci. Inst. Math., Warsaw, 2020.
+[30] F.-V. Kuhlmann. Invariance group and invariance valuation ring of a cut. In preparation, preliminary version
+avaliable at: https://math.usask.ca/~fvk/CUTS.pdf, Unknown.
+[31] K. Kuhlmann, A. Linzi, and H. Stojałowska. Orderings and valuations in hyperfields. J. Algebra, 611:399–421,
+2022.
+[32] J. Lee. Hyperfields, truncated DVRs, and valued fields. J. Number Theory, 212:40–71, 2020.
+[33] A. Linzi. Algebraic hyperstructures in the model theory of valued fields. PhD thesis, University of Szczecin
+(Uniwersytet Szczeciński), https://bip.usz.edu.pl/doktorat-habilitacja/16282/alessandro-linzi, 2022.
+[34] A. Linzi and P. Touchard. On the hyperfields associated to valued fields. Submitted. Preprint available at
+arXiv:2211.05082, 2022.
+[35] F. Marty. Sur une généralization de la notion de groupe. In Huitiéme Congrès des Mathématiciens Scandenaves
+Stockholm, pages 45–49, 1934.
+[36] F. Marty. Rôle de la notion de hypergroupe dans l’étude de groupes non abéliens. C. R. Acad. Sci. (Paris),
+201:636–638, 1935.
+[37] F. Marty. Sur les groupes et hypergroupes attaches à une fraction rationnelle. Ann. Sci. École Norm. Sup. (3),
+53:83–123, 1936.
+[38] C. Massouros and G. Massouros. An overview of the foundations of the hypergroup theory. Mathematics, 9(9):
+, 2021.
+[39] C. G. Massouros. Methods of constructing hyperfields. Internat. J. Math. Math. Sci., 8(4):725–728, 1985.
+
+30
+LINZI, A.
+[40] C. G. Massouros. On the theory of hyperrings and hyperfields. Algebra i Logika, 24(6):728–742, 749, 1985.
+[41] J. Mittas. Hypergroupes canoniques valués et hypervalués. Math. Balkanica, 1:181–185, 1971.
+[42] J. Mittas. Sur les hyperanneaux et les hypercorps. Math. Balkanica, 3:368–382, 1973.
+[43] A. Prestel and C. N. Delzell. Mathematical logic and model theory. Universitext. Springer, London, 2011. A brief
+introduction, Expanded translation of the 1986 German original.
+[44] A. Prestel and P. Roquette. Formally p-adic fields, volume 1050 of Lecture Notes in Mathematics. Springer-
+Verlag, Berlin, 1984.
+[45] R. Rota. Multiplicative hyperrings (Sugli iperanelli moltiplicativi). Rend. Mat. (7), 2(4):711–724 (1983), 1982.
+[46] J. Tolliver. An equivalence between two approaches to limits of local fields. J. Number Theory, 166:473–492,
+2016.
+[47] O. Viro. Hyperfields for tropical geometry I. hyperfields and dequantization. arXiv:1006.3034, 2010.
+[48] M. Vuković. Remembering Professor Marc Krasner. Sarajevo J. Math., 12(25)(2, suppl.):283–298, 2016.
+Center for Information Technologies and Applied Mathematics, University of Nova Gorica, Slove-
+nia.
+Email address: alessandro.linzi@ung.si
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf,len=1700
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='08639v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='AC]  20 Jan 2023  NOTES ON VALUATION THEORY FOR KRASNER HYPERFIELDS  ALESSANDRO LINZI ID  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' The main aim of this article is to study and develop valuation theory for Krasner  hyperfields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' In analogy with classical valuation theory for fields, we generalise the formalism of  valuation rings to describe equivalence of valuations on hyperfields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' After proving basic results  and discussing several examples, we focus on the valued hyperfields that Krasner originally defined  in 1957.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' We find that these must have a particular additive structure which in turns implies the  existence of a valuation a’la Krasner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' We note that given such a valued hyperfield (F, v), the  valuation induced by its additive structure does not have to be equivalent to v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' We discuss the  cases in which it does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Introduction  In 1957, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Krasner in [24] (the article is included in Krasner’s collected works [4, pages 413–  490]) formulated for the first time an axiomatisation of structures that generalise fields by allowing  the additive operation to be multivalued (see [4, pages 407–490] for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' He called these  hyperfields (in french hypercorps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  On the one hand, he found his inspiration in the work of Marty [35, 36, 37] which is considered  as the starting point of hypergroup theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' In fact, the additive part of a hyperfield is a hypergroup  (see also [48, 38] for a more detailed historical overview).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  On the other hand, Krasner was motivated by his interest in valuation theory (in connection with  p-adic numbers) and he sensed the importance of some hyperfields which are canonically associated  to any valued field (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let us briefly recall some basic notions of classical valuation  theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Let K be a field and Γ a linearly ordered abelian group (written additively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' A surjective map  v : K → Γ ∪ {∞}  is called a (Krull) valuation on K if it satisfies for all x, y ∈ K:  – v(x) = ∞ if and only if x = 0,  – v(xy) = v(x) + v(y),  – v(x + y) ≥ min{v(x), v(y)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Here, ∞ is a symbol such that γ + ∞ = ∞ + γ = ∞ > γ for all γ ∈ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' If a valuation v on a field  K is given, then (K, v) is called a valued field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' One usually denotes Γ by vK and call it the value  group of (K, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' The value v(x) of x ∈ K will often be written as vx, if there is no risk of confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Primary: 12J20, 20N20 Secondary: 13A18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Hyperfield, multifield, hyperring, valuation, ordered abelian group, tropical hyperfield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  The author would like to spend a few words to thank H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Stojałowska, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Touchard, Franz-Viktor and Katarzyna  Kuhlmann, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Cristea as well as Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Massouros, who all, in one way or another, contributed to the realisation of the  final version of this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  1  2  LINZI, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  If (K, v) is a valued field, then  Ov := {x ∈ K | vx ≥ 0}  is a subring of K, called the valuation ring of (K, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' It determines the valuation map v up to  equivalence, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=', up to composition with an order preserving isomorphism of the value group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Any  valuation ring has a unique maximal ideal  Mv := {x ∈ K | vx > 0}  and the field Kv := Ov/Mv is called the residue field of (K, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' For further details, let us mention  [10, 44] as general references on classical valuation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  While it is quite natural to generalise the concept of valuation to hyperfields (see Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='1),  Krasner noticed that the structures which attracted his attention come equipped with a map similar  to a valuation which satisfies two additional properties (see Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' These properties would  be vacuous if postulated for valued fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Thus, Krasner included them in his axiomatisation of  valued hyperfields (hypercorps valué).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Krasner’s motivation was not model theoretical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Nevertheless, it turns out that the structures he  studied play an important role in the model theory of valued fields;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' specifically for the problem of  quantifier elimination for henselian valued fields (of characteristic 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' In this setting, these objects  are known as RV-structures or leading-term structures and have been considered, independently of  Krasner, by Flenner in [11] (see also [3, 27, 33, 34] for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' In addition, an interesting  application of hyperfields in the model theory of valued fields recently appeared in [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  The interest for valued hyperfields may also be motivated by the following observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Clas-  sically, real algebra, which studies real fields (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=', linearly ordered fields, with an order compatible  with the operations) and was developed by E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Artin in his solution of Hilbert’s seventeenth problem,  is in relation with valuation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' After the works of Marshall and Gładki [15, 16] on real hyper-  fields, it is to expect that a development of valuation theory for hyperfields would be beneficial for  this line of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Significant developments have already been achieved with the generalisation  of classical results, such as the Baer–Krull Theorem, to the multivalued setting (see [26, 31]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' The  reader interested in these aspects may look also at [12, 13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Furthermore, the unpublished work  of Viro [47], followed up by Jun and Jell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' [20, 18], indicate applications in the realm of trop-  icalization maps and analytification, topics that are as well, classically, in relation with valuation  theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  We believe that developing valuation theory for hyperfields will eventually lead to further appli-  cations back to the classical theory of valuations for fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  The switch from singlevalued operations to multivalued ones may be not difficult to describe  and understand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Nevertheless, the consequences of this switch have to be handled very carefully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  For instance, valuation theorists are accostumed to the fact that v(x − y) always defines a metric  (in fact, an ultrametric, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='1) on a valued field (K, v), but this clearly ceases to be true  (in general) if the operation of addition (and hence of difference) is multivalued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Indeed, in the  latter case, the distance map may depend on the choice of some element of the set x − y and this  choice is not canonical, in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' This obstacle is overtaken using one of Krasner’s postulates  which forces all the elements of x − y, for x ̸= y, to have the same value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' However, one may expect  the latter being a quite strong requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' In fact, the restrictions due to Krasner’s axioms for  valued hyperfields are so strong that, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=', the trivial valuation on a hyperfield F (Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='2)  satisfies them only if F is actually a field (Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' For this and other reasons (which we  will discuss later in the paper), it makes sense to consider also valuations on hyperfields which do  not satisfy the two additional properties imposed by Krasner (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='17) and it turns out  NOTES ON VALUATION THEORY FOR KRASNER HYPERFIELDS  3  that this choice is more beneficial than harmful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' As a consequence of these observations, the term  “valued hyperfield” will be used in this paper in a broader sense than the original one;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' the valued  hyperfields satisfying in addition Krasner’s axioms will be called “Krasner valued hyperfields”and  their valuations “Krasner valuations”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  The manuscript is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' In Section 2 we introduce the necessary concepts and  terminology from hypercompositional algebra, taking the opportunity to discuss several examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  In Section 3, we show that the formalism of valuation rings to describe valuations up to equivalence  generalises without major modifications to the multivalued framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' We use this formalism to  state and prove one of our main results Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='22 on Krasner valued hyperfields in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Our main theorem states that the existence of a Krasner valuation on a hyperfield F implies that  the additive structure of F satisfies certain additional axioms (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Conversely, if  the additive structure of a hyperfield F satisfies those axioms, then F admits a Krasner valuation  (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let v be a Krasner valuation on a hyperfield F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Then the additive structure  of F induces a Krasner valuation w on F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' We observe that, in general, w is not equivalent to  v (Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' After Section 5, where we put into our context the notion of coarsening of a  valuation, which turns out to be necessary for our discussion, we describe situations in which v and  w are equivalent in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' The final Section 7 of this manuscript contains possible further lines  of investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Besides presenting some new results, the article is also thought to serve as a unified reference for  basic hyperring and hyperfield theory and related terminology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' On the one hand, these hypercom-  positional structures have recently attracted increasing interest from the mathematical community  (including top mathematicians such as A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Connes [7, 8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' On the other hand, we found the relevant  literature on hyperrings and hyperfields to be quite fragmented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' After the time and effort spent  during the PhD studies and thanks to the fruitful connections with some of the mathematicians who  are part of the early history of hypercompositional algebra, we felt that we could give a contribution  from this point of view as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Preliminaries  Let H be a non-empty set and P(H) its power set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' A multivalued operation + on H is a function  which associates to every pair (x, y) ∈ H × H an element of P(H), denoted by x + y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' If + is a  multivalued operation on H ̸= ∅, then for x ∈ H and A, B ⊆ H we set  A + B :=  �  a∈A,b∈B  a + b,  A + x := A + {x} and x + A := {x} + A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' If A or B is empty, then so is A + B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  A hypergroup can be defined as a non-empty set H with a multivalued operation + which is  associative (see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='3 (CH1) below) and reproductive on H (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=', x+H = H +x = H for all  x ∈ H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' This notion was first considered by F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Marty in [35, 36, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' The theory of hypergroups,  with a detailed historical overview, is presented in [38], where an extensive bibliography is also  provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' A hyperoperation + on H is a multivalued operation such that x + y ̸= ∅ for all  x, y ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='2 (Theorem 12 in [38]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' If (H, +) is a hypergroup, then + is a hyperoperation on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Aiming for a contradiction, suppose that x + y = ∅ for some x, y ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Then  H = x + H = x + (y + H) = (x + y) + H = ∅ + H = ∅,  4  LINZI, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  which is excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  □  The following special class of hypergroups (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='4 below) will be of interest for us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' A canonical hypergroup is a triple (H, +, 0), where H ̸= ∅, + is a multivalued  operation on H and 0 is an element of H such that the following axioms hold:  (CH1) + is associative, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=', (x + y) + z = x + (y + z) for all x, y, z ∈ H,  (CH2) x + y = y + x for all x, y ∈ H,  (CH3) for every x ∈ H there exists a unique x′ ∈ H such that 0 ∈ x + x′ (the element x′ will be  denoted by −x),  (CH4) z ∈ x + y implies y ∈ z − x := z + (−x) for all x, y, z ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let (H, +, 0) be a canonical hypergroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Then the multivalued operation + is repro-  ductive on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' In particular, (H, +) is a hypergroup and + is a hyperoperation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Fix a ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' For all x ∈ H + a there exists y ∈ H such that x ∈ y + a ⊆ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Therefore,  H + a ⊆ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' For the other inclusion, observe that for all x ∈ H we have that  x ∈ x + 0 ⊆ x + (a − a) = (x − a) + a,  so there exists y ∈ x − a ⊆ H such that x ∈ y + a ⊆ H + a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' We have proved that H = H + a  for an arbitrary a ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Now the conclusions of the lemma follow from (CH2), the definition of  hypergroups and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  □  Let (H, +, 0) be a canonical hypergroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Then 0 is called the neutral element for + in H because  of the following observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='5 (Section III, (b) in [39]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let (H, +, 0) be a canonical hypergroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Then x + 0 = {x}  for all x ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Take x ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' If y ∈ x + 0, then 0 ∈ y − x follows by (CH4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Now y = x follows from the  uniqueness required in (CH3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  □  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Note that an abelian group (G, ∗, 0) is not a priori a canonical hypergroup, because the  operation on G is not a multivalued operation, as it takes values in G and not in P(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' However,  it can be turned into a canonical hypergroup (G, +, 0) by setting x + y := {x ∗ y} for all x, y ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Conversely, if a canonical hypergroup (H, +, 0) satisfies that x + y is a singleton for all x, y ∈ H,  then one can define on H a standard operation which makes it an abelian group with identity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' In  these cases, we sometimes abusively say that (G, ∗, 0) is a canonical hypergroup or that (H, +, 0)  is a group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Hyperrings and hyperfields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let us now define the structures of main interest in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' A (commutative) hyperring is a tuple (R, +, ·, 0) which satisfies the following  axioms:  (HR1) (R, +, 0) is a canonical hypergroup,  (HR2) (R, ·) is a (commutative) semigroup and 0 is an absorbing element, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=', 0 · x = x · 0 = 0, for  all x ∈ R,  (HR3) the operation · is distributive with respect to +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' That is, for all x, y, z ∈ R,  x(y + z) = xy + xz,  NOTES ON VALUATION THEORY FOR KRASNER HYPERFIELDS  5  where, for x ∈ R and A ⊆ R, we have set  xA := {xa | a ∈ A}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  If the operation · has a neutral element 1 ̸= 0, then we say that (R, +, ·, 0, 1) is a hyperring with unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  If (R, +, ·, 0, 1) is a hyperring with unity and (R \\ {0}, ·, 1) is an abelian group, then (R, +, ·, 0, 1) is  called a hyperfield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' If R is a hyperring with unity, then we denote by R× the multiplicative group  of the units of R, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=',  R× := {x ∈ R | ∃y ∈ R : xy = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  In particular, if R is a hyperfield, then R× = R \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Since we will only consider the commutative case, in the following sections we will call commu-  tative hyperrings simply hyperrings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' In the literature, the name “hyperring” can be found for structures (R, +, ·) where +  is an operation and · is a multivalued operation or where both + and · are multivalued operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Some authors refer to the structures (R, +, ·, 0) where only the addition is multivalued as Krasner  hyperrings (some more historical remarks on this are given in in [17, Section 4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Structures (R, +, ·, 0) where + is an operation and · is a multivalued operation (satisfying similar  axioms) were introduced in [45] and called multiplicative hyperrings (in italian iperanelli moltiplica-  tivi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Structures (R, +, ·, 0) with both + and · multivalued operations (satisfying similar axioms) are  instead called general hyperrings (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' [9, Section 2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  In this paper, the name “hyperring” will be used for Krasner hyperrings exclusively, as indicated  in the above definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  In the literature, one may also find the term multiring referring to a structure (R, +, ·, 0), where  + is a multivalued operation and · is an operation, satisfying (HR1), (HR2) and the following  weaker version of (HR3):  (MR) x · (y + z) ⊆ x · y + x · z, for all x, y, z ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Thus, in this case, the different name reflects a difference in the axioms rather than in the structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Similarly, multifields are defined as hyperfields satisfying (MR) instead of (HR3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Multirings and  multifields have been considered for instance in [12], where, among other facts, it is observed that  all multifields are hyperfields, while there are several meaningful examples of multirings that are  not hyperrings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  In this paper, we preferred to use the name “hyperfield” solely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  We say that a hyperring (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' hyperfield) R is a ring (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' field) if the additive canonical  hypergroup of R is a group (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' The next observation gives a necessary condition for  a hyperring with unity to be a ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' This fact is an immediate corollary of a result already noted  in [42, page 369] (see also Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='45 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' We wish to state it for later reference and we take  the opportunity to write a quick proof .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='9 ([42]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' A hyperring with unity R is a ring if and only if 1 − 1 = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' If R is a field, then 1 − 1 = {0} holds trivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Conversely, by axiom (HR3) 1 − 1 = {0}  implies x − x = {0} for all x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Take a, b ∈ R and x, y ∈ a + b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' We have that  x − y ⊆ (a + b) − (a + b) = (a − a) + (b − b) = {0}  In particular, 0 ∈ x − y and x = y follows fro (CH3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' We have proved that a + b is a singleton for  all a, b ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  □  6  LINZI, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' The K-hyperfield K is the set {0, 1} with the hyperoperation + which has 0 as its  neutral element and satisfies 1 + 1 = {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' The multiplication is the obvious one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' It is customary among some authors to call the above hyperfield the Krasner hy-  perfield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' We prefer to avoid that terminology which, in view of Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='8 above, might lead to  unnecessary confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' The sign hyperfield S is the set {−1, 0, 1} with the hyperoperation + which has  0 as its neutral element and satisfies x + x = {x} for x ∈ {−1, 1} and 1 − 1 = {−1, 0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' The  multiplication is the obvious one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' The weak sign hyperfield W is the set {−1, 0, 1} with the hyperoperation + which  has 0 as its neutral element and satisfies x + x = {x, −x} for x ∈ {−1, 1} and 1 − 1 = {−1, 0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  The multiplication is the obvious one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Other examples of finite hyperfields are described e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let us now consider some infinite  examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let (Γ, +, <, 0) be an ordered abelian group and ∞ be a symbol such that γ+∞ =  ∞+γ = ∞ > γ for all γ ∈ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' For x, y ∈ Γ∪{∞} such that x ≤ y (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=', x < y or x = y), let us denote  by [x, y] the set consisting of all z ∈ Γ ∪ {∞} such that x ≤ z ≤ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Consider the hyperoperation ⊞  defined on T (Γ) := Γ ∪ {∞} as x ⊞ ∞ = ∞ ⊞ x = {x} for all x ∈ T (Γ) and:  x ⊞ y :=  �  {min{x, y}}  if x ̸= y,  [x, ∞]  if x = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  (x, y ∈ T (Γ))  It is not difficult to check that (T (Γ), ⊞, +, ∞, 0) is a hyperfield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' The hyperfield T (R, +, 0, >),  where > denotes the standard order of the real numbers, is known as the tropical hyperfield (see  [47, Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Therefore, we call the hyperfields of the form T (Γ) generalised tropical hyperfields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let (Γ, +, <, 0) be an ordered abelian group and ∞ be a symbol such that γ+∞ =  ∞ + γ = ∞ > γ for all γ ∈ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' For x, y ∈ Γ ∪ {∞} such that x < y, let us denote by (x, y] the set  consisting of all z ∈ Γ∪{∞} such that x < z ≤ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Consider the hyperoperation ⊞′ defined on T (Γ)  as x + ∞ = ∞ + x = {x} for all x ∈ T (Γ) and:  x ⊞′ y :=  �  {min{x, y}}  if x ̸= y,  (x, ∞]  if x = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  (x, y ∈ T (Γ))  It is not difficult to check that (T (Γ), ⊞′, +, ∞, 0) is a hyperfield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' We will denote it by T ′(Γ) and  call it the strict generalised tropical hyperfield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  The above examples are (up to isomorphism) all instances of a general construction which yields  a hyperring from a ring and a subgroup of its multiplicative semigroup (see Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='17 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  This construction was first described by Krasner in [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' We briefly recall it in the following example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let A be a commutative ring and T a subgroup of the commutative semigroup  (A, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let AT denote the set of all multiplicative cosets xT for x ∈ A, in particular 0T = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Krasner showed that by setting  xT + yT := {(x + yt)T | t ∈ T }  and  xT · yT := xyT  NOTES ON VALUATION THEORY FOR KRASNER HYPERFIELDS  7  we have that (AT , +, ·, 0T ) is a hyperring where −(xT ) = (−x)T for all x ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' In addition, if A  is a field, then this construction always yields a hyperfield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' This is called the factor (or quotient)  hyperring (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' hyperfield) of A modulo T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  For the next proposition we employ the notion of isomorphism of hyperfields (see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='33  below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' The following assertions hold:  (i) For any field K with more than two elements, we have that K is isomorphic to KK×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  (ii) For any ordered field K with positive cone P, we have that S is isomorphic to KP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  (iii) If p > 3 is a prime such that p ≡ 3 mod 4, then W is isomorphic to (Fp)(F×  p )2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  (iv) Let Γ be an ordered abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' For any field k with more than two elements, let K :=  k((tΓ)) denote the Hahn series field and let v be its canonical t-adic valuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Then KO×  v  is isomorphic to T (Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  (v) Let Γ be an ordered abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let K := F2((tΓ)) denote the Hahn series field and let  v be its canonical t-adic valuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Then KO×  v is isomorphic to T ′(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let K be a field with more than two elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Then KK× contains precisely two elements,  the coset 0K× and the coset 1K×, the latter containing all non-zero elements of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' If x ∈ K×, then  −x ∈ K× as well and thus 0K× belongs to 1K× + 1K×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Since by assumption there are x, y ∈ K×  with x ̸= y, also (x − y)K× ̸= 0K× is an element of 1K× + 1K×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' This shows (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Let K be an ordered field with positive cone P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Here, this means that P is a subset of K× such  that P + P, P · P ⊆ P and K× is the disjoint union of P and −P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' It follows that KP contains  precisely three1 elements: 0P, 1P (containing all the elements of P) and −1P (containing all the  elements of −P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Since P + P ⊆ P, we have that 1P + 1P only contains 1P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' By definition, if  x ∈ P, then −x ∈ −P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Thus, 1P − 1P contains 0P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' It does also contain 1P and −1P since e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  1 + 1 ∈ P (because 1 ∈ P) and (1 + 1) − 1 = 1 ∈ P while 1 − (1 + 1) = −1 ∈ −P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' This shows (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Let K be Fp (the finite field with p elements) for some prime number p > 3 such that p ≡ 3  mod 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' The prime number p is certainly odd and thus the cardinality of (K×)2 is p−1  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' If −1 is  a square in K, then K× contains an element of order 4 but this is excluded by the assumption  p ≡ 3 mod 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' We have that x ∈ K× is a square if and only if −x is not a square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' It follows that  K(K×)2 contains precisely three elements: 0(K×)2, 1(K×)2 (containing all the non-zero squares)  and −1(K×)2 (containing all the non-squares).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Moreover, 1(K×)2 − 1(K×)2 contains 0(K×)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  This shows that −1(K×)2 is the hyperinverse of 1(K×)2 in K(K×)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Since 1 + 1 ̸= 0 in K and  1 = (1 + 1) − 1, we have that either 1(K×)2 or −1(K×)2 (depending on which among 1 + 1 and  −(1 + 1) is a square) belongs to 1(K×)2 − 1(K×)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' By the symmetry of 1(K×)2 − 1(K×)2 under  the application of − (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=', multiplication by −1(K×)2), we obtain that both 1(K×)2 and −1(K×)2  must be elements of 1(K×)2 −1(K×)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' In a finite field, any non-square is the sum of two (non-zero)  squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Thus, −1(K×)2 is an element of 1(K×)2 + 1(K×)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' In order to show that 1(K×)2 is an  element of 1(K×)2+1(K×)2 as well, we distinguish two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' If 1+1 is a square in K, then 1(K×)2  is in 1(K×)2 + 1(K×)2 trivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Otherwise, 1 + 1 is not a square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' In this case, either 1 + 1 + 1 is a  square, in which case 1 = (1+1+1)−(1+1) shows that 1(K×)2 is an element of 1(K×)2 +1(K×)2,  or −(1 + 1 + 1) = −((1 + 1) + 1)) is a square, in which case −(1 + 1) = −(1 + 1 + 1) + 1 implies  that 1(K×)2 is an element of 1(K×)2 + 1(K×)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' This proves (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  1In the literature, sometimes positive cones are defined as “non-negative cones”, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=', containing 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Notice that for  our statement to hold it is necessary to exclude 0 from positive cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  8  LINZI, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Let k be a field with more thatn 2 elements and consider the Hahn series field K = k((tΓ))  equipped with the t-adic valuation v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' A non-zero element x of K× is represented as a formal series  x =  �  γ∈Γ  aγtγ  with well-ordered support, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=', {γ ∈ Γ | aγ ̸= 0} is a well-ordered set with respect to the order of  Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' The t-adic value of x under v is defined to be the minimum of the support of x (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' [10, Exercise  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  The value group of v is thus Γ and it follows from general valuation theory that K×/O×  v is  isomorphic to Γ as an ordered abelian group (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' [10, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='1] or Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='20 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  For  x ∈ K×, we have that xO×  v contains all the elements of K with the same value of x under v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' If  x, y ∈ K are such that vx ̸= vy (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=', xO×  v ̸= yO×  v ), then  v(x + yt) = v(x + y) = min{vx, vy}  for any t ∈ O×  v (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=', any t with vt = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' This follows from [10, Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='3 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='4)] (or Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='5  (iv) below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' We now note that if x ∈ K has any positive value under v, then v(x − 1) = 0 and thus  vx = v(1 + (x − 1)) is the value of some element of K generating a coset in KO×  v which belongs to  1O×  v +1O×  v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Moreover, by the assumption on the cardinality of k, there exists a ∈ k \\{1}, so by the  definition of the t-adic valuation, we have that va = 0 and v(1 − a) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Since v(1) = v(−1) = 0,  we conclude that the image of 1O×  v − aO×  v = 1O×  v + 1O×  v under v is [0, ∞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' By distributivity,  this suffices to show that T (Γ) ≃ KO×  v (see also Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='3 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' The proof of (iv) is now also  complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  The proof for (iv) can easily be adapted to prove (v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  □  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let us mention at this point that not all hyperfields are factor hyperfields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' This fact  was proved by Massouros in [39] who then further improved his results in [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' According to [21,  Theorem 6] finite hyperfields such that 1 /∈ 1 + 1 and with the property that 0 does not belong to  any finite sum of 1 with itself are also not quotient hyperfields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Subhyperrings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' To choose a good notion of subhyperring is not as straightforward as it might  seem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Two options are presented in the next definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let (R, +, ·, 0) be a hyperring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' A subset S of R is a relational subhyperring of R  if it is multiplicatively closed and with the induced multivalued operation:  x +S y := (x + y) ∩ S  (x, y ∈ S)  we have that (S, +S, ·, 0) is a hyperring as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  A subset S of R is a (traditional) subhyperring of R if 0 ∈ S and for all x, y ∈ S we have that  x − y ⊆ S and xy ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  If R is a hyperring with unity 1, then we say that a relational subhyperring S of R is a relational  subhyperfield of R if 1 ∈ S and (S, +S, ·, 0, 1) is a hyperfield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' In addition, a subhyperring S of R is  called a (traditional) subhyperfield if 1 ∈ S and (S \\ {0}, ·, 1) is an abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' The adjective “relational” has been chosen for the following reason.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' A multivalued  operation can be encoded in a first-order language via the ternary relation z ∈ x + y (as noticed  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' in [32]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' If we consider a first-order language L with a constant symbol for 0, a binary function  symbol for · and a ternary relation symbol for +, then a hyperring naturally becomes a structure on  L, which is a model of all the axioms of Definitions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='7 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='3 (written as sentences in L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Under  this interpretation, the relational subhyperrings of a hyperring R are precisely the submodels of R,  NOTES ON VALUATION THEORY FOR KRASNER HYPERFIELDS  9  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=', the substructures of R (see [43, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='3]) which satisfy the axioms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' For more details on the  model theoretical point of view let us refer the reader to [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Our choices regarding terminology are also motivated by historical reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' In fact, subhyper-  groups (and consequently subhyperfields) have been defined long time ago (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Definition 2 and  the subsequent remark in [22]2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' It is clear that a subhyperring S of a hyperring (R, +, ·, 0) is a relational subhyperring  of R and that +S = + in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  On the other hand, not all relational subhyperrings are  subhyperrings, as the following example shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Consider an ordered abelian group (Γ, +, 0, <).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let R := T (Γ) be the hyperfield  obtained as in Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='14 and consider the subset S := {∞, 0} of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' It is straightforward to check  that S is a relational subhyperfield of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' However, S is not a subhyperfield of R since 0⊞0 = [0, ∞]  is not a subset of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let Γ be an ordered abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' It is not difficult to see that the only subhyperfield  of T (Γ) is T (Γ) itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' On the other hand, if ∆ is a subgroup of Γ, then T (∆) is a relational  subhyperfield of T (Γ) (Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='22 above corresponds to the case ∆ = {0}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Homomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Next, let us consider a notion of homomorphism for hyperrings and hy-  perfields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let (R, +R, ·R, 0R) and (S, +S, ·S, 0S) be hyperrings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' We say that a map σ :  R → S is a homomorphism of hyperrings if  (HH1) σ(0R) = 0S,  (HH2) σ(x ·R y) = σ(x) ·S σ(y), for all x, y ∈ R,  (HH3) σ(x +R y) ⊆ σ(x) +S σ(y), for all x, y ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  If (R, +R, ·R, 0R, 1R) and (S, +S, ·S, 0S, 1S) are hyperfields, then we say that a homomorphism of  hyperrings σ : R → S is a homomorphism of hyperfields when the following properties  (HH4) σ(1R) = 1S,  (HH5) σ(x−1) = σ(x)−1, for all x ∈ R \\ {0},  hold as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let R be a hyperring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' The map defined by σ(0) := 0 and σ(x) := 1 for x ∈ R\\{0}  is a homomorphism of hyperrings R → K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let (Γ, <, +, 0Γ) be an ordered abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' The map  ι : K → T (Γ)  0K �→ ∞  1K �→ 0Γ  is a homomorphism of hyperfields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let K be a field and T a subgroup of K×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' The function K → KT mapping x ∈ K  to [x]T ∈ KT is a homomorphism of hyperfields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let σ : R → S be a homomorphism of hyperrings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Then the kernel of σ  ker σ := {x ∈ R | σ(x) = 0S}  2This article is included in Krasner’s collected works [4, pages 280–406]  10  LINZI, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  is a subhyperring of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Indeed, for x, y ∈ ker σ, we have that  σ(x − y) ⊆ σ(x) − σ(y) = 0S − 0S = {0S}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Thus, σ(z) = 0S for all z ∈ x − y, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=', x − y ⊆ ker σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let (R, +, ·, 0, 1) be a hyperring and S ⊆ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Consider the inclusion map ι : S ֒→ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  It is straightforward to verify that if S is a relational subhyperring of R, then ι is a homomoprhism  of hyperrings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  On the other hand, if (R, +R, ·R, 0R) and (S, +S, ·S, 0S) are hyperrings with S ⊆ R and the  inclusion ι : S ֒→ R is a homomorphism of hyperrings, then one cannot conclude that S is a  relational subhyperring of R, as the following example shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' The identity S → W is a homomorphism of hyperrings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' However, S is not a  subhyperring of W as the hypersum of 1 with itself in S is {1}, while the hypersum of 1 with itself  in W intersected with S is {−1, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  The above example motivates the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let (R, +R, ·R, 0R) and (S, +S, ·S, 0S) be hyperrings (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' hyperfields with uni-  ties 1R and 1S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' An injective homomormphism of hyperrings (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' hyperfields) σ : R → S is called  an embedding of hyperrings (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' hyperfields) if  (EM1) σ(x +R y) = (σ(x) +S σ(y)) ∩ Im σ, for all x, y ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  We leave the straightforward proof of following lemma to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' If S is a relational subhyperring of a hyperring R, then the inclusion map S ֒→ R is  an embedding of hyperrings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Conversely, If R and S are hyperrings with S ⊆ R and the inclusion  map S ֒→ R is an embedding of hyperrings, then S is a relational subhyperring of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' A homomorphism of hyperrings (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' hyperfields) σ : R → S is called an  isomorphism of hyperrings (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' hyperfields) if it is a bijective map and σ−1 : S → R is also a  homomorphism of hyperrings (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' hyperfields).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' We say that two hyperrings (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' hyperfields) R  and S are isomorphic and we write R ≃ S if there exists an isomorphism σ : R → S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let R and S be hyperrings (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' hyperfields) and denote by +R and +S their  hyperoperations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' A map σ : R → S is an isomorphism of hyperrings (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' hyperfields)  if and only if σ is a surjective embedding of hyperrings (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' hyperfields).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' If σ is an isomorphism, then it is bijective by definition, in particular Im σ = S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Since σ is  a homomorphism, we have that σ(x +R y) ⊆ σ(x) +S σ(y) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' On the other hand, since σ−1  satisfies (HH3) as well, we obtain that  σ−1(σ(x) +S σ(y)) ⊆ x +R y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Applying σ to both sides then yields σ(x +R y) ⊇ σ(x) +S σ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Assume now that σ is a surjective embedding and let us show that σ−1 is a homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Clearly, (HH1) and (HH2) hold for σ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Since by (EM1) and the surjectivity we have that  σ(x +R y) = σ(x) +S σ(y), property (HH3) follows applying σ−1 to both sides and using again  the surjectivity of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  □  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Note that a bijective homomorphism is not necessarily an isomorphism as it can be  deduced from Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  NOTES ON VALUATION THEORY FOR KRASNER HYPERFIELDS  11  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let (Γ, <, +, 0Γ) be an ordered abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Consider the relational subhyper-  field S := {∞, 0} of T (Γ) as in Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Then (S, +S, ·, ∞, 0Γ) is isomorphic to K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  As usual we identify isomorphic structures: Examples 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='22 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='36 can be expressed by saying  that K is a subhyperfield of T (Γ) for any ordered abelian group Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Hyperideals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' We now briefly discuss how the classical ring theory notion of ideal generalises  in the multivalued setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='37 (Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='1 in [5] and Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='9 in [31]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let R be a hyperring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' A subhy-  perring I of R is a hyperideal of R if it satisfies  (HID1) xy ∈ I, for all x ∈ R and y ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let R be a hyperring with no zero divisors, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=', xy = 0R implies x = 0R or y = 0R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  A relational subhyperring I of R satisfying (HID1) is a hyperideal of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Pick x, y ∈ I and let z ∈ x − y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' It suffices to prove that z ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' If x = 0R or y = 0R,  then there is nothing to show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Otherwise, by distributivity in R we have that zy ∈ xy − y2 and by  (HID1) we obtain that zy ∈ I, since y ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Therefore, zy ∈ (xy − y2) ∩ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Now, distributivity in I  (with respect to the induced multivalued operation +I) yields zy ∈  �  (x − y) ∩ I  �  y, so there exists  z′ ∈ (x − y) ∩ I such that zy = z′y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Hence, 0R ∈ zy − z′y = (z − z′)y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Since R has no zero divisors,  this implies that 0R ∈ z − z′ and hence z = z′ ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  □  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' We wish to justify the choice of requiring hyperideals to be traditional subhyperrings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  In [19, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='1], Jun provides a generalisation of the classical quotient construction of a ring  modulo an ideal in the multivalued setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' One property that is certainly desirable to preserve is for  the canonical epimorphism from a hyperring R to the quotient hyperring R/I modulo a hyperideal  I to be a homomorphism of hyperrings R → R/I having I as its kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' We have no examples  of hyperrings with a relational subhyperring satisfying (HID1), but at the same time we did not  succeed in proving Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='38 without the assumption on zero divisors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' If such an example would  exist, then that relational subhyperring could not be the kernel of a homomorphism of hyperrings  by Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='28 and so we should not call it a hyperideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let σ : R → S be a homomorphism of hyperrings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Then it is easy to show that ker σ  is a hyperideal of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Conversely, for all hyperideals I of a hyperring R the map σ : R → K defined  as  σ(x) :=  �  0  if x ∈ I,  1  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  is a homomorphism of hyperrings such that ker σ = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  The following statements can be shown to hold as in the classical theory of rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='41 (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='12 in [31]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' If a hyperideal I of a hyperring R contains a unit, then I = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='42 (Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='13 in [31]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' The only hyperideals of a hyperfield F are {0} and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='43 (Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='14 (ii) in [31]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let I be a hyperideal of a hyperring R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Then I is  called maximal if I ⊊ R and for all hyperideals J of R we have that I ⊊ J implies J = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  We recall without proof the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='44 (Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='16 (ii) in [31]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Assume that R is a hyperring with unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let I  be a hyperideal R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Then I is maximal if and only if R/I is a hyperfield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  12  LINZI, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='45 ([42]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let R be a hyperring with unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' An element s of R is called a scalar if  x + s is a singleton for all x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' As in the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='9, it is not difficult to show that  s ∈ R is a scalar if and only if s − s = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Interestingly, the set of all scalar elements of R forms a  hyperideal of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Assume now that R is a hyperring with unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' If 1 is a scalar, then by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='41 the hyperideal  of scalars is R and thus R is a ring (as all of its elements are scalars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Moreover, we deduce from  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='42 that if F is a hyperfield, but not a field, then the only scalar of F is 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=', x − x is  not a singleton for all x ∈ F ×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Valued hyperfields  Some of the results presented in this section can be found in [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Some proofs have been  improved, others we will repeat for the convenience of the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Valuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' The next definition is a straightforward generalisation of the definition of valua-  tion for fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='1 (Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='1 in [31]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Take a hyperfield F and an ordered abelian group Γ (written  additively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' A surjective map v : F → Γ ∪ {∞} is called a valuation on F if it has the following  properties:  (V1) vx = ∞ ⇐⇒ x = 0, for all x ∈ F,  (V2) v(xy) = vx + vy, for all x, y ∈ F,  (V3) z ∈ x + y =⇒ vz ≥ min{vx, vy}, for all x, y, z ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  If v is a valuation on a hyperfield F we call (F, v) a valued hyperfield and we denote by vF the  ordered abelian group v(F ×), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=', the value group of (F, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let F be a hyperfield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' The trivial valuation vx = 0 for all x ∈ F × is always a  valuation on F with value group {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  The next result provides further examples of valued hyperfields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='3 (Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='2 in [31]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let (K, v) be a valued field and take a subgroup T of K×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Denote by O×  v the group of units of the valuation ring of K, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=', O×  v := {x ∈ K | vx = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' If  T ⊆ O×  v , then  vT : KT → vK ∪ {∞}  xT �→ vx  is a valuation on KT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' The assumption T ⊆ O×  v guarantees that the map vT is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Moreover, surjectivity,  (V1) and (V2) follow immediately from the corresponding properties of v and the definition of the  factor hyperfield KT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' It remains to verify (V3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Take x, y ∈ K and t ∈ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' We obtain that  v(x + yt) ≥ min{vx, v(yt)} = min{vx, vy + vt} = min{vx, vy}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  From this it easily follows that (V3) holds for (KT , vT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  □  The next lemma gives an alternative definition of valuation on (hyper)fields (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' [2, Example  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let F be a hyperfield and Γ an ordered abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Then (F, v) is a valued hyperfield  with vF = Γ if and only if the map v : F → T (Γ) is a surjective homomorphism of hyperfields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  NOTES ON VALUATION THEORY FOR KRASNER HYPERFIELDS  13  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Assume that (F, v) is a valued hyperfield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Then v : F → T (vF) is a surjective map, (HH1)  follows from (V1), (HH2) follows from (V2), (HH3) follows from (V3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Property (HH4) follows  because vx = vx + v(1) for all x ∈ F and (HH5) follows since, for x ∈ F ×, we have that  0 = v(1) = v(xx−1) = vx + v(x−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Thus, v is a homomorphism of hyperfields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Conversely, let F be a hyperfield and v : F → T (Γ) a surjective homomoprhism of hyperfields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Property (V2) for v follows from (HH2) and (V3) follows from (HH3) together with the definition  of the hyperoperation of T (Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Property (V1) follows from Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='42 since v(1) = 0 ̸= ∞ and  thus ker v = {x ∈ F | vx = ∞} is a proper hyperideal of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' We proved that v is a valuation on F  with vF = Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  □  On the basis of the previous lemma, the following result is almost immediate to verify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Thus,  we omit its proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='5 (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='5 in [31]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let v : F → Γ ∪ {∞} be a valuation on a hyperfield F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Then:  (i) v(1) = v(−1) = 0,  (ii) v(−x) = vx for all x ∈ F,  (iii) vx−1 = −vx for all x ∈ F ×,  (iv) if vx ̸= vy, then vz = min{vx, vy}, for every x, y ∈ F and z ∈ x + y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let (KT , w) be a valued hyperfield which is a factor hyperfield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Then there exists a  valuation v on K such that T ⊆ O×  v and w = vT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let Γ be the value group of (KT , w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Since K → KT , x �→ [x]T and w : KT → T (Γ) are  surjective homomorphisms of hyperfields, their composition v : K → T (Γ) is a valuation on K  satisfying the conditions of the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  □  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='17 (iv) we have that T (Γ) is isomorphic to all factor hyperfields of  the form k((tΓ))O×  vt for some field k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' By the above lemmas, the isomorphism σ : k((tΓ))O×  vt → T (Γ)  is a valuation on k((tΓ))O×  vt and the identity map id = σ ◦ σ−1 : T (Γ) → T (Γ) is a valuation on  T (Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Valuation hyperrings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' The next definition is inspired by classical valuation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='8 (Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='6 in [31]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let F be a hyperfield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' A relational subhyperring O of F  is called a valuation hyperring in F if for all x ∈ F × we have that either x ∈ O or x−1 ∈ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Observe that, by definition, it follows that 1 ∈ O for any valuation hyperring O in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let us  now prove some more basic properties of valuation hyperrings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='9 (Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='7 in [31]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' A valuation hyperring O in a hyperfield F is a subhyperring  of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' It suffices to show that a − b ⊆ O for all a, b ∈ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Take a, b ∈ O and x ∈ a − b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' If x ∈ O,  then there is nothing to show (note that this case also includes x = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Otherwise, we have that  x−1 ∈ O and thus ax−1, bx−1 ∈ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Since x ∈ a − b we obtain from (CH4) that a ∈ x + b, so, using  axiom (HR3),  ax−1 ∈ (x + b)x−1 = 1 + bx−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  14  LINZI, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  We have obtained that ax−1 ∈ (1 + bx−1) ∩ O = 1 +O bx−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' By (CH4) and (HR3) applied to the  hyperring (O, +O, ·, 0), it follows that  xx−1 = 1 ∈ ax−1 +O (−bx−1) = (a +O (−b))x−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Therefore, x ∈ a +O (−b) ⊆ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' This shows that a − b ⊆ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  □  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='10 (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='8 in [31]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let O be a valuation hyperring in a hyperfield F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Then  M := O \\ O× is the unique maximal hyperideal of O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Take a ∈ M and c ∈ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' If ca is invertible in O, then there exists x ∈ O such that x(ca) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Hence (xc)a = 1 and a−1 = xc ∈ O contradicting a ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' This proves that ca ∈ M and shows that  M satisfies (HID1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Take a, b ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' We may assume that ab−1 ∈ O (otherwise ba−1 ∈ O and we can interchange the  roles of a and b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Since O is a subhyperring of F (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='9), we obtain that 1 − ab−1 ⊆ O  and therefore, using what we have just proved, we conclude that  b − a = b(1 − ab−1) ⊆ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  We have shown that M is a hyperideal of O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Since, by the definition of M, we have that O\\M = O×, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='41, every proper hyperideal  of O must be contained in M, showing that M is the unique maximal hyperideal of O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Residue hyperfield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Any valuation on a hyperfield F induces a valuation hyperring in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='11 (Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='11 in [31]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let v : F → Γ ∪ {∞} be a valuation on a hyperfield  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Then  Ov := {x ∈ F | vx ≥ 0}  is a valuation hyperring in F and  Mv := {x ∈ F | vx > 0}  is its unique maximal hyperideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' We first prove that Ov is a subhyperring of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Take a, b ∈ Ov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' By (V3), for all c ∈ a − b we  have vc ≥ min{va, v(−b)} = min{va, vb} ≥ 0, so a − b ⊆ Ov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Further, we have ab ∈ Ov by (V2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  By Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='5 (iii) we conclude that if x /∈ Ov, then x−1 ∈ Ov so Ov is a valuation hyperring in  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Next we show that Mv is the unique maximal hyperideal of Ov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Observe that, by virtue of  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='5 (iii),  O×  v = {x ∈ Ov | vx = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Hence, Mv = Ov \\ O×  v and then Mv is the unique maximal hyperideal of Ov by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  □  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let (F, v) be a valued hyperfield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Note that Ov can be described in terms of the  hyperoperation ⊞ of T (vF) as the preimage in F of v(1) ⊞ v(1) under v:  Ov = v−1(v(1) ⊞ v(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  It follows from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='44 that, for a valuation hyperring O with maximal hyperideal M,  the quotient hyperring  O/M = {x + M | x ∈ O},  with the multivalued operation ⊕ defined as  (x + M) ⊕ (y + M) := {z + M | z ∈ x + y},  NOTES ON VALUATION THEORY FOR KRASNER HYPERFIELDS  15  is a hyperfield (see [19, Section 3] and Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='39 above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' If (F, v) is a valued hyperfield, then we call the hyperfield Ov/Mv the residue  hyperfield of (F, v) and we denote it by Fv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' For an element x ∈ Ov, we denote by xv its natural  image in Fv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let (K, v) be a valued field and T ⊆ O×  v a subgroup of K×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Then KT vT ≃  (Kv)(T v), where T v := {tv | t ∈ T }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' By definition vT [x]T = vx for all x ∈ K, thus OvT = (Ov)T and MvT = (Mv)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' It follows  that  KT vT = OvT /MvT = (Ov)T /(Mv)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Let us define  σ : (Ov)T /(Mv)T → (Kv)(T v)  [x]T + (Mv)T �→ [xv]T v  and show that σ is an isomorphism of hyperfields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' By [19, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='3] we have that [x]T +(Mv)T =  [y]T + (Mv)T if and only if v(x − yt) > 0 for some t ∈ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' This implies that  0 = (x − yt)v = xv − yv · tv ∈ [xv]T v − [yv]T v ,  where we have used the assumption T ⊆ O×  v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' This shows that σ is well-defined and injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' The  surjectivity of σ is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Moreover, σ is easily seen to be a homomorphism of the corresponding  multiplicative groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Finally, we observe that  σ  �  [x]T + (Mv)T ⊕ ([y]T + (Mv)T )  �  = {σ  �  [z]T + (Mv)T  �  | [z]T ∈ [x]T + [y]T }  = {[zv]T v | z = x + yt for some t ∈ T }  = {[xv + yv · tv]T v | t ∈ T }  = [xv]T v + [yv]T v  hence σ is an isomorphism of hyperfields by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  □  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='17, T (Γ) ≃ KO×  v , where K = k((tΓ)) for some field k with more  than two elements and v is its canonical t-adic valuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Since Kv = k and O×  v v = k×, by the  above proposition and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='17 (i) we have that the residue field of T (Γ) with respect to  its valuation given by the identity map is isomorphic to kk× ≃ K which, moreover, is a relational  subhyperfield of T (Γ) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Examples 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='22 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  For a valued field (K, v) we denote by 1 + Mv the set of 1-units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' That is, those x ∈ K such that  v(x − 1) > 0 or equivalently, xv = 1v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let (K, v) be a valued field and T ⊆ O×  v a subgroup of K×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Then the map  ι : KT vT → KT  [xv]T v �→ [x]T  is an embedding of hyperfields if and only if 1 + Mv ⊆ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' If 1 + Mv ⊆ T and x, y ∈ O×  v , then [xv]T v = [yv]T v if and only if xv = yv · tv = (yt)v for  some t ∈ T if and only if v(1 − ytx−1) = v(x − yt) > 0 if and only if ytx−1 ∈ 1 + Mv ⊆ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' It  follows that [y]T [x]−1  T  = [ytx−1]T = [1]T and ι is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' On the other hand if [x]T = [y]T for  16  LINZI, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  some x, y ∈ O×  v , then for some t ∈ T we have that x = yt and thus xv = (yt)v = yv · tv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' It follows  that ι is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Moreover, we have that  ι  �  [xv]T v · [yv]−1  T v  �  = ι  �  [xv · y−1v]T v  �  = [xy−1]T = [x]T [y]−1  T  and hence ι satisfies (HH2) and (HH5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' It clearly satisfies (HH1) and (HH4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' It remains to show  that (EM1) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' First observe that Im ι consists of classes [x]T where x = 0 or x ∈ R ⊆ O×  v ,  where R is a set of some selected representatives for Kv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Take again x, y ∈ O×  v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' We have that  ι ([xv]T v + [yv]T v) = {[x + yt]T | x, y ∈ R, t ∈ T ∩ R} = ([x]T + [y]T ) ∩ Im ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  This completes the proof of one implication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  If a ∈ 1 + Mv is not in T , then av = 1v and thus [av]T v = [1v]T v holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' On the other hand, we  have that  ι[av] = [a]T ̸= [1]T = ι[1v]T  and thus ι is not well-defined and in particular not an embedding of hyperfields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Equivalence of valuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' In analogy with classical valuation theory, our aim is now to  show that valuation hyperrings can be used to describe valuations up to composition with an order  preserving isomorphism of the value group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='17 (Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='12 in [31]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let F be a hyperfield and O a valuation hyperring  in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Consider the multiplicative group Γ := F ×/O× and define a relation ≤ on Γ as follows:  aO× ≤ bO× ⇐⇒ ba−1 ∈ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Then (Γ, ·, ≤) is an ordered abelian group and the canonical projection  π : F → Γ ∪ {∞},  extended so that π(0F ) = ∞, is a valuation on F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Furthermore, Oπ = O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' First we show that ≤ is an ordering for (Γ, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Since aa−1 = 1F ∈ O, reflexivity is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' If  ab−1, ba−1 ∈ O, then ab−1 ∈ O× so aO× = bO×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Hence ≤ is antisymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' If ab−1, bc−1 ∈ O,  then ac−1 = ab−1bc−1 ∈ O, showing that ≤ is transitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Take now a, b ∈ F × such that aO× ≤ bO×  and c ∈ F ×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' We have that bc(ac)−1 = bcc−1a−1 = ba−1 ∈ O, whence acO× ≤ bcO×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' This shows  that ≤ is compatible with the operation of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Finally, ≤ is a total order since O is a valuation  hyperring, so that ab−1 ∈ O or ba−1 ∈ O for all a, b ∈ F ×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  We now show that π is a valuation on F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Clearly, π is a surjective map, onto the ordered abelian  group Γ with ∞ and (V1) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Since π is a homomorphism of groups we obtain (V2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' It remains  to show that (V3) holds for π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Take x, y ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' If one of them is 0F , then (V3) is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  We may then assume that x, y ∈ F × and that xO× ≤ yO×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Take z ∈ x + y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' We wish to show that  zx−1 ∈ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' By assumption we have that yx−1 ∈ O, thus  zx−1 ∈ (x + y)x−1 = 1 + yx−1 ⊆ O,  where we used Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Finally, we observe that, by definition  Oπ = {x ∈ F | πx ≥ 1} = {x ∈ F | xO× ≥ 1O×} = {x ∈ F | x ∈ O} = O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  □  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let (Γi, <i) be (partially) ordered sets for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' A map σ : Γ1 → Γ2 is said  to be order preserving if γ1 ≤1 γ2 implies that σ(γ1) ≤2 σ(γ2) for all γ1, γ2 ∈ Γ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  NOTES ON VALUATION THEORY FOR KRASNER HYPERFIELDS  17  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' For i = 1, 2 let vi : F → Γi ∪ {∞} be valuations on a hyperfield F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' We say  that v1 and v2 are equivalent if there exists an isomorphism of groups σ : Γ1 → Γ2 which is order  preserving and such that v2 = σ ◦ v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let v : F → Γ ∪ {∞} be a valuation on a hyperfield F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Then Γ ≃ F ×/O×  v with an  isomorphism of groups which is order preserving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' We consider F ×/O×  v as an ordered abelian group with the ordering defined in Proposition  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Using the surjectivity of v, we define a map  σ : Γ → F ×/O×  v  by σ(va) = aO×  v for all a ∈ F ×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' This is well-defined since if va = vb, then va − vb = v(ab−1) = 0  so that ab−1 ∈ O×  v and then aO×  v = bO×  v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Using property (V2) of valuations, we obtain that σ is  a homomorphism of groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Further, if va ≤ vb, then ba−1 ∈ Ov which means that σ(va) ≤ σ(vb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Thus, σ is order preserving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' It is clear that σ is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' It therefore remains to show that σ is  injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' To this end, assume that aO×  v = bO×  v for some a, b ∈ F ×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Then there exists c ∈ O×  v such  that a = bc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Since vc = 0, by (V2) we obtain that va = vb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  □  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Observe that by construction of σ in the above proof, we have that σ ◦ v = π where  π is the canonical epimorphism F × → F ×/O×  v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' For i = 1, 2 let vi : F → Γi ∪ {∞} be valuations on a hyperfield F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Then v1 and  v2 are equivalent if and only if Ov1 = Ov2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' By the previous lemma we obtain for i = 1, 2 that Γi ≃ F ×/O×  vi as ordered abelian groups  with isomorphisms σi such that σi ◦ vi = πi where πi : F × → F ×/O×  vi is the canonical projection  for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Thus, if Ov1 = Ov2, then π1 = π2 and σ := σ−1  2  σ1 is an isomorphism of ordered  abelian groups Γ1 → Γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Further we have that  σ ◦ v1 = σ−1  2  (σ1 ◦ v1) = σ−1  2  π2 = v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Hence, v1 and v2 are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  On the other hand, if v1 and v2 are equivalent, then we obtain that F ×/O×  v1 ≃ F ×/O×  v2 as ordered  abelian groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' In particular, for a ∈ F × we have that 1O×  v1 ≤ aO×  v1 if and only if 1O×  v2 ≤ aO×  v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Using the definition of the ordering in F ×/O×  vi we see that this means that a ∈ Ov1 if and only if  a ∈ Ov2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Since 0 ∈ Ovi for i = 1, 2, we conclude that Ov1 = Ov2 as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  □  In the following sections we will always consider valuations on hyperfields up to equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Krasner valued hyperfields  In this section, we focus our attention on those valued hyperfields which satisfy the original more  restrictive axioms of Krasner and were called by him hypercorps valué.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' These valued hyperfields  still attract the attention of mathematicians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' For instance, they have recently been considered by  Tolliver and Lee (see [46, 32]) who called them simply “valued hyperfields”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Ultrametric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' We begin by presenting some basic theory of ultrametric spaces, a no-  tion as well studied by Krasner (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' [23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' The axioms for an ultrametric distance can be formulated  using just the linear order of non-negative real numbers where 0 is a bottom element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Since nothing  but the order is used from the structure of real numbers, we will use the term ultrametric space in a  broader sense allowing ultrametric distances to take values in any linearly ordered set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' In addition,  since the value group of a valuation has a top element, our definition of ultrametric space below  18  LINZI, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  is a modification which better fits into our context (the same approach can be found e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' in [28]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  In the case of real-valued distances, this modification corresponds to a replacement of the linearly  ordered set (R≥0, <) with (R ∪ {∞}, >).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' An ultrametric distance (or simply an ultrametric) on a set X is a function  d : X × X → Γ ∪ {∞}, where (Γ, <) is a linearly ordered set and ∞ satisfies γ < ∞ for all γ ∈ Γ,  such that for all x, y, z ∈ X  (U1) d(x, y) = ∞ if and only if x = y,  (U2) d(x, y) = d(y, x),  (U3) d(x, z) ≥ min{d(x, y), d(y, z)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  We call (X, d) an ultrametric space whenever d is an ultrametric on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' We call the set dX :=  {d(x, y) | x, y ∈ X, x ̸= y} ⊆ Γ the value set of d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let (K, v) be a valued field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Then the function  K × K → Γ ∪ {∞}  (x, y) �→ v(x − y)  is an ultrametric on K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let (X, d) be an ultrametric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' A subset B ⊆ X is called a ball if for all  y, z ∈ B and all x ∈ X we have that the following implication  d(x, y) ≥ d(y, z)  =⇒  x ∈ B  holds for all x, y, z ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' A subset ρ of a linearly ordered set (Γ, <) is called an initial segment (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' final  segment) if for all δ ∈ ρ and all γ ∈ Γ if γ < δ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' γ > δ), then γ ∈ ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' For every element x of an ultrametric space (X, d) and every final segment ρ of  dX ∪ {∞}, we have that  Bρ(x) := {y ∈ X | d(x, y) ∈ ρ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  is a ball in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Conversely, if B is a ball in X and ρ is the smallest (with respect to inclusion) final  segment of dX ∪ {∞} containing d(y, z) for all y, z ∈ B, then for every x ∈ B,  B = Bρ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  In particular, Bρ(x) = Bρ(y) for every y ∈ Bρ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Assume that y, z ∈ Bρ(x), that is, d(x, y) ∈ ρ and d(x, z) ∈ ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  If t ∈ X is such that  d(y, t) ≥ d(y, z), then the inequalities  d(x, t) ≥ min{d(x, y), d(y, t)} ≥ min{d(x, y), d(y, z)} ≥ min{d(x, y), d(y, x), d(x, z)} ∈ ρ  follow from (U2) and (U3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Since ρ is a final segment of dX ∪ {∞}, we conclude that d(x, t) ∈ ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Hence, t ∈ Bρ(x)and Bρ(x) is a ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  For the converse, assume that B is a ball and let ρ be as in the assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Further, let x be any  element in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' If y ∈ B, then d(x, y) ∈ ρ and thus y ∈ Bρ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' On the other hand, if y ∈ Bρ(x),  then d(x, y) ∈ ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' So by definition of ρ, there is some z ∈ B such that d(x, z) ≤ d(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Since B is  a ball, it follows that y ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' We have proved that B = Bρ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  □  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let (X, d) be an ultrametric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Every two balls with non-empty intersection  are comparable by inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  NOTES ON VALUATION THEORY FOR KRASNER HYPERFIELDS  19  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Take two balls B and B′ and suppose that z ∈ B ∩ B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='5 there are final  segments ρ, ς of dX ∪ {∞} such that B = Bρ(z) and B′ = Bς(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Since ρ and ς are final segments,  we must have ρ ⊆ ς or ς ⊆ ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Hence, B ⊆ B′ or B′ ⊆ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Krasner valuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let (Γ, <, +, 0) be an ordered abelian group, ρ be an initial segment of  Γ and γ an element of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' We will sometimes write γ > ρ to indicate that γ /∈ ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  The following class of valued hyperfields is of special interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='7 (Section 3 of [24], Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='4 in [46] and Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='4 in [32]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' We call a valued  hyperfield (F, v) a Krasner valued hyperfield if  (KVH1) For all x, y ∈ F, v(x + y) is a singleton unless 0 ∈ x + y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  (KVH2) There exists an initial segment ρv of vF such that 0 ∈ ρv and for all x, y, z, t ∈ F we have  that z ∈ x + y implies that t ∈ x + y if and only if vs > ρv + min{vx, vy} for all s ∈ z − t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  The initial segment ρv is called the norm of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' We will also say that v is a Krasner valuation on F  when (F, v) is a Krasner valued hyperfield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let (K, v) be a valued field and consider K as a hyperfield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Then v is a Krasner  valuation on K with norm vK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let (F, v) be a Krasner valued hyperfield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' For all x, y ∈ F such that x ̸= y, by  axiom (KVH1), v(x−y) contains a unique element γx,y ∈ vF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Define a map dv : F ×F → Γ∪{∞}  as  dv(x, y) :=  �  γx,y  if x ̸= y,  ∞  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Then dv is an ultrametric on F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Moreover, for all x, y ∈ F and any z ∈ x + y, in the ultrametric  space (F, dv) we have that  x + y = Bρv+min{vx,vy}(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Axiom (U1) follows from axiom (V1), axiom (U2) follows from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='5 (ii) and axiom  (U3) is a direct consequence of axiom (V3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' The last statement is just a reformulation of axiom  (KVH2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  □  We call the ultrametric dv on a Krasner valued hyperfield (F, v) the ultrametric induced by v on  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let F be a hyperfield and let v be the trivial valuation on F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' If v is a Krasner  valuation of norm ρv, then by (KVH2) for all x, y ∈ F × we have that x ∈ 1 − 1 if and only if  0 = vx > ρv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Since 0 ∈ ρv, it follows that x − x = {0} for all x ∈ F and then F is a field by Lemma  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Conversely, if K is a field, then the map v(0) := ∞ and vx := 0 for all x ∈ K× is a Krasner  valuation on K with value group vK = {0} and norm vK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Krasner’s main motivation probably came from the following example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' For a valued field (K, v) and an initial segment ρ ⊆ vK containing 0, let us  consider the (multiplicative) group of 1-units of level ρ:  1 + Mρ  v = {x ∈ K | v(x − 1) > ρ} ⊆ O×  v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Then vρ := v1+Mρ  v is a Krasner valuation on Kρ := K1+Mρ  v and the norm of vρ is ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  20  LINZI, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Actually, any Krasner valued hyperfield which is a factor hyperfield is of this form, as we show  below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let F be a factor hyperfield admitting a Krasner valuation w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Then F = Kρ  and w is vρ for some valued field (K, v) and some initial segment ρ of vK = wF containing 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='6 there is a valued field (K, v) and a subgroup T ⊆ O×  w of K× such that  vT = w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Suppose that t ∈ T is not a 1-unit in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Then vt = 0 and v(t − 1) = 0 must hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  now, on the one hand, since w = vT is a Krasner valuation, for all [x]T ∈ [1]T − [1]T we have that  vx = vT [x]T > 0 by (KVH2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' On the other hand, since t ∈ T we have that [t − 1]T ∈ [1]T − [1]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  This contradiction proves that T ⊆ 1 + Mv must hold and the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  □  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' We do not know if all Krasner valued hyperfields are factor hyperfields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Some results  connected to this problem are provided in [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Consider the Hahn series field K := F2((tΓ)) for some non-trivial ordered abelian  group Γ and let v denote its canonical t-adic valuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' In this case, since Kv ≃ F2 we have that  O×  v = 1 + Mv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' We conclude that  T ′(Γ) ≃ Kρ,  where ρ = {γ ∈ Γ | γ ≤ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' We have that the identity map vρ is the identity map on T (Γ) = T ′(Γ)  and it is a Krasner valuation on T ′(Γ) with norm ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Note that, the same map is not a Krasner  valuation on T (Γ) as 0 ∈ [0, ∞] = 0 ⊞ 0 violates (KVH2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  We now study the residue hyperfield of a Krasner valued hyperfield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' The residue hyperfield Fv of a Krasner valued hyperfield (F, v) is a field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Take xv ∈ 1v − 1v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' This means that x ∈ 1 − 1 and by axiom (KVH2), we deduce that  vx > ρv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Since 0 ∈ ρv, it follows that xv = 0v and therefore Fv is a field by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  □  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let (K, v) be a valued field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' It follows from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='14 that, for any initial  segment ρ of vK containing 0, the residue hyperfield of Kρ is a field isomorphic to Kv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Let us now consider a valued hyperfield which is not a Krasner valued hyperfield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Consider the rational function field K := Q(X) over the rationals, and its mul-  tiplicative subgroup T := Q×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Under the canonical X-adic valuation v, we have that T ⊆ O×  v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Hence, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='3 yields a valued hyperfield (KT , vT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='14 and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='17  (i), we have that  KT vT ≃ QQ× ≃ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Since K is not a field, the Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='15 implies that (KT , vT ) is not a Krasner valued hyperfield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Mittas and superiorly canonical hypergroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Mittas was a student of Krasner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' He  introduced superiorly canonical hypergroups (defined below) and published (sometimes without  proofs) some results which connect them to Krasner valuations (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' [41]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Some of the contents  of this subsection were inspired by his work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='18 ([41] and page 81 of [38]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' A canonical hypergroup H is called superiorly canonical  if  (SCH1) For all x, y ∈ H, if x ∈ x + y, then x + y = {x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  (SCH2) For all x, y, z, t ∈ H, if (x + y) ∩ (z + t) ̸= ∅, then x + y ⊆ z + t or z + t ⊆ x + y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  (SCH3) For all x, y ∈ H such that x ̸= y we have that z, t ∈ x − y implies z − z = t − t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  NOTES ON VALUATION THEORY FOR KRASNER HYPERFIELDS  21  (SCH4) For all x, y, z ∈ H, if x ∈ z − z and y /∈ z − z, then x − x ⊆ y − y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Any abelian group is a superiorly canonical hypergroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Other examples of superiorly canonical hypergroups are provided by the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let (F, v) be a Krasner valued hyperfield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Then the additive canonical hyper-  group of F is superiorly canonical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' We verify the axioms one by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Assume that x ∈ x+ y for some x, y ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' By reversibility,  y ∈ x − x, so the inequalities vy > ρv + vx ≥ vx follow from axiom (KVH2) and 0 ∈ ρv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Therefore,  if z ∈ x + y, then vx = vz by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='5 (iv) and since by reversibility y ∈ z − x and dv(y, 0) =  vy > ρv + min{vx, vz}, axiom (KVH2) implies that 0 ∈ z − x and so x = z must hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' This shows  (SCH1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Axiom (SCH2) follows from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='9 and Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  For (SCH3), take x, y ∈ F and assume that x ̸= y, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=', 0 /∈ x − y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Take z, t ∈ x − y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' We  claim that vz = vt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Suppose that vz < vt, then by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='5 (iv) we have that va = vz for all  a ∈ z − t, thus  dv(z, 0) = vz = dv(z, t) > ρv + min{vx, vy}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Axiom (KVH2) now implies that 0 ∈ x − y, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Therefore, vz ≥ vt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Symmetrically,  vt ≥ vz and so vz = vt as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Now, by (KVH2) we have that a ∈ z − z if and only if  va > ρv + vz and a ∈ t − t if and only if va > ρv + vt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Since vz = vt we conclude that z − z = t − t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  This shows that (SCH3) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  For (SCH4), take x ∈ z − z and y /∈ z − z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' By axiom (KVH2) it follows that vx > ρv + vz and  vy ∈ ρv +vz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' In particular, vx > vy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Now, again by axiom (KVH2), if a ∈ x−x, then va > ρv +vx  which implies va > ρv + vy and so a ∈ y − y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Hence, x − x ⊆ y − y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  □  The theory of superiorly canonical hypergroups and the theory of Krasner valued hyperfields are  even more deeply related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let F be a hyperfield with a superiorly canonical additive hypergroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Then  O := {x ∈ F | x − x ⊆ 1 − 1}  is a valuation hyperring in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Moreover, if v is the valuation such that Ov = O, then (F, v) is a  Krasner valued hyperfield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' If F is a field, then O = F and thus v is the trivial valuation on F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Since F is a field, (F, v)  is a Krasner valued hyperfield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' We can then assume that F is not a field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  If x − x ⊆ 1 − 1 and y − y ⊆ 1 − 1, then  xy − xy = (x − x)y ⊆ (1 − 1)y = y − y ⊆ 1 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Hence, O is multiplicatively closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  If x − x ⊈ 1 − 1, then 1 − 1 ⊊ x − x by axiom (SCH2), since 0 ∈ (x − x) ∩ (1 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Multiplying  by x−1 we obtain that x−1 − x−1 ⊊ 1 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Therefore, x−1 ∈ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Assume that x − x ⊆ 1 − 1 and y − y ⊆ 1 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' We claim that if z ∈ x + y, then z − z ⊆ 1 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Pick a ∈ z − z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Since z ∈ x + y we have that  a ∈ z − z ⊆ (x − x) + (y − y) ⊆ (1 − 1) + (1 − 1),  so there exists x′ ∈ 1 − 1 and y′ ∈ 1 − 1 such that a ∈ x′ + y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Hence, 1 ∈ x′ + 1 by reversibility, and  a ∈ x′ + y′ ⊆ (x′ + 1) − 1 = 1 − 1,  22  LINZI, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  where we have used axiom (SCH1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' We proved that O is a valuation hyperring in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Clearly, O× = {x ∈ F | x−x = 1−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Set vF := F ×/O× and let v : F × → vF be the canonical  epimorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' We have to show that (F, v) is a Krasner valued hyperfield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' It is a valued hyperfield  by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let us now verify the validity of axioms (KVH1) and (KVH2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Take x, y ∈ F × and assume that 0 /∈ x + y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Pick z, t ∈ x + y and suppose that vz < vt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' This  means that tz−1 ∈ O \\ O×, so  tz−1 − tz−1 ⊊ 1 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  But then t − t ⊊ z − z which contradicts axiom (SCH3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' We have proved that (KVH1) holds for  (F, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  We now have to verify (KVH2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Our first claim is that v(1 − 1) is a final segment of vF which  does not contain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Pick a ∈ 1 − 1 and γ > va.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let b ∈ F × be such that vb = γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Since vb > va, we  have that b − b ⊊ a − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Now, if b /∈ 1 − 1, then axiom (SCH4) implies a − a ⊆ b − b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Therefore,  b ∈ 1 − 1 must hold and thus γ = vb ∈ v(1 − 1) and v(1 − 1) is a final segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  For x ∈ F ×, if x ∈ x − x, then x − x = {x} by axiom (SCH1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' However, since 0 ∈ x − x, this  cannot be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' An element x ∈ F × has value 0 if and only if x − x = 1 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Hence it cannot belong to  1 − 1 as x /∈ x − x for all x ∈ F ×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' This shows that v(1 − 1) does not contain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  We let ρv be the complement in vF of v(1 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Then ρv is an initial segment of vF which  contains 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Observe that for all x ∈ F we have that a ∈ x − x = (1 − 1)x if and only if there exists y ∈ 1 − 1  such that a = yx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' This implies that va = vy + vx > ρv + vx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Conversely, if va > ρv + vx, then  v(ax−1) ∈ v(1 − 1), so there exists b ∈ 1 − 1 such that vb = v(ax−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Therefore,  b ∈ 1 − 1 = bxa−1 − bxa−1 = b(xa−1 − xa−1),  so 1 ∈ xa−1 − xa−1 implying that a ∈ x − x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Take now x, y ∈ F such that x ̸= y and vx ≤ vy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Fix z ∈ x − y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' We have to show that t ∈ x − y  if and only if dv(z, t) > ρv + vx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Assume first that t ∈ x − y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' If t = z there is nothing to show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Otherwise, it suffices to show that  z − t ⊆ x − x by what we have already shown above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Take a ∈ z − t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Since z ∈ x − y, we have that  a ∈ z − t ⊆ x − (y + t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Hence, there exists b ∈ y+t such that a ∈ x−b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' We obtain that b ∈ (x−a)∩(y+t), so y+t ⊆ x−a  or x − a ⊆ y + t by axiom (SCH2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' In the first case, since x ∈ y + t we have that x ∈ x − a and  a ∈ x − x follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' It remains to deal with the case x − a ⊊ y + t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' In this case, since t ∈ x − y and  y − y ⊆ x − x, we obtain that  x − a ⊊ y + t ⊆ y + x − y ⊆ x + (x − x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Take c ∈ x − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' There exists x′ ∈ x − x such that c ∈ x + x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' This implies that  a ∈ x − c ⊆ x − (x + x′) = x − x  where we have used axiom (CH4) and axiom (SCH1) to conclude that x + x′ = {x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Now, assume that dv(z, t) > ρv + vx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' We have to prove that t ∈ x − y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' If z = t, then there is  nothing to show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Hence, we can assume that z ̸= t and so z − t ⊊ x − x must hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' We have that  t ∈ t + 0 ⊆ t + z − z ⊊ x − x + z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Hence, there is b ∈ z − x such that t ∈ x + b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' We conclude that b ∈ (z − x) ∩ (t − x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' By axiom  (SCH2) we then obtain that z − x ⊆ t − x or t − x ⊆ z − x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' In the first case, −y ∈ z − x ⊆ t − x  NOTES ON VALUATION THEORY FOR KRASNER HYPERFIELDS  23  and t ∈ x − y follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' It remains to deal with the case t − x ⊊ z − x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' In this case, since z ∈ x − y,  we have that  t − x ⊊ z − x ⊆ x − y − x = x − x − y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Take a ∈ t − x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' There exists x′ ∈ x − x such that a ∈ x′ − y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Now, by reversibility and since  a ∈ t − x, we have that  −y ∈ a − x′ ⊆ t − (x + x′) = t − x,  where for the last equality we have used axiom (SCH1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Now, t ∈ x−y follows by axiom (CH4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  □  Directly from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='20 and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='21 above, we deduce the following characteri-  zation theorem for the hyperfields which admit a Krasner valuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let F be a hyperfield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Then F admits a Krasner valuation if and only if the  additive hypergroup of F is superiorly canonical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' The additive hypergroup of the hyperfield that we have considered in Example  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='17 above is not superiorly canonical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Indeed,  1T + 1T = {0T, 1T }  and thus (SCH1) fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' It follows from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='22 that this hyperfield does not admit Krasner  valuations at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Consider a generalised tropical hyperfield T (Γ), where Γ is some non-trivial ordered  abelian group (see Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='17 (iv) and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='3 we conclude that T (Γ)  is a valued hyperfield (see also the discussion after Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Nevertheless, by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='22,  T (Γ) does not admit Krasner valuations as its additive hypergroup does not satisfy (SCH1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Let us now analyse another example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Take K = Q(X) with the valuation v := vp ◦ vX where vX denotes the X-adic  valuation on Q(X) and vp denotes the p-adic valuation on Q, for some prime number p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' More  explicitly, for a polynomial f(X) = �n  i=0 aiXi in Q[X], we have  vf(X) =  �  vXf(X), vp(avXf(X))  �  ∈ Z × Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  The order relation in vK is the lexicographic order of Z × Z, that is, (n1, n2) < (m1, m2) if and  only if n1 < m1 ∨ (n1 = m1 ∧ n2 < m2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Consider the Krasner valued hyperfield F := Kρ where ρ is the smallest initial segment of Z × Z  which contains {0} × Z (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let us denote its Krasner valuation vρ by w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Note that, if (m1, m2) ∈ ρ and m1 > 0, then, since ρ is an initial segment, we have that  {0} × Z ⊆ {(k1, k2) ∈ Z × Z | (k1, k2) ≤ (m1 − 1, m2)} ⊊ ρ,  in contradiction with the minimality of ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Therefore, m1 ≤ 0 for all (m1, m2) ∈ ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Conversely, if  m1 ≤ 0, then (m1, m2) ≤ (0, m2) for all m2 ∈ Z and therefore (m1, m2) ∈ ρ since ρ is an initial  segment containing {0} × Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' It follows that  (1)  ρ = {(m1, m2) ∈ Z × Z | m1 ≤ 0} = {(m1, m2 + n) ∈ Z × Z | m1 ≤ 0} = ρ + (0, n)  for any (0, n) ∈ {0} × Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='20, the additive hypergroup of F is superiorly canonical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Therefore, by Propo-  sition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='21, we have the valuation hyperring  Ou = {x ∈ F | x − x ⊆ 1 − 1}  24  LINZI, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  for some Krasner valuation u on F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let us now show that w and u are not equivalent valuations  on F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  In fact, we will show that Ow ⊊ Ou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Pick x ∈ Ow, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=', wx ≥ (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' By axiom (KVH2) we have  that y ∈ x − x if and only if wy > ρ + wx ≥ ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Another application of axiom (KVH2) shows that  x − x ⊆ 1 − 1 so that x ∈ Ou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Now, consider e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' the element x of F corresponding to the rational  number p−1 in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Since vp(p−1) = −1 we have that wx = (0, −1) < (0, 0) so that x /∈ Ow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' On  the other hand, since wx = (0, −1) ∈ {0} × Z, we have that ρ + wx = ρ by (1) and thus using  (KVH2), we obtain that y ∈ x − x if and only if wy > ρ if and only if y ∈ 1 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' This implies that  x − x ⊆ 1 − 1 and thus x ∈ Ou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  The above example constitutes the starting point for the discussion that we will have in Section  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' More on ordered abelian groups  In this section, we characterise generalised tropical hyperfields and discuss the concept of coars-  ening of a valuation in the multivalued setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' We derive some conclusions on the relation between  ordered abelian groups and generalised tropical hyperfields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Characterisation of generalised tropical hyperfields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' To characterise generalised tropi-  cal hyperfields, we will apply another remarkable characterisation result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' A hyperfield is stringent  if x + y is a singleton whenever 0 /∈ x + y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Bowler and Su in [6] characterised stringent hyperfields  and we will now briefly recall their result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  In [6, Section 4] a natural construction of a hyperfield arising from a short exact sequence  (2)  {1}  F ×  H×  Γ  {0}  ϕ  ψ  where Γ is an ordered abelian group and F is any hyperfield is described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  The thus obtained  hyperfield has H× as multiplicative group and is called the Γ-layering of F along the short exact  sequence (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' After that the following theorem is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='1 (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='10 in [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' A hyperfield H is stringent if and only if it is the Γ-layering  of F along the short exact sequence (2) with F being (isomorphic to) either K, S or a field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  From the details of the construction (which we omit for brevity) it is not difficult to verify that  the extension of the map ϕ sending 0F to 0H is in any case an embedding of hyperfields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  We say that a hyperfield has characteristic 2 if 0 ∈ 1 + 1 and that it has C-characteristic 1 if  1 ∈ 1 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' For example, S satisfies the latter but not the former while K satisfies both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' For more  information on characteristic and C-characteristic of hyperfields the reader can see [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  We are now ready to state and prove our characterisation theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='2 (Characterisation of generalised tropical hyperfields).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Generalised tropical hyper-  fields are precisely stringent hyperfields of characteristic 2 and C-characteristic 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' The reader can easily verify from the relevant definitions that a generalised tropical hyperfield  T (Γ) is a stringent hyperfield of characteristic 2 and C-characteristic 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  For the converse, let H be a stringent hyperfield of characteristic 2 and C-characteristic 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' By  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='1 and our observations on the extension of the map ϕ above, we have that either K, S or  a field embed into H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Since H has characteristic 2 we have that 1 = −1, thus S cannot embed into  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' On the other hand, any field cannot embed into H neither.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Indeed, our assumption 0, 1 ∈ 1 + 1  NOTES ON VALUATION THEORY FOR KRASNER HYPERFIELDS  25  implies that {0, 1} = (1 + 1) ∩ {0, 1} ⊆ (1 + 1) ∩ F = 1 +F 1 is not a singleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' It follows that H is  the Γ-layering of K along a short exact sequence  {1}  K×  H×  Γ  {0}  ϕ  ψ  for some ordered abelian group Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Now, since K× = {1}, we obtain that the multiplicative group  of H is isomorphic to Γ and the details of the construction given in [6, Section 4] show that the  hyperoperation of H is the one of T (Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' We conclude that H is a generalised tropical hyperfield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  □  An analogous reasoning yields to the following (but maybe less interesting) characterisation of  strict generalised tropical hyperfields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Generalised tropical hyperfields are precisely the Γ-layerings of F along the short  exact sequence (2), with F = F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Coarsenings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let (Γ, <, +, 0) be an ordered abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' A subgroup ∆ of Γ is convex if for  all γ ∈ Γ we have that if there exist δ1, δ2 ∈ ∆ such that δ1 < γ < δ2, then γ ∈ ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  It is not difficult to see that the intersection of a family of convex subgroups of an ordered  abelian group is again a convex subgroup and that the collection of all convex subgroups of an  ordered abelian group is linearly ordered by inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let us recall another basic fact which makes  convex subgroups important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Fact 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let (Γ, <, +, 0) be an ordered abelian group and ∆ a convex subgroup of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Then (Γ/∆, ≺  , +, 0) is an ordered abelian group, where  x + ∆ ≺ y + ∆ ⇐⇒ x < y and y − x /∈ ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  In particular, the canonical epimorphism Γ → Γ/∆ is order preserving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  The following notion will play a fundamental role later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let (Γ, <, +, 0) be an ordered abelian group and ρ an initial segment of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' We  call the set  ig(ρ) := {γ ∈ Γ | ρ + γ = ρ}  the invariance group of ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' If ig(ρ) = {0}, then we say that ρ has trivial invariance group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' In [29, 30] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='-V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Kuhlmann proves several results about invariance groups associated  to initial segments in ordered abelian groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' In particular, it follows from [30, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='3] that  the invariance group of an initial segment ρ of an ordered abelian group Γ such that 0 ∈ ρ is a  convex subgroup of Γ contained in ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let v, w be two valuations on a hyperfield F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' We say that w is a coarsening of v  if Ov ⊆ Ow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  In classical valuation theory for fields, it is well-known that to any convex subgroup of the value  group one can associate a coarsening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' We note that this result generalises to valued hyperfields,  actually with a much more conceptual proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let (F, v) be a valued hyperfield and let ∆ be a convex subgroup of vF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Then  v∆ : F → (vF/∆) ∪ {∞}  x �→ vx + ∆  is a valuation on F which is a coarsening of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  26  LINZI, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='4, v : F → T (vF) is a surjective homomorphism of hyperfields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' In addition,  the canonical epimorphism vF → vF/∆ extends to a surjective map T (vF) → T (vF/∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' From the  fact that the latter is order preserving, it easily follows that it is a homomorphism of hyperfields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  We conclude that the composition v∆ : F → T (vF/∆) of v with the latter map is a surjective  homomorphism of hyperfields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' This proofs the first part of the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Now, since  Ov∆ = {x ∈ F | v∆x ⪰ 0vF/∆}  = {x ∈ F | vx + ∆ ⪰ ∆}  = {x ∈ F | vx ≥ 0 or vx ∈ ∆}  ⊇ {x ∈ F | vx ≥ 0} = Ov,  we conclude that v∆ is a coarsening of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  □  In the above proof, we have seen that if ∆ is a convex subgroup of an ordered abelian group Γ,  then the natural map  π∆ : T (Γ) → T (Γ/∆)  is a surjective homomorphism of hyperfields, which, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='4, is a valuation on T (Γ) with  value group Γ/∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' The valuation hyperring of this valuation is  O∆ = {γ ∈ Γ | γ + ∆ ⪰ 0 + ∆} = {γ ∈ Γ | γ ∈ ∆ or γ > ∆}  and its unique maximal hypeideal is  M∆ = {γ ∈ Γ | γ > ∆}  It follows that the residue hyperfield T (Γ)π∆ = O∆/M∆ of T (Γ) with respect to the valuation π∆  is isomorphic to T (∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' An isomorphism is given by the map  T (∆) → T (Γ)π∆  ∞ �→ ∞  δ �→ δπ∆  In the next result we show that the valuations of T (Γ) are (up to equivalence) all of this form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let Γ be an ordered abelian group and assume that v : T (Γ) → T (Γ′) is a  valuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Then there exists a convex subgroup ∆ of Γ such that Γ/∆ ≃ Γ′ via an order preserving  isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let 0′ ∈ Γ′ denote the neutral element of the group Γ′, ⊞ the hyperoperation of T (Γ) and ⊞′  the hyperoperation of T (Γ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Set ∆ := v−1(0′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Since v is a homomorphism of groups Γ → Γ′, we  have that ∆ is a subgroup of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Moreover, since v is surjective, we have that Γ/∆ ≃ Γ′ as groups  by the first homomorphism theorem for groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Assume that δ1, δ2 ∈ ∆ and that γ ∈ Γ is such  that δ1 ≤ γ ≤ δ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Hence, δ2 ∈ γ ⊞ γ and so 0′ = v(δ2) ∈ v(γ) ⊞′ v(γ) since v is a homomorphism  of hyperfields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' On the other hand, γ ∈ δ1 ⊞ δ1 so that v(γ) ∈ 0′ ⊞′ 0′ = [0′, ∞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' If v(γ) > 0′, then  0′ /∈ [v(γ), ∞] = v(γ) ⊞′ v(γ), a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' We conclude that v(γ) = 0′ must hold and thus  γ ∈ ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' We have shown that ∆ is a convex subgroup of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' It remains to show that the isomorphism  of groups σ : γ + ∆ �→ v(γ) is order preserving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Assume that γ1 + ∆ ≺ γ2 + ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' This means that  γ1 < γ2 and γ2 −γ1 /∈ ∆ hold in Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Thus, {γ1} = γ1 ⊞γ2 and then v(γ1) ∈ v(γ1)⊞′ v(γ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Since σ is  bijective and γ1 ̸= γ2, we have that v(γ1) ̸= v(γ2), so v(γ1) ∈ v(γ1) ⊞′ v(γ2) =  �  min{v(γ1), v(γ2)}  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  This shows that v(γ1) < v(γ2) must hold in Γ′ and thus σ is order preserving and the proof is  complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  □  NOTES ON VALUATION THEORY FOR KRASNER HYPERFIELDS  27  The next result now follows from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let Γ be an ordered abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' There is a bijective correspondence between  convex subgroups of Γ and valuation hyperrings of T (Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Krasner valuations induced by the additive structure  In this section we apply some of the results obtained in the previous section to Krasner valued  hyperfields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' We begin by observing that coarsenings of Krasner valuations might not be Krasner  valuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let Γ be a non-trivial ordered abelian group with a non-trivial convex subgroup ∆  and consider the identity map as a Krasner valuation v : T ′(Γ) → T (Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' As a map, the coarsening  v∆ : T ′(Γ) → T (Γ/∆)  is just the canonical epimorphism Γ → Γ/∆ extended to send ∞ to ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Suppose that v∆ is a  Krasner valuation of norm ρ for some initial segment ρ of Γ/∆ containing 0Γ/∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Then by (KVH2)  for all γ ∈ Γ we would have that  γ > 0 ⇐⇒ γ ∈ 0 ⊞′ 0 ⇐⇒ γ + ∆ > ρ + (0 + ∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  On the other hand, since 0Γ/∆ ∈ ρ , any positive γ ∈ ∆ would violate this equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Nevertheless, we have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let (F, v) be a Krasner valued hyperfield and let w be the Krasner valuation on F  determined by  Ow = {x ∈ F | x − x ⊆ 1 − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Then w is equivalent to the coarsening of v corresponding to ig(ρv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' In particular, the coarsening  of a Krasner valuation v corresponding to ig(ρv) is a Krasner valuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' We show that these two valuations have the same valuation hyperring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Indeed, for all x ∈ F  we have that vx + ig(ρv) ⪰ 0vF/ ig(ρv) if and only if vx ∈ ig(ρv) or vx > ig(ρv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' In both cases by  (KVH2) applied to v we have that  y ∈ x − x ⇐⇒ vy > ρv + vx ⊇ ρv =⇒ y ∈ 1 − 1,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=', y ∈ Ow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Conversely, vx + ig(ρv) ≺ 0vF/ ig(ρv) if and only if vx /∈ ig(ρv) and vx < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Therefore,  ρv + vx ⊊ ρv and there exists y ∈ F such that vy ∈ ρv and vy /∈ ρv + vx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' By (KVH2) applied to v,  the former implies y /∈ 1 − 1 while the latter is equivalent to y ∈ x − x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' We conclude that x /∈ Ow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  We have proved that vig(ρv) and w have the same valuation hyperring in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' The theorem follows  from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  □  Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let F be a hyperfield admitting two Krasner valuations v1 and v2 of norm ρ1 and  ρ2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Then the coarsening of v1 corresponding to ig(ρ1) is equivalent to the coarsening  of v2 corresponding to ig(ρ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let (F, v) be a Krasner valued hyperfield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' If ρv has trivial invariance group, then  Ov = {x ∈ F | x − x ⊆ 1 − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let F be a hyperfield with a superiorly canonical additive hypergroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Then the  norm of the Krasner valuation v determined on F by the valuation hyperring  Ov = {x ∈ F | x − x ⊆ 1 − 1}  has trivial invariance group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  28  LINZI, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Let F be a hyperfield with a superiorly canonical additive hypergroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' There is a  unique (up to equivalence) Krasner valuation v on F such that ig(ρv) = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' In the model theory of valued fields, the RV-structure (mentioned in the introduction)  of level γ of a valued field (K, v), where γ is a non-negative element of vK, is essentially the Krasner  valued hyperfield3 Kγ := Kργ, where ργ := {δ ∈ vK | δ ≤ γ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Since ig(ργ) = {0}, it follows, as a  consequence of Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='4, that the valuation hyperring of the Krasner valuation on Kγ induced  by v is always definable in the language of hyperfields, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=', using the ternary relation z ∈ x + y to  encode the multivalued operation +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Further research  There are at least three interesting points that have been touched but not developed further in  this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Below we briefly describe them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  (1) In the multivalued setting, (F, v), vF and Fv can all be described as (valued) hyperfields  and they are not distinct structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' This applies in particular when F is a field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  (2) In the literature there are not many examples of infinite hyperfields that are not factor  hyperfields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Essentially, only the work of Massouros [39, 40] provides such examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' In  all those examples M we have by definition x + x = M \\ {x} for all x ∈ M ×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' If v is a  valuation on M and x ̸= 1, then x, x−1 ∈ 1 + 1, so by (V3) and Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='5 (iii) we have  that vx = 0 must hold for all x ∈ M ×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' It follows that M only admits the trivial valuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Is it true that any hyperfield admitting a non-trivial valuation is a factor hyperfield?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  (3) In view of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='2, a non-trivial valuation on a hyperfield F is a surjective homomor-  phism onto a hyperfield H satisfying the following three conditions:  H has characteristic 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  H has C-characteristic 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  H is stringent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  From this point of view, the image of a valuation is a quite special hyperfield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' For example,  one could think that responsible for the fact that finite (hyper)fields only admit the trivial  valuation is the third property of H, since there are many examples of finite hyperfields  satisfying 0, 1 ∈ 1+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' We speculate that investigating weakenings of the stringency property  for H may lead to a notion of generalised valuation which have the potential to be applied  also in the classical singlevalued setting e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' to use (generalised) valuation-theoretic methods  over finite fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  References  [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Ameri, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Eyvazi, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Hoskova-Mayerova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Advanced results in enumeration of hyperfields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' AIMS Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=',  5(6):6552–6579, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  [2] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Baker and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Lorscheid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Descartes’ rule of signs, Newton polygons, and polynomials over hyperfields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Algebra, 569:416–441, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  [3] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Basarab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Relative elimination of quantifiers for Henselian valued fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Pure Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Logic, 53(1):51–74,  1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  [4] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Bélair and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Poizat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Oeuvres Choisies de Marc Krasner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Editions L’Harmattan, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  [5] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Bordbar and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Cristea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Height of prime hyperideals in Krasner hyperrings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Filomat, 31(19):6153–6163, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  [6] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Bowler and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Su.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Classification of doubly distributive skew hyperfields and stringent hypergroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Algebra,  574:669–698, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  [7] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Connes and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Consani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' From monoids to hyperstructures: in search of an absolute arithmetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' In Casimir  force, Casimir operators and the Riemann hypothesis, pages 147–198.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Walter de Gruyter, Berlin, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  3This hyperfield is called the valued γ-hyperfield in [32]  NOTES ON VALUATION THEORY FOR KRASNER HYPERFIELDS  29  [8] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Connes and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Consani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' The hyperring of adèle classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Number Theory, 131(2):159–194, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  [9] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Cristea and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Jančić-Rašović.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Composition hyperrings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Ştiinţ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' “Ovidius” Constanţa Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=',  21(2):81–94, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  [10] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Engler and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Prestel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Valued fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Springer Monographs in Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Springer-Verlag, Berlin, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  [11] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Flenner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Relative decidability and definability in Henselian valued fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Symbolic Logic, 76(4):1240–1260,  2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  [12] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Gładki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Orderings of higher level in multifields and multirings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Sil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=', (24):15–25, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  [13] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Gładki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Witt equivalence of fields: a survey with a special emphasis on applications of hyperfields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' In Ordered  algebraic structures and related topics, volume 697 of Contemp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=', pages 169–185.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=',  Providence, RI, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  [14] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Gładki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  A  survey  on  multirings,  hyperrings  and  hyperfields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Notes  available  at:  http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
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+page_content='pdf, Unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  [15] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Gładki and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Marshall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Orderings and signatures of higher level on multirings and hyperfields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' K-Theory,  10(3):489–518, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  [16] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Gładki and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Marshall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Witt equivalence of function fields over global fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
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+page_content=',  369(11):7861–7881, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  [17] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Golzio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' A brief historical survey on hyperstructures in algebra and logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
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+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=', 4(1):91–119,  2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  [18] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Jell, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Scheiderer, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Real tropicalization and analytification of semialgebraic sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
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+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' IMRN, (2):928–958, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' [Initially appeared as 2021, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' 24, 19178–19208].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  [19] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Jun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Algebraic geometry over hyperrings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=', 323:142–192, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  [20] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Jun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Geometry of hyperfields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Algebra, 569:220–257, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  [21] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Kędzierski, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Linzi, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Stojałowska.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Characteristic, C-characterisitic and positive cones in hyperfields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Submitted, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  [22] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Krasner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Sur la primitivité des corps P-adiques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Mathematika (Cluj), 13(4):72–191, 1937.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  [23] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Krasner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Nombres semi-réels et espaces ultramétriques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Paris, 219:433–435, 1944.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  [24] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Krasner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Approximation des corps valués complets de caractéristique p ̸= 0 par ceux de caractéristique 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  In Colloque d’algèbre supérieure, tenu à Bruxelles du 19 au 22 décembre 1956, Centre Belge de Recherches  Mathématiques, pages 129–206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Établissements Ceuterick, Louvain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Librairie Gauthier-Villars, Paris, 1957.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  [25] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Krasner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' A class of hyperrings and hyperfields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Internat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=', 6(2):307–311, 1983.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  [26] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Krull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Allgemeine Bewertungstheorie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Reine Angew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=', 167:160–196, 1932.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  [27] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='-V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Kuhlmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Quantifier elimination for Henselian fields relative to additive and multiplicative congruences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Israel J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=', 85(1-3):277–306, 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  [28] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='-V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Kuhlmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Maps on ultrametric spaces, Hensel’s lemma, and differential equations over valued fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Algebra, 39(5):1730–1776, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  [29] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='-V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Kuhlmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Selected methods for the classification of cuts, and their applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' In Algebra, logic and  number theory, volume 121 of Banach Center Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=', pages 85–106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Polish Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=', Warsaw, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  [30] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='-V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Kuhlmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Invariance group and invariance valuation ring of a cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' In preparation, preliminary version  avaliable at: https://math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='usask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='ca/~fvk/CUTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='pdf, Unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  [31] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Kuhlmann, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Linzi, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Stojałowska.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Orderings and valuations in hyperfields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Algebra, 611:399–421,  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  [32] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Lee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Hyperfields, truncated DVRs, and valued fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Number Theory, 212:40–71, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  [33] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Linzi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Algebraic hyperstructures in the model theory of valued fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' PhD thesis, University of Szczecin  (Uniwersytet Szczeciński), https://bip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='usz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='pl/doktorat-habilitacja/16282/alessandro-linzi, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  [34] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Linzi and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Touchard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' On the hyperfields associated to valued fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Submitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Preprint available at  arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='05082, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  [35] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Marty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Sur une généralization de la notion de groupe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' In Huitiéme Congrès des Mathématiciens Scandenaves  Stockholm, pages 45–49, 1934.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  [36] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Marty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Rôle de la notion de hypergroupe dans l’étude de groupes non abéliens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' (Paris),  201:636–638, 1935.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  [37] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Marty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Sur les groupes et hypergroupes attaches à une fraction rationnelle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' École Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Sup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' (3),  53:83–123, 1936.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  [38] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Massouros and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Massouros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' An overview of the foundations of the hypergroup theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Mathematics, 9(9):  , 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  [39] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Massouros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Methods of constructing hyperfields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Internat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=', 8(4):725–728, 1985.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  30  LINZI, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  [40] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Massouros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' On the theory of hyperrings and hyperfields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Algebra i Logika, 24(6):728–742, 749, 1985.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  [41] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Mittas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Hypergroupes canoniques valués et hypervalués.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Balkanica, 1:181–185, 1971.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  [42] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Mittas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Sur les hyperanneaux et les hypercorps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Balkanica, 3:368–382, 1973.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  [43] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Prestel and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Delzell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Mathematical logic and model theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Universitext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Springer, London, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' A brief  introduction, Expanded translation of the 1986 German original.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  [44] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Prestel and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Roquette.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Formally p-adic fields, volume 1050 of Lecture Notes in Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Springer-  Verlag, Berlin, 1984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  [45] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Rota.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Multiplicative hyperrings (Sugli iperanelli moltiplicativi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Rend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' (7), 2(4):711–724 (1983), 1982.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  [46] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Tolliver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' An equivalence between two approaches to limits of local fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Number Theory, 166:473–492,  2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  [47] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Viro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Hyperfields for tropical geometry I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' hyperfields and dequantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' arXiv:1006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='3034, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  [48] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Vuković.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Remembering Professor Marc Krasner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Sarajevo J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=', 12(25)(2, suppl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content=' ):283–298, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Center for Information Technologies and Applied Mathematics, University of Nova Gorica, Slove-  nia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='  Email address: alessandro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='linzi@ung.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
+page_content='si' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FAT4oBgHgl3EQfpR2f/content/2301.08639v1.pdf'}
diff --git a/8dAyT4oBgHgl3EQfQvbW/content/tmp_files/2301.00054v1.pdf.txt b/8dAyT4oBgHgl3EQfQvbW/content/tmp_files/2301.00054v1.pdf.txt
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+Fault-tolerant error correction for a universal non-Abelian topological quantum
+computer at finite temperature
+Alexis Schotte,1, ∗ Lander Burgelman,2 and Guanyu Zhu1, 3, †
+1IBM Quantum, IBM Almaden Research Center, San Jose, CA 95120, USA
+2Department of Physics and Astronomy, Ghent University, Krijgslaan 281, 9000 Gent, Belgium
+3IBM T. J. Watson Research Center, Yorktown Heights, NY 10598, USA
+We study fault-tolerant error correction in a quantum memory constructed as a two-dimensional model of Fi-
+bonacci anyons on a torus, in the presence of thermal noise represented by pair-creation processes and measure-
+ment errors. The correction procedure is based on the cellular automaton decoders originating in the works of
+G´acs [1] and Harrington [2]. Through numerical simulations, we observe that this code behaves fault-tolerantly
+and that threshold behavior is likely present. Hence, we provide strong evidence for the existence of a fault-
+tolerant universal non-Abelian topological quantum computer.
+I.
+INTRODUCTION
+Anyons are emergent quasi-particles that exist in
+two-dimensional condensed matter systems and whose
+exchange statistics generalize that of Bosons and
+Fermions. These particles have spurred much interest
+due to their potential applications for quantum compu-
+tation. In particular, it was found that with certain types
+of non-Abelian anyons, a universal quantum compu-
+tation can be performed by braiding and fusing these
+particles [3–5]. An intriguing benefit of this paradigm
+is that, due to their topological nature, computations
+are intrinsically robust to perturbations at zero tem-
+perature. At non-zero temperature, however, thermal
+anyonic excitations can corrupt the computation by
+performing non-trivial braids with the computational
+anyons. Since systems exhibiting anyonic excitations
+have a spectral gap ∆, this source of errors can be sup-
+pressed to some extent at temperatures T ≪ ∆/kB
+as the density of thermal anyons scales as e−∆/kBT .
+Alas, this passive protection does not suffice, because
+the presence of thermal anyons is unavoidable at non-
+zero temperatures when scaling up the size of com-
+putations. Therefore, proficient active error correction
+schemes for non-Abelian models are paramount for the
+realization of topological quantum computers.
+Besides their envisaged use for topological quantum
+computation, topologically ordered systems (i.e., those
+that support anyonic excitations on top of their ground
+space) are also of much interest for quantum error cor-
+rection. In particular, one of the characteristics of such
+systems is a robust ground space degeneracy, which al-
+∗ alexis.schotte@posteo.net
+† guanyu.zhu@ibm.com
+lows one to use their ground space as the code space of
+an error correcting code. This realization led to the dis-
+covery of topological quantum error correcting codes,
+which encode logical quantum states in topologically
+ordered states of a system of qudits (typically arranged
+on a two-dimensional lattice). Since their discovery
+in the 90s, most research has focused exclusively on
+Abelian topological codes such as the surface code and
+the color code [5–14], which admit an elegant charac-
+terization in terms of the stabilizer formalism [15]. Due
+to their geometrical locality and high error thresholds,
+these codes are considered to be promising candidates
+for protecting quantum information from noise in error-
+corrected quantum computers. One of the drawbacks
+of Abelian topological codes, however, is that they do
+not allow one to execute a universal set of logical gates
+in a protected fashion in two dimensions. Hence, they
+must be supplemented with additional protocols such
+as magic state distillation [16] or code switching to
+higher-dimensional codes [17, 18] , which introduce a
+large space-time overhead [19]. Alternatively, there ex-
+ist non-Abelian topological codes which do not suffer
+from this inherent limitation, and are able to perform
+a universal gate set natively within their code space in
+two dimensions [3]. The trade-off is that such codes go
+beyond the stabilizer formalism and are therefore very
+hard to simulate classically.
+While active error correction in Abelian anyon mod-
+els and Abelian topological codes has been studied
+extensively, quantum error correction based on non-
+Abelian anyon models has not enjoyed the same fo-
+cus. Nevertheless, important progress has been made
+over the last decade, including both analytical proofs
+and numerical demonstrations of threshold behavior
+for various non-Abelian topological error correcting
+codes [20–24]. Moreover, syndrome extraction circuits
+for such non-Abelian string-net codes have been devel-
+arXiv:2301.00054v1  [quant-ph]  30 Dec 2022
+
+2
+oped in recent years [24, 25]. In addition, state prepa-
+ration for non-Abelian codes based on the Kitaev quan-
+tum double models via measurements has also been
+proposed recently for the experimental implementa-
+tion on qubit lattices [26, 27], although further devel-
+opment is still needed in the context of fault-tolerant
+state preparation. Notably, previous studies in this field
+already include codes based on the Fibonacci anyon
+model, which is universal for quantum computation
+[23, 24]. In particular, a quantum memory of qubits
+supporting doubled Fibonacci anyonic excitations was
+found to have a threshold that lies remarkably close to
+that of the surface code under similar assumptions [24].
+These results, however, all assume perfect syndrome
+measurements, which are topological charge measure-
+ments in this context. As we aim to model more re-
+alistic scenarios, we must take faulty measurements
+into consideration. Again, much is known in the case
+of Abelian topological codes [2, 7, 28–31]. For their
+non-Abelian counterparts, one key result stands out: in
+Ref. [32] a proof was formulated that topological codes
+based on non-cyclic anyon models admit a error cor-
+rection thresholds with faulty topological charge mea-
+surements. While this result is remarkable, non-cyclic
+anyon models are not universal for quantum computa-
+tion, and it remains an open question whether similar
+claims can be made for universal models.
+In this work, we take a step towards demonstrat-
+ing that fault-tolerance is indeed possible for universal
+non-Abelian topological codes. To this end, we define
+a quantum memory constructed as a two-dimensional
+model of Fibonacci anyons on a torus. We study ac-
+tive continuous quantum error correction on this model
+in the presence of thermal noise represented by pair-
+creation processes, and with faulty syndrome measure-
+ments. The correction procedure is based on the cel-
+lular automaton decoders originating in the works of
+G´acs [1] and Harrington [2], and further studied in the
+context of non-Abelian models in Ref. [32]. Through
+numerical simulations, we study how the average mem-
+ory lifetime changes with the error rate. The results in-
+dicate that this code is indeed fault-tolerant, which is
+strong evidence for the existence of fault-tolerant uni-
+versal non-Abelian codes.
+The structure of this work is as follows. In Sec. II
+we introduce the topological Fibonacci code. We then
+describe the details of the noise model in Sec. III and
+introduce the cellular automaton decoder in Sec. IV.
+We proceed by giving an outline of the numerical sim-
+ulations performed in this work in Sec. V. Finally, we
+present the corresponding numerical results in Sec. VI
+and conclude with a discussion in Sec. VII.
+II.
+THE FIBONACCI CODE
+We consider a two-dimensional model comprised of
+hexagonal tiles laid out on the surface of a torus. The
+resulting geometry can be represented as an L × L
+hexagonal lattice with periodic boundary conditions in
+both directions (Fig. 2). Each of these hexagonal tiles
+can contain an excitation known as a Fibonacci anyon.
+Anyons are point-like quasi-particle excitations
+which can be characterized algebraically in terms of a
+unitary modular tensor category (UMTC). A thorough
+description of anyon models using UMTCs goes be-
+yond the scope of this work, however, some details are
+given in Sec. A. For now, it is sufficient to state that
+an anyon model specifies a set of anyon labels, also
+referred to as particle types, which can fuse according
+to a specific set of fusion rules. The Fibonacci anyon
+model considered in this work contains two labels, 1
+and τ, which obey the fusion rules
+1×1 = 1 ,
+1×τ = τ×1 = τ ,
+τ×τ = 1+τ . (1)
+In general, one can associate a vector space to a
+given set of anyons, where the basis vectors are la-
+beled by the different ways in which the anyons can
+fuse. This fusion space has a topological degeneracy,
+and can therefore be used to robustly encode quan-
+tum information. In particular, for the Fibonacci anyon
+model the anyonic vacuum on a two-dimensional torus
+has a twofold degeneracy [33]. Starting from our two-
+dimensional model, we can therefore define an error
+correcting code whose code space is identified with the
+anyonic vacuum on the torus and which encodes a sin-
+gle logical qubit. A basis for this code space can be
+defined using Wilson line operators along the homo-
+logically non-trivial cycles x and y shown in Fig. 1:
+W a
+x |1⟩x = Sa1
+S11 |1⟩x ,
+W a
+x |τ⟩x = Saτ
+S1τ |τ⟩x ,
+a ∈ {1, τ} ,
+(2)
+We note that a different basis, {|0⟩y , |1⟩y}, can be
+defined analogously by swapping the x and y labels
+above, where the two bases are related through the
+modular S matrix Sab =
+y⟨a|b⟩x. For the Fibonacci
+anyon model its numerical values are
+S =
+1
+�
+1 + φ2
+�
+1
+φ
+φ −1
+�
+,
+(3)
+where φ = 1 +
+√
+5
+2
+.
+The action of the mapping class group on the any-
+onic vacuum then corresponds to unitary operations on
+
+3
+x
+y
+Figure 1: The two homologically non-trivial cycles on
+a torus.
+the code space. For the Fibonacci category, any logical
+unitary operator can be realized in this way, up to arbi-
+trary precision [3]. Therefore, the quantum error cor-
+recting code defined above natively supports universal
+quantum computation.
+Errors in this code appear as spurious anyonic ex-
+citations, which can corrupt the encoded informa-
+tion if their world lines between creation and re-
+annihilation are topologically non-trivial, i.e., form a
+non-contractible cycle 1. The objective of error cor-
+rection is then to systematically remove these spurious
+excitations, without corrupting the quantum memory in
+the process. This correction is performed in an active
+and continuous manner, and can be broken down into a
+series of discrete steps. At each step, a suitable recov-
+ery operation is performed based on a measured list of
+positions and types of the excitations, called the error
+syndrome.
+We conclude this section by noting that the numer-
+ical simulation of the error-correction process requires
+the introduction of some additional manipulations on
+fusion states of multiple Fibonacci anyons. As these
+are technical details that do not contribute to the in-
+tuition of the procedure, we defer their definition to
+Sec. A.
+III.
+NOISE MODEL AND CORRECTABILITY
+Having defined our model and code space, we now
+turn to the description of the noise model used in our
+1 Note that a pair of Fibonacci anyons can also fuse to a single non-
+trivial anyon when one member of the pair has been transported
+along a non-trivial cycle. Since the resulting state is no longer in
+the code space, this does not constitute a logical operation on the
+encoded information. However, one can show that the encoded
+information is irrevocably lost in case of such an event [21].
+simulations.
+We model continuous active error cor-
+rection in our Fibonacci code as a sequence of time
+steps, where each time step itself consists of three parts:
+the application of pair-creation noise, faulty syndrome
+measurement, and error correction respectively.
+At each time step, first, for each edge of the hexago-
+nal lattice graph a pair of anyons is created across this
+edge with a probability p. Immediately after each pair
+creation event, the resulting charge in the two affected
+tiles is sampled, effectively collapsing all superposi-
+tions of anyonic charge within each tile to either 1 or
+τ. After the pair creation noise has been applied, faulty
+syndrome extraction is simulated by generating a list of
+the anyon charge in all tiles, and flipping each outcome
+individually with a probability q. In addition to the
+charges that are correctly detected, the resulting faulty
+syndrome can contain both “ghost defects” (indicating
+a non-trivial charge when none is truly present) and
+“missing defects” (failing to report a true non-trivial
+charge). Finally, this faulty syndrome is passed to a de-
+coder, introduced in the following section, which then
+performs a set of local operations based on the current
+(and past) syndrome information in an attempt to move
+the system back towards the initial state.
+After each time step, the current state of the sys-
+tem is copied and it is checked whether it is still cor-
+rectable. This is done by passing the copy to a clus-
+tering decoder [22–24] and simulating a decoding pro-
+cedure with perfect syndrome measurements starting
+from this given initial state. If this perfect decoding
+is successful, the memory is considered intact and the
+simulation is continued. If perfect decoding is unsuc-
+cessful, the memory is considered corrupted and the
+simulation is aborted. The memory lifetime is then de-
+fined as the number of time steps after which a perfect
+clustering decoder can no longer successfully restore
+the initial state.
+For a given pair of tiles which share an edge, the
+process of pair creation across this edge corresponds to
+the matrix elements
+⟨a′, b′; c′| Upc |a, b; c⟩ = δc,c′F aa1
+ττa′F a′τa
+b c b′ ,
+(4)
+where we have used the F-symbols of the Fibonacci
+category, given in (A5). Here, |a, b; c⟩ represents the
+state where the affected tiles have anyon charges a
+and b, respectively, with total charge c. This then de-
+fines the probability distribution according to which
+outcomes a′ and b′ are sampled.
+Since our noise
+model does not allow any superposition in the any-
+onic charge of individual tiles, it should be considered
+semi-classical rather than fully quantum-mechanical.
+
+4
+(a)
+(b)
+Figure 2: (a) Pair-creation events creating anyonic
+excitations in neighboring tiles. The dotted ellipse
+represents a collapse to the total charge of the anyons
+it contains. A ghost defect is shown in blue, a missing
+defect is highlighted in orange with a cross. (b) The
+outcome of this noise process represented on the
+decoding graph. Note that the missing defect
+(highlighted in orange) will (by definition) not be
+visible in the syndrome.
+Note, however, that this does not render our model
+completely classical. Indeed, superpositions in the total
+charge c of the affected tiles are an inherent part of the
+state evolution that cannot be captured faithfully by any
+classical probabilistic process. Furthermore, while the
+extreme decoherence assumption for the anyon charge
+in individual tiles greatly simplifies the numerical sim-
+ulation outlined in this work, it was argued in Ref. [22]
+that this decoherence is unlikely to have any tangible
+influence on the observed memory lifetimes, as the es-
+sential topological nature of the noise processes is still
+captured correctly.
+We note that this type of noise can be understood as
+originating from the connection to a thermal bath with
+inverse temperature β = 1/(kBT) determined by the
+error rate p through the relation
+p
+1 − p = e−β∆ .
+(5)
+Here, ∆ represents the energy required to create a pair
+of anyonic excitations and place them in neighboring
+tiles.
+To conclude this section, we emphasize that in the
+case of non-Abelian error correction, even decoding
+with perfect syndrome measurements is still an inher-
+ently stochastic procedure due to the indeterminacy of
+anyonic charge measurements. This means that perfect
+decoding can sometimes either be successful or unsuc-
+cessful even starting from the same initial state. Our
+definition of the memory lifetime therefore simply cor-
+responds to a statistical estimate of the actual memory
+lifetime. Furthermore, it is not known which decoder
+is optimal for the Fibonacci code. Hence, the choice
+for the clustering decoder to verify the correctability of
+states is, in a way, an arbitrary one. This choice, how-
+ever, is motivated by the recent discovery that the clus-
+tering decoder yields high thresholds for a related er-
+ror correcting code exhibiting doubled Fibonacci any-
+onic excitations, and performed significantly better in
+this context than decoders based on a perfect matching
+strategy [24]. In any case, one should keep in mind that
+the memory lifetime as defined above, does not repre-
+sent the true memory lifetime. Instead the sub-optimal
+verification process entails that it merely provides us
+with a lower bound on the true value.
+IV.
+HARRINGTON’S CELLULAR AUTOMATON
+DECODER
+The model described above is paired with a decoder
+which is a straight-forward adaptation of the cellu-
+lar automaton decoder introduced in Ref. [2]. Previ-
+ously, this decoder has also been used for a similar
+phenomenological model of Ising anyons in Ref. [32],
+where the existence of an error correction threshold
+was proven analytically. At each time step during the
+error correction simulation, based on the reported mea-
+surement outcomes in the faulty syndrome, the decod-
+ing algorithm will apply local transition rules to fuse
+neighboring anyons or to move anyons to neighboring
+tiles.
+Intuitively these transition rules work as follows.
+The lattice is divided into square colonies of size Q×Q.
+At each time step, the transition rules will attempt to
+fuse neighboring non-trivial anyons, as observed in
+the faulty syndrome.
+If a non-trivial anyon has no
+neighbors, the transition rules will move it to the cen-
+ter of its colony.
+At larger timescales, higher-level
+transition rules are applied on a renormalized lattice
+where anyons located at colony centers will be fused
+with anyons at neighboring colony centers, or moved
+toward the center of their respective super-colonies,
+which consist of Q × Q colonies. This renormalization
+scheme is then continued at higher levels until eventu-
+ally the Qn × Qn super-colony covers the entire lattice
+for some integer n. To ensure the latter is possible, we
+will always assume that the linear lattice size satisfies
+L = Qn for some integer n. An example of these pro-
+cesses is shown in Fig. 4.
+To describe the action of the decoding algorithm
+more precisely, we will define its action at different
+renormalization levels k. The level-0 transition rules
+
+5
+are those already discussed above and are applied at
+every time step based on the reported faulty syndrome
+obtained from the most recent round of faulty measure-
+ments. The transition rules are applied to one location
+at a time and take into consideration only the anyon
+content of that site and of its eight neighbors. A de-
+tailed definition of these rules is given in App. C. When
+an anyon is moved from a site l to a neighboring site
+l′, the (true) anyon content of site l is fused with that
+of site l′ and the resulting charge is placed on site l′
+while the charge of site l is restored to the vacuum.
+This happens irrespective of whether or not the syn-
+drome for both sites was correct. Hence, when the de-
+coder attempts to move a ghost defect (a trivial charge
+misidentified as a non-trivial one) to a neighboring site,
+this process does not create additional excitations. This
+does not, however, mean that mistaking a trivial charge
+for a non-trivial one has no negative consequences. In-
+deed, these wrong syndromes may cause the decoder
+to stretch out existing errors.
+The level-1 transition rules are not applied in every
+time step, but only when t is a multiple of a param-
+eter called the work period, which we will denote by
+U. We require that U = b2 for some positive integer
+b. One should think of U as the time scale at which a
+coarse-graining is performed. Level-1 transition rules
+act at on a coarse-grained lattice where the sites corre-
+spond to the centers of the level-0 colonies, and these
+are grouped into level-1 colonies of size Q2 × Q2.
+Hence, the actions determined by the level-1 transition
+rules involve pairs of level-0 colony centers separated
+by a distance Q. An example of such a move is pro-
+vided in Fig. 4(d). The transition rules themselves are
+nearly identical to the level-0 rules, but are based on
+two sets of level-1 syndromes s1,c and s1,n (defined
+below) rather than one. For a site l (which is a level-
+0 colony center), the transition rules use s1,c(l) as the
+anyon content of site, while the anyon content of its
+neighbors (that is, the neighboring level-0 colony cen-
+ters) is taken to be s1,n(l′).
+The definitions of the level-1 syndromes s1,c and
+s1,n require a pair of variables fc, fn ∈ [0, 1].
+In-
+tuitively these variables serve as detection thresholds
+for the level-1 syndromes by determining the fraction
+of measurements that must return a non-trivial out-
+come at a site before it qualifies as a non-trivial level-1
+syndrome. The proper definition, however, is slightly
+more complicated and uses a coarse-grained counting
+method. Below, we give the precise definition of s1,c,
+the definition of s1,n is entirely analogous (using fn
+instead of fc). We start by dividing the work period
+U = b2, into b intervals of b time steps each. For
+each of these intervals, we say a non-trivial syndrome
+is present at a colony center l if a non-trivial charge
+was reported there for at least fcb of the b time steps in
+the interval. When at least fcb of the b intervals have
+a non-trivial syndrome, s1,c(l) is set to one. A visual
+example of this coarse-grained counting procedure is
+shown in Fig. 3.
+≥ fc · b
+< fc · b
+≥ fc · b
+Figure 3: A visual example of the coarse-grained
+counting procedure to determine the level-1 syndrome
+for a level-0 colony center. The row of dots represent
+U time steps, divided in b intervals of size b. The time
+steps during which a non-trivial measurement
+outcome was reported are indicated by the colored
+dots. The crosses in the second row indicate in which
+intervals the fraction of non-trivial measurement
+outcomes is equal to or higher than fc.
+The motivation for using two types of syndromes for
+k > 0 is as follows. Suppose that an error spans across
+two neighboring colonies, which we will label ρ and ρ′.
+The level-0 transition rules will transport all resulting
+anyons to the respective colony centers, where they can
+now be acted upon by level-1 transition rules at the end
+of the work period. Imagine that a non-trivial anyon
+is now present at both colony centers. When consid-
+ering the level-1 transition rules acting on ρ, there are
+four possible scenarios for the syndromes s1,c(ρ) and
+s1,n(ρ′). In case s1,c(ρ) = 0, the transition rules act
+trivially on ρ. If both s1,c(ρ) = 1 and s1,n(ρ′) = 1
+then the transition rules will be applied correctly and
+the anyons will be fused. However, if s1,c(ρ) = 1 but
+s1,n(ρ′) = 0, the transition rules may move the anyon
+in ρ away from ρ′, thereby increasing the weight of
+the error. Hence, it is desirable to set fc > fn to de-
+crease the odds that when a level-k syndrome reports
+an non-trivial anyon at a colony center, the level-k syn-
+drome for its neighbors are false negatives. We must
+be careful not to set fc too high or fn too low, how-
+ever. If we choose fc to high, low-weight errors could
+cause s1,c to never report any non-trivial charges, de-
+laying any necessary corrections. Similarly, setting fn
+too low will result in low-weight error triggering many
+false positives for s1,n, which can cause the decoder to
+make wrong decisions.
+
+6
+Level-k
+transition
+rules
+are
+applied
+when
+t
+mod U k = 0. They operate on a renormalized lattice
+that uses the centers of level-(k − 1) colonies as sites,
+and groups these into level-k colonies of size Qk ×Qk.
+The level-k syndromes sk,c and sk,n are determined by
+the coarse-grained counting method described above,
+using bk intervals of bk time steps each. For linear sys-
+tem size L, k ranges from 0 to kmax = logQ(L).
+It is important to note that non-Abelian anyons do
+not allow for instantaneous moves. Indeed, while one
+can construct a unitary string operator for Abelian
+anyons, no such operator can be constructed for the
+non-Abelian case. This discrepancy can be traced back
+to the fact that fusion outcomes are non-deterministic
+for non-Abelian anyons, implying it is not possible to
+move an non-Abelian anyon by annihilating it with one
+member of a particle-antiparticle pair (as is done in
+e.g., the surface code).
+Therefore, the actions determined by level-k transi-
+tion rules, for k > 0 cannot be applied withing a sin-
+gle time step. Instead, they will be broken up into a
+sequence of moves involving only pairs of neighbor-
+ing sites which will be applied in Qk consecutive time
+steps. We further limit the model by requiring that the
+number recovery operations affecting a single tile in the
+lattice (or site in the decoding graph), is no greater than
+one in each time step. This allows all recovery opera-
+tions applied in one time step to be performed in par-
+allel. Hence, we must define a hierarchy determining
+which actions (moves or fusions between neighboring
+tiles) get prioritized based on the renormalization level
+from which they originated. In our case, it was opted to
+always prioritize correction processes from the highest
+renormalization level 2
+It was argued in [32] that the prohibition of instanta-
+neous corrections would likely not influence the thresh-
+old behavior other than slightly lowering the memory
+lifetimes relative to a hypothetical case where this re-
+striction is dropped. We explicitly verify this claim for
+our Fibonacci model below in Sec. VI.
+V.
+OUTLINE OF THE SIMULATION
+The goal of this work is to numerically determine
+a fault-tolerant error threshold for the error correcting
+2 Note that if one were to prioritize the level-0 corrections, higher-
+level correction could never be completed, as they would be un-
+done immediately after their first action is applied.
+(a)
+(b)
+(c)
+(d)
+Figure 4: Illustration of the transition rules on the
+decoding graph. The gray disks represent non-trivial
+syndromes, and the blue arrows represent the actions
+suggested by the decoder. The blue dotted lines
+represent the 3 × 3 colonies. (a-c) show a sequence of
+level-0 transition rules and possible outcomes of those
+actions. In (d) non-trivial anyons have been
+transported to two neighboring colony centers, the
+blue arrow represent a level-1 transition which could
+be applied at the end of the work period.
+code defined in Sec. II with pair-creation noise and
+measurement noise as outlined in Sec. III, and with the
+cellular automaton decoder introduced in Sec. IV. This
+is achieved by performing Monte-Carlo simulations to
+determine the average memory lifetime for a range of
+system sizes and error rates. These results then allow
+one to estimate the value of the error threshold.
+A single Monte-Carlo sample (with some fixed val-
+ues for the noise strength p and the measurement er-
+ror rate q) is obtained as follows. First, the state of
+the system is initialized as a ground state (i.e.: con-
+taining no anyons). Then a sequence of time steps is
+performed consisting of the application of pair-creation
+noise with rate p, a round of faulty syndrome measure-
+ments with error probability q, and finally a sequence
+of recovery operations. At the end of each time step,
+it is verified whether or not the state is considered cor-
+rectable, according to the criteria specified in Sec. III.
+
+7
+(a)
+(b)
+(c)
+Figure 5: (a) Level-0 colonies of size Q × Q. (b)
+Level-1 colonies defined as Q × Q level-0 colonies.
+(c) Renormalized lattice used for the level-1 transition
+rules.
+This sequence of time steps is continued until one of
+the following three outcomes occurs: (1) The largest
+connected group of anyons grows too large, rendering
+its classical simulation intractable 3; (2) A noise pro-
+cess or recovery operation induces a logical error by
+fusing a pair of anyons along a path that forms a non-
+contractible loop when combined with their fusion tree;
+(3) The verification procedure at the end of a time step
+fails. The memory lifetime is then set as the number of
+time steps that were completed. The course of a single
+Monte-Carlo sample in the simulation is summarized
+as pseudo-code in Alg. 1.
+3 Note that such cases are likely to correspond configurations in
+which the initial state cannot be recovered.
+Algorithm 1 Numerical simulation
+initialize state
+t = 0
+while correctable with clustering decoder & no logical
+errors made do
+t ← t + 1
+apply pair-creation noise
+perform faulty measurements
+for k = 0 : kmax do
+if t mod U k = 0 then
+update level-k syndromes
+apply level-k transition rules
+end if
+end for
+end while
+memory lifetime = t
+VI.
+NUMERICAL RESULTS
+The Monte Carlo simulation described above were
+performed for various system sizes with p = q. The
+following parameters were used:
+Q = 3 ,
+b = 7 ,
+Fc = 0.7 ,
+Fn = 0.2 .
+The resulting average memory lifetimes for L = 3,
+L = 9 and L = 27 are shown below in Fig. 6(a).
+These results clearly indicate that the code pre-
+sented in this work is indeed fault-tolerant. Further-
+more, while the current data is not sufficient to demon-
+strate a clear-cut fault-tolerant threshold, it still ex-
+hibits threshold behavior and is remarkably similar to
+the results previously obtained for the toric code [2]
+and the Ising topological code [32]. We estimate that
+the fault-tolerant threshold for the Fibonacci topologi-
+cal with pair-creation noise and measurement noise lies
+between p = 10−4 and p = 5 · 10−4, which corre-
+sponds to an inverse temperature between β = 9.2/∆
+and β = 7.6/∆. This is comparable to the threshold
+found for the Ising topological code [32], and only one
+order of magnitude below that for the surface code un-
+der similar circumstances [2]. For physical error rates
+near p = q = 10−4, corresponding to a temperature
+1/β one order of magnitude below the spectral gap, a
+code of linear size L = 27 yields logical error rates of
+the order 10−8.
+
+8
+(a) Average memory lifetime in function of the error strength,
+with p = q, for various system sizes. Each data point
+represents the average over 1000 Monte Carlo samples. The
+blue line shows the coherence time of a single physical qubit.
+The average memory lifetimes for p ≤ 10−3 were fitted to a
+function of the form f(p) ∼ p−a. The results for L = 3,
+L = 9 and L = 27 are shown as the green, yellow and red
+lines respectively.
+(b) Average lifetime in function of the error strength with
+p = q for various system sizes and (unphysical)
+instantaneous corrections. Each data point represents the
+average over 1000 Monte Carlo samples. The blue line shows
+the coherence time of a single physical qubit.
+Figure 6
+A second round of simulations was performed to de-
+termine average memory lifetimes with the assumption
+that all corrections happen instantaneously. While this
+is akin to the Abelian topological codes, where dis-
+tant anyons can be fused using unitary string-operators,
+this scenario is unphysical for non-Abelian anyons as
+they do not admit unitary string-operators.
+Never-
+theless, it is worth studying to which extend the re-
+sults in Fig. 6(a) are influenced by the restriction to
+non-instantaneous recovery operations.
+In Ref. [32]
+it was conjectured that allowing instantaneous correc-
+tions does not significantly change the qualitative be-
+havior of the average memory lifetimes as a function
+of the error rate, but mostly just increases the memory
+lifetimes. Our results, shown in Fig. 6(b), confirm this
+hypothesis.
+VII.
+DISCUSSION AND OUTLOOK
+The results presented in this work demonstrate that
+fault-tolerant error correction is possible for non-
+Abelian topological quantum error correcting codes
+supporting a universal logical gate set within their code
+space.
+For a code consisting of Fibonacci anyons
+in hexagonal tiles on a two-dimensional torus, sub-
+jected to pair creation noise and measurement noise,
+we demonstrated that the cellular automaton decoder
+detailed in this work is fault-tolerant. In particular, for
+physical error rates p ≤ 10−3, it was found that the
+logical memory lifetime surpasses the physical coher-
+ence time for all system sizes. When interpreting the
+pair-creation noise as resulting from a non-zero tem-
+perature, this pseudo-threshold corresponds to an in-
+verse temperature β = 6.9/∆, where ∆ is the energy
+required to create a pair of Fibonacci anyons. Further-
+more, our results suggest that this code admits a fault-
+tolerant quantum error correction threshold around p =
+10−4, or β = 9.2/∆, which is similar to the fault-
+tolerant threshold found for the Ising topological code
+[32].
+Several future research directions present them-
+selves. First, more research on a possible fault-tolerant
+threshold is necessary. Wile the numeric results pre-
+sented in this work provide a strong indication that a
+fault-tolerant error correction threshold exists, they do
+not conclusively prove its existence, nor do they pro-
+vide a precise estimate of its value. Hence, an impor-
+tant open problem is the formulation of a mathematical
+proof of its existence. Such proofs were previously for-
+mulated for the toric code [2] and for non-cyclic non-
+Abelian anyon models such as in the Ising topological
+
+9
+code [32]. Due to the cyclic nature of Fibonacci anyons
+(or any universal anyon model), however, the existing
+proofs are not sufficient.
+Second, it would be interesting to study different
+decoders in an identical setting.
+This includes both
+different cellular-automaton decoders such as those in
+Refs. [31, 34], as well as new decoders tailored to the
+Fibonacci topological code.
+Third, while this work demonstrates that the Fi-
+bonacci topological code can be operated fault-
+tolerantly as a quantum memory, results regarding
+its use for fault-tolerant quantum computing are still
+lacking.
+We envisage that fault-tolerant topological
+quantum computing at non-zero temperatures could be
+achieved by combining the code and decoding proce-
+dure presented in this work with the scheme for per-
+forming Dehn twists presented in Refs. [35–37]. Alter-
+natively, one can also perform transversal logical gates
+in a folded Fibonacci code [38].
+Finally, it would be of great interest to expand the
+current results to microscopic models for non-Abelian
+topological quantum error correction, such as the Fi-
+bonacci Turaev-Viro code [24].
+ACKNOWLEDGMENTS
+The authors would like to thank Guillaume Dauphi-
+nais and Jim Harrington for enlightening discussions
+on the cellular automaton decoder. The computational
+resources (Stevin Supercomputer Infrastructure) and
+services used in this work were provided by the Flem-
+ish Supercomputer Center (VSC), funded by Ghent
+University, the Research Foundation Flanders (FWO),
+and the Flemish Government. AS was supported by a
+fellowship of the Belgian American Educational Foun-
+dation. LB was supported by a PhD fellowship from
+the FWO. GZ was supported by the U.S. Department
+of Energy, Office of Science, National Quantum In-
+formation Science Research Centers, Co-design Center
+for Quantum Advantage (C2QA) under contract num-
+ber DE-SC0012704.
+Appendix A: Fibonacci anyons
+Below, we give with a brief overview of the topolog-
+ical aspects of our model. A thorough exposition of the
+theory of anyon models [39–42] is beyond the scope
+of this work, and we restrict to a basic description of
+the aspects of the Fibonacci model that are required for
+the specific purpose of our simulations. We refer to
+Ref. [33] for an in-depth discussion of anyonic fusion
+states on the torus.
+The Fibonacci anyon model has two particle types,
+1 and τ, that satisfy the fusion rules
+1 × 1 = 1 ,
+1 × τ = τ × 1 = τ ,
+τ × τ = 1 + τ .
+(A1)
+It is a non-Abelian anyon model, as the fusion of two
+τ-anyons can yield two distinct outcomes. In this case,
+the fusion space V c
+ab associated to the fusion of anyons
+a and b to c is one-dimensional, and is spanned by the
+state vector |a, b; c⟩ which we will represent graphi-
+cally as
+|a, b; c⟩ →
+b
+a
+c
+.
+(A2)
+When fusing several anyons, their total charge is a col-
+lective property of the anyons that does not depend on
+the specific order in which they are fused. Mathemati-
+cally, this is expressed through the associativity of the
+fusion rules,
+(a × b) × c = a × (b × c) .
+(A3)
+If we consider the case of three anyons a, b and c that
+fuse to a total charge d then this fusion may be car-
+ried out in two distinct ways, implying the existence
+of two equivalent decompositions of the associated fu-
+sion space V d
+abc in terms of fusion states (A2). These
+equivalent decompositions are related through a uni-
+tary transformation called an F-move, which is repre-
+sented graphically as
+b
+c
+e
+d
+a
+=
+�
+f
+F abe
+cdf
+f
+d
+b
+c
+a
+.
+(A4)
+The coefficients F abe
+cdf in this expression are called F-
+symbols of the anyon model. For the Fibonacci model
+they are given by
+F ττ1
+ττ1 = 1
+φ ,
+F τττ
+ττ1 = F ττ1
+τττ =
+1
+√φ ,
+F τττ
+τττ = − 1
+φ ,
+(A5)
+where all other F-symbols consistent with the fusion
+rules (A1) are equal to 1, and 0 otherwise.
+In addition to this recoupling one can also consider
+the exchange or braiding of pairs of anyons, which pre-
+serves their total charge.
+At the level of the fusion
+
+10
+space, such an exchange corresponds to a basis trans-
+formation to a basis associated to a different linear or-
+dering of the anyons4. Within V c
+ab there are two such
+possible basis transformations,
+b
+a
+c
+= Rab
+c
+c
+a
+b
+=
+�
+Rba
+c
+�∗
+c
+b
+a
+,
+(A6)
+which we will refer to as a clockwise and a counter-
+clockwise swap respectively. For the Fibonacci model
+the R-symbols appearing in these expressions are given
+by
+Rττ
+1
+= e
+4πi
+5 ,
+Rττ
+τ
+= e− 3πi
+5 ,
+(A7)
+where all other Rab
+c
+allowed by the fusion rules are
+equal to 1, and 0 otherwise.
+As we are dealing with a system that allows for an
+extensive amount of anyonic excitations, we will be in-
+terested in fusion states of many anyons, a1, a2, ..., an,
+with some total charge c.
+This gives rise to an ex-
+ponentially large topological Hilbert space V c
+a1a2···an
+spanned basis states of the form
+a1
+a2
+c
+b1
+a3
+an−1
+an
+b2
+bn−3
+bn−2
+.
+(A8)
+When dealing with anyonic fusion states on surfaces of
+higher genus, one must take into account additional de-
+grees of freedom in these states that are related to the
+anyonic charge that runs along non-contractible cycles.
+On a torus, this results in two distinct descriptions of
+fusion states, which are related through a basis change.
+These are known as the inside and outside bases, and
+are depicted in Fig. 7. A detailed discussion can be
+found in Ref. [33]. We will simply refer to these ad-
+ditional degrees of freedom as the handle labels of the
+state.
+4 As opposed to the more conventional view of braiding as an active
+transformation that maps between different fusion spaces, we have
+opted for the equivalent framework of braiding as a passive basis
+transformation, as the latter is more appropriate in view of our
+specific model and numerical simulations.
+(a)
+(b)
+Figure 7: Two possible basis choices for anyons on a
+torus: a) the inside basis, b) the outside basis.
+For our current purpose, we won’t need the full de-
+scription of said handle labels. It is sufficient for us to
+pick one basis, and subsequently set the total charge of
+all anyons to the vacuum. We are left with only a single
+label (1 or τ) representing the anyonic charge flowing
+along the non-contractible cycle associated with our
+basis choice. This is precisely the origin of the twofold
+degeneracy of the anyonic vacuum on the torus, which
+we have taken to be our code space.
+In the this work we will always start from the code
+state corresponding to a trivial handle label.
+As a
+change in handle label at any point during the error cor-
+rection procedure must be the consequence of a topo-
+logically non-trivial process which constitutes a logi-
+cal error, any simulation is aborted at the occurrence
+of such an event, meaning that all handle labels can be
+safely ignored in the remainder of our discussion.
+A basic functionality required for the simulation of
+the error correction procedure is the ability to correctly
+sample a measurement of the total charge of a given set
+of anyons. Starting from a given fusion state, this can
+be achieved by first transforming to a basis in which the
+relevant anyons are fused sequentially. For a system
+of many anyons these reordering basis transformations
+are obtained by combining Eqs. (A4) and (A6), giving
+
+11
+rise to the clockwise swap
+aj
+b3
+b2
+b1
+aj+1
+=
+�
+b′
+2
+Bb1ajb2
+aj+1b3b′
+2
+aj
+aj+1
+b3
+b2
+b1
+,
+(A9)
+and the counterclockwise swap
+aj
+b3
+b2
+b1
+aj+1
+=
+�
+b′
+2
+�
+Bb1ajb2
+aj+1b3b′
+2
+�∗
+aj
+aj+1
+b3
+b2
+b1
+,
+(A10)
+where
+Bb1ajb2
+aj+1b3b′
+2 =
+�
+c
+F aj+1ajc
+b3b1b′
+2
+Rajaj+1
+c
+F b1ajb2
+aj+1b3c .
+(A11)
+By performing a certain set of these basis transforma-
+tions, the fusion order can always be made consistent
+with the group of anyons of which we want to mea-
+sure the total charge. Subsequently, the fusion state is
+recoupled such that the relevant group of anyons is con-
+nected to the rest of the state by a single edge c. For a
+charge measurement of a pair of anyons this recoupling
+takes the form
+aj
+b3
+b2
+b1
+aj+1
+=
+�
+c
+F b1ajb2
+aj+1b3c
+aj
+b3
+b1
+aj+1
+c
+,
+(A12)
+and charge measurements of larger groups of anyons
+simply require consecutive applications of the recou-
+pling identity (A4). Finally, the charge outcome is ob-
+tained sampling the total charge c from the probability
+distribution corresponding to the resulting superposi-
+tion of fusion states.
+Appendix B: Classical simulatbility
+It is well known that Fibonacci anyons are universal
+for quantum computation [3]. One might therefore be
+tempted to conclude that the classical simulation of the
+topological Fibonacci code described in Sec. II with
+pair-creation noise is unlikely to succeed. However,
+as was noted in Ref. [23], the simulation of noise and
+error correction processes does not require the simula-
+tion of general anyon dynamics. In particular, individ-
+ual noise processes create distinct connected groups of
+anyons with vacuum total charge (or extend such exist-
+ing groups). These groups correspond to anyons that
+have interacted at some point during their lifetime, and
+must thus only be merged whenever a noise or error
+correction process involves two members from discon-
+nected groups. Since each connected group has a triv-
+ial total charge, braiding between disconnected groups
+is trivial. Hence, the total fusion space factorizes into a
+tensor product of fusion spaces of individual connected
+groups, and we are only required to simulate anyon dy-
+namics within each of these groups separately. This
+factorization of the fusion space is illustrated in Fig. 8.
+The creation and subsequent merging of discon-
+nected groups of anyons by noise and recovery pro-
+cesses can be thought of as a kind of percolation pro-
+cess.
+Hence, below the percolation threshold, one
+expects that the size of the largest connected group
+scales as O(log(L)) (with variance O(1)), where L
+is the linear system size [43]. As this is a probabilis-
+tic statement, there will be instances where the largest
+connected group has a size larger than O(log(L), but
+the probability of such events is suppressed exponen-
+tially with the system size L. This logarithmic scal-
+ing of the largest cluster size s = O(log(L)) coun-
+ters the exponential scaling of the dimension of the fu-
+sion space d = O(exp(s)) for individual connected
+groups. Therefore, the fusion spaces of individual con-
+nect groups will have dimension dim = O(poly(L)),
+meaning that the dynamics within connected groups
+can be simulated efficiently.
+Exploiting the tensor product structure within the to-
+tal fusion space, requires the use of basis for the any-
+onic fusion space which reflects this structure and can
+be updated dynamically to keep track of noise and re-
+covery processes. This is best achieved by using the
+framework of curve diagrams, which were introduced
+in Ref. [23] (also see Ref. [44] for a more rigorous
+treatment using the language of modular functors), and
+also discussed extensively in Ref. [24]. Since these are
+merely a technical tool for keeping track of the most
+appropriate basis during the numerical simulations, we
+will refrain from discussing them here. Interested read-
+ers are referred to the aforementioned references for
+details.
+
+12
+V ab
+1 ⊗ V cde
+1
+⊗ V fg
+1
+V ab
+1 ⊗ V cde
+1
+⊗ V fg
+1
+(a)
+a
+b
+c
+d
+e
+f
+g
+x
+y
+V abcdefg
+1
+= ⊕x,y
+�
+V ab
+x ⊗ V x cde
+y
+⊗ V y fg
+1
+�
+(b)
+V ab
+1 ⊗ V cde
+1
+⊗ V fg
+1
+a
+b
+c
+d
+e
+f
+g
+(c)
+Figure 8: (a) A set of noise processes. (b) A generic
+state in the full fusion space of all anyons created by
+the noise processes involves superpositions in the
+labels x and y. (c) The factorized Hilbert space which
+is sufficient to represent the state.
+Appendix C: Transition rules for Q = 3
+Below, we give a full definition of the transition rules
+for Q = 3. It is possible to define more general tran-
+sition rules that apply for any colony size, examples of
+such rules can be found in Refs. [2] and [32]. However,
+since the numerical simulations performed in this work
+were performed to Q = 3 we have taken the freedom
+to tailor the transition rules to this case specifically.
+• North-West
+if sk,c(ρ) = 0, do nothing;
+else if sk,n(ρ + (0, −1)) ̸= 0, do nothing;
+else if sk,n(ρ + (−1, 0)) ̸= 0, do nothing;
+else if sk,n(ρ + (0, 1)) ̸= 0, move east;
+else if sk,n(ρ + (1, 0)) ̸= 0, move south;
+else if sk,n(ρ + (1, −1)) ̸= 0, do nothing;
+else if sk,n(ρ + (−1, −1)) ̸= 0, do nothing;
+else if sk,n(ρ + (−1, 1)) ̸= 0, do nothing;
+else move south;
+• North
+if sk,c(ρ) = 0, do nothing;
+else if sk,n(ρ + (−1, 0)) ̸= 0, do nothing;
+else if sk,n(ρ + (0, −1)) ̸= 0, do nothing;
+else if sk,n(ρ + (0, 1)) ̸= 0, do nothing;
+else if sk,n(ρ + (−1, 1)) ̸= 0, do nothing;
+else if sk,n(ρ + (−1, −1)) ̸= 0, do nothing;
+else move south;
+• North-East
+if sk,c(ρ) = 0, do nothing;
+else if sk,n(ρ + (0, 1)) ̸= 0, move east;
+else if sk,n(ρ + (−1, 1)) ̸= 0, move north-east;
+else if sk,n(ρ + (1, 1)) ̸= 0, move south;
+else if sk,n(ρ + (−1, 0)) ̸= 0, do nothing;
+else if sk,n(ρ + (0, −1)) ̸= 0, move west;
+else if sk,n(ρ + (1, 0)) ̸= 0, move south;
+else if sk,n(ρ + (−1, −1)) ̸= 0, do nothing;
+else move south-west;
+• East
+if sk,c(ρ) = 0, do nothing;
+else if sk,n(ρ + (0, 1)) ̸= 0, move east;
+else if sk,n(ρ + (−1, 1)) ̸= 0, move north-east;
+else if sk,n(ρ + (1, 1)) ̸= 0, move south;
+else if sk,n(ρ + (−1, 0)) ̸= 0, do nothing;
+else if sk,n(ρ + (1, 0)) ̸= 0, do nothing;
+else move west;
+• South-East
+if sk,c(ρ) = 0, do nothing;
+else if sk,n(ρ + (0, 1)) ̸= 0, move east;
+else if sk,n(ρ + (1, 0)) ̸= 0, move south;
+else if sk,n(ρ + (−1, 1)) ̸= 0, move north-east;
+else if sk,n(ρ + (1, −1)) ̸= 0, move south-west;
+else if sk,n(ρ + (1, 1)) ̸= 0, move south;
+else if sk,n(ρ + (0, −1)) ̸= 0, move west;
+else move north;
+
+13
+• South
+if sk,c(ρ) = 0, do nothing;
+else if sk,n(ρ + (1, 0)) ̸= 0, move south;
+else if sk,n(ρ + (1, −1)) ̸= 0, move south-west;
+else if sk,n(ρ + (1, 1)) ̸= 0, move east;
+else if sk,n(ρ + (0, −1)) ̸= 0, do nothing;
+else if sk,n(ρ + (0, 1)) ̸= 0, do nothing;
+else move north;
+• South-West
+if sk,c(ρ) = 0, do nothing;
+else if sk,n(ρ + (1, 0)) ̸= 0, move south;
+else if sk,n(ρ + (1, −1)) ̸= 0, move south-west;
+else if sk,n(ρ + (1, 1)) ̸= 0, move east;
+else if sk,n(ρ + (0, −1)) ̸= 0, do nothing;
+else if sk,n(ρ + (−1, 0)) ̸= 0, move north;
+else if sk,n(ρ + (0, 1)) ̸= 0, move east;
+else if sk,n(ρ + (−1, −1)) ̸= 0, do nothing;
+else move north-east;
+• West
+if sk,c(ρ) = 0, do nothing;
+else if sk,n(ρ + (0, −1)) ̸= 0, do nothing;
+else if sk,n(ρ + (−1, 0)) ̸= 0, do nothing;
+else if sk,n(ρ + (1, 0)) ̸= 0, do nothing;
+else if sk,n(ρ + (1, −1)) ̸= 0, do nothing;
+else if sk,n(ρ + (−1, −1)) ̸= 0, do nothing;
+else move east;
+[1] P. G´acs, Reliable computation with cellular automata,
+Journal of Computer and System Sciences 32, 15
+(1986).
+[2] J. W. Harrington, Analysis of quantum error-correcting
+codes: symplectic lattice codes and toric codes, Ph.D.
+thesis, California Institute of Technology (2004).
+[3] M. H. Freedman, M. Larsen, and Z. Wang, A Modu-
+lar Functor Which is Universal for Quantum Compu-
+tation, Communications in Mathematical Physics 227,
+605 (2002).
+[4] M. H. Freedman, A. Kitaev, and Z. Wang, Simulation of
+topological field theories by quantum computers, Com-
+munications in Mathematical Physics 227, 587 (2002).
+[5] A. Kitaev, Fault-tolerant quantum computation by
+anyons, Annals of Physics 303, 2 (2003).
+[6] S. B. Bravyi and A. Y. Kitaev, Quantum codes on a
+lattice with boundary, arXiv preprint quant-ph/9811052
+(1998).
+[7] E. Dennis, A. Kitaev, A. Landahl, and J. Preskill,
+Topological quantum memory, Journal of Mathematical
+Physics 43, 4452 (2002).
+[8] D. S. Wang, A. G. Fowler, A. M. Stephens, and L. C.
+Hollenberg, Threshold error rates for the toric and sur-
+face codes, arXiv preprint arXiv:0905.0531 (2009).
+[9] A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N.
+Cleland, Surface codes: Towards practical large-scale
+quantum computation, Phys. Rev. A 86, 032324 (2012).
+[10] J. R. Wootton and D. Loss, High threshold error correc-
+tion for the surface code, Physical review letters 109,
+160503 (2012).
+[11] H. Anwar, B. J. Brown, E. T. Campbell, and D. E.
+Browne, Fast decoders for qudit topological codes, New
+Journal of Physics 16, 063038 (2014).
+[12] S. Bravyi, M. Suchara, and A. Vargo, Efficient algo-
+rithms for maximum likelihood decoding in the surface
+code, Physical Review A 90, 032326 (2014).
+[13] J. Wootton, A simple decoder for topological codes, En-
+tropy 17, 1946 (2015).
+[14] R. S. Andrist, J. R. Wootton, and H. G. Katzgraber,
+Error thresholds for abelian quantum double models:
+Increasing the bit-flip stability of topological quantum
+memory, Physical Review A 91, 042331 (2015).
+[15] D. E. Gottesman, Stabilizer Codes and Quantum Error
+Correction, Ph.D. thesis, California Institute of Tech-
+nology (1997).
+[16] S. Bravyi and A. Kitaev, Universal quantum computa-
+tion with ideal clifford gates and noisy ancillas, Physical
+Review A 71, 022316 (2005).
+[17] A. Kubica and M. E. Beverland, Universal transversal
+gates with color codes: A simplified approach, Physical
+Review A 91, 032330 (2015).
+[18] H. Bomb´ın, Dimensional jump in quantum error correc-
+tion, New Journal of Physics 18, 043038 (2016).
+[19] E. T. Campbell, B. M. Terhal, and C. Vuillot, Roads
+towards fault-tolerant universal quantum computation,
+Nature 549, 172 (2017), 1612.07330.
+[20] A. Hutter and J. R. Wootton, Continuous error correc-
+tion for ising anyons, Physical Review A 93, 042327
+(2016).
+[21] J. R. Wootton, J. Burri, S. Iblisdir, and D. Loss, Error
+correction for non-abelian topological quantum compu-
+tation, Phys. Rev. X 4, 011051 (2014).
+[22] C. G. Brell, S. Burton, G. Dauphinais, S. T. Flammia,
+and D. Poulin, Thermalization, error correction, and
+memory lifetime for ising anyon systems, Phys. Rev. X
+4, 031058 (2014).
+[23] S. Burton, C. G. Brell, and S. T. Flammia, Classical sim-
+ulation of quantum error correction in a fibonacci anyon
+code, Phys. Rev. A 95, 022309 (2017).
+[24] A. Schotte, G. Zhu, L. Burgelman, and F. Verstraete,
+Quantum Error Correction Thresholds for the Univer-
+sal Fibonacci Turaev-Viro Code, Physical Review X 12,
+
+14
+021012 (2022).
+[25] N. E. Bonesteel and D. P. DiVincenzo, Quantum circuits
+for measuring Levin-Wen operators, Physical Review B
+86, 165113 (2012).
+[26] R.
+Verresen,
+N.
+Tantivasadakarn,
+and
+A.
+Vish-
+wanath, Efficiently preparing ghz, topological and frac-
+ton states by measuring cold atoms, arXiv preprint
+arXiv:2112.03061 (2021).
+[27] N. Tantivasadakarn, R. Verresen, and A. Vishwanath,
+The shortest route to non-abelian topological order on
+a quantum processor, arXiv preprint arXiv:2209.03964
+(2022).
+[28] R. Raussendorf and J. Harrington, Fault-tolerant quan-
+tum computation with high threshold in two dimen-
+sions, Physical review letters 98, 190504 (2007).
+[29] A. G. Fowler, A. M. Stephens, and P. Groszkowski,
+High-threshold universal quantum computation on the
+surface code, Physical Review A 80, 052312 (2009).
+[30] F. H. Watson, H. Anwar, and D. E. Browne, Fast fault-
+tolerant decoder for qubit and qudit surface codes, Phys-
+ical Review A 92, 032309 (2015).
+[31] M. Herold, M. J. Kastoryano, E. T. Campbell, and
+J. Eisert, Cellular automaton decoders of topological
+quantum memories in the fault tolerant setting, New
+Journal of Physics 19, 063012 (2017).
+[32] G. Dauphinais and D. Poulin, Fault-tolerant quantum
+error correction for non-abelian anyons, Comm. Math.
+Phys. 355, 519 (2017).
+[33] R. N. Pfeifer, O. Buerschaper, S. Trebst, A. W. Ludwig,
+M. Troyer, and G. Vidal, Translation invariance, topol-
+ogy, and protection of criticality in chains of interacting
+anyons, Phys. Rev. B 86, 155111 (2012).
+[34] M. Herold, E. T. Campbell, J. Eisert, and M. J. Kas-
+toryano, Cellular-automaton decoders for topological
+quantum memories, npj Quantum information 1, 1
+(2015).
+[35] G. Zhu, A. Lavasani, and M. Barkeshli, Universal logi-
+cal gates on topologically encoded qubits via constant-
+depth unitary circuits, Phys. Rev. Lett. 125, 050502
+(2020).
+[36] G. Zhu, A. Lavasani, and M. Barkeshli, Instantaneous
+braids and Dehn twists in topologically ordered states,
+Phys. Rev. B 102, 075105 (2020).
+[37] A. Lavasani, G. Zhu, and M. Barkeshli, Universal log-
+ical gates with constant overhead: instantaneous dehn
+twists for hyperbolic quantum codes, Quantum 3, 180
+(2019).
+[38] G. Zhu, M. Hafezi, and M. Barkeshli, Quantum origami:
+Transversal gates for quantum computation and mea-
+surement of topological order, Phys. Rev. Research 2,
+013285 (2020).
+[39] V. G. Turaev and O. Y. Viro, State sum invariants of 3-
+manifolds and quantum 6j-symbols, Topology 31, 865
+(1992).
+[40] A. Kitaev, Anyons in an exactly solved model and be-
+yond, Annals of Physics 321, 2 (2006), arXiv:cond-
+mat/0506438.
+[41] Z. Wang, Topological Quantum Computation (Ameri-
+can Mathematical Soc., 2010).
+[42] V. Turaev and A. Virelizier, Monoidal Categories and
+Topological Field Theory, Progress in Mathematics
+(Birkh¨auser Basel, 2017).
+[43] M. Z. Bazant, Largest cluster in subcritical percolation,
+Phys. Rev. E 62, 1660 (2000).
+[44] S. Burton, A short guide to anyons and modular functors
+(2016), arxiv:1610.05384 [quant-ph].
+
diff --git a/8dAyT4oBgHgl3EQfQvbW/content/tmp_files/load_file.txt b/8dAyT4oBgHgl3EQfQvbW/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..f57061efbdb9d226a058e72d12237874d43003c1
--- /dev/null
+++ b/8dAyT4oBgHgl3EQfQvbW/content/tmp_files/load_file.txt
@@ -0,0 +1,673 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf,len=672
+page_content='Fault-tolerant error correction for a universal non-Abelian topological quantum  computer at finite temperature  Alexis Schotte,1, ∗ Lander Burgelman,2 and Guanyu Zhu1, 3, †  1IBM Quantum, IBM Almaden Research Center, San Jose, CA 95120, USA  2Department of Physics and Astronomy, Ghent University, Krijgslaan 281, 9000 Gent, Belgium  3IBM T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Watson Research Center, Yorktown Heights, NY 10598, USA  We study fault-tolerant error correction in a quantum memory constructed as a two-dimensional model of Fi-  bonacci anyons on a torus, in the presence of thermal noise represented by pair-creation processes and measure-  ment errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' The correction procedure is based on the cellular automaton decoders originating in the works of  G´acs [1] and Harrington [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Through numerical simulations, we observe that this code behaves fault-tolerantly  and that threshold behavior is likely present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Hence, we provide strong evidence for the existence of a fault-  tolerant universal non-Abelian topological quantum computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  INTRODUCTION  Anyons are emergent quasi-particles that exist in  two-dimensional condensed matter systems and whose  exchange statistics generalize that of Bosons and  Fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' These particles have spurred much interest  due to their potential applications for quantum compu-  tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' In particular, it was found that with certain types  of non-Abelian anyons, a universal quantum compu-  tation can be performed by braiding and fusing these  particles [3–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' An intriguing benefit of this paradigm  is that, due to their topological nature, computations  are intrinsically robust to perturbations at zero tem-  perature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' At non-zero temperature, however, thermal  anyonic excitations can corrupt the computation by  performing non-trivial braids with the computational  anyons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Since systems exhibiting anyonic excitations  have a spectral gap ∆, this source of errors can be sup-  pressed to some extent at temperatures T ≪ ∆/kB  as the density of thermal anyons scales as e−∆/kBT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  Alas, this passive protection does not suffice, because  the presence of thermal anyons is unavoidable at non-  zero temperatures when scaling up the size of com-  putations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Therefore, proficient active error correction  schemes for non-Abelian models are paramount for the  realization of topological quantum computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  Besides their envisaged use for topological quantum  computation, topologically ordered systems (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=', those  that support anyonic excitations on top of their ground  space) are also of much interest for quantum error cor-  rection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' In particular, one of the characteristics of such  systems is a robust ground space degeneracy, which al-  ∗ alexis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='schotte@posteo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='net  † guanyu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='zhu@ibm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='com  lows one to use their ground space as the code space of  an error correcting code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' This realization led to the dis-  covery of topological quantum error correcting codes,  which encode logical quantum states in topologically  ordered states of a system of qudits (typically arranged  on a two-dimensional lattice).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Since their discovery  in the 90s, most research has focused exclusively on  Abelian topological codes such as the surface code and  the color code [5–14], which admit an elegant charac-  terization in terms of the stabilizer formalism [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Due  to their geometrical locality and high error thresholds,  these codes are considered to be promising candidates  for protecting quantum information from noise in error-  corrected quantum computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' One of the drawbacks  of Abelian topological codes, however, is that they do  not allow one to execute a universal set of logical gates  in a protected fashion in two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Hence, they  must be supplemented with additional protocols such  as magic state distillation [16] or code switching to  higher-dimensional codes [17, 18] , which introduce a  large space-time overhead [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Alternatively, there ex-  ist non-Abelian topological codes which do not suffer  from this inherent limitation, and are able to perform  a universal gate set natively within their code space in  two dimensions [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' The trade-off is that such codes go  beyond the stabilizer formalism and are therefore very  hard to simulate classically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  While active error correction in Abelian anyon mod-  els and Abelian topological codes has been studied  extensively, quantum error correction based on non-  Abelian anyon models has not enjoyed the same fo-  cus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Nevertheless, important progress has been made  over the last decade, including both analytical proofs  and numerical demonstrations of threshold behavior  for various non-Abelian topological error correcting  codes [20–24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Moreover, syndrome extraction circuits  for such non-Abelian string-net codes have been devel-  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='00054v1  [quant-ph]  30 Dec 2022  2  oped in recent years [24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' In addition, state prepa-  ration for non-Abelian codes based on the Kitaev quan-  tum double models via measurements has also been  proposed recently for the experimental implementa-  tion on qubit lattices [26, 27], although further devel-  opment is still needed in the context of fault-tolerant  state preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Notably, previous studies in this field  already include codes based on the Fibonacci anyon  model, which is universal for quantum computation  [23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' In particular, a quantum memory of qubits  supporting doubled Fibonacci anyonic excitations was  found to have a threshold that lies remarkably close to  that of the surface code under similar assumptions [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  These results, however, all assume perfect syndrome  measurements, which are topological charge measure-  ments in this context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' As we aim to model more re-  alistic scenarios, we must take faulty measurements  into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Again, much is known in the case  of Abelian topological codes [2, 7, 28–31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' For their  non-Abelian counterparts, one key result stands out: in  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' [32] a proof was formulated that topological codes  based on non-cyclic anyon models admit a error cor-  rection thresholds with faulty topological charge mea-  surements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' While this result is remarkable, non-cyclic  anyon models are not universal for quantum computa-  tion, and it remains an open question whether similar  claims can be made for universal models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  In this work, we take a step towards demonstrat-  ing that fault-tolerance is indeed possible for universal  non-Abelian topological codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' To this end, we define  a quantum memory constructed as a two-dimensional  model of Fibonacci anyons on a torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' We study ac-  tive continuous quantum error correction on this model  in the presence of thermal noise represented by pair-  creation processes, and with faulty syndrome measure-  ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' The correction procedure is based on the cel-  lular automaton decoders originating in the works of  G´acs [1] and Harrington [2], and further studied in the  context of non-Abelian models in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Through  numerical simulations, we study how the average mem-  ory lifetime changes with the error rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' The results in-  dicate that this code is indeed fault-tolerant, which is  strong evidence for the existence of fault-tolerant uni-  versal non-Abelian codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  The structure of this work is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' II  we introduce the topological Fibonacci code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' We then  describe the details of the noise model in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' III and  introduce the cellular automaton decoder in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  We proceed by giving an outline of the numerical sim-  ulations performed in this work in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Finally, we  present the corresponding numerical results in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' VI  and conclude with a discussion in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  THE FIBONACCI CODE  We consider a two-dimensional model comprised of  hexagonal tiles laid out on the surface of a torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' The  resulting geometry can be represented as an L × L  hexagonal lattice with periodic boundary conditions in  both directions (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Each of these hexagonal tiles  can contain an excitation known as a Fibonacci anyon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  Anyons are point-like quasi-particle excitations  which can be characterized algebraically in terms of a  unitary modular tensor category (UMTC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' A thorough  description of anyon models using UMTCs goes be-  yond the scope of this work, however, some details are  given in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' For now, it is sufficient to state that  an anyon model specifies a set of anyon labels, also  referred to as particle types, which can fuse according  to a specific set of fusion rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' The Fibonacci anyon  model considered in this work contains two labels, 1  and τ, which obey the fusion rules  1×1 = 1 ,  1×τ = τ×1 = τ ,  τ×τ = 1+τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' (1)  In general, one can associate a vector space to a  given set of anyons, where the basis vectors are la-  beled by the different ways in which the anyons can  fuse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' This fusion space has a topological degeneracy,  and can therefore be used to robustly encode quan-  tum information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' In particular, for the Fibonacci anyon  model the anyonic vacuum on a two-dimensional torus  has a twofold degeneracy [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Starting from our two-  dimensional model, we can therefore define an error  correcting code whose code space is identified with the  anyonic vacuum on the torus and which encodes a sin-  gle logical qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' A basis for this code space can be  defined using Wilson line operators along the homo-  logically non-trivial cycles x and y shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' 1:  W a  x |1⟩x = Sa1  S11 |1⟩x ,  W a  x |τ⟩x = Saτ  S1τ |τ⟩x ,  a ∈ {1, τ} ,  (2)  We note that a different basis, {|0⟩y , |1⟩y}, can be  defined analogously by swapping the x and y labels  above, where the two bases are related through the  modular S matrix Sab =  y⟨a|b⟩x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' For the Fibonacci  anyon model its numerical values are  S =  1  �  1 + φ2  �  1  φ  φ −1  �  ,  (3)  where φ = 1 +  √  5  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  The action of the mapping class group on the any-  onic vacuum then corresponds to unitary operations on  3  x  y  Figure 1: The two homologically non-trivial cycles on  a torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  the code space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' For the Fibonacci category, any logical  unitary operator can be realized in this way, up to arbi-  trary precision [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Therefore, the quantum error cor-  recting code defined above natively supports universal  quantum computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  Errors in this code appear as spurious anyonic ex-  citations, which can corrupt the encoded informa-  tion if their world lines between creation and re-  annihilation are topologically non-trivial, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=', form a  non-contractible cycle 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' The objective of error cor-  rection is then to systematically remove these spurious  excitations, without corrupting the quantum memory in  the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' This correction is performed in an active  and continuous manner, and can be broken down into a  series of discrete steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' At each step, a suitable recov-  ery operation is performed based on a measured list of  positions and types of the excitations, called the error  syndrome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  We conclude this section by noting that the numer-  ical simulation of the error-correction process requires  the introduction of some additional manipulations on  fusion states of multiple Fibonacci anyons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' As these  are technical details that do not contribute to the in-  tuition of the procedure, we defer their definition to  Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  NOISE MODEL AND CORRECTABILITY  Having defined our model and code space, we now  turn to the description of the noise model used in our  1 Note that a pair of Fibonacci anyons can also fuse to a single non-  trivial anyon when one member of the pair has been transported  along a non-trivial cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Since the resulting state is no longer in  the code space, this does not constitute a logical operation on the  encoded information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' However, one can show that the encoded  information is irrevocably lost in case of such an event [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  We model continuous active error cor-  rection in our Fibonacci code as a sequence of time  steps, where each time step itself consists of three parts:  the application of pair-creation noise, faulty syndrome  measurement, and error correction respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  At each time step, first, for each edge of the hexago-  nal lattice graph a pair of anyons is created across this  edge with a probability p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Immediately after each pair  creation event, the resulting charge in the two affected  tiles is sampled, effectively collapsing all superposi-  tions of anyonic charge within each tile to either 1 or  τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' After the pair creation noise has been applied, faulty  syndrome extraction is simulated by generating a list of  the anyon charge in all tiles, and flipping each outcome  individually with a probability q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' In addition to the  charges that are correctly detected, the resulting faulty  syndrome can contain both “ghost defects” (indicating  a non-trivial charge when none is truly present) and  “missing defects” (failing to report a true non-trivial  charge).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Finally, this faulty syndrome is passed to a de-  coder, introduced in the following section, which then  performs a set of local operations based on the current  (and past) syndrome information in an attempt to move  the system back towards the initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  After each time step, the current state of the sys-  tem is copied and it is checked whether it is still cor-  rectable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' This is done by passing the copy to a clus-  tering decoder [22–24] and simulating a decoding pro-  cedure with perfect syndrome measurements starting  from this given initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' If this perfect decoding  is successful, the memory is considered intact and the  simulation is continued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' If perfect decoding is unsuc-  cessful, the memory is considered corrupted and the  simulation is aborted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' The memory lifetime is then de-  fined as the number of time steps after which a perfect  clustering decoder can no longer successfully restore  the initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  For a given pair of tiles which share an edge, the  process of pair creation across this edge corresponds to  the matrix elements  ⟨a′, b′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' c′| Upc |a, b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' c⟩ = δc,c′F aa1  ττa′F a′τa  b c b′ ,  (4)  where we have used the F-symbols of the Fibonacci  category, given in (A5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Here, |a, b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' c⟩ represents the  state where the affected tiles have anyon charges a  and b, respectively, with total charge c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' This then de-  fines the probability distribution according to which  outcomes a′ and b′ are sampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  Since our noise  model does not allow any superposition in the any-  onic charge of individual tiles, it should be considered  semi-classical rather than fully quantum-mechanical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  4  (a)  (b)  Figure 2: (a) Pair-creation events creating anyonic  excitations in neighboring tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' The dotted ellipse  represents a collapse to the total charge of the anyons  it contains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' A ghost defect is shown in blue, a missing  defect is highlighted in orange with a cross.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' (b) The  outcome of this noise process represented on the  decoding graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Note that the missing defect  (highlighted in orange) will (by definition) not be  visible in the syndrome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  Note, however, that this does not render our model  completely classical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Indeed, superpositions in the total  charge c of the affected tiles are an inherent part of the  state evolution that cannot be captured faithfully by any  classical probabilistic process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Furthermore, while the  extreme decoherence assumption for the anyon charge  in individual tiles greatly simplifies the numerical sim-  ulation outlined in this work, it was argued in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' [22]  that this decoherence is unlikely to have any tangible  influence on the observed memory lifetimes, as the es-  sential topological nature of the noise processes is still  captured correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  We note that this type of noise can be understood as  originating from the connection to a thermal bath with  inverse temperature β = 1/(kBT) determined by the  error rate p through the relation  p  1 − p = e−β∆ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  (5)  Here, ∆ represents the energy required to create a pair  of anyonic excitations and place them in neighboring  tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  To conclude this section, we emphasize that in the  case of non-Abelian error correction, even decoding  with perfect syndrome measurements is still an inher-  ently stochastic procedure due to the indeterminacy of  anyonic charge measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' This means that perfect  decoding can sometimes either be successful or unsuc-  cessful even starting from the same initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Our  definition of the memory lifetime therefore simply cor-  responds to a statistical estimate of the actual memory  lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Furthermore, it is not known which decoder  is optimal for the Fibonacci code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Hence, the choice  for the clustering decoder to verify the correctability of  states is, in a way, an arbitrary one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' This choice, how-  ever, is motivated by the recent discovery that the clus-  tering decoder yields high thresholds for a related er-  ror correcting code exhibiting doubled Fibonacci any-  onic excitations, and performed significantly better in  this context than decoders based on a perfect matching  strategy [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' In any case, one should keep in mind that  the memory lifetime as defined above, does not repre-  sent the true memory lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Instead the sub-optimal  verification process entails that it merely provides us  with a lower bound on the true value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  HARRINGTON’S CELLULAR AUTOMATON  DECODER  The model described above is paired with a decoder  which is a straight-forward adaptation of the cellu-  lar automaton decoder introduced in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Previ-  ously, this decoder has also been used for a similar  phenomenological model of Ising anyons in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' [32],  where the existence of an error correction threshold  was proven analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' At each time step during the  error correction simulation, based on the reported mea-  surement outcomes in the faulty syndrome, the decod-  ing algorithm will apply local transition rules to fuse  neighboring anyons or to move anyons to neighboring  tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  Intuitively these transition rules work as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  The lattice is divided into square colonies of size Q×Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  At each time step, the transition rules will attempt to  fuse neighboring non-trivial anyons, as observed in  the faulty syndrome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  If a non-trivial anyon has no  neighbors, the transition rules will move it to the cen-  ter of its colony.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  At larger timescales, higher-level  transition rules are applied on a renormalized lattice  where anyons located at colony centers will be fused  with anyons at neighboring colony centers, or moved  toward the center of their respective super-colonies,  which consist of Q × Q colonies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' This renormalization  scheme is then continued at higher levels until eventu-  ally the Qn × Qn super-colony covers the entire lattice  for some integer n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' To ensure the latter is possible, we  will always assume that the linear lattice size satisfies  L = Qn for some integer n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' An example of these pro-  cesses is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  To describe the action of the decoding algorithm  more precisely, we will define its action at different  renormalization levels k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' The level-0 transition rules  5  are those already discussed above and are applied at  every time step based on the reported faulty syndrome  obtained from the most recent round of faulty measure-  ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' The transition rules are applied to one location  at a time and take into consideration only the anyon  content of that site and of its eight neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' A de-  tailed definition of these rules is given in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' When  an anyon is moved from a site l to a neighboring site  l′, the (true) anyon content of site l is fused with that  of site l′ and the resulting charge is placed on site l′  while the charge of site l is restored to the vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  This happens irrespective of whether or not the syn-  drome for both sites was correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Hence, when the de-  coder attempts to move a ghost defect (a trivial charge  misidentified as a non-trivial one) to a neighboring site,  this process does not create additional excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' This  does not, however, mean that mistaking a trivial charge  for a non-trivial one has no negative consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' In-  deed, these wrong syndromes may cause the decoder  to stretch out existing errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  The level-1 transition rules are not applied in every  time step, but only when t is a multiple of a param-  eter called the work period, which we will denote by  U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' We require that U = b2 for some positive integer  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' One should think of U as the time scale at which a  coarse-graining is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Level-1 transition rules  act at on a coarse-grained lattice where the sites corre-  spond to the centers of the level-0 colonies, and these  are grouped into level-1 colonies of size Q2 × Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  Hence, the actions determined by the level-1 transition  rules involve pairs of level-0 colony centers separated  by a distance Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' An example of such a move is pro-  vided in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' 4(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' The transition rules themselves are  nearly identical to the level-0 rules, but are based on  two sets of level-1 syndromes s1,c and s1,n (defined  below) rather than one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' For a site l (which is a level-  0 colony center), the transition rules use s1,c(l) as the  anyon content of site, while the anyon content of its  neighbors (that is, the neighboring level-0 colony cen-  ters) is taken to be s1,n(l′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  The definitions of the level-1 syndromes s1,c and  s1,n require a pair of variables fc, fn ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  In-  tuitively these variables serve as detection thresholds  for the level-1 syndromes by determining the fraction  of measurements that must return a non-trivial out-  come at a site before it qualifies as a non-trivial level-1  syndrome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' The proper definition, however, is slightly  more complicated and uses a coarse-grained counting  method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Below, we give the precise definition of s1,c,  the definition of s1,n is entirely analogous (using fn  instead of fc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' We start by dividing the work period  U = b2, into b intervals of b time steps each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' For  each of these intervals, we say a non-trivial syndrome  is present at a colony center l if a non-trivial charge  was reported there for at least fcb of the b time steps in  the interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' When at least fcb of the b intervals have  a non-trivial syndrome, s1,c(l) is set to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' A visual  example of this coarse-grained counting procedure is  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  ≥ fc · b  < fc · b  ≥ fc · b  Figure 3: A visual example of the coarse-grained  counting procedure to determine the level-1 syndrome  for a level-0 colony center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' The row of dots represent  U time steps, divided in b intervals of size b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' The time  steps during which a non-trivial measurement  outcome was reported are indicated by the colored  dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' The crosses in the second row indicate in which  intervals the fraction of non-trivial measurement  outcomes is equal to or higher than fc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  The motivation for using two types of syndromes for  k > 0 is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Suppose that an error spans across  two neighboring colonies, which we will label ρ and ρ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  The level-0 transition rules will transport all resulting  anyons to the respective colony centers, where they can  now be acted upon by level-1 transition rules at the end  of the work period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Imagine that a non-trivial anyon  is now present at both colony centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' When consid-  ering the level-1 transition rules acting on ρ, there are  four possible scenarios for the syndromes s1,c(ρ) and  s1,n(ρ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' In case s1,c(ρ) = 0, the transition rules act  trivially on ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' If both s1,c(ρ) = 1 and s1,n(ρ′) = 1  then the transition rules will be applied correctly and  the anyons will be fused.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' However, if s1,c(ρ) = 1 but  s1,n(ρ′) = 0, the transition rules may move the anyon  in ρ away from ρ′, thereby increasing the weight of  the error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Hence, it is desirable to set fc > fn to de-  crease the odds that when a level-k syndrome reports  an non-trivial anyon at a colony center, the level-k syn-  drome for its neighbors are false negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' We must  be careful not to set fc too high or fn too low, how-  ever.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' If we choose fc to high, low-weight errors could  cause s1,c to never report any non-trivial charges, de-  laying any necessary corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Similarly, setting fn  too low will result in low-weight error triggering many  false positives for s1,n, which can cause the decoder to  make wrong decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  6  Level-k  transition  rules  are  applied  when  t  mod U k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' They operate on a renormalized lattice  that uses the centers of level-(k − 1) colonies as sites,  and groups these into level-k colonies of size Qk ×Qk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  The level-k syndromes sk,c and sk,n are determined by  the coarse-grained counting method described above,  using bk intervals of bk time steps each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' For linear sys-  tem size L, k ranges from 0 to kmax = logQ(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  It is important to note that non-Abelian anyons do  not allow for instantaneous moves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Indeed, while one  can construct a unitary string operator for Abelian  anyons, no such operator can be constructed for the  non-Abelian case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' This discrepancy can be traced back  to the fact that fusion outcomes are non-deterministic  for non-Abelian anyons, implying it is not possible to  move an non-Abelian anyon by annihilating it with one  member of a particle-antiparticle pair (as is done in  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=', the surface code).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  Therefore, the actions determined by level-k transi-  tion rules, for k > 0 cannot be applied withing a sin-  gle time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Instead, they will be broken up into a  sequence of moves involving only pairs of neighbor-  ing sites which will be applied in Qk consecutive time  steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' We further limit the model by requiring that the  number recovery operations affecting a single tile in the  lattice (or site in the decoding graph), is no greater than  one in each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' This allows all recovery opera-  tions applied in one time step to be performed in par-  allel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Hence, we must define a hierarchy determining  which actions (moves or fusions between neighboring  tiles) get prioritized based on the renormalization level  from which they originated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' In our case, it was opted to  always prioritize correction processes from the highest  renormalization level 2  It was argued in [32] that the prohibition of instanta-  neous corrections would likely not influence the thresh-  old behavior other than slightly lowering the memory  lifetimes relative to a hypothetical case where this re-  striction is dropped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' We explicitly verify this claim for  our Fibonacci model below in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  OUTLINE OF THE SIMULATION  The goal of this work is to numerically determine  a fault-tolerant error threshold for the error correcting  2 Note that if one were to prioritize the level-0 corrections, higher-  level correction could never be completed, as they would be un-  done immediately after their first action is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  (a)  (b)  (c)  (d)  Figure 4: Illustration of the transition rules on the  decoding graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' The gray disks represent non-trivial  syndromes, and the blue arrows represent the actions  suggested by the decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' The blue dotted lines  represent the 3 × 3 colonies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' (a-c) show a sequence of  level-0 transition rules and possible outcomes of those  actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' In (d) non-trivial anyons have been  transported to two neighboring colony centers, the  blue arrow represent a level-1 transition which could  be applied at the end of the work period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  code defined in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' II with pair-creation noise and  measurement noise as outlined in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' III, and with the  cellular automaton decoder introduced in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' This  is achieved by performing Monte-Carlo simulations to  determine the average memory lifetime for a range of  system sizes and error rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' These results then allow  one to estimate the value of the error threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  A single Monte-Carlo sample (with some fixed val-  ues for the noise strength p and the measurement er-  ror rate q) is obtained as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' First, the state of  the system is initialized as a ground state (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' : con-  taining no anyons).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Then a sequence of time steps is  performed consisting of the application of pair-creation  noise with rate p, a round of faulty syndrome measure-  ments with error probability q, and finally a sequence  of recovery operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' At the end of each time step,  it is verified whether or not the state is considered cor-  rectable, according to the criteria specified in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  7  (a)  (b)  (c)  Figure 5: (a) Level-0 colonies of size Q × Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' (b)  Level-1 colonies defined as Q × Q level-0 colonies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  (c) Renormalized lattice used for the level-1 transition  rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  This sequence of time steps is continued until one of  the following three outcomes occurs: (1) The largest  connected group of anyons grows too large, rendering  its classical simulation intractable 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' (2) A noise pro-  cess or recovery operation induces a logical error by  fusing a pair of anyons along a path that forms a non-  contractible loop when combined with their fusion tree;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  (3) The verification procedure at the end of a time step  fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' The memory lifetime is then set as the number of  time steps that were completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' The course of a single  Monte-Carlo sample in the simulation is summarized  as pseudo-code in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  3 Note that such cases are likely to correspond configurations in  which the initial state cannot be recovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  Algorithm 1 Numerical simulation  initialize state  t = 0  while correctable with clustering decoder & no logical  errors made do  t ← t + 1  apply pair-creation noise  perform faulty measurements  for k = 0 : kmax do  if t mod U k = 0 then  update level-k syndromes  apply level-k transition rules  end if  end for  end while  memory lifetime = t  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  NUMERICAL RESULTS  The Monte Carlo simulation described above were  performed for various system sizes with p = q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' The  following parameters were used:  Q = 3 ,  b = 7 ,  Fc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='7 ,  Fn = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  The resulting average memory lifetimes for L = 3,  L = 9 and L = 27 are shown below in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' 6(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  These results clearly indicate that the code pre-  sented in this work is indeed fault-tolerant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Further-  more, while the current data is not sufficient to demon-  strate a clear-cut fault-tolerant threshold, it still ex-  hibits threshold behavior and is remarkably similar to  the results previously obtained for the toric code [2]  and the Ising topological code [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' We estimate that  the fault-tolerant threshold for the Fibonacci topologi-  cal with pair-creation noise and measurement noise lies  between p = 10−4 and p = 5 · 10−4, which corre-  sponds to an inverse temperature between β = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='2/∆  and β = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='6/∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' This is comparable to the threshold  found for the Ising topological code [32], and only one  order of magnitude below that for the surface code un-  der similar circumstances [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' For physical error rates  near p = q = 10−4, corresponding to a temperature  1/β one order of magnitude below the spectral gap, a  code of linear size L = 27 yields logical error rates of  the order 10−8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  8  (a) Average memory lifetime in function of the error strength,  with p = q, for various system sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Each data point  represents the average over 1000 Monte Carlo samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' The  blue line shows the coherence time of a single physical qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  The average memory lifetimes for p ≤ 10−3 were fitted to a  function of the form f(p) ∼ p−a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' The results for L = 3,  L = 9 and L = 27 are shown as the green, yellow and red  lines respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  (b) Average lifetime in function of the error strength with  p = q for various system sizes and (unphysical)  instantaneous corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Each data point represents the  average over 1000 Monte Carlo samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' The blue line shows  the coherence time of a single physical qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  Figure 6  A second round of simulations was performed to de-  termine average memory lifetimes with the assumption  that all corrections happen instantaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' While this  is akin to the Abelian topological codes, where dis-  tant anyons can be fused using unitary string-operators,  this scenario is unphysical for non-Abelian anyons as  they do not admit unitary string-operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  Never-  theless, it is worth studying to which extend the re-  sults in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' 6(a) are influenced by the restriction to  non-instantaneous recovery operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' [32]  it was conjectured that allowing instantaneous correc-  tions does not significantly change the qualitative be-  havior of the average memory lifetimes as a function  of the error rate, but mostly just increases the memory  lifetimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Our results, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' 6(b), confirm this  hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  DISCUSSION AND OUTLOOK  The results presented in this work demonstrate that  fault-tolerant error correction is possible for non-  Abelian topological quantum error correcting codes  supporting a universal logical gate set within their code  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  For a code consisting of Fibonacci anyons  in hexagonal tiles on a two-dimensional torus, sub-  jected to pair creation noise and measurement noise,  we demonstrated that the cellular automaton decoder  detailed in this work is fault-tolerant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' In particular, for  physical error rates p ≤ 10−3, it was found that the  logical memory lifetime surpasses the physical coher-  ence time for all system sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' When interpreting the  pair-creation noise as resulting from a non-zero tem-  perature, this pseudo-threshold corresponds to an in-  verse temperature β = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='9/∆, where ∆ is the energy  required to create a pair of Fibonacci anyons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Further-  more, our results suggest that this code admits a fault-  tolerant quantum error correction threshold around p =  10−4, or β = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='2/∆, which is similar to the fault-  tolerant threshold found for the Ising topological code  [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  Several future research directions present them-  selves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' First, more research on a possible fault-tolerant  threshold is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Wile the numeric results pre-  sented in this work provide a strong indication that a  fault-tolerant error correction threshold exists, they do  not conclusively prove its existence, nor do they pro-  vide a precise estimate of its value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Hence, an impor-  tant open problem is the formulation of a mathematical  proof of its existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Such proofs were previously for-  mulated for the toric code [2] and for non-cyclic non-  Abelian anyon models such as in the Ising topological  9  code [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Due to the cyclic nature of Fibonacci anyons  (or any universal anyon model), however, the existing  proofs are not sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  Second, it would be interesting to study different  decoders in an identical setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  This includes both  different cellular-automaton decoders such as those in  Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' [31, 34], as well as new decoders tailored to the  Fibonacci topological code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  Third, while this work demonstrates that the Fi-  bonacci topological code can be operated fault-  tolerantly as a quantum memory, results regarding  its use for fault-tolerant quantum computing are still  lacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  We envisage that fault-tolerant topological  quantum computing at non-zero temperatures could be  achieved by combining the code and decoding proce-  dure presented in this work with the scheme for per-  forming Dehn twists presented in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' [35–37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Alter-  natively, one can also perform transversal logical gates  in a folded Fibonacci code [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  Finally, it would be of great interest to expand the  current results to microscopic models for non-Abelian  topological quantum error correction, such as the Fi-  bonacci Turaev-Viro code [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  The authors would like to thank Guillaume Dauphi-  nais and Jim Harrington for enlightening discussions  on the cellular automaton decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' The computational  resources (Stevin Supercomputer Infrastructure) and  services used in this work were provided by the Flem-  ish Supercomputer Center (VSC), funded by Ghent  University, the Research Foundation Flanders (FWO),  and the Flemish Government.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' AS was supported by a  fellowship of the Belgian American Educational Foun-  dation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' LB was supported by a PhD fellowship from  the FWO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' GZ was supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Department  of Energy, Office of Science, National Quantum In-  formation Science Research Centers, Co-design Center  for Quantum Advantage (C2QA) under contract num-  ber DE-SC0012704.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  Appendix A: Fibonacci anyons  Below, we give with a brief overview of the topolog-  ical aspects of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' A thorough exposition of the  theory of anyon models [39–42] is beyond the scope  of this work, and we restrict to a basic description of  the aspects of the Fibonacci model that are required for  the specific purpose of our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' We refer to  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' [33] for an in-depth discussion of anyonic fusion  states on the torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  The Fibonacci anyon model has two particle types,  1 and τ, that satisfy the fusion rules  1 × 1 = 1 ,  1 × τ = τ × 1 = τ ,  τ × τ = 1 + τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  (A1)  It is a non-Abelian anyon model, as the fusion of two  τ-anyons can yield two distinct outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' In this case,  the fusion space V c  ab associated to the fusion of anyons  a and b to c is one-dimensional, and is spanned by the  state vector |a, b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' c⟩ which we will represent graphi-  cally as  |a, b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' c⟩ →  b  a  c  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  (A2)  When fusing several anyons, their total charge is a col-  lective property of the anyons that does not depend on  the specific order in which they are fused.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Mathemati-  cally, this is expressed through the associativity of the  fusion rules,  (a × b) × c = a × (b × c) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  (A3)  If we consider the case of three anyons a, b and c that  fuse to a total charge d then this fusion may be car-  ried out in two distinct ways, implying the existence  of two equivalent decompositions of the associated fu-  sion space V d  abc in terms of fusion states (A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' These  equivalent decompositions are related through a uni-  tary transformation called an F-move, which is repre-  sented graphically as  b  c  e  d  a  =  �  f  F abe  cdf  f  d  b  c  a  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  (A4)  The coefficients F abe  cdf in this expression are called F-  symbols of the anyon model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' For the Fibonacci model  they are given by  F ττ1  ττ1 = 1  φ ,  F τττ  ττ1 = F ττ1  τττ =  1  √φ ,  F τττ  τττ = − 1  φ ,  (A5)  where all other F-symbols consistent with the fusion  rules (A1) are equal to 1, and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  In addition to this recoupling one can also consider  the exchange or braiding of pairs of anyons, which pre-  serves their total charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  At the level of the fusion  10  space, such an exchange corresponds to a basis trans-  formation to a basis associated to a different linear or-  dering of the anyons4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Within V c  ab there are two such  possible basis transformations,  b  a  c  = Rab  c  c  a  b  =  �  Rba  c  �∗  c  b  a  ,  (A6)  which we will refer to as a clockwise and a counter-  clockwise swap respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' For the Fibonacci model  the R-symbols appearing in these expressions are given  by  Rττ  1  = e  4πi  5 ,  Rττ  τ  = e− 3πi  5 ,  (A7)  where all other Rab  c  allowed by the fusion rules are  equal to 1, and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  As we are dealing with a system that allows for an  extensive amount of anyonic excitations, we will be in-  terested in fusion states of many anyons, a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=', an,  with some total charge c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  This gives rise to an ex-  ponentially large topological Hilbert space V c  a1a2···an  spanned basis states of the form  a1  a2  c  b1  a3  an−1  an  b2  bn−3  bn−2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  (A8)  When dealing with anyonic fusion states on surfaces of  higher genus, one must take into account additional de-  grees of freedom in these states that are related to the  anyonic charge that runs along non-contractible cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  On a torus, this results in two distinct descriptions of  fusion states, which are related through a basis change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  These are known as the inside and outside bases, and  are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' A detailed discussion can be  found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' We will simply refer to these ad-  ditional degrees of freedom as the handle labels of the  state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  4 As opposed to the more conventional view of braiding as an active  transformation that maps between different fusion spaces, we have  opted for the equivalent framework of braiding as a passive basis  transformation, as the latter is more appropriate in view of our  specific model and numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  (a)  (b)  Figure 7: Two possible basis choices for anyons on a  torus: a) the inside basis, b) the outside basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  For our current purpose, we won’t need the full de-  scription of said handle labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' It is sufficient for us to  pick one basis, and subsequently set the total charge of  all anyons to the vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' We are left with only a single  label (1 or τ) representing the anyonic charge flowing  along the non-contractible cycle associated with our  basis choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' This is precisely the origin of the twofold  degeneracy of the anyonic vacuum on the torus, which  we have taken to be our code space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  In the this work we will always start from the code  state corresponding to a trivial handle label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  As a  change in handle label at any point during the error cor-  rection procedure must be the consequence of a topo-  logically non-trivial process which constitutes a logi-  cal error, any simulation is aborted at the occurrence  of such an event, meaning that all handle labels can be  safely ignored in the remainder of our discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  A basic functionality required for the simulation of  the error correction procedure is the ability to correctly  sample a measurement of the total charge of a given set  of anyons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Starting from a given fusion state, this can  be achieved by first transforming to a basis in which the  relevant anyons are fused sequentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' For a system  of many anyons these reordering basis transformations  are obtained by combining Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' (A4) and (A6), giving  11  rise to the clockwise swap  aj  b3  b2  b1  aj+1  =  �  b′  2  Bb1ajb2  aj+1b3b′  2  aj  aj+1  b3  b2  b1  ,  (A9)  and the counterclockwise swap  aj  b3  b2  b1  aj+1  =  �  b′  2  �  Bb1ajb2  aj+1b3b′  2  �∗  aj  aj+1  b3  b2  b1  ,  (A10)  where  Bb1ajb2  aj+1b3b′  2 =  �  c  F aj+1ajc  b3b1b′  2  Rajaj+1  c  F b1ajb2  aj+1b3c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  (A11)  By performing a certain set of these basis transforma-  tions, the fusion order can always be made consistent  with the group of anyons of which we want to mea-  sure the total charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Subsequently, the fusion state is  recoupled such that the relevant group of anyons is con-  nected to the rest of the state by a single edge c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' For a  charge measurement of a pair of anyons this recoupling  takes the form  aj  b3  b2  b1  aj+1  =  �  c  F b1ajb2  aj+1b3c  aj  b3  b1  aj+1  c  ,  (A12)  and charge measurements of larger groups of anyons  simply require consecutive applications of the recou-  pling identity (A4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Finally, the charge outcome is ob-  tained sampling the total charge c from the probability  distribution corresponding to the resulting superposi-  tion of fusion states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  Appendix B: Classical simulatbility  It is well known that Fibonacci anyons are universal  for quantum computation [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' One might therefore be  tempted to conclude that the classical simulation of the  topological Fibonacci code described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' II with  pair-creation noise is unlikely to succeed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' However,  as was noted in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' [23], the simulation of noise and  error correction processes does not require the simula-  tion of general anyon dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' In particular, individ-  ual noise processes create distinct connected groups of  anyons with vacuum total charge (or extend such exist-  ing groups).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' These groups correspond to anyons that  have interacted at some point during their lifetime, and  must thus only be merged whenever a noise or error  correction process involves two members from discon-  nected groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Since each connected group has a triv-  ial total charge, braiding between disconnected groups  is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Hence, the total fusion space factorizes into a  tensor product of fusion spaces of individual connected  groups, and we are only required to simulate anyon dy-  namics within each of these groups separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' This  factorization of the fusion space is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  The creation and subsequent merging of discon-  nected groups of anyons by noise and recovery pro-  cesses can be thought of as a kind of percolation pro-  cess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  Hence, below the percolation threshold, one  expects that the size of the largest connected group  scales as O(log(L)) (with variance O(1)), where L  is the linear system size [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' As this is a probabilis-  tic statement, there will be instances where the largest  connected group has a size larger than O(log(L), but  the probability of such events is suppressed exponen-  tially with the system size L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' This logarithmic scal-  ing of the largest cluster size s = O(log(L)) coun-  ters the exponential scaling of the dimension of the fu-  sion space d = O(exp(s)) for individual connected  groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Therefore, the fusion spaces of individual con-  nect groups will have dimension dim = O(poly(L)),  meaning that the dynamics within connected groups  can be simulated efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  Exploiting the tensor product structure within the to-  tal fusion space, requires the use of basis for the any-  onic fusion space which reflects this structure and can  be updated dynamically to keep track of noise and re-  covery processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' This is best achieved by using the  framework of curve diagrams, which were introduced  in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' [23] (also see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' [44] for a more rigorous  treatment using the language of modular functors), and  also discussed extensively in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Since these are  merely a technical tool for keeping track of the most  appropriate basis during the numerical simulations, we  will refrain from discussing them here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Interested read-  ers are referred to the aforementioned references for  details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  12  V ab  1 ⊗ V cde  1  ⊗ V fg  1  V ab  1 ⊗ V cde  1  ⊗ V fg  1  (a)  a  b  c  d  e  f  g  x  y  V abcdefg  1  = ⊕x,y  �  V ab  x ⊗ V x cde  y  ⊗ V y fg  1  �  (b)  V ab  1 ⊗ V cde  1  ⊗ V fg  1  a  b  c  d  e  f  g  (c)  Figure 8: (a) A set of noise processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' (b) A generic  state in the full fusion space of all anyons created by  the noise processes involves superpositions in the  labels x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' (c) The factorized Hilbert space which  is sufficient to represent the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  Appendix C: Transition rules for Q = 3  Below, we give a full definition of the transition rules  for Q = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' It is possible to define more general tran-  sition rules that apply for any colony size, examples of  such rules can be found in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' [2] and [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' However,  since the numerical simulations performed in this work  were performed to Q = 3 we have taken the freedom  to tailor the transition rules to this case specifically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  North-West  if sk,c(ρ) = 0, do nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  else if sk,n(ρ + (0, −1)) ̸= 0, do nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  else if sk,n(ρ + (−1, 0)) ̸= 0, do nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  else if sk,n(ρ + (0, 1)) ̸= 0, move east;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  else if sk,n(ρ + (1, 0)) ̸= 0, move south;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  else if sk,n(ρ + (1, −1)) ̸= 0, do nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  else if sk,n(ρ + (−1, −1)) ̸= 0, do nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  else if sk,n(ρ + (−1, 1)) ̸= 0, do nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  else move south;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  North  if sk,c(ρ) = 0, do nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  else if sk,n(ρ + (−1, 0)) ̸= 0, do nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  else if sk,n(ρ + (0, −1)) ̸= 0, do nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  else if sk,n(ρ + (0, 1)) ̸= 0, do nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  else if sk,n(ρ + (−1, 1)) ̸= 0, do nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  else if sk,n(ρ + (−1, −1)) ̸= 0, do nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  else move south;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  North-East  if sk,c(ρ) = 0, do nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  else if sk,n(ρ + (0, 1)) ̸= 0, move east;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  else if sk,n(ρ + (−1, 1)) ̸= 0, move north-east;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  else if sk,n(ρ + (1, 1)) ̸= 0, move south;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  else if sk,n(ρ + (−1, 0)) ̸= 0, do nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  else if sk,n(ρ + (0, −1)) ̸= 0, move west;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  else if sk,n(ρ + (1, 0)) ̸= 0, move south;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  else if sk,n(ρ + (−1, −1)) ̸= 0, do nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  else move south-west;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  East  if sk,c(ρ) = 0, do nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  else if sk,n(ρ + (0, 1)) ̸= 0, move east;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  else if sk,n(ρ + (−1, 1)) ̸= 0, move north-east;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  else if sk,n(ρ + (1, 1)) ̸= 0, move south;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  else if sk,n(ρ + (−1, 0)) ̸= 0, do nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  else if sk,n(ρ + (1, 0)) ̸= 0, do nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  else move west;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  South-East  if sk,c(ρ) = 0, do nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  else if sk,n(ρ + (0, 1)) ̸= 0, move east;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  else if sk,n(ρ + (1, 0)) ̸= 0, move south;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  else if sk,n(ρ + (−1, 1)) ̸= 0, move north-east;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  else if sk,n(ρ + (1, −1)) ̸= 0, move south-west;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  else if sk,n(ρ + (1, 1)) ̸= 0, move south;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  else if sk,n(ρ + (0, −1)) ̸= 0, move west;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  else move north;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  13  South  if sk,c(ρ) = 0, do nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  else if sk,n(ρ + (1, 0)) ̸= 0, move south;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  else if sk,n(ρ + (1, −1)) ̸= 0, move south-west;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  else if sk,n(ρ + (1, 1)) ̸= 0, move east;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  else if sk,n(ρ + (0, −1)) ̸= 0, do nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  else if sk,n(ρ + (0, 1)) ̸= 0, do nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  else move north;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  South-West  if sk,c(ρ) = 0, do nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  else if sk,n(ρ + (1, 0)) ̸= 0, move south;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  else if sk,n(ρ + (1, −1)) ̸= 0, move south-west;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  else if sk,n(ρ + (1, 1)) ̸= 0, move east;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  else if sk,n(ρ + (0, −1)) ̸= 0, do nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  else if sk,n(ρ + (−1, 0)) ̸= 0, move north;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  else if sk,n(ρ + (0, 1)) ̸= 0, move east;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  else if sk,n(ρ + (−1, −1)) ̸= 0, do nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  else move north-east;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  West  if sk,c(ρ) = 0, do nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  else if sk,n(ρ + (0, −1)) ̸= 0, do nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  else if sk,n(ρ + (−1, 0)) ̸= 0, do nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  else if sk,n(ρ + (1, 0)) ̸= 0, do nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  else if sk,n(ρ + (1, −1)) ̸= 0, do nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  else if sk,n(ρ + (−1, −1)) ̸= 0, do nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  else move east;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  [1] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' G´acs, Reliable computation with cellular automata,  Journal of Computer and System Sciences 32, 15  (1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  [2] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Harrington, Analysis of quantum error-correcting  codes: symplectic lattice codes and toric codes, Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  thesis, California Institute of Technology (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  [3] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Freedman, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Larsen, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Wang, A Modu-  lar Functor Which is Universal for Quantum Compu-  tation, Communications in Mathematical Physics 227,  605 (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  [4] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Freedman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Kitaev, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Wang, Simulation of  topological field theories by quantum computers, Com-  munications in Mathematical Physics 227, 587 (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  [5] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Kitaev, Fault-tolerant quantum computation by  anyons, Annals of Physics 303, 2 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  [6] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Bravyi and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Kitaev, Quantum codes on a  lattice with boundary, arXiv preprint quant-ph/9811052  (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  [7] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Dennis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Kitaev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Landahl, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Preskill,  Topological quantum memory, Journal of Mathematical  Physics 43, 4452 (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  [8] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Wang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Fowler, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Stephens, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  Hollenberg, Threshold error rates for the toric and sur-  face codes, arXiv preprint arXiv:0905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='0531 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  [9] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Fowler, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Mariantoni, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Martinis, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  Cleland, Surface codes: Towards practical large-scale  quantum computation, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' A 86, 032324 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  [10] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Wootton and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Loss, High threshold error correc-  tion for the surface code, Physical review letters 109,  160503 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  [11] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Anwar, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Brown, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Campbell, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  Browne, Fast decoders for qudit topological codes, New  Journal of Physics 16, 063038 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  [12] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Bravyi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Suchara, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Vargo, Efficient algo-  rithms for maximum likelihood decoding in the surface  code, Physical Review A 90, 032326 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  [13] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Wootton, A simple decoder for topological codes, En-  tropy 17, 1946 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  [14] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Andrist, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Wootton, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Katzgraber,  Error thresholds for abelian quantum double models:  Increasing the bit-flip stability of topological quantum  memory, Physical Review A 91, 042331 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  [15] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Gottesman, Stabilizer Codes and Quantum Error  Correction, Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' thesis, California Institute of Tech-  nology (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  [16] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Bravyi and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Kitaev, Universal quantum computa-  tion with ideal clifford gates and noisy ancillas, Physical  Review A 71, 022316 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  [17] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Kubica and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Beverland, Universal transversal  gates with color codes: A simplified approach, Physical  Review A 91, 032330 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  [18] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Bomb´ın, Dimensional jump in quantum error correc-  tion, New Journal of Physics 18, 043038 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  [19] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Campbell, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Terhal, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Vuillot, Roads  towards fault-tolerant universal quantum computation,  Nature 549, 172 (2017), 1612.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='07330.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  [20] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Hutter and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Wootton, Continuous error correc-  tion for ising anyons, Physical Review A 93, 042327  (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  [21] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Wootton, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Burri, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Iblisdir, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Loss, Error  correction for non-abelian topological quantum compu-  tation, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' X 4, 011051 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  [22] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Brell, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Burton, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Dauphinais, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Flammia,  and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Poulin, Thermalization, error correction, and  memory lifetime for ising anyon systems, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' X  4, 031058 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  [23] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Burton, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Brell, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Flammia, Classical sim-  ulation of quantum error correction in a fibonacci anyon  code, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' A 95, 022309 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  [24] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Schotte, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Zhu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Burgelman, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Verstraete,  Quantum Error Correction Thresholds for the Univer-  sal Fibonacci Turaev-Viro Code, Physical Review X 12,  14  021012 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  [25] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Bonesteel and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' DiVincenzo, Quantum circuits  for measuring Levin-Wen operators, Physical Review B  86, 165113 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  [26] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  Verresen,  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  Tantivasadakarn,  and  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  Vish-  wanath, Efficiently preparing ghz, topological and frac-  ton states by measuring cold atoms, arXiv preprint  arXiv:2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='03061 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  [27] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Tantivasadakarn, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Verresen, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Vishwanath,  The shortest route to non-abelian topological order on  a quantum processor, arXiv preprint arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='03964  (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  [28] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Raussendorf and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Harrington, Fault-tolerant quan-  tum computation with high threshold in two dimen-  sions, Physical review letters 98, 190504 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  [29] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Fowler, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Stephens, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Groszkowski,  High-threshold universal quantum computation on the  surface code, Physical Review A 80, 052312 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  [30] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Watson, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Anwar, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Browne, Fast fault-  tolerant decoder for qubit and qudit surface codes, Phys-  ical Review A 92, 032309 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  [31] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Herold, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Kastoryano, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Campbell, and  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Eisert, Cellular automaton decoders of topological  quantum memories in the fault tolerant setting, New  Journal of Physics 19, 063012 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  [32] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Dauphinais and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Poulin, Fault-tolerant quantum  error correction for non-abelian anyons, Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' 355, 519 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  [33] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Pfeifer, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Buerschaper, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Trebst, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Ludwig,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Troyer, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Vidal, Translation invariance, topol-  ogy, and protection of criticality in chains of interacting  anyons, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' B 86, 155111 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  [34] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Herold, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Campbell, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Eisert, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Kas-  toryano, Cellular-automaton decoders for topological  quantum memories, npj Quantum information 1, 1  (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  [35] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Zhu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Lavasani, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Barkeshli, Universal logi-  cal gates on topologically encoded qubits via constant-  depth unitary circuits, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' 125, 050502  (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  [36] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Zhu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Lavasani, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Barkeshli, Instantaneous  braids and Dehn twists in topologically ordered states,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' B 102, 075105 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  [37] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Lavasani, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Zhu, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Barkeshli, Universal log-  ical gates with constant overhead: instantaneous dehn  twists for hyperbolic quantum codes, Quantum 3, 180  (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  [38] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Zhu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Hafezi, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Barkeshli, Quantum origami:  Transversal gates for quantum computation and mea-  surement of topological order, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Research 2,  013285 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  [39] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Turaev and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Viro, State sum invariants of 3-  manifolds and quantum 6j-symbols, Topology 31, 865  (1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  [40] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Kitaev, Anyons in an exactly solved model and be-  yond, Annals of Physics 321, 2 (2006), arXiv:cond-  mat/0506438.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  [41] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Wang, Topological Quantum Computation (Ameri-  can Mathematical Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  [42] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Turaev and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Virelizier, Monoidal Categories and  Topological Field Theory, Progress in Mathematics  (Birkh¨auser Basel, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  [43] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Bazant, Largest cluster in subcritical percolation,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' E 62, 1660 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='  [44] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content=' Burton, A short guide to anyons and modular functors  (2016), arxiv:1610.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
+page_content='05384 [quant-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'}
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+arXiv:2301.03101v1  [cs.IT]  8 Jan 2023
+1
+Massive MIMO and NOMA
+Bits-per-Antenna Efficiency under Power
+Allocation Policies
+Thiago A. Bruza Alves, Taufik Abrão
+State University of Londrina (UEL), Department of Electrical Engineering, Londrina-PR, Brazil.
+Abstract—A comparative resource allocation analysis in terms
+of received bits-per-antenna spectral efficiency (SE) and energy
+efficiency (EE) in downlink (DL) single-cell massive multiple-input
+multiple-output (mMIMO) and non-orthogonal multiple access
+(NOMA) systems considering a BS equipped with many (M)
+antennas, while K devices operate with a single-antenna, and the
+loading of devices ρ = K
+M ranging in 0 < ρ ≤ 2 is carried out under
+three different Power Allocations (PA) strategies: the inverse
+of the channel power allocation (PICPA), a modified water-
+filling (∆-WF) allocation method, and the equal power allocation
+(EPA) reference method. Since the two devices per cluster are
+overlapped in the power domain in the NOMA system, the channel
+matrix requires transformation to perform the zero-forcing (ZF)
+precoding adopted in mMIMO. Hence, NOMA operating under
+many antennas can favor a group of devices with higher array gain,
+overcoming the mMIMO and operating conveniently in the higher
+loading range 0.6 < ρ < 2.0. In such a scenario, a more realistic
+and helpful metric consists in evaluating the area under SE and
+EE curves, by measuring the bit-per-antenna and bit-per-antenna-
+per-watt efficiency, respectively. Our numerical results confirm a
+superiority of NOMA w.r.t. mMIMO of an order of 3x for the
+SE-area and 2x for the EE-area metric.
+Index Terms—Non-Orthogonal Multiple Access (NOMA); mas-
+sive Multiple-Input Multiple-Output (mMIMO); Energy Efficiency
+(EE); Spectral Efficiency (SE).
+I. Introduction
+The beyond Fifth Generation (5G) of wireless communica-
+tion systems must allow ultra-dense connections with vastly
+heterogeneous requirements. The challenges in networks per-
+sist, including the Spectral Efficiency (SE) and the Energy
+Efficiency (EE) joint improvement, the increase in the SE-
+EE trade-off, and Quality of Service (QoS), always aiming
+to meet the growing number of devices connected to the
+network. Among the proposals to solve these challenges, the
+massive Multiple-Input Multiple-Output (mMIMO) system is
+the primary proposed system that allows the increase of the
+link capacity, exploring the propagation of multiple paths with
+the use of a large number of antennas at the Base Station
+(BS) [1], [2]. Another relevant enabling technology is the
+Non-Orthogonal Multiple Access (NOMA), which explores the
+power domain as an alternative way in terms of multiple ac-
+cess technology, helping to mitigate the spectrum exhaustion
+problem and serving more than one device per resource block
+[3].
+Although in many works mMIMO is classified as an or-
+thogonal technique, allocating the signal from devices in the
+same resource block, possible by spatial diversity, allows us to
+This work was partly supported by The National Council for Scien-
+tific and Technological Development (CNPq) of Brazil under Grants
+310681/2019-7, partly by the CAPES- Brazil - Finance Code 001, and
+the Londrina State University - Paraná State Government (UEL).
+T. A. Bruza Alves and Taufik Abrão are with the State University of
+Londrina (UEL), Department of Electrical Engineering, Londrina-PR, E-
+mails: thiagobruza@hotmail.com, taufik@uel.br
+classify it as a non-orthogonal technique too [4]. There is a
+vast literature demonstrating the superior performance of the
+Spectral Efficiency of NOMA when compared to Orthogonal
+Multiple Access (OMA) techniques [5]. Previous aims to
+improve the communication system performance by combining
+MIMO (with a small number of antennas M) and NOMA have
+been discussed in [6]–[9].
+Studies comparing NOMA and mMIMO in a single cell are
+proposed in [4], [10], [11]. The acquisition of Channel State
+Information (CSI) through pilot acquisition to NOMA system
+is proposed in [10]. In [11], the application of NOMA in the
+mMIMO scheme is proposed, and better results are achieved
+in the proposed comparative. Moreover, in [4] is analyzed the
+performance of NOMA and mMIMO in line of sight and non-
+line of sight.
+The canonical mMIMO refers to the systems with BSs
+formed by a large number of antennas M when compared
+to the number of actives devices, K, succinctly M ≫ K is
+considered a mMIMO setup. The typical NOMA improves the
+SE by superposing the signals of the selected devices to form
+a cluster in the power domain, multiplexing it over the same
+signal and served by the same beamforming. Nonetheless,
+the success of NOMA depends on the Successive Interference
+Cancellation (SIC).
+Power-domain NOMA can be a candidate technology in
+dense networks [12]. To improve performance and minimize
+the impact to assume the perfect SIC [13], devices are divided
+into two groups. After grouping in pairs and forming a cluster,
+each pair forms a cluster with a high difference between
+channel conditions. The device with a higher channel condition
+can decode the signal sent to the device with the lower channel
+condition. The interference can thus be eliminated by SIC. The
+use of NOMA in BS equipped with a large number of antennas
+was investigated in terms of SE [4], [10] we propose in a similar
+configuration system increasing the loading up to 2 times the
+number of antennas in BS and analyze the SE, EE, and SE-EE
+trade-off.
+The EE metric is a popular figure of merit employed to
+analyze the balance between power consumption and data
+rate. The EE is the ratio between the effectively transmitted
+data rate and the total power expended during the transmis-
+sion process, including instantaneous and static components.
+With the EE metric, it is possible to evaluate the efficiency
+with which a system uses the limited energy resource to
+communicate data and optimize this ratio. Can show the
+tendency of energy consumption in the case of seeking justice
+among devices.
+The Zero Forcing (ZF) is simple and popular alternative
+interference suppression beamforming under perfect CSI con-
+dition and achieving a satisfactory condition in real situations
+when imperfect CSI, in this work, we adopt perfect CSI, for
+
+2
+that the pilots are needed. The adoption of NOMA system
+with a large number of antennas requires a defined equivalent
+channel to be deployed for interference mitigation; and ac-
+cording to the NOMA principle, makes the equivalent channel
+matrix smaller than the original one due to the exploration of
+power domain in NOMA.
+Various transmission topologies already deal with the EE
+problem in mMIMO, finding the optimal number of antennas,
+number of devices in a cell, and the maximal EE [2], [14]. The
+EE analysis in the NOMA system is carried out in [15], and its
+superiority is demonstrated when compared with conventional
+orthogonal multiple access (OMA) systems. Recent researches
+seek to improve the NOMA performance, e.g. in [16], the
+minimum pairing distance is defined and compared to the
+OMA, while in [17] it’s presented a comparison between OMA
+and cell-free system equipped with mMIMO-NOMA. An EE
+analysis in Terahertz (THz)-NOMA-Multiple-Input Multiple-
+Output (MIMO) was proposed in [18]. Still, the number of
+active devices is much smaller than the number of antennas
+in the BS, and [19] is a survey about Power Domain NOMA
+and makes clear the vacuum of EE analysis and comparison
+between NOMA with many antennas and mMIMO.
+Recent works propose the deployment of NOMA combined
+with other techniques a more effective transmission scheme;
+e.g., in [20] NOMA and mMIMO are jointly considered in a
+two-tier network for accommodating colossal traffic. Further-
+more, in [21], authors apply NOMA in Distributed Antenna
+Systems (DAS), aiming to achieve better performance when
+compared to the conventional NOMA or DAS technique alone.
+While [22] shows an in-depth survey of the state-of-the-art
+of power-domain NOMA variants; moreover, several open
+issues and research challenges of NOMA-based applications
+are systematized. The NOMA system presents drawbacks,
+such as hardware (including SIC) complexity, channel feed-
+back, receiver design, and careful power and pilot allocation
+strategies [12], [19], [23].
+This work focus on revealing the advantages of applying
+the mMIMO scheme versus NOMA scheme with a massive
+number of BS antennas, and varying the loading of devices,
+i.e., the ratio of the number of mobile devices to the number
+of BS antennas, ρ =
+K
+M , while we change the PA strategy.
+Besides, we adopt a realistic model for the system’s power
+consumption as in [2] but adapted to our needs, aiming at
+providing a suitable analysis of the system resource allocation.
+Contributions: the contributions of this work are fourfold.
+a) an extensive and comparative analysis on the spectral
+efficiency (SE) performance of mMIMO system against NOMA
+system, varying the system loading under specific (three dif-
+ferent) power allocation methods and making use of the area
+under the SE (Ssyst) curve of the system as an effective, useful
+and fair metric of performance and efficiency; b) we develop
+an energy efficiency (EE) analysis using a detailed model of
+energy consumption, with fixed and variable terms related to
+circuitry power consumption with number of antennas and
+devices, respectively, providing an extensive and comparative
+analysis on both the NOMA and mMIMO systems under
+realistic operation scenarios and making use of the area under
+the EE (Esyst) curve of system; c) an analysis on the SE-EE
+trade-off is developed considering a wide range of loading of
+devices, verifying the fairness between devices; d) finally, under
+mild conditions, we provide evidences for the NOMA’s ability
+to serve a greater number of devices than mMIMO system.
+The remainder of the paper is organized as follows. Section
+II describes the system models for NOMA and mMIMO
+adopted in this work. In Section III we present the proposed
+EE-SE formulation for NOMA and massive MIMO systems.
+Numerical results are analyzed in IV. Section V concludes the
+paper.
+Notation. In this work, boldface lower case and upper case
+characters denote vectors and matrices, respectively. The
+operator (x)+ = max(0, x). The operators [·]T, E[·] and
+|·| denote transpose, expectation and cardinality, respectively.
+A random vector x ∼ CN {0, Im} is circularly symmetric
+Gaussian distributed with mean 0 and covariance matrix Im.
+Im is m × m identity matrix.
+II. System Models
+Let us consider a multi-user single-cell Downlink (DL)
+transmission operating in a Time Division Duplex (TDD) with
+K single-antenna actives devices, communicating with one BS,
+which is equipped with M transmit antennas in Non-Line-Of-
+Sight (NLOS). The set K is formed by K devices, these devices
+are randomly distributed in a radius disk dmax, the disk area
+is formed by two sub-disk with the same number of devices
+in each sub-area; both subsets are identified as KH and KL.
+In the first subset, KH represents devices’ indexes having the
+higher channel coefficient and sort in descending order, while
+the other subset KL are formed by the devices with lower
+channel coefficient and sort in ascending order; the indexes
+k ∈ KH and k ∈ KL such that:
+K = KH ∪ KL,
+where
+KH = {1, ..., K/2} and KL = {K/2 + 1, ..., K}.
+(1)
+The channel vector modeling of device k can be described liked
+as:
+hk =
+�
+βkh′
+k,
+k = 1, ..., K,
+(2)
+where βk is the large-scale fading coefficient and satisfy
+βj > βi,
+∀j ∈ KH, ∀i ∈ KL.
+(3)
+Herein, the pathloss model in [dB] is defined as:
+βk = β0 + 10 · ξ · log10(dk),
+(4)
+where dk is the distance of user k to BS, ξ is the pathloss
+coefficient, and β0 is the attenuation at the distance of
+reference.
+In each coherence interval, h′
+k in (2) for device k is an
+independent random small-scale fading realization from an in-
+dependent Rayleigh fading distribution, h′
+k ∼ CN(0, IM), k =
+1, ..., K . The transmitted signal xk ∈ CM is the beamformed
+data symbol of device k:
+xk = gk
+√pksk,
+(5)
+where gk is a normalized beamforming vector, pk normalized
+transmission power and sk ∼ CN(0, 1) is the data symbol of
+device k, and period Ts. The signal received at the k-th device:
+yk =
+K
+�
+k′=1
+hT
+k xk′ + nk,
+=
+�
+βkh′T
+k
+K
+�
+k′=1
+gk′√pk′sk′ + nk,
+(6)
+=
+�
+βkpkh′T
+k gksk +
+�
+βkh′T
+k
+K
+�
+k′̸=k
+k′=1
+gk′√pk′sk′ + nk,
+where nk ∼ CN(0, 1) is the additive noise. Notice that this
+modeling applies to both NOMA and mMIMO systems, but
+beamforming is selected differently, and this topic will be
+addressed in the next sections.
+
+3
+A. Prior Actions
+Because the BS needs to know a priori crucial information
+related to the channel and devices distributed in the cell,
+including device location, rate demanded, and channel coeffi-
+cient, such required a priori information may differ depending
+on the multiple access scheme considered [12].
+The option for the mMIMO and NOMA systems was carried
+out with the guarantee that the needs of the devices would
+be met, and this initial step was carried out successfully.
+Subsection III-E briefly discusses the preliminary information
+required to proceed with different Power Allocation (PA)
+procedures in both mMIMO and NOMA systems.
+B. Pilot Overhead for Channel State Information
+Fig. 1 compares the pilot-data transmission structure along
+the one-channel coherence interval for both mMIMO and
+NOMA systems considered. Notice that Ts is the time required
+to transmit a data symbol (Data). Moreover, the channel
+coherence time interval T is assumed to be a multiple of the
+Data symbol period, T = ι · Ts. The power allocated to each
+pilot in the training step is enough. In contrast, the number of
+pilots, and the dedicated portion of coherence interval for data
+transmission are assumed to be the same in both systems.
+!"
+!"
+#"
+$%&'(
+#)()
+#"
+#"
+*+,+-
+.-+/
+-0123'41510612,0(157)&
+(%*1
+Figure 1: Coherence time interval structure: the training and
+data transmission structure for mMIMO and NOMA schemes
+under TDD NLOS setup.
+Notice that in NOMA transmission, the pilot transmission
+step is split into two portions, half for the UL transmission
+pilots and receive the DL pilot confirmation; this happens
+because to perform SIC, the cell-center devices need to learn
+the effective channels that are established by the beamforming.
+Additionally, the beamforming vectors are based on cell-center
+devices, producing limited rates achieved in cell-edge devices.
+On the other hand, in the mMIMO scheme, a significant
+advantage is that there is no need for DL pilots since the
+effective channels created by the beamforming are highly
+predictable, i.e. nearly deterministic gain and phase due to
+channel hardening effect [4].
+Assumption 1: In the NOMA system, the power allocated to
+each downlink pilot is sufficient to reach the destination device.
+C. Beamforming for NOMA and mMIMO systems
+At the mMIMO system, each device is served by a sin-
+gle beamforming vector. The ZF technique is a popular
+interference-suppressing beamforming scheme in the mMIMO
+system since it eliminates all inter-user interference using
+individual beamforming for each device, while the favorable
+propagation facilitates such interference suppressing in massive
+MIMO configurations. Besides, to perform ZF precoding in
+NOMA system, it is essential to understand the NOMA user-
+pairing concept.
+User-pairing: Inherent to the NOMA system, user clustering
+can be performed in several ways after the user-sorting and
+the user classification in center-users and edge-users subsets.
+Because we know that the SE of NOMA is directly proportional
+to the difference between the pathloss of the users, a natural
+choice consists in pairing users with as higher as possible
+pathloss differences [6]:
+∆βk = βk − βK+1−k,
+(7)
+forming the cluster k for k = 1, ..., K/2. With the pair formed,
+carefully beamforming vectors selection is required. Hence, in
+NOMA we assume that the beamforming vector for paired
+users is the same, i.e., gk = gK+1−k for all k = 1, ..., K/2.
+Assumption 2: In user-pairing procedure, we assume that
+the paired users are aligned with the BS so that the same
+beamforming can serve all paired users simultaneously. Hence,
+by admitting that each pair of devices is spatially aligned with
+the BS, and using localizing tools described, for instance, in
+[23], [24], one should assume further a priori user-pairing step
+in NOMA systems.
+Assumption 3: In NOMA system, beamforming serves more
+than one aligned device simultaneously; specifically, in this
+paper, two aligned devices per cluster are admitted according
+to the user-pairing step, while eliminating the inter-cluster
+interference (favorable propagation) under adopted perfect
+CSI conditions.
+In this work we adopt the linear ZF precoding as defined by
+the vector:
+gk = h′k(h′T
+k h′k)−1,
+(8)
+and satisfying h′T
+i gk
+=
+0, ∀ i
+̸=
+k, i.e., the favorable
+propagation effect between users belonging to distinct clusters.
+III. SE-EE in NOMA and mMIMO systems
+We discuss the SE and EE configurations in the NOMA and
+mMIMO systems. The operation of the NOMA system requires
+pairing devices so that the channel coefficients of the devices
+in the same cluster must be appropriately different, enabling
+power domain usage. As already mentioned, the interference
+cancellation process via beamforming presents problems that
+we will demonstrate below.
+A. Data Rates in NOMA with ZF
+Devices are divided into two sets like described in Eq. (1),
+and these groups are represented by Eq. (9) by their large-
+scale fading coefficient and are grouped into pairs forming a
+cluster as Fig. 2, the cluster k is formed by one device in cell
+center set KH and one device in cell edge set KL. Hence, the
+devices are grouped into two subsets:
+KH ={β1 > β2 > ... > βk/2},
+(center devices set)
+(9)
+KL ={βK < βk−1 < ... < βk/2+1}. (edge devices set)
+The user-pairing adopted in Eq. (9) is the same as proposed
+in [6], creating the largest possible difference in channel
+coefficients for devices not yet paired.
+Assumption 4: In this paper, we assume perfect SIC, and only
+one perfect SIC stage per cluster is performed, since just 2
+devices per cluster are admitted.
+
+4
+!"
+Figure 2: System Model indicating the Paring formation in
+NOMA system. Both mMIMO and NOMA systems deploy the
+same massive number of antennas at base-station, M.
+The instantaneous Signal to Interference plus Noise Ratio
+(SINR) of devices in cluster k is defined as:
+SINRk =
+βkpk|h′T
+k gk|2
+βk
+�K
+k′̸=k pk′|h′T
+k gk′|2 + 1
+.
+(10)
+In each cluster, the cell-edge devices treat the interference as
+noise and decode their data symbols, whereas the cell-center
+device can decode the data symbols of the cell-edge device
+and perform SIC, hence effectively removing the interference
+due to the cell-edge device under Assumption 3.
+To perform SIC, the cell-center device needs to be able to
+decode data signal intended for the cell-edge device, i.e., the
+ergodic SINR of the cell-edge device, sinrK+1−k, at device
+k, defined as sinrk,K+1−k, must be greater than or equal
+to the ergodic SINR of the k-th cell-center device. Hence,
+given the uplink (mMIMO and NOMA) and downlink (NOMA)
+pilot overhead and assuming perfect CSI in all receivers, and
+admitting Assumption 4, the following condition must be
+satisfied [4], [10]:
+E[SINRk,K+1−k] ≥ E[SINRk],
+(11)
+where
+SINRk,K+1−k =
+βkpK+1−k|h′T
+k gk|2
+βk
+�K
+k′̸=K+1−k pk′|h′T
+k gk′|2 + 1
+.
+(12)
+Herein, the condition in (11) must be satisfied by selecting the
+transmit powers appropriately.
+The achievable ergodic rate of devices in cluster k, i.e.
+device k in KH subset and device K+1−k in KL subset, under
+Assumptions 1–4, is given by the ergodic rate contribution of
+user-center device:
+Rnoma
+k
+= τE [log2 (1 + SINRk)] ,
+∀k ∈ KH
+(13)
+in [bits/s/Hz], and for the user-edge device:
+Rnoma
+K+1−k = τE [log2 (1 + SINRK+1−k)] ,
+∀k ∈ KL (14)
+where τ = (1− K·Ts
+T ), is the portion of each channel coherence
+interval (T) that is used for data transmission.
+Assuming perfect channel state information, ZF precoding
+for inter-clusters interference elimination, and using random
+matrix theory results [25], the k-th cluster NOMA achievable
+rate is obtained plugging eq. (10), (13) and (14):
+Rnoma
+cl-k
+= τE
+�
+log2
+�
+1 + ¯
+Mβkpk
+��
++
+(15)
+τE
+�
+log2
+�
+1 + βK+1−kpK+1−k
+βK+1−kpk + 1
+�
+,
+�
+∀k ∈ K and ¯
+M = M + 1 − K/2. Hence, the NOMA system
+can operate until K < 2M − 1. A detailed derivation of the
+expressions on this section can be found in [4] and [10].
+B. Data Rates in mMIMO with ZF
+In the mMIMO system with ZF precoding the ergodic
+achievable rate for device k is given by:
+Rm-mimo
+k
+= τE [log2 (1 + SINRk)] ,
+[bits/s/Hz]
+(16)
+where SINRk is defined in (10). Hence, the above mMIMO
+achievable rate equation becomes:
+Rm-mimo
+k
+= τE [log2 (1 + (M − K)pkβk)] ,
+[bits/s/Hz],
+(17)
+where (M − K) is obtained using random matrix theory,
+representing the coherent array gain of the received signal [25].
+Under linear precoding and combiners, the mMIMO system
+operates consistently when K < M. Finally, the average
+system sum-rate (avg-sum-rate) is defined simply by:
+Rm-mimo =
+K
+�
+k=1
+Rm-mimo
+k
+and
+Rnoma =
+KH
+�
+k=1
+Rnoma
+cl-k .
+The mMIMO system equations have been thoroughly investi-
+gated in literature and can be found in [25] and [26].
+C. Energy Efficiency
+EE metric is the ratio of the number of effective bits of
+information received over the total energy consumed by the
+overall system to transmit and receive/decode such informa-
+tion. The system data rate can determine the number of
+effective information bits received at the destination. Power
+consumption required for processing the signal at the trans-
+mitter and receiver side is often neglected; in this sense, it
+is calculated just as proportional to the radiated transmitted
+power. The growth in the number of antennas in the BS and
+the increased number of devices in 5G systems can lead to
+unattainable EE goals. In general, the average EE can be
+expressed as:
+EE =
+�K
+k=1 Rk
+Ptot
+,
+[bits/Joule/Hz],
+(18)
+where Ptot is the total power consumption across the com-
+munication system. It should consider transmission power
+consumption, such as RF power amplifier inefficiency, base-
+band signal processing, and cooling, among others. Therefore,
+a more realistic and detailed energy consumption model is
+required.
+Based on [2], the adopted power consumption model in our
+work considers two power terms: a) fixed-term; b) terms scaled
+with the number of antennas M and the number of devices
+K. The scaled terms occur because of the transceiver chains,
+coding/decoding, channel estimation, and precoding. Let the
+computational efficiency be L operations per joule in BS. We
+describe it as follows:
+RF Power: Prf is the power consumed to transmit the signal
+to active devices achieved the SINR target and 0 < ̟ ≤ 1 is
+the efficiency of the power amplifier.
+
+β1dcenl
+50mβ2K/2
+β
+K
++1
+2KOK-1
+umac5
+Fixed consumption: Pfixed is the power consumed at the
+BS which is independent of the number of transmit antennas
+and devices in the cell, is formed of term P0 including the
+power consumption of backhaul infrastructure, control signal-
+ing, baseband processor, and term Psyn a single oscillator used
+in all BS.
+Pfixed = P0 + Psyn
+Dependence only on K: PK is formed by the consumption
+to coding and modulation of information symbols to devices,
+represented by Pcod, the consumption to BS decoded the K
+sequences of information symbols, defined by Pdec, and the
+received power, represented by PRX, still composes this term,
+multiplied by K as well. In addition, a portion of the ZF
+precoding cost [27] depends only on K3.
+PK = K(Pcod + Pdec + PRX) + K3
+2
+3LT
+Dependence only on M. The term PM represents the
+transmitted power (PTX), hence
+PM = MPTX
+Dependence on K and M: the term PKM is the cost of the
+ZF precoding (due to LU-based matrix inversion) [27], which
+depends on the number of devices, the number of antennas,
+and the vector information symbol.
+PKM = MK 3 + T
+T L
++ MK2 2
+T L
+Adding the portions, we obtain the overall power consump-
+tion of the system:
+Ptot = Prf
+̟ + Pfixed + PK + PM + PKM
+[W].
+(19)
+D. Power Allocation Strategies
+In the sequel, we present three well-known and frequently
+applied strategies for power allocation. Still, due to the in-
+herent characteristics of NOMA, we propose modifications on
+the classical water-filling (WF) algorithm to enable application
+in the NOMA system. Such modifications, namely ∆-WF,
+ensure that none of the paired devices are dropped-out without
+undoing the pairing of devices. To guarantee a certain level of
+power disparities in each paired device, the power allocation ∆-
+WF procedure in the NOMA system has two steps: a) first, we
+allocated power for the clusters; b) we allocate power between
+paired devices. Thereby, we could analyze the behavior of the
+systems and compare their results.
+Notice that both mMIMO and NOMA systems deploy the
+same massive number of antennas at base-station, M. Hence,
+due to the channel hardening effect [1], [25] inherent to
+massive MIMO configuration, the small-scale fading vanishes
+across the M antennas equipped with a linear ZF precoding
+with vector as eq. (8). Hence, one can consider just the
+pathloss coefficients βk as the main parameter in the power
+allocation policies of systems based on a massive number of
+antennas.
+1) Equal Power Allocation (EPA): Equal Power Allocation
+(EPA) power allocation is deployed as a simple, naive strategy,
+where all devices are served with the same power. In mMIMO,
+all devices are served with the same transmission power
+regardless of their distance from the BS. In NOMA, power
+allocation has two steps. In the first step, the power is allocated
+equally between the pairs, and then we allocate each device’s
+power equally. The EPA strategy applied to mMIMO can be
+defined by:
+pk = Prf
+K
+[W],
+∀ k ∈ K.
+(20)
+In the case of EPA procedure applied to NOMA, it is composed
+of two steps: in the first step, power reference to each cluster
+can be defined simply as:
+pcl
+ref = 2 · Prf
+K
+[W],
+∀ k ∈ KH.
+(21)
+In the second step, the power allocation among the devices in
+the same cluster is defined as:
+pcl-k
+k
+= pcl-k
+K+1−k = pcl
+ref
+2 .
+(22)
+2) Proportional
+Channel
+Inversion
+Power
+Allocation
+(PICPA): is another power allocation technique adopted in
+this study. Unlike the EPA technique, which applies the same
+power to all devices, PICPA applies more power to devices
+with the worst channel conditions, favoring fairness across
+the devices. Such power allocation penalizes the average sum
+rate in favor of fairness among all users.
+The Proportional to the Inverse of the Channel Power Al-
+location (PICPA) strategy applied to mMIMO can be defined
+as:
+pk = Prf
+β−1
+k
+�K
+k=1 β−1
+k
+[W],
+∀k ∈ K,
+(23)
+while the PICPA procedure applied to NOMA follows two
+steps; in the first step, the power is allocated equally among
+the K/2 clusters:
+pcl
+ref = 2 · Prf
+K
+[W],
+∀k ∈ KH,
+(24)
+after that, the power of each device within the k-th cluster is
+defined by allocating more power to the device with smaller
+large-scale fading βk:
+pcl-k
+k
+= pcl
+ref
+βK+1−k
+βk − βK+1−k
+,
+and
+pcl-k
+K+1−k = pcl
+ref − pcl-k
+k
+,
+(25)
+where pcl-k
+k
+is the power allocated to the device k in the cl-k
+cluster, and pcl-k
+K+1−k is the power allocated to the device (K +
+1 − k), also belonging to the k-th cluster.
+3) Classical Water-Filling (WF) Algorithm: The application
+of the Water-Filling (WF) algorithm in mMIMO system results
+in an optimal (maximum) system sum-rate solution. However,
+some devices are dropped out of the service due to the dete-
+riorated channel condition. The WF power allocation strategy
+for mMIMO is described as:
+µ = 1
+|K|
+
+Prf +
+|K|
+�
+k=1,k∈K
+1
+βk
+
+ ,
+(26)
+Prf =
+|K|
+�
+k=1,k∈K
+pk ,
+where
+pk =
+�
+µ − 1
+βk
+�+
+, ∀k ∈ K
+and
+p = [p1, p2, ..., p|K|],
+with the operator (z)+ = max(0, z). Notice that the con-
+strained value for the total power available is set to Prf [W].
+The Algorithm 1 describes the classical WF procedure.
+On the other hand, the direct application of WF algorithm
+in the NOMA system implies harming the pair formation, i.e.,
+devices present in the KL set are effectively dropped-out of
+the service, undoing the pair. We propose a modification in
+classical WF like the following to allow some comparison.
+
+6
+Algorithm 1: Classical Water Filling (WF) for mMIMO
+Input: K,Prf, K = |K|
+1 NP ̸= ⊘;
+2 while (NP ̸= ⊘) do
+3
+solve Eq. (26) → p;
+4
+NP ← identify null positions in p;
+5
+K/{k}NP ← exclude from p devices labeled as NP
+6 end
+Output: p = [p1, p2, . . . , p|K|]
+4) ∆-WF for NOMA: The application of classical WF in
+NOMA in the same way as it is applied to mMIMO causes
+some formed pairings to be broken, since after WF algorithm
+application, some devices are dropped-out from the system,
+making the NOMA power difference (∆) in the devices of the
+same cluster vanish. Hence, we suggest modifying the classical
+WF procedure to be applied to NOMA accordingly. The steps
+of the ∆-WF algorithm are described as follows.
+In NOMA, the power allocation has two steps, in the first
+step, the allocation is between clusters. Hence, to prevent the
+formed pairs from being broken, we propose the application of
+WF based on the large-scale fading differences of the paired
+devices, as defined in eq. (7): ∆βk = (βk − βK+1−k) inside
+each KL and KH subsets, eq. (9). In the second step of
+the procedure, the power is allocated between the devices
+inside the group, assuming perfect successive interference
+cancellation (SIC); for this to be possible, the condition in Eq.
+(11) must be satisfied. The new water-level in the modified
+∆-WF power allocation strategy for NOMA is defined by
+µ = 2
+KH
+
+Prf +
+|KH|
+�
+k=1,k∈KH
+1
+∆βk
+
+ ,
+(27)
+Prf =
+|KH|
+�
+k=1,k∈KH
+pcl-k,
+∀k ∈ KH
+where
+pcl-k =
+�
+µ −
+1
+∆βk
+�+
+,
+and
+pcl-k = [p1, p2, ..., p|KH|],
+In the second step, the power allocation to both devices in the
+k-th cluster is defined as:
+pcl-k
+K+1−k = pcl-k
+k
+= pcl-k
+2
+(28)
+Algorithm 2 summarize the proposed ∆-WF power allocation
+procedure, aiming to improve the SE of NOMA systems.
+Algorithm 2: ∆-WF (modified) for NOMA systems
+Input: KH, KL, Prf
+1 NP ̸= ⊘;
+2 while (NP ̸= ⊘) do
+3
+solve Eq. (27) → pcl-k;
+4
+NP ← identify null position in pcl-k;
+5
+KH/{k}NP ← exclude from pcl-k devices labeled as
+NP
+6 end
+Output: pcl-k = [p1, p2, . . . , p|KH|]
+Complexity analysis: In a comparative analysis of complexity,
+the ∆-WF algorithm for power allocation in NOMA system
+(Algorithm 2) performs two simple additional operations com-
+pared to the classical WF procedure (Algorithm 1): a) in eq.
+(27) the subtraction in (βk − βK+1−k); and b) the division
+by two in (28). Besides, NOMA vs. mMIMO systems require
+different a priori information to proceed accordingly with the
+PA procedure.
+E. Prior Information for Power Allocation Step
+For implementing the PA policies, prior information is re-
+quired at the BS, as defined in Table I. Some of this necessary
+information can be obtained through the dedicated pilot trans-
+mission step, at the cost of some overhead, as described in
+Section II-B. Moreover, a preliminary step is known, in which
+the spatial localization and path loss estimation of all devices
+must be realized. With such a priori information availability,
+the user-sorting and user-pairing steps can be performed.
+Table I: Prior information required to PA procedure
+PA
+βk
+h′
+k
+K
+KH
+KL
+pk, eq.
+EPA
+mMIMO
+−
+−
+▲∗
+−
+−
+(20)
+NOMA
+−
+−
+▲
+−
+−
+(21), (22)
+PICPA
+mMIMO
+▲∗
+−
+−
+▲∗
+▲∗
+(23)
+NOMA
+▲
+−
+−
+▲∗
+▲
+(24), (25)
+WF
+mMIMO
+▲∗
+−
+−
+▲∗
+▲∗
+(26), Alg. 1
+∆-WF
+NOMA
+▲
+−
+−
+▲∗
+▲
+(27, 28), Alg. 2
+▲ information needed a prior
+∗ obtained via Pilot Overhead
+− Information not needed
+IV. Numerical Results
+The numerical evaluations for the proposed analyses of
+NOMA and mMIMO systems are presented in this section.
+The simulation system and channel parameter values de-
+ployed along this section are depicted in Table II. The BS
+is located at the center of cell and equipped with massive
+M BS transmit antennas in typical non-line-of-sight (NLOS)
+channel propagation scenario. At the same time, the devices
+are randomly distributed in the cell area and split into two
+subsets, KL and KH, as illustrated in Fig. 2. In all simulations,
+we consider a block fading model where the time-frequency
+resources are divided into coherence time intervals (T), in
+which the channels remain constant and frequency flat, and
+it is measured in multiple of symbol transmit period (Ts).
+The system and channel scenarios have been simulated using
+Matlab 2019 software running under one Intel HD Graphics
+6000 GPU, Intel(R) Dual-Core(TM) I5 CPU @ 1.6 GHz, and
+8 GB RAM.
+Table II:
+Simulation Parameters
+Parameter
+Value
+BS antennas
+M = 64, 128 and 256
+Max. # Devices in the cell
+K = ζ · M (NOMA)
+K = M (mMIMO)
+Cell loading
+ρ = K/M
+Total RF power available
+Prf = 1W
+Pairs NOMA / Clusters
+N = K/ζ = K/2
+NOMA devices per cluster
+ζ = 2
+# antennas per device
+1
+Cell edge length
+dmax = 350 m
+Strong device position
+[dmin; d1] ∈ [50; 100] m
+Weak device position
+[d2; dmax] ∈ [150; 350] m
+Array gain MIMO device
+M − K
+Array gain NOMA kH
+M + 1 − K/2
+Data symbol period
+Ts
+Coherence time interval
+T = 512 · Ts,
+ι = 512
+Channel
+Pathloss exponent
+ξ = 3.78
+Attenuation at a d0 reference
+β0 = 130 [dB]
+# Monte-Carlo realizations
+1000
+
+7
+A. Spectral Efficiency Comparison
+The mMIMO and NOMA SE performance analysis is carried
+out in this subsection, by increasing the number of devices
+two by two until the loading limit ρ = 2. The results consider
+M = 64, 128 and 256 BS antennas. In Fig. 3. (a) the results
+of SE are achieved when the RF power available is allocated
+following the EPA strategy, where each device receives the
+same PA values. The mMIMO system overcame NOMA in all
+situations when the loading ρ < 0.6. However, the NOMA
+system achieves a higher SE than the mMIMO for each M
+scenario when the loading of devices increases, ρ > 0.6. The
+maximum avg-SE is 373 [bits/s/Hz], being attained with ZF-
+NOMA M = 256 antennas and ρ ≈ 0.76. Besides, one can
+infer that the mMIMO does not work with a loading higher
+than 1, due to the array gain reaching 0 at full loading of
+devices, while NOMA works suitably until the loading attains
+M · ζ, where ζ is number of devices per cluster.
+Fig. 3.(b) depicts the results of SE achieved in the mMIMO
+and NOMA systems when the PICPA method is applied to
+allocate the available RF power per device along the BS an-
+tennas. The maximum avg-SE in mMIMO system overcomes
+the NOMA counterpart until the loading ρ exceeds ≈ 0.62 for
+the three BS antenna configurations, M = 64, 128, and 256.
+This PA technique provides more power to devices with the
+worst channel condition, making the SE result reach maximum
+values below the EPA.
+Fig. 3.(c) depicts the conventional WF algorithm applied
+to mMIMO. Under such a power allocation approach, we
+highlight that forming pairs is unfeasible in the NOMA sys-
+tem. Indeed, the WF algorithm can maximize the system SE
+since it allocates more power to devices with better channel
+conditions. In contrast, devices under bad channel conditions
+(below the water level) are dropped out of the service.
+The classical WF algorithm has been adapted to the NOMA
+system dropped-out always a pair of devices. Such adaptation
+reveals substantial improvements of avg-sum-rate when M
+is low compared to classical WF PA in mMIMO. The ∆-
+WF power allocation procedure preserves the pairs clustering
+formation in the NOMA system, allocating more power to
+the cluster with a higher difference between coefficients of
+large-scale fading. Fig. 3.(c) shows that the maximum avg-SE
+≈ 361 [bits/s/Hz], which is achieved under ρ = 1 (K ≈ 256
+devices) when the modified WF is deployed in NOMA system.
+Moreover, when the number of BS antennas is lower (M = 64
+or 128), the NOMA achieved a peak higher than mMIMO,
+e.g., for M = 64 antennas, the peak of SE mMIMO occurs
+at loading ρ ≈ 0.7, while the NOMA SE peaks at ρ ≈ 1.2.
+However, as the number of BS antennas grows, the NOMA
+SE advantage decreases.
+Number of active devices after PA procedure. Fig. 4 shows
+the number of actives device after applying PA methods: in
+the EPA and PICPA algorithms, all devices remain activated.
+However, in classical WF mMIMO system when the number
+of device increases beyond ρ ≈ 0.25, half of the devices are
+dropped-out; while in ∆-WF NOMA PA, the percentage of
+active devices is always higher, e.g. higher than 70% for ρ ≈
+1.1 and M = 256 antennas, the worst case.
+Fig. 5 summarizes avg-sum-rate surfaces in terms of SE
+×ρ × M results achieved by NOMA with EPA, mMIMO
+with WF, and NOMA with modified ∆-WF. In the initial
+loading part, ρ < 0.65, the classical WF PA in ZF-mMIMO
+achieves better results until the loading (pink surface). When
+the number of antennas is low as M = 64 and ρ is in
+between 0.7 and 1.8, the EPA PA applied to NOMA (green
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+Loading  = K/M
+0
+50
+100
+150
+200
+250
+300
+350
+400
+Average Sum Rate (bits/s/Hz)
+ZF-mMIMO
+ZF-NOMA
+M = 256
+M = 128
+M = 64
+(a) EPA
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+Loading  = K/M
+0
+30
+60
+90
+120
+150
+180
+Average Sum Rate (bits/s/Hz)
+ZF-mMIMO
+ZF-NOMA
+M = 256
+M = 128
+M = 64
+(b) PICPA
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+Loading  = K/M
+0
+50
+100
+150
+200
+250
+300
+350
+400
+450
+Average Sum Rate (bits/s/Hz)
+ZF-mMIMO-WF
+ZF-NOMA-
+-WF
+M = 256
+M = 128
+M = 64
+(c) Classical WF in ZF-mMIMO and ∆-WF in ZF-NOMA
+Figure 3: The average sum-rate with the loading of devices 0 <
+ρ ≤ 2, considering four power allocation methods: EPA, PICPA,
+WF, and ∆-WF The average is obtained over 1000 random devices
+locations.
+
+8
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+1.1
+1.2
+1.3
+1.4
+1.5
+1.6
+1.7
+1.8
+1.9
+2
+Loading  = K/M
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+Actives Users
+ZF-mMIMO-WF - M = 64
+ZF-NOMA-
+-WF - M = 64
+ZF-mMIMO-WF - M = 128
+ZF-NOMA-
+-WF - M = 128
+ZF-mMIMO-WF - M = 256
+ZF-NOMA-
+-WF - M = 256
+ALL-EPA
+ALL-PICPA
+Figure 4: The average of active devices after PA procedure
+versus loading of devices in the range 0 < ρ ≤ 2. The average
+is obtained over 1000 random devices locations.
+Figure 5: Average sum rate with loading ρ and M.
+surface) achieved superior results. Moreover, when M = 128
+and 0.8 < ρ < 1.6, the ZF-NOMA-EPA achieve superior
+results (green surface). For a higher number of antennas in
+BS, i.e., M = 256 only in a short loading of devices range,
+0.86 < ρ < 0.97, the ZF-NOMA-EPA achieves superior SE
+results. Finally, when ρ > 0.97, the modified ∆-WF achieves
+competitive results (blue surface).
+B. Jain’s Fairness Index
+Another critical analysis developed was to analyze the
+fairness between the devices, i.e., to know the difference in
+the transmission rate achieved by active devices in the cell. For
+this measure, we use the Jain’s Fairness index like described
+in [28] and can be defined as:
+Fsyst
+m
+=
+��M
+k=1 Rk
+�2
+M �M
+k1 R2
+k
+.
+(29)
+The Fig. 6. depicts the fairness curves attainable by NOMA
+and mMIMO systems with EPA, PICPA, WF, and ∆-WF PA
+procedures when the loading of devices grows until ρ = 2. Fig.
+6.(a) shows the Jain’s Fairness Index when EPA policy is used,
+the mMIMO system performs better F results than NOMA for
+ρ < 1, on the other hand, NOMA can attain Fnoma
+m
+≈ 0.5 in
+almost every loading of devices, independent of M.
+Fig. 6.(b) reveals the Jain’s Fairness Index when the PICPA
+method is applied, despite the SE result being lower in this PA
+method, the mMIMO obtains the best fairness result, keeping
+the Fmmimo
+m
+consistently above 0.85, still the NOMA under
+0.5.
+The Jain’s Fairness Index when WF and δ-WF are depicted
+in Fig.6(c), It is intrinsic to these algorithms to allocate more
+power to devices with better channel conditions, which causes
+fairness between devices to be impaired. In this method, it is
+possible to observe a significant influence of the number of
+antennas M in the BS and the fairness result.
+C. Energy Efficiency Comparison
+Energy efficiency (EE) is another important figure of merit
+used to compare systems’ performance. In this section, a power
+consumption model based on fixed circuitry power part and
+that one varying according to the number of antennas M and
+the number of devices K has been adopted, following eq. (19).
+Table III [2] presents the adopted parameter values for the
+EE analysis and comparison discussed in this subsection. Fig.
+7 depicts the performance of EE with EPA, PICPA, WF, and
+∆-WF PA procedures, considering the exact three quantities
+of antennas. Fig. 7.(a) shows the EE performance with EPA. In
+this method, all devices receive the same power. The avg-EE
+mMIMO overcomes the NOMA around 13% to 20%. It was
+possible to observe that adding antennas in the BS increases
+the power consumption, harming the EE result. Again, under
+loading of devices 0.6 < ρ ≤ 2.0, the NOMA overcomes the
+mMIMO system.
+Table III: Adopted Parameters values for EE analysis [2]
+Parameter
+Value
+Backhaul Infrastruture
+P0 = 2 W
+Single oscillator
+Psyn = 2 W
+Coding and modulation
+PCOD = 4 W per device
+Decoding and demodulation
+PDEC = 0.5 W per device
+Receive power
+PRX = 0.3 W per device
+Transmitted power
+PT X = 1 W per antenna
+Efficiency of Power Amplifier
+̟ = 0.3
+Operations/Joule
+L = 109 oper. per joule
+Fig. 7.(b) reveals the EE performance with PICPA. In this
+method, more power is allocated to devices with poor channel
+coefficients, resulting in reduced poor EE performance for both
+NOMA and M-MIMO systems, attaining a maximum of 0.25
+and 0.16 [bits/W] for mMIMO and NOMA, respectively. The
+maximum EE attained by mMIMO is generally around 50%
+higher than NOMA. However, for loading of devices ρ ≥ 0.65,
+NOMA overcomes mMIMO EE performance.
+Fig. 7.(c) depicts the EE performance in the mMIMO with
+classical WF and in the NOMA with ∆-WF algorithm. It
+is possible to confirm the superiority of energy efficiency of
+mMIMO within the range where it operates consistently, i.e.,
+0 < ρ < 1. Notice that the maximum EE achieved by mMIMO
+is about 70 % higher than NOMA for different BS antennas.
+Finally, NOMA becomes more energy efficient than mMIMO
+only when the loading of devices is high, ρ > 0.95.
+In all analyzed system scenarios, the mMIMO equipped with
+classical WF PA procedure achieves higher maximum EE. The
+mMIMO attains better EE results than NOMA for ρ < 1. On
+the other hand, NOMA can serve a more significant number
+of devices (twice) than mMIMO.
+Fig. 8 summarizes the best EE results in a surface plotting
+for the mMIMO with WF overcoming NOMA across the entire
+
+IOMA-Z-VWF
+ZF-NOMA-EPA450
+n Rate (bits/s/Hz)
+400
+350
+300
+200256128
+M
+64Average Sur
+150
+100
+50
+0
+2
+1.8
+1.6
+1.4
+1.2
+p
+0.8
+0.6
+0.4
+0.29
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+Loading  = K/M
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+Jain Fairness Index
+ZF-mMIMO - M = 64
+ZF-NOMA - M = 64
+ZF-mMIMO - M = 128
+ZF-NOMA - M = 128
+ZF-mMIMO - M = 256
+ZF-NOMA - M = 256
+(a) EPA
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+Loading  = K/M
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+Jain Fairness Index
+ZF-mMIMO - M = 64
+ZF-NOMA - M = 64
+ZF-mMIMO - M = 128
+ZF-NOMA - M = 128
+ZF-mMIMO - M = 256
+ZF-NOMA - M = 256
+(b) PICPA
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+Loading  = K/M
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+Jain Fairness Index
+ZF-mMIMO - M = 64
+ZF-NOMA - M = 64
+ZF-mMIMO - M = 128
+ZF-NOMA - M = 128
+ZF-mMIMO - M = 256
+ZF-NOMA - M = 256
+(c) WF and ∆-WF
+Figure 6: The fairness of NOMA and mMIMO system under three
+power allocation procedures: (a) EPA; b) PICPA; (c) WF and ∆-
+WF. The average is obtained over 1000 random device locations.
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+Loading  = K/M
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+EE (bits/Joule/Hz)
+ZF-mMIMO - M=64
+ZF-NOMA - M=64
+ZF-mMIMO - M=128
+ZF-NOMA - M=128
+ZF-mMIMO - M=256
+ZF-NOMA - M=256
+(a) EPA
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+Loading  = K/M
+0.03
+0.06
+0.09
+0.12
+0.15
+0.18
+0.21
+0.24
+0.27
+EE (bits/Joule/Hz)
+ZF-mMIMO - M=64
+ZF-NOMA - M=64
+ZF-mMIMO - M=128
+ZF-NOMA - M=128
+ZF-mMIMO - M=256
+ZF-NOMA - M=256
+(b) PICPA
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+Loading  = K/M
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+EE (bits/Joule/Hz)
+ZF-mMIMO-WF - M=64
+ZF-NOMA-
+-WF - M=64
+ZF-mMIMO-WF - M=128
+ZF-NOMA-
+-WF - M=128
+ZF-mMIMO-WF - M=256
+ZF-NOMA-
+-WF - M=256
+(c) WF and ∆-WF
+Figure 7: EE for NOMA vs. mMIMO under three power allocation
+procedures: (a) EPA; b) PICPA; (c) WF and ∆-WF. Average EE
+obtained over 1000 random devices locations.
+
+10
+loading range where it operates consistently. For device loading
+ρ > 1, the NOMA operates under lower EE until the loading
+ρ = 2. Moreover, considering the smallest number of antennas
+in the BS, the NOMA with EPA overcame the NOMA with
+modified ∆-WF; despite that, as the number of antennas in
+the BS increases, the NOMA with ∆-WF achieves marginal
+superior EE results.
+Figure 8: EE with loading ρ and M.
+D. Area Under Curves SE and EE
+For a fair comparison, one can consider a wide range
+of average SE and EE along the loading of devices, and
+normalized per antenna, which can be attainable by NOMA
+and mMIMO systems. Hence, let us consider the corresponding
+areas under the SE and EE curves in Fig. 3 and Fig. 7, such
+that:
+Ssyst
+M
+= 1
+M ·
+� ρ=2
+0
+SE(ρ) dρ
+�bits/antenna
+s · Hz
+�
+and
+Esyst
+M
+= 1
+M ·
+� ρ=2
+0
+EE(ρ) dρ
+�bits/antenna
+Joule · Hz
+�
+,
+respectively, where SE(ρ) is the average overall system sum-
+rate, and EE(ρ) is the average overall system energy efficiency
+achieved under specific loading of devices ρ. Hence, comparing
+the values of corresponding areas under the SE and EE curves
+of Fig. 3 and Fig. 7, we obtained Fig. 9.
+From the SE perspective, and considering EPA policy, the
+higher area-under-SE-curve ratio gain is achieved when the
+number of BS antennas is M = 64:
+Snoma
+M=64 ≈ 2.7 · SmMIMO
+M=64 .
+Notice that when the number of antennas M grows, the ratio
+above decreases. In the same way, considering WF policy, the
+gain trend remains. In contrast, considering the PICPA policy,
+the ratio practically remains the same.
+Furthermore, considering now the EE perspective, under
+EPA policy, the higher ratio is achieved when the BS is
+equipped with M = 256 antennas:
+Enoma
+M=256 ≈ 1.8 · EmMIMO
+M=256 .
+As one can conclude, in almost all scenarios, NOMA is more
+spectrally and energetically efficient than mMIMO over an
+extensive range of loading of devices 0 < ρ ≤ 2, roughly,
+in average, 80% in terms of energy efficiency, and 170% more
+efficient in terms of spectral efficiency.
+EPA-64
+EPA-128
+EPA-256
+WF-64
+WF-128
+WF-256
+PICPA-64
+PICPA-128
+PICPA-256
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4
+ZF-NOMA
+mMIMO
+EPA
+WF
+PICPA
+(a) S - Bar chart of Area Under SE curves.
+EPA-64
+EPA-128
+EPA-256
+WF-64
+WF-128
+WF-256
+PICPA-64
+PICPA-128
+PICPA-256
+0
+0.002
+0.004
+0.006
+0.008
+0.01
+0.012
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+ZF-NOMA
+mMIMO
+EPA
+WF
+PICPA
+(b) E - Bar chart of Area Under EE curves.
+Figure 9: The Area Under curve of NOMA and mMIMO
+system under three power allocation procedures: (a) SE curves;
+b) EE curves. The average is obtained over 1000 random
+devices locations.
+E. Resource Efficiency (SE-EE Trade-off)
+The NOMA and mMIMO are analyzed in terms of SE
+and EE trade-off, namely resource efficiency (RE), considering
+loading of devices increasing up to 2. From Fig. 10, one
+can find graphically the best loading of devices range that
+maximizes the SE-EE trade-off for each BS antenna configu-
+ration M. The left y-axis depicts avg-SE, and the right y-axis
+shows the avg-EE. Table IV summarizes the optimal loading
+of devices that maximizes the SE-EE trade-off and shows the
+percentage of active users after Power Allocations and Jain’s
+Fairness Index. Fig. 10.(a) reveals the results when M = 64,
+NOMA with EPA achieved SE-EE trade-off with the highest
+loading of devices and the highest SE in trade-off; on the
+other hand, mMIMO with classical WF achieved the highest
+SE-EE trade-off; however, the percentage of actives devices is
+around 0.5. Fig. 10.(b) depicts results when M = 128, NOMA
+with EPA achieved SE-EE trade-off with the highest loading
+of devices, in contrast, mMIMO with classical WF with lower
+loading of devices achieved higher values of SE and EE in
+trade-off with half of the active devices. Fig 10.(c) showed
+the results when M=256, NOMA with ∆-WF achieved SE-EE
+trade-off with the highest loading of devices, and one more
+time mMIMO with WF achieved higher SE-EE trade-off with
+47% of active devices. It is possible to demonstrate that the
+
+ZE.I0.8
+0.7256128
+M
+64
+0.6
+0.4
+0.2
+0EE(bits/Joule
+0.4
+0.3
+0.2
+0.1 ~
+0
+2
+1.8
+1.6
+1.4
+1.2
+1
+0.8
+p11
+increase of antennas in the BS improves the SE result, on the
+other hand, it worsens the EE result.
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+0
+134
+0
+0.781
+ZF-mMIMO-WF
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+Loading  = K/M
+0
+158
+0
+0.475
+ZF-NOMA-
+-WF
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+0
+163
+Average Sum rate (bits/s/Hz)
+0
+0.476
+EE (bits/Joule/Hz)
+ZF-NOMA-EPA
+(a) M = 64 BS antennas
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+0
+249
+0
+0.767
+ZF-mMIMO-WF
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+Loading  = K/M
+0
+251
+0
+0.449
+ZF-NOMA-
+-WF
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+0
+267
+Average Sum rate (bits/s/Hz)
+0
+0.453
+EE (bits/Joule/Hz)
+ZF-NOMA-EPA
+(b) M = 128 BS antennas
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+0
+412
+0
+0.706
+ZF-mMIMO-WF
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+Loading  = K/M
+0
+361
+0
+0.405
+ZF-NOMA-
+-WF
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+0
+373
+Average Sum rate (bits/s/Hz)
+0
+0.412
+EE (bits/Joule/Hz)
+ZF-NOMA-EPA
+(c) M = 256 BS antennas
+Figure 10: SE-EE trade-off points when M = 64, 128 and 256.
+V. Conclusion and Future Works
+This work proposes a comparative SE and EE analysis in
+DL single-cell between mMIMO and NOMA with BS equipped
+with three antenna configurations. Under the SE perspective,
+mMIMO with the classical WF algorithm achieved better low-
+and medium-loading results. On the other hand, when the
+Table IV: SE-EE Trade-off
+M = 64
+ρ
+SE
+EE
+Actives Users
+F
+mMIMO-WF
+0.652
+131.15
+0.762
+.50
+.485
+NOMA-EPA
+0.875
+147.45
+0.428
+1.0
+.485
+NOMA-∆-WF
+0.844
+143.48
+0.441
+1.0
+.475
+M = 128
+ρ
+SE
+EE
+Actives Users
+F
+mMIMO-WF
+0.625
+243.21
+0.745
+.50
+.475
+NOMA-EPA
+0.734
+241.54
+0.411
+1.0
+.49
+NOMA-∆-WF
+0.703
+226.32
+0.405
+.98
+.45
+M =256
+ρ
+SE
+EE
+Actives Users
+F
+mMIMO-WF
+0.578
+404.84
+0.691
+.47
+.43
+NOMA-EPA
+0.523
+344.70
+0.380
+1.0
+.495
+NOMA-∆-WF
+0.594
+333.31
+0.375
+.85
+.398
+system loading is higher as ρ > 0.6 the NOMA achieves better
+results in the range 0.6 < ρ ≤ 2.
+The analyzed PA methods applied to the NOMA system,
+(EPA, PICPA, and ∆-WF) result in different SE performance.
+Indeed, when the channel hardening condition is fully attained,
+and the amount of BS antennas increases(M = 128 and 256),
+the best SE results are attained with the proposed ∆-WF
+algorithm, but, as expected, the fairness index is harmed.
+Under the EE perspective, the mMIMO achieved better
+results when employing the three EPA, PICPA, and WF PA
+methods under K < M. However, the NOMA can operate
+under higher system loading, i.e., K < 2M − 1.
+In terms of area-under-SE-curve and EE-curve metrics, S
+and E, respectively, the NOMA system attained better results,
+due to its ability to serve a larger number of users than
+mMIMO. Such numerical results confirm NOMA’s ability to
+operate with high loading of devices. On the other hand,
+achieving high fairness with NOMA is impossible.
+From the perspective of SE-EE trade-off, mMIMO achieved
+the best results, because of the superiority in EE; always
+achieved in loading of devices ρ = 0.6 in all M setups.
+NOMA systems present exciting features and have been
+intensively investigated as a promising technique in devising
+future wireless generations. As future works, hybrid NOMA
+systems and alternative techniques such as rate-splitting mul-
+tiple access (RSMA) can improve the overall EE of massive
+MIMO systems.
+Acknowledgement
+This work was partly supported by The National Council for
+Scientific and Technological Development (CNPq) of Brazil
+under Grants 310681/2019-7, partly by the CAPES- Brazil -
+Finance Code 001, and the Londrina State University - Paraná
+State Government (UEL).
+References
+[1] E. G. Larsson, O. Edfors, F. Tufvesson, and T. L. Marzetta, “Massive
+MIMO for next generation wireless systems,” IEEE Communications
+Magazine, vol. 52, no. 2, pp. 186–195, 2014.
+[2] E. Bjornson, L. Sanguinetti, J. Hoydis, and M. Debbah, “Designing
+multi-user MIMO for energy efficiency: When is massive MIMO
+the answer?,” 2014 IEEE Wireless Communications and Networking
+Conference (WCNC), pp. 242–247, apr 2014.
+[3] Z. Ding and H. V. Poor, “Design of Massive-MIMO-NOMA with
+Limited Feedback,” IEEE Signal Processing Letters, vol. 23, pp. 629–
+633, may 2016.
+[4] K. Senel, H. V. Cheng, E. Bjornson, and E. G. Larsson, “What Role
+Can NOMA Play in Massive MIMO?,” IEEE Journal on Selected
+Topics in Signal Processing, vol. 4553, no. c, pp. 1–16, 2019.
+[5] A. Anwar, B.-C. Seet, M. A. Hasan, and X. J. Li, “A Survey on
+Application of Non-Orthogonal Multiple Access to Different Wireless
+Networks,” Electronics, vol. 8, p. 1355, nov 2019.
+
+12
+[6] M. S. Ali, E. Hossain, and D. I. Kim, “Non-Orthogonal Multiple
+Access (NOMA) for Downlink Multiuser MIMO Systems: User Clus-
+tering, Beamforming, and Power Allocation,” IEEE Access, vol. 5,
+pp. 565–577, 2017.
+[7] M. Zeng, A. Yadav, O. A. Dobre, G. I. Tsiropoulos, and H. V.
+Poor, “Capacity Comparison Between MIMO-NOMA and MIMO-
+OMA With Multiple Users in a Cluster,” IEEE Journal on Selected
+Areas in Communications, vol. 35, pp. 2413–2424, oct 2017.
+[8] S. M. R. Islam, M. Zeng, O. A. Dobre, and K.-S. Kwak, “Resource
+Allocation for Downlink NOMA Systems: Key Techniques and Open
+Issues,” IEEE Wireless Communications, vol. 25, pp. 40–47, apr
+2018.
+[9] X. Chen, F.-K. Gong, G. Li, H. Zhang, and P. Song, “User Pairing
+and Pair Scheduling in Massive MIMO-NOMA Systems,” IEEE
+Communications Letters, vol. 22, pp. 788–791, apr 2018.
+[10] H. V. Cheng, E. Bjornson, and E. G. Larsson, “Performance Anal-
+ysis of NOMA in Training-Based Multiuser MIMO Systems,” IEEE
+Transactions on Wireless Communications, vol. 17, pp. 372–385, jan
+2018.
+[11] L. Dai, B. Wang, M. Peng, and S. Chen, “Hybrid Precoding-
+Based Millimeter-Wave Massive MIMO-NOMA With Simultaneous
+Wireless Information and Power Transfer,” IEEE Journal on Selected
+Areas in Communications, vol. 37, pp. 131–141, jan 2019.
+[12] B. Makki, K. Chitti, A. Behravan, and M.-S. Alouini, “A Survey of
+NOMA: Current Status and Open Research Challenges,” IEEE Open
+Journal of the Communications Society, vol. 1, no. December 2019,
+pp. 179–189, 2020.
+[13] M. R. Usman, A. Khan, M. A. Usman, Y. S. Jang, and S. Y. Shin,
+“On the performance of perfect and imperfect SIC in downlink non
+orthogonal multiple access (NOMA),” 2016 International Conference
+on Smart Green Technology in Electrical and Information Systems
+(ICSGTEIS), pp. 102–106, oct 2016.
+[14] E. Bjornson and E. G. Larsson, “How Energy-Efficient Can a Wireless
+Communication System Become?,” Conference Record - Asilomar
+Conference on Signals, Systems and Computers, vol. 2018-Octob,
+pp. 1252–1256, 2019.
+[15] Y. Zhang, H. M. Wang, T. X. Zheng, and Q. Yang, “Energy-
+Efficient Transmission Design in Non-orthogonal Multiple Access,”
+IEEE Transactions on Vehicular Technology, vol. 66, no. 3, pp. 2852–
+2857, 2017.
+[16] S. Mounchili and S. Hamouda, “Pairing Distance Resolution and
+Power Control for Massive Connectivity Improvement in NOMA
+Systems,” IEEE Transactions on Vehicular Technology, vol. 69, no. 4,
+pp. 4093–4103, 2020.
+[17] F. Rezaei, C. Tellambura, A. Tadaion, and A. R. Heidarpour, “Rate
+analysis of cell-free massive MIMO-NOMA with three linear pre-
+coders,” IEEE Transactions on Communications, vol. 68, no. 6,
+pp. 3480–3494, 2020.
+[18] H. Zhang, H. Zhang, W. Liu, K. Long, J. Dong, and V. C. Leung,
+“Energy Efficient User Clustering, Hybrid Precoding and Power
+Optimization in Terahertz MIMO-NOMA Systems,” arXiv, vol. 38,
+no. 9, pp. 2074–2085, 2020.
+[19] O. Maraqa, A. S. Rajasekaran, S. Al-Ahmadi, H. Yanikomeroglu,
+and S. M. Sait, “A Survey of Rate-Optimal Power Domain NOMA
+with Enabling Technologies of Future Wireless Networks,” IEEE
+Communications Surveys and Tutorials, vol. 22, no. 4, pp. 2192–
+2235, 2020.
+[20] S. Rajoria, A. Trivedi, and W. W. Godfrey, “Sum-rate optimization
+for NOMA based two-tier hetnets with massive MIMO enabled
+wireless backhauling,” AEU - International Journal of Electronics and
+Communications, vol. 132, no. October 2020, p. 153626, 2021.
+[21] D. Kim and M. Choi, “Non-Orthogonal Multiple Access in Dis-
+tributed Antenna Systems for Max-Min Fairness and Max-Sum-
+Rate,” IEEE Access, vol. 9, pp. 69467–69480, 2021.
+[22] I. Budhiraja, N. Kumar, S. Tyagi, S. Tanwar, Z. Han, D. Y. Suh,
+and M. J. Piran, A Systematic Review on NOMA Variants for 5G
+and Beyond, vol. 9. IEEE, 2021.
+[23] F. Zafari, A. Gkelias, and K. K. Leung, “A Survey of Indoor Local-
+ization Systems and Technologies,” IEEE Communications Surveys
+and Tutorials, vol. 21, no. 3, pp. 2568–2599, 2019.
+[24] E. M. Mohamed, “Joint users selection and beamforming in downlink
+milimetre-wave NOMA based on users positioning,” IET Communi-
+cations, vol. 14, no. 8, pp. 1234–1240, 2020.
+[25] T. L. Marzetta, E. G. Larsson, H. Yang, and H. Q. Ngo, Fundamen-
+tals of Massive MIMO. Cambridge University Press, 2016.
+[26] H. Yang and T. L. Marzetta, “Performance of conjugate and zero-
+forcing beamforming in large-scale antenna systems,” IEEE Journal
+on Selected Areas in Communications, vol. 31, no. 2, pp. 172–179,
+2013.
+[27] S.
+Boyd
+and
+L.
+Vandenberghe,
+“Numerical
+lin-
+ear
+algebra
+background,”
+[On
+Line]
+-
+Available:
+https://web.stanford.edu/class/ee364a/lectures/num-lin-alg.pdf.
+[28] J. L. Jacob and T. Abrão, “Nonorthogonal multiple access systems
+optimization to ensure maximum fairness to users,” Transactions on
+Emerging Telecommunications Technologies, vol. 31, no. 4, pp. 1–
+14, 2020.
+
diff --git a/AtE1T4oBgHgl3EQfVQTf/content/tmp_files/load_file.txt b/AtE1T4oBgHgl3EQfVQTf/content/tmp_files/load_file.txt
new file mode 100644
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf,len=1075
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='03101v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='IT]  8 Jan 2023  1  Massive MIMO and NOMA  Bits-per-Antenna Efficiency under Power  Allocation Policies  Thiago A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Bruza Alves, Taufik Abrão  State University of Londrina (UEL), Department of Electrical Engineering, Londrina-PR, Brazil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  Abstract—A comparative resource allocation analysis in terms  of received bits-per-antenna spectral efficiency (SE) and energy  efficiency (EE) in downlink (DL) single-cell massive multiple-input  multiple-output (mMIMO) and non-orthogonal multiple access  (NOMA) systems considering a BS equipped with many (M)  antennas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' while K devices operate with a single-antenna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' and the  loading of devices ρ = K  M ranging in 0 < ρ ≤ 2 is carried out under  three different Power Allocations (PA) strategies: the inverse  of the channel power allocation (PICPA),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' a modified water-  filling (∆-WF) allocation method,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' and the equal power allocation  (EPA) reference method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Since the two devices per cluster are  overlapped in the power domain in the NOMA system, the channel  matrix requires transformation to perform the zero-forcing (ZF)  precoding adopted in mMIMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Hence, NOMA operating under  many antennas can favor a group of devices with higher array gain,  overcoming the mMIMO and operating conveniently in the higher  loading range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='6 < ρ < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' In such a scenario, a more realistic  and helpful metric consists in evaluating the area under SE and  EE curves, by measuring the bit-per-antenna and bit-per-antenna-  per-watt efficiency, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Our numerical results confirm a  superiority of NOMA w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' mMIMO of an order of 3x for the  SE-area and 2x for the EE-area metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  Index Terms—Non-Orthogonal Multiple Access (NOMA);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' mas-  sive Multiple-Input Multiple-Output (mMIMO);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Energy Efficiency  (EE);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Spectral Efficiency (SE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Introduction  The beyond Fifth Generation (5G) of wireless communica-  tion systems must allow ultra-dense connections with vastly  heterogeneous requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' The challenges in networks per-  sist, including the Spectral Efficiency (SE) and the Energy  Efficiency (EE) joint improvement, the increase in the SE-  EE trade-off, and Quality of Service (QoS), always aiming  to meet the growing number of devices connected to the  network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Among the proposals to solve these challenges, the  massive Multiple-Input Multiple-Output (mMIMO) system is  the primary proposed system that allows the increase of the  link capacity, exploring the propagation of multiple paths with  the use of a large number of antennas at the Base Station  (BS) [1], [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Another relevant enabling technology is the  Non-Orthogonal Multiple Access (NOMA), which explores the  power domain as an alternative way in terms of multiple ac-  cess technology, helping to mitigate the spectrum exhaustion  problem and serving more than one device per resource block  [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  Although in many works mMIMO is classified as an or-  thogonal technique, allocating the signal from devices in the  same resource block, possible by spatial diversity, allows us to  This work was partly supported by The National Council for Scien-  tific and Technological Development (CNPq) of Brazil under Grants  310681/2019-7, partly by the CAPES- Brazil - Finance Code 001, and  the Londrina State University - Paraná State Government (UEL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Bruza Alves and Taufik Abrão are with the State University of  Londrina (UEL), Department of Electrical Engineering, Londrina-PR, E-  mails: thiagobruza@hotmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='com, taufik@uel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='br  classify it as a non-orthogonal technique too [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' There is a  vast literature demonstrating the superior performance of the  Spectral Efficiency of NOMA when compared to Orthogonal  Multiple Access (OMA) techniques [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Previous aims to  improve the communication system performance by combining  MIMO (with a small number of antennas M) and NOMA have  been discussed in [6]–[9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  Studies comparing NOMA and mMIMO in a single cell are  proposed in [4], [10], [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' The acquisition of Channel State  Information (CSI) through pilot acquisition to NOMA system  is proposed in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' In [11], the application of NOMA in the  mMIMO scheme is proposed, and better results are achieved  in the proposed comparative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Moreover, in [4] is analyzed the  performance of NOMA and mMIMO in line of sight and non-  line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  The canonical mMIMO refers to the systems with BSs  formed by a large number of antennas M when compared  to the number of actives devices, K, succinctly M ≫ K is  considered a mMIMO setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' The typical NOMA improves the  SE by superposing the signals of the selected devices to form  a cluster in the power domain, multiplexing it over the same  signal and served by the same beamforming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Nonetheless,  the success of NOMA depends on the Successive Interference  Cancellation (SIC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  Power-domain NOMA can be a candidate technology in  dense networks [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' To improve performance and minimize  the impact to assume the perfect SIC [13], devices are divided  into two groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' After grouping in pairs and forming a cluster,  each pair forms a cluster with a high difference between  channel conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' The device with a higher channel condition  can decode the signal sent to the device with the lower channel  condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' The interference can thus be eliminated by SIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' The  use of NOMA in BS equipped with a large number of antennas  was investigated in terms of SE [4], [10] we propose in a similar  configuration system increasing the loading up to 2 times the  number of antennas in BS and analyze the SE, EE, and SE-EE  trade-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  The EE metric is a popular figure of merit employed to  analyze the balance between power consumption and data  rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' The EE is the ratio between the effectively transmitted  data rate and the total power expended during the transmis-  sion process, including instantaneous and static components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  With the EE metric, it is possible to evaluate the efficiency  with which a system uses the limited energy resource to  communicate data and optimize this ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Can show the  tendency of energy consumption in the case of seeking justice  among devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  The Zero Forcing (ZF) is simple and popular alternative  interference suppression beamforming under perfect CSI con-  dition and achieving a satisfactory condition in real situations  when imperfect CSI, in this work, we adopt perfect CSI, for  2  that the pilots are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' The adoption of NOMA system  with a large number of antennas requires a defined equivalent  channel to be deployed for interference mitigation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' and ac-  cording to the NOMA principle, makes the equivalent channel  matrix smaller than the original one due to the exploration of  power domain in NOMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  Various transmission topologies already deal with the EE  problem in mMIMO, finding the optimal number of antennas,  number of devices in a cell, and the maximal EE [2], [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' The  EE analysis in the NOMA system is carried out in [15], and its  superiority is demonstrated when compared with conventional  orthogonal multiple access (OMA) systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Recent researches  seek to improve the NOMA performance, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' in [16], the  minimum pairing distance is defined and compared to the  OMA, while in [17] it’s presented a comparison between OMA  and cell-free system equipped with mMIMO-NOMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' An EE  analysis in Terahertz (THz)-NOMA-Multiple-Input Multiple-  Output (MIMO) was proposed in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Still, the number of  active devices is much smaller than the number of antennas  in the BS, and [19] is a survey about Power Domain NOMA  and makes clear the vacuum of EE analysis and comparison  between NOMA with many antennas and mMIMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  Recent works propose the deployment of NOMA combined  with other techniques a more effective transmission scheme;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=', in [20] NOMA and mMIMO are jointly considered in a  two-tier network for accommodating colossal traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Further-  more, in [21], authors apply NOMA in Distributed Antenna  Systems (DAS), aiming to achieve better performance when  compared to the conventional NOMA or DAS technique alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  While [22] shows an in-depth survey of the state-of-the-art  of power-domain NOMA variants;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' moreover, several open  issues and research challenges of NOMA-based applications  are systematized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' The NOMA system presents drawbacks,  such as hardware (including SIC) complexity, channel feed-  back, receiver design, and careful power and pilot allocation  strategies [12], [19], [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  This work focus on revealing the advantages of applying  the mMIMO scheme versus NOMA scheme with a massive  number of BS antennas, and varying the loading of devices,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=', the ratio of the number of mobile devices to the number  of BS antennas, ρ =  K  M , while we change the PA strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  Besides, we adopt a realistic model for the system’s power  consumption as in [2] but adapted to our needs, aiming at  providing a suitable analysis of the system resource allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  Contributions: the contributions of this work are fourfold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  a) an extensive and comparative analysis on the spectral  efficiency (SE) performance of mMIMO system against NOMA  system, varying the system loading under specific (three dif-  ferent) power allocation methods and making use of the area  under the SE (Ssyst) curve of the system as an effective, useful  and fair metric of performance and efficiency;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' b) we develop  an energy efficiency (EE) analysis using a detailed model of  energy consumption, with fixed and variable terms related to  circuitry power consumption with number of antennas and  devices, respectively, providing an extensive and comparative  analysis on both the NOMA and mMIMO systems under  realistic operation scenarios and making use of the area under  the EE (Esyst) curve of system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' c) an analysis on the SE-EE  trade-off is developed considering a wide range of loading of  devices, verifying the fairness between devices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' d) finally, under  mild conditions, we provide evidences for the NOMA’s ability  to serve a greater number of devices than mMIMO system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  The remainder of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Section  II describes the system models for NOMA and mMIMO  adopted in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' In Section III we present the proposed  EE-SE formulation for NOMA and massive MIMO systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  Numerical results are analyzed in IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Section V concludes the  paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' In this work, boldface lower case and upper case  characters denote vectors and matrices, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' The  operator (x)+ = max(0, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' The operators [·]T, E[·] and  |·| denote transpose, expectation and cardinality, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  A random vector x ∼ CN {0, Im} is circularly symmetric  Gaussian distributed with mean 0 and covariance matrix Im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  Im is m × m identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' System Models  Let us consider a multi-user single-cell Downlink (DL)  transmission operating in a Time Division Duplex (TDD) with  K single-antenna actives devices, communicating with one BS,  which is equipped with M transmit antennas in Non-Line-Of-  Sight (NLOS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' The set K is formed by K devices, these devices  are randomly distributed in a radius disk dmax, the disk area  is formed by two sub-disk with the same number of devices  in each sub-area;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' both subsets are identified as KH and KL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  In the first subset, KH represents devices’ indexes having the  higher channel coefficient and sort in descending order, while  the other subset KL are formed by the devices with lower  channel coefficient and sort in ascending order;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' the indexes  k ∈ KH and k ∈ KL such that:  K = KH ∪ KL,  where  KH = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=', K/2} and KL = {K/2 + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=', K}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  (1)  The channel vector modeling of device k can be described liked  as:  hk =  �  βkh′  k,  k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=', K,  (2)  where βk is the large-scale fading coefficient and satisfy  βj > βi,  ∀j ∈ KH, ∀i ∈ KL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  (3)  Herein, the pathloss model in [dB] is defined as:  βk = β0 + 10 · ξ · log10(dk),  (4)  where dk is the distance of user k to BS, ξ is the pathloss  coefficient, and β0 is the attenuation at the distance of  reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  In each coherence interval, h′  k in (2) for device k is an  independent random small-scale fading realization from an in-  dependent Rayleigh fading distribution, h′  k ∼ CN(0, IM), k =  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=', K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' The transmitted signal xk ∈ CM is the beamformed  data symbol of device k:  xk = gk  √pksk,  (5)  where gk is a normalized beamforming vector, pk normalized  transmission power and sk ∼ CN(0, 1) is the data symbol of  device k, and period Ts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' The signal received at the k-th device:  yk =  K  �  k′=1  hT  k xk′ + nk,  =  �  βkh′T  k  K  �  k′=1  gk′√pk′sk′ + nk,  (6)  =  �  βkpkh′T  k gksk +  �  βkh′T  k  K  �  k′̸=k  k′=1  gk′√pk′sk′ + nk,  where nk ∼ CN(0, 1) is the additive noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Notice that this  modeling applies to both NOMA and mMIMO systems, but  beamforming is selected differently, and this topic will be  addressed in the next sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  3  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Prior Actions  Because the BS needs to know a priori crucial information  related to the channel and devices distributed in the cell,  including device location, rate demanded, and channel coeffi-  cient, such required a priori information may differ depending  on the multiple access scheme considered [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  The option for the mMIMO and NOMA systems was carried  out with the guarantee that the needs of the devices would  be met, and this initial step was carried out successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  Subsection III-E briefly discusses the preliminary information  required to proceed with different Power Allocation (PA)  procedures in both mMIMO and NOMA systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Pilot Overhead for Channel State Information  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 1 compares the pilot-data transmission structure along  the one-channel coherence interval for both mMIMO and  NOMA systems considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Notice that Ts is the time required  to transmit a data symbol (Data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Moreover, the channel  coherence time interval T is assumed to be a multiple of the  Data symbol period, T = ι · Ts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' The power allocated to each  pilot in the training step is enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' In contrast, the number of  pilots, and the dedicated portion of coherence interval for data  transmission are assumed to be the same in both systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='"  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='"  #"  $%&\'(  #)()  #"  #"  +,+-  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content="-+/  0123'41510612,0(157)&  (%*1  Figure 1: Coherence time interval structure: the training and  data transmission structure for mMIMO and NOMA schemes  under TDD NLOS setup." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  Notice that in NOMA transmission, the pilot transmission  step is split into two portions, half for the UL transmission  pilots and receive the DL pilot confirmation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' this happens  because to perform SIC, the cell-center devices need to learn  the effective channels that are established by the beamforming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  Additionally, the beamforming vectors are based on cell-center  devices, producing limited rates achieved in cell-edge devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  On the other hand, in the mMIMO scheme, a significant  advantage is that there is no need for DL pilots since the  effective channels created by the beamforming are highly  predictable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' nearly deterministic gain and phase due to  channel hardening effect [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  Assumption 1: In the NOMA system, the power allocated to  each downlink pilot is sufficient to reach the destination device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Beamforming for NOMA and mMIMO systems  At the mMIMO system, each device is served by a sin-  gle beamforming vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' The ZF technique is a popular  interference-suppressing beamforming scheme in the mMIMO  system since it eliminates all inter-user interference using  individual beamforming for each device, while the favorable  propagation facilitates such interference suppressing in massive  MIMO configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Besides, to perform ZF precoding in  NOMA system, it is essential to understand the NOMA user-  pairing concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  User-pairing: Inherent to the NOMA system, user clustering  can be performed in several ways after the user-sorting and  the user classification in center-users and edge-users subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  Because we know that the SE of NOMA is directly proportional  to the difference between the pathloss of the users, a natural  choice consists in pairing users with as higher as possible  pathloss differences [6]:  ∆βk = βk − βK+1−k,  (7)  forming the cluster k for k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=', K/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' With the pair formed,  carefully beamforming vectors selection is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Hence, in  NOMA we assume that the beamforming vector for paired  users is the same, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=', gk = gK+1−k for all k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=', K/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  Assumption 2: In user-pairing procedure, we assume that  the paired users are aligned with the BS so that the same  beamforming can serve all paired users simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Hence,  by admitting that each pair of devices is spatially aligned with  the BS, and using localizing tools described, for instance, in  [23], [24], one should assume further a priori user-pairing step  in NOMA systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  Assumption 3: In NOMA system, beamforming serves more  than one aligned device simultaneously;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' specifically, in this  paper, two aligned devices per cluster are admitted according  to the user-pairing step, while eliminating the inter-cluster  interference (favorable propagation) under adopted perfect  CSI conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  In this work we adopt the linear ZF precoding as defined by  the vector:  gk = h′k(h′T  k h′k)−1,  (8)  and satisfying h′T  i gk  =  0, ∀ i  ̸=  k, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=', the favorable  propagation effect between users belonging to distinct clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' SE-EE in NOMA and mMIMO systems  We discuss the SE and EE configurations in the NOMA and  mMIMO systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' The operation of the NOMA system requires  pairing devices so that the channel coefficients of the devices  in the same cluster must be appropriately different, enabling  power domain usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' As already mentioned, the interference  cancellation process via beamforming presents problems that  we will demonstrate below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Data Rates in NOMA with ZF  Devices are divided into two sets like described in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' (1),  and these groups are represented by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' (9) by their large-  scale fading coefficient and are grouped into pairs forming a  cluster as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 2, the cluster k is formed by one device in cell  center set KH and one device in cell edge set KL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Hence, the  devices are grouped into two subsets:  KH ={β1 > β2 > .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' > βk/2},  (center devices set)  (9)  KL ={βK < βk−1 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' < βk/2+1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' (edge devices set)  The user-pairing adopted in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' (9) is the same as proposed  in [6], creating the largest possible difference in channel  coefficients for devices not yet paired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  Assumption 4: In this paper, we assume perfect SIC, and only  one perfect SIC stage per cluster is performed, since just 2  devices per cluster are admitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  4  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='"  Figure 2: System Model indicating the Paring formation in  NOMA system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Both mMIMO and NOMA systems deploy the  same massive number of antennas at base-station, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  The instantaneous Signal to Interference plus Noise Ratio  (SINR) of devices in cluster k is defined as:  SINRk =  βkpk|h′T  k gk|2  βk  �K  k′̸=k pk′|h′T  k gk′|2 + 1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  (10)  In each cluster, the cell-edge devices treat the interference as  noise and decode their data symbols, whereas the cell-center  device can decode the data symbols of the cell-edge device  and perform SIC, hence effectively removing the interference  due to the cell-edge device under Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  To perform SIC, the cell-center device needs to be able to  decode data signal intended for the cell-edge device, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=', the  ergodic SINR of the cell-edge device, sinrK+1−k, at device  k, defined as sinrk,K+1−k, must be greater than or equal  to the ergodic SINR of the k-th cell-center device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Hence,  given the uplink (mMIMO and NOMA) and downlink (NOMA)  pilot overhead and assuming perfect CSI in all receivers, and  admitting Assumption 4, the following condition must be  satisfied [4], [10]:  E[SINRk,K+1−k] ≥ E[SINRk],  (11)  where  SINRk,K+1−k =  βkpK+1−k|h′T  k gk|2  βk  �K  k′̸=K+1−k pk′|h′T  k gk′|2 + 1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  (12)  Herein, the condition in (11) must be satisfied by selecting the  transmit powers appropriately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  The achievable ergodic rate of devices in cluster k, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  device k in KH subset and device K+1−k in KL subset, under  Assumptions 1–4, is given by the ergodic rate contribution of  user-center device:  Rnoma  k  = τE [log2 (1 + SINRk)] ,  ∀k ∈ KH  (13)  in [bits/s/Hz], and for the user-edge device:  Rnoma  K+1−k = τE [log2 (1 + SINRK+1−k)] ,  ∀k ∈ KL (14)  where τ = (1− K·Ts  T ), is the portion of each channel coherence  interval (T) that is used for data transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  Assuming perfect channel state information, ZF precoding  for inter-clusters interference elimination, and using random  matrix theory results [25], the k-th cluster NOMA achievable  rate is obtained plugging eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' (10), (13) and (14):  Rnoma  cl-k  = τE  �  log2  �  1 + ¯  Mβkpk  ��  +  (15)  τE  �  log2  �  1 + βK+1−kpK+1−k  βK+1−kpk + 1  �  ,  �  ∀k ∈ K and ¯  M = M + 1 − K/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Hence, the NOMA system  can operate until K < 2M − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' A detailed derivation of the  expressions on this section can be found in [4] and [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Data Rates in mMIMO with ZF  In the mMIMO system with ZF precoding the ergodic  achievable rate for device k is given by:  Rm-mimo  k  = τE [log2 (1 + SINRk)] ,  [bits/s/Hz]  (16)  where SINRk is defined in (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Hence, the above mMIMO  achievable rate equation becomes:  Rm-mimo  k  = τE [log2 (1 + (M − K)pkβk)] ,  [bits/s/Hz],  (17)  where (M − K) is obtained using random matrix theory,  representing the coherent array gain of the received signal [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  Under linear precoding and combiners, the mMIMO system  operates consistently when K < M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Finally, the average  system sum-rate (avg-sum-rate) is defined simply by:  Rm-mimo =  K  �  k=1  Rm-mimo  k  and  Rnoma =  KH  �  k=1  Rnoma  cl-k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  The mMIMO system equations have been thoroughly investi-  gated in literature and can be found in [25] and [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Energy Efficiency  EE metric is the ratio of the number of effective bits of  information received over the total energy consumed by the  overall system to transmit and receive/decode such informa-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' The system data rate can determine the number of  effective information bits received at the destination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Power  consumption required for processing the signal at the trans-  mitter and receiver side is often neglected;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' in this sense, it  is calculated just as proportional to the radiated transmitted  power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' The growth in the number of antennas in the BS and  the increased number of devices in 5G systems can lead to  unattainable EE goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' In general, the average EE can be  expressed as:  EE =  �K  k=1 Rk  Ptot  ,  [bits/Joule/Hz],  (18)  where Ptot is the total power consumption across the com-  munication system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' It should consider transmission power  consumption, such as RF power amplifier inefficiency, base-  band signal processing, and cooling, among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Therefore,  a more realistic and detailed energy consumption model is  required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  Based on [2], the adopted power consumption model in our  work considers two power terms: a) fixed-term;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' b) terms scaled  with the number of antennas M and the number of devices  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' The scaled terms occur because of the transceiver chains,  coding/decoding, channel estimation, and precoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Let the  computational efficiency be L operations per joule in BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' We  describe it as follows:  RF Power: Prf is the power consumed to transmit the signal  to active devices achieved the SINR target and 0 < ̟ ≤ 1 is  the efficiency of the power amplifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  β1dcenl  50mβ2K/2  β  K  +1  2KOK-1  umac5  Fixed consumption: Pfixed is the power consumed at the  BS which is independent of the number of transmit antennas  and devices in the cell, is formed of term P0 including the  power consumption of backhaul infrastructure, control signal-  ing, baseband processor, and term Psyn a single oscillator used  in all BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  Pfixed = P0 + Psyn  Dependence only on K: PK is formed by the consumption  to coding and modulation of information symbols to devices,  represented by Pcod, the consumption to BS decoded the K  sequences of information symbols, defined by Pdec, and the  received power, represented by PRX, still composes this term,  multiplied by K as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' In addition, a portion of the ZF  precoding cost [27] depends only on K3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  PK = K(Pcod + Pdec + PRX) + K3  2  3LT  Dependence only on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' The term PM represents the  transmitted power (PTX), hence  PM = MPTX  Dependence on K and M: the term PKM is the cost of the  ZF precoding (due to LU-based matrix inversion) [27], which  depends on the number of devices, the number of antennas,  and the vector information symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  PKM = MK 3 + T  T L  + MK2 2  T L  Adding the portions, we obtain the overall power consump-  tion of the system:  Ptot = Prf  ̟ + Pfixed + PK + PM + PKM  [W].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  (19)  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Power Allocation Strategies  In the sequel, we present three well-known and frequently  applied strategies for power allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Still, due to the in-  herent characteristics of NOMA, we propose modifications on  the classical water-filling (WF) algorithm to enable application  in the NOMA system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Such modifications, namely ∆-WF,  ensure that none of the paired devices are dropped-out without  undoing the pairing of devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' To guarantee a certain level of  power disparities in each paired device, the power allocation ∆-  WF procedure in the NOMA system has two steps: a) first, we  allocated power for the clusters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' b) we allocate power between  paired devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Thereby, we could analyze the behavior of the  systems and compare their results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  Notice that both mMIMO and NOMA systems deploy the  same massive number of antennas at base-station, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Hence,  due to the channel hardening effect [1], [25] inherent to  massive MIMO configuration, the small-scale fading vanishes  across the M antennas equipped with a linear ZF precoding  with vector as eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Hence, one can consider just the  pathloss coefficients βk as the main parameter in the power  allocation policies of systems based on a massive number of  antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  1) Equal Power Allocation (EPA): Equal Power Allocation  (EPA) power allocation is deployed as a simple, naive strategy,  where all devices are served with the same power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' In mMIMO,  all devices are served with the same transmission power  regardless of their distance from the BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' In NOMA, power  allocation has two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' In the first step, the power is allocated  equally between the pairs, and then we allocate each device’s  power equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' The EPA strategy applied to mMIMO can be  defined by:  pk = Prf  K  [W],  ∀ k ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  (20)  In the case of EPA procedure applied to NOMA, it is composed  of two steps: in the first step, power reference to each cluster  can be defined simply as:  pcl  ref = 2 · Prf  K  [W],  ∀ k ∈ KH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  (21)  In the second step, the power allocation among the devices in  the same cluster is defined as:  pcl-k  k  = pcl-k  K+1−k = pcl  ref  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  (22)  2) Proportional  Channel  Inversion  Power  Allocation  (PICPA): is another power allocation technique adopted in  this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Unlike the EPA technique, which applies the same  power to all devices, PICPA applies more power to devices  with the worst channel conditions, favoring fairness across  the devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Such power allocation penalizes the average sum  rate in favor of fairness among all users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  The Proportional to the Inverse of the Channel Power Al-  location (PICPA) strategy applied to mMIMO can be defined  as:  pk = Prf  β−1  k  �K  k=1 β−1  k  [W],  ∀k ∈ K,  (23)  while the PICPA procedure applied to NOMA follows two  steps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' in the first step,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' the power is allocated equally among  the K/2 clusters:  pcl  ref = 2 · Prf  K  [W],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  ∀k ∈ KH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  (24)  after that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' the power of each device within the k-th cluster is  defined by allocating more power to the device with smaller  large-scale fading βk:  pcl-k  k  = pcl  ref  βK+1−k  βk − βK+1−k  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  and  pcl-k  K+1−k = pcl  ref − pcl-k  k  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  (25)  where pcl-k  k  is the power allocated to the device k in the cl-k  cluster,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' and pcl-k  K+1−k is the power allocated to the device (K +  1 − k),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' also belonging to the k-th cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  3) Classical Water-Filling (WF) Algorithm: The application  of the Water-Filling (WF) algorithm in mMIMO system results  in an optimal (maximum) system sum-rate solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' However,  some devices are dropped out of the service due to the dete-  riorated channel condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' The WF power allocation strategy  for mMIMO is described as:  µ = 1  |K|  \uf8eb  \uf8edPrf +  |K|  �  k=1,k∈K  1  βk  \uf8f6  \uf8f8 ,  (26)  Prf =  |K|  �  k=1,k∈K  pk ,  where  pk =  �  µ − 1  βk  �+  , ∀k ∈ K  and  p = [p1, p2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=', p|K|],  with the operator (z)+ = max(0, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Notice that the con-  strained value for the total power available is set to Prf [W].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  The Algorithm 1 describes the classical WF procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  On the other hand, the direct application of WF algorithm  in the NOMA system implies harming the pair formation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=',  devices present in the KL set are effectively dropped-out of  the service, undoing the pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' We propose a modification in  classical WF like the following to allow some comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  6  Algorithm 1: Classical Water Filling (WF) for mMIMO  Input: K,Prf, K = |K|  1 NP ̸= ⊘;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  2 while (NP ̸= ⊘) do  3  solve Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' (26) → p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  4  NP ← identify null positions in p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  5  K/{k}NP ← exclude from p devices labeled as NP  6 end  Output: p = [p1, p2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' , p|K|]  4) ∆-WF for NOMA: The application of classical WF in  NOMA in the same way as it is applied to mMIMO causes  some formed pairings to be broken, since after WF algorithm  application, some devices are dropped-out from the system,  making the NOMA power difference (∆) in the devices of the  same cluster vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Hence, we suggest modifying the classical  WF procedure to be applied to NOMA accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' The steps  of the ∆-WF algorithm are described as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  In NOMA, the power allocation has two steps, in the first  step, the allocation is between clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Hence, to prevent the  formed pairs from being broken, we propose the application of  WF based on the large-scale fading differences of the paired  devices, as defined in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' (7): ∆βk = (βk − βK+1−k) inside  each KL and KH subsets, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' In the second step of  the procedure, the power is allocated between the devices  inside the group, assuming perfect successive interference  cancellation (SIC);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' for this to be possible, the condition in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  (11) must be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' The new water-level in the modified  ∆-WF power allocation strategy for NOMA is defined by  µ = 2  KH  \uf8eb  \uf8edPrf +  |KH|  �  k=1,k∈KH  1  ∆βk  \uf8f6  \uf8f8 ,  (27)  Prf =  |KH|  �  k=1,k∈KH  pcl-k,  ∀k ∈ KH  where  pcl-k =  �  µ −  1  ∆βk  �+  ,  and  pcl-k = [p1, p2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=', p|KH|],  In the second step, the power allocation to both devices in the  k-th cluster is defined as:  pcl-k  K+1−k = pcl-k  k  = pcl-k  2  (28)  Algorithm 2 summarize the proposed ∆-WF power allocation  procedure, aiming to improve the SE of NOMA systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  Algorithm 2: ∆-WF (modified) for NOMA systems  Input: KH, KL, Prf  1 NP ̸= ⊘;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  2 while (NP ̸= ⊘) do  3  solve Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' (27) → pcl-k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  4  NP ← identify null position in pcl-k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  5  KH/{k}NP ← exclude from pcl-k devices labeled as  NP  6 end  Output: pcl-k = [p1, p2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' , p|KH|]  Complexity analysis: In a comparative analysis of complexity,  the ∆-WF algorithm for power allocation in NOMA system  (Algorithm 2) performs two simple additional operations com-  pared to the classical WF procedure (Algorithm 1): a) in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  (27) the subtraction in (βk − βK+1−k);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' and b) the division  by two in (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Besides, NOMA vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' mMIMO systems require  different a priori information to proceed accordingly with the  PA procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Prior Information for Power Allocation Step  For implementing the PA policies, prior information is re-  quired at the BS, as defined in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Some of this necessary  information can be obtained through the dedicated pilot trans-  mission step, at the cost of some overhead, as described in  Section II-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Moreover, a preliminary step is known, in which  the spatial localization and path loss estimation of all devices  must be realized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' With such a priori information availability,  the user-sorting and user-pairing steps can be performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  Table I: Prior information required to PA procedure  PA  βk  h′  k  K  KH  KL  pk, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  EPA  mMIMO  −  −  ▲∗  −  −  (20)  NOMA  −  −  ▲  −  −  (21), (22)  PICPA  mMIMO  ▲∗  −  −  ▲∗  ▲∗  (23)  NOMA  ▲  −  −  ▲∗  ▲  (24), (25)  WF  mMIMO  ▲∗  −  −  ▲∗  ▲∗  (26), Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 1  ∆-WF  NOMA  ▲  −  −  ▲∗  ▲  (27, 28), Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 2  ▲ information needed a prior  ∗ obtained via Pilot Overhead  − Information not needed  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Numerical Results  The numerical evaluations for the proposed analyses of  NOMA and mMIMO systems are presented in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  The simulation system and channel parameter values de-  ployed along this section are depicted in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' The BS  is located at the center of cell and equipped with massive  M BS transmit antennas in typical non-line-of-sight (NLOS)  channel propagation scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' At the same time, the devices  are randomly distributed in the cell area and split into two  subsets, KL and KH, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' In all simulations,  we consider a block fading model where the time-frequency  resources are divided into coherence time intervals (T), in  which the channels remain constant and frequency flat, and  it is measured in multiple of symbol transmit period (Ts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  The system and channel scenarios have been simulated using  Matlab 2019 software running under one Intel HD Graphics  6000 GPU, Intel(R) Dual-Core(TM) I5 CPU @ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='6 GHz, and  8 GB RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  Table II:  Simulation Parameters  Parameter  Value  BS antennas  M = 64, 128 and 256  Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' # Devices in the cell  K = ζ · M (NOMA)  K = M (mMIMO)  Cell loading  ρ = K/M  Total RF power available  Prf = 1W  Pairs NOMA / Clusters  N = K/ζ = K/2  NOMA devices per cluster  ζ = 2  # antennas per device  1  Cell edge length  dmax = 350 m  Strong device position  [dmin;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' d1] ∈ [50;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 100] m  Weak device position  [d2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' dmax] ∈ [150;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 350] m  Array gain MIMO device  M − K  Array gain NOMA kH  M + 1 − K/2  Data symbol period  Ts  Coherence time interval  T = 512 · Ts,  ι = 512  Channel  Pathloss exponent  ξ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='78  Attenuation at a d0 reference  β0 = 130 [dB]  # Monte-Carlo realizations  1000  7  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Spectral Efficiency Comparison  The mMIMO and NOMA SE performance analysis is carried  out in this subsection, by increasing the number of devices  two by two until the loading limit ρ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' The results consider  M = 64, 128 and 256 BS antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' (a) the results  of SE are achieved when the RF power available is allocated  following the EPA strategy, where each device receives the  same PA values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' The mMIMO system overcame NOMA in all  situations when the loading ρ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' However, the NOMA  system achieves a higher SE than the mMIMO for each M  scenario when the loading of devices increases, ρ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' The  maximum avg-SE is 373 [bits/s/Hz], being attained with ZF-  NOMA M = 256 antennas and ρ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Besides, one can  infer that the mMIMO does not work with a loading higher  than 1, due to the array gain reaching 0 at full loading of  devices, while NOMA works suitably until the loading attains  M · ζ, where ζ is number of devices per cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' (b) depicts the results of SE achieved in the mMIMO  and NOMA systems when the PICPA method is applied to  allocate the available RF power per device along the BS an-  tennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' The maximum avg-SE in mMIMO system overcomes  the NOMA counterpart until the loading ρ exceeds ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='62 for  the three BS antenna configurations, M = 64, 128, and 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  This PA technique provides more power to devices with the  worst channel condition, making the SE result reach maximum  values below the EPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' (c) depicts the conventional WF algorithm applied  to mMIMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Under such a power allocation approach, we  highlight that forming pairs is unfeasible in the NOMA sys-  tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Indeed, the WF algorithm can maximize the system SE  since it allocates more power to devices with better channel  conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' In contrast, devices under bad channel conditions  (below the water level) are dropped out of the service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  The classical WF algorithm has been adapted to the NOMA  system dropped-out always a pair of devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Such adaptation  reveals substantial improvements of avg-sum-rate when M  is low compared to classical WF PA in mMIMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' The ∆-  WF power allocation procedure preserves the pairs clustering  formation in the NOMA system, allocating more power to  the cluster with a higher difference between coefficients of  large-scale fading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' (c) shows that the maximum avg-SE  ≈ 361 [bits/s/Hz], which is achieved under ρ = 1 (K ≈ 256  devices) when the modified WF is deployed in NOMA system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  Moreover, when the number of BS antennas is lower (M = 64  or 128), the NOMA achieved a peak higher than mMIMO,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=', for M = 64 antennas, the peak of SE mMIMO occurs  at loading ρ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='7, while the NOMA SE peaks at ρ ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  However, as the number of BS antennas grows, the NOMA  SE advantage decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  Number of active devices after PA procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 4 shows  the number of actives device after applying PA methods: in  the EPA and PICPA algorithms, all devices remain activated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  However, in classical WF mMIMO system when the number  of device increases beyond ρ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='25, half of the devices are  dropped-out;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' while in ∆-WF NOMA PA, the percentage of  active devices is always higher, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' higher than 70% for ρ ≈  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='1 and M = 256 antennas, the worst case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 5 summarizes avg-sum-rate surfaces in terms of SE  ×ρ × M results achieved by NOMA with EPA, mMIMO  with WF, and NOMA with modified ∆-WF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' In the initial  loading part, ρ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='65, the classical WF PA in ZF-mMIMO  achieves better results until the loading (pink surface).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' When  the number of antennas is low as M = 64 and ρ is in  between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='7 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='8, the EPA PA applied to NOMA (green  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='8  2  Loading  = K/M  0  50  100  150  200  250  300  350  400  Average Sum Rate (bits/s/Hz)  ZF-mMIMO  ZF-NOMA  M = 256  M = 128  M = 64  (a) EPA  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='8  2  Loading  = K/M  0  30  60  90  120  150  180  Average Sum Rate (bits/s/Hz)  ZF-mMIMO  ZF-NOMA  M = 256  M = 128  M = 64  (b) PICPA  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='8  2  Loading  = K/M  0  50  100  150  200  250  300  350  400  450  Average Sum Rate (bits/s/Hz)  ZF-mMIMO-WF  ZF-NOMA-  WF  M = 256  M = 128  M = 64  (c) Classical WF in ZF-mMIMO and ∆-WF in ZF-NOMA  Figure 3: The average sum-rate with the loading of devices 0 <  ρ ≤ 2, considering four power allocation methods: EPA, PICPA,  WF, and ∆-WF The average is obtained over 1000 random devices  locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  8  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
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+page_content='9  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
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+page_content='9  2  Loading  = K/M  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
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+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='9  1  Actives Users  ZF-mMIMO-WF - M = 64  ZF-NOMA-  WF - M = 64  ZF-mMIMO-WF - M = 128  ZF-NOMA-  WF - M = 128  ZF-mMIMO-WF - M = 256  ZF-NOMA-  WF - M = 256  ALL-EPA  ALL-PICPA  Figure 4: The average of active devices after PA procedure  versus loading of devices in the range 0 < ρ ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' The average  is obtained over 1000 random devices locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  Figure 5: Average sum rate with loading ρ and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  surface) achieved superior results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Moreover, when M = 128  and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='8 < ρ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='6, the ZF-NOMA-EPA achieve superior  results (green surface).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' For a higher number of antennas in  BS, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=', M = 256 only in a short loading of devices range,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='86 < ρ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='97, the ZF-NOMA-EPA achieves superior SE  results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Finally, when ρ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='97, the modified ∆-WF achieves  competitive results (blue surface).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Jain’s Fairness Index  Another critical analysis developed was to analyze the  fairness between the devices, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=', to know the difference in  the transmission rate achieved by active devices in the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' For  this measure, we use the Jain’s Fairness index like described  in [28] and can be defined as:  Fsyst  m  =  ��M  k=1 Rk  �2  M �M  k1 R2  k  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  (29)  The Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' depicts the fairness curves attainable by NOMA  and mMIMO systems with EPA, PICPA, WF, and ∆-WF PA  procedures when the loading of devices grows until ρ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' (a) shows the Jain’s Fairness Index when EPA policy is used,  the mMIMO system performs better F results than NOMA for  ρ < 1, on the other hand, NOMA can attain Fnoma  m  ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='5 in  almost every loading of devices, independent of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' (b) reveals the Jain’s Fairness Index when the PICPA  method is applied, despite the SE result being lower in this PA  method, the mMIMO obtains the best fairness result, keeping  the Fmmimo  m  consistently above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='85, still the NOMA under  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  The Jain’s Fairness Index when WF and δ-WF are depicted  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='6(c), It is intrinsic to these algorithms to allocate more  power to devices with better channel conditions, which causes  fairness between devices to be impaired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' In this method, it is  possible to observe a significant influence of the number of  antennas M in the BS and the fairness result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Energy Efficiency Comparison  Energy efficiency (EE) is another important figure of merit  used to compare systems’ performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' In this section, a power  consumption model based on fixed circuitry power part and  that one varying according to the number of antennas M and  the number of devices K has been adopted, following eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  Table III [2] presents the adopted parameter values for the  EE analysis and comparison discussed in this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  7 depicts the performance of EE with EPA, PICPA, WF, and  ∆-WF PA procedures, considering the exact three quantities  of antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' (a) shows the EE performance with EPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' In  this method, all devices receive the same power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' The avg-EE  mMIMO overcomes the NOMA around 13% to 20%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' It was  possible to observe that adding antennas in the BS increases  the power consumption, harming the EE result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Again, under  loading of devices 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='6 < ρ ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='0, the NOMA overcomes the  mMIMO system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  Table III: Adopted Parameters values for EE analysis [2]  Parameter  Value  Backhaul Infrastruture  P0 = 2 W  Single oscillator  Psyn = 2 W  Coding and modulation  PCOD = 4 W per device  Decoding and demodulation  PDEC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='5 W per device  Receive power  PRX = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='3 W per device  Transmitted power  PT X = 1 W per antenna  Efficiency of Power Amplifier  ̟ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='3  Operations/Joule  L = 109 oper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' per joule  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' (b) reveals the EE performance with PICPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' In this  method, more power is allocated to devices with poor channel  coefficients, resulting in reduced poor EE performance for both  NOMA and M-MIMO systems, attaining a maximum of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='25  and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='16 [bits/W] for mMIMO and NOMA, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' The  maximum EE attained by mMIMO is generally around 50%  higher than NOMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' However, for loading of devices ρ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='65,  NOMA overcomes mMIMO EE performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' (c) depicts the EE performance in the mMIMO with  classical WF and in the NOMA with ∆-WF algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' It  is possible to confirm the superiority of energy efficiency of  mMIMO within the range where it operates consistently, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=',  0 < ρ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Notice that the maximum EE achieved by mMIMO  is about 70 % higher than NOMA for different BS antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  Finally, NOMA becomes more energy efficient than mMIMO  only when the loading of devices is high, ρ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  In all analyzed system scenarios, the mMIMO equipped with  classical WF PA procedure achieves higher maximum EE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' The  mMIMO attains better EE results than NOMA for ρ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' On  the other hand, NOMA can serve a more significant number  of devices (twice) than mMIMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 8 summarizes the best EE results in a surface plotting  for the mMIMO with WF overcoming NOMA across the entire  IOMA-Z-VWF  ZF-NOMA-EPA450  n Rate (bits/s/Hz)  400  350  300  200256128  M  64Average Sur  150  100  50  0  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='2  p  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='29  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='8  2  Loading  = K/M  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='9  1  Jain Fairness Index  ZF-mMIMO - M = 64  ZF-NOMA - M = 64  ZF-mMIMO - M = 128  ZF-NOMA - M = 128  ZF-mMIMO - M = 256  ZF-NOMA - M = 256  (a) EPA  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
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+page_content='9  1  Jain Fairness Index  ZF-mMIMO - M = 64  ZF-NOMA - M = 64  ZF-mMIMO - M = 128  ZF-NOMA - M = 128  ZF-mMIMO - M = 256  ZF-NOMA - M = 256  (b) PICPA  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
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+page_content='9  1  Jain Fairness Index  ZF-mMIMO - M = 64  ZF-NOMA - M = 64  ZF-mMIMO - M = 128  ZF-NOMA - M = 128  ZF-mMIMO - M = 256  ZF-NOMA - M = 256  (c) WF and ∆-WF  Figure 6: The fairness of NOMA and mMIMO system under three  power allocation procedures: (a) EPA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' b) PICPA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' (c) WF and ∆-  WF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' The average is obtained over 1000 random device locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
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+page_content='8  2  Loading  = K/M  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
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+page_content='6  EE (bits/Joule/Hz)  ZF-mMIMO - M=64  ZF-NOMA - M=64  ZF-mMIMO - M=128  ZF-NOMA - M=128  ZF-mMIMO - M=256  ZF-NOMA - M=256  (a) EPA  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
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+page_content='27  EE (bits/Joule/Hz)  ZF-mMIMO - M=64  ZF-NOMA - M=64  ZF-mMIMO - M=128  ZF-NOMA - M=128  ZF-mMIMO - M=256  ZF-NOMA - M=256  (b) PICPA  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
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+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
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+page_content='8  2  Loading  = K/M  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
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+page_content='8  EE (bits/Joule/Hz)  ZF-mMIMO-WF - M=64  ZF-NOMA-  WF - M=64  ZF-mMIMO-WF - M=128  ZF-NOMA-  WF - M=128  ZF-mMIMO-WF - M=256  ZF-NOMA-  WF - M=256  (c) WF and ∆-WF  Figure 7: EE for NOMA vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' mMIMO under three power allocation  procedures: (a) EPA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' b) PICPA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' (c) WF and ∆-WF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Average EE  obtained over 1000 random devices locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  10  loading range where it operates consistently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' For device loading  ρ > 1, the NOMA operates under lower EE until the loading  ρ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Moreover, considering the smallest number of antennas  in the BS, the NOMA with EPA overcame the NOMA with  modified ∆-WF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' despite that, as the number of antennas in  the BS increases, the NOMA with ∆-WF achieves marginal  superior EE results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  Figure 8: EE with loading ρ and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Area Under Curves SE and EE  For a fair comparison, one can consider a wide range  of average SE and EE along the loading of devices, and  normalized per antenna, which can be attainable by NOMA  and mMIMO systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Hence, let us consider the corresponding  areas under the SE and EE curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 3 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 7, such  that:  Ssyst  M  = 1  M ·  � ρ=2  0  SE(ρ) dρ  �bits/antenna  s · Hz  �  and  Esyst  M  = 1  M ·  � ρ=2  0  EE(ρ) dρ  �bits/antenna  Joule · Hz  �  ,  respectively, where SE(ρ) is the average overall system sum-  rate, and EE(ρ) is the average overall system energy efficiency  achieved under specific loading of devices ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Hence, comparing  the values of corresponding areas under the SE and EE curves  of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 3 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 7, we obtained Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  From the SE perspective, and considering EPA policy, the  higher area-under-SE-curve ratio gain is achieved when the  number of BS antennas is M = 64:  Snoma  M=64 ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='7 · SmMIMO  M=64 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  Notice that when the number of antennas M grows, the ratio  above decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' In the same way, considering WF policy, the  gain trend remains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' In contrast, considering the PICPA policy,  the ratio practically remains the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  Furthermore, considering now the EE perspective, under  EPA policy, the higher ratio is achieved when the BS is  equipped with M = 256 antennas:  Enoma  M=256 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='8 · EmMIMO  M=256 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  As one can conclude, in almost all scenarios, NOMA is more  spectrally and energetically efficient than mMIMO over an  extensive range of loading of devices 0 < ρ ≤ 2, roughly,  in average, 80% in terms of energy efficiency, and 170% more  efficient in terms of spectral efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  EPA-64  EPA-128  EPA-256  WF-64  WF-128  WF-256  PICPA-64  PICPA-128  PICPA-256  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='5  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='5  4  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='5  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='5  4  ZF-NOMA  mMIMO  EPA  WF  PICPA  (a) S - Bar chart of Area Under SE curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  EPA-64  EPA-128  EPA-256  WF-64  WF-128  WF-256  PICPA-64  PICPA-128  PICPA-256  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='004  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='006  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='008  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
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+page_content='012  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
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+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='8  2  ZF-NOMA  mMIMO  EPA  WF  PICPA  (b) E - Bar chart of Area Under EE curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  Figure 9: The Area Under curve of NOMA and mMIMO  system under three power allocation procedures: (a) SE curves;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  b) EE curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' The average is obtained over 1000 random  devices locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Resource Efficiency (SE-EE Trade-off)  The NOMA and mMIMO are analyzed in terms of SE  and EE trade-off, namely resource efficiency (RE), considering  loading of devices increasing up to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 10, one  can find graphically the best loading of devices range that  maximizes the SE-EE trade-off for each BS antenna configu-  ration M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' The left y-axis depicts avg-SE, and the right y-axis  shows the avg-EE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Table IV summarizes the optimal loading  of devices that maximizes the SE-EE trade-off and shows the  percentage of active users after Power Allocations and Jain’s  Fairness Index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' (a) reveals the results when M = 64,  NOMA with EPA achieved SE-EE trade-off with the highest  loading of devices and the highest SE in trade-off;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' on the  other hand, mMIMO with classical WF achieved the highest  SE-EE trade-off;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' however, the percentage of actives devices is  around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' (b) depicts results when M = 128, NOMA  with EPA achieved SE-EE trade-off with the highest loading  of devices, in contrast, mMIMO with classical WF with lower  loading of devices achieved higher values of SE and EE in  trade-off with half of the active devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Fig 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' (c) showed  the results when M=256, NOMA with ∆-WF achieved SE-EE  trade-off with the highest loading of devices, and one more  time mMIMO with WF achieved higher SE-EE trade-off with  47% of active devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' It is possible to demonstrate that the  ZE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='I0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='7256128  M  64  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
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+page_content='1 ~  0  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
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+page_content='8  p11  increase of antennas in the BS improves the SE result, on the  other hand, it worsens the EE result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
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+page_content='8  2  Loading  = K/M  0  361  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='405  ZF-NOMA-  WF  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
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+page_content='8  2  0  373  Average Sum rate (bits/s/Hz)  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='412  EE (bits/Joule/Hz)  ZF-NOMA-EPA  (c) M = 256 BS antennas  Figure 10: SE-EE trade-off points when M = 64, 128 and 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Conclusion and Future Works  This work proposes a comparative SE and EE analysis in  DL single-cell between mMIMO and NOMA with BS equipped  with three antenna configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Under the SE perspective,  mMIMO with the classical WF algorithm achieved better low-  and medium-loading results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' On the other hand, when the  Table IV: SE-EE Trade-off  M = 64  ρ  SE  EE  Actives Users  F  mMIMO-WF  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
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+page_content='45  M =256  ρ  SE  EE  Actives Users  F  mMIMO-WF  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
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+page_content='398  system loading is higher as ρ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='6 the NOMA achieves better  results in the range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='6 < ρ ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  The analyzed PA methods applied to the NOMA system,  (EPA, PICPA, and ∆-WF) result in different SE performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  Indeed, when the channel hardening condition is fully attained,  and the amount of BS antennas increases(M = 128 and 256),  the best SE results are attained with the proposed ∆-WF  algorithm, but, as expected, the fairness index is harmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  Under the EE perspective, the mMIMO achieved better  results when employing the three EPA, PICPA, and WF PA  methods under K < M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' However, the NOMA can operate  under higher system loading, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=', K < 2M − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  In terms of area-under-SE-curve and EE-curve metrics, S  and E, respectively, the NOMA system attained better results,  due to its ability to serve a larger number of users than  mMIMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Such numerical results confirm NOMA’s ability to  operate with high loading of devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' On the other hand,  achieving high fairness with NOMA is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  From the perspective of SE-EE trade-off, mMIMO achieved  the best results, because of the superiority in EE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' always  achieved in loading of devices ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='6 in all M setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  NOMA systems present exciting features and have been  intensively investigated as a promising technique in devising  future wireless generations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' As future works, hybrid NOMA  systems and alternative techniques such as rate-splitting mul-  tiple access (RSMA) can improve the overall EE of massive  MIMO systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  Acknowledgement  This work was partly supported by The National Council for  Scientific and Technological Development (CNPq) of Brazil  under Grants 310681/2019-7, partly by the CAPES- Brazil -  Finance Code 001, and the Londrina State University - Paraná  State Government (UEL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  References  [1] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Larsson, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Edfors, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Tufvesson, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Marzetta, “Massive  MIMO for next generation wireless systems,” IEEE Communications  Magazine, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 52, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 186–195, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  [2] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Bjornson, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Sanguinetti, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Hoydis, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Debbah, “Designing  multi-user MIMO for energy efficiency: When is massive MIMO  the answer?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=',” 2014 IEEE Wireless Communications and Networking  Conference (WCNC), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 242–247, apr 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  [3] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Ding and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Poor, “Design of Massive-MIMO-NOMA with  Limited Feedback,” IEEE Signal Processing Letters, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 23, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 629–  633, may 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  [4] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Senel, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Cheng, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Bjornson, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Larsson, “What Role  Can NOMA Play in Massive MIMO?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=',” IEEE Journal on Selected  Topics in Signal Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 4553, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' c, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 1–16, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  [5] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Anwar, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Seet, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Hasan, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Li, “A Survey on  Application of Non-Orthogonal Multiple Access to Different Wireless  Networks,” Electronics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 8, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 1355, nov 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  12  [6] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Ali, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Hossain, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Kim, “Non-Orthogonal Multiple  Access (NOMA) for Downlink Multiuser MIMO Systems: User Clus-  tering, Beamforming, and Power Allocation,” IEEE Access, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 5,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 565–577, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  [7] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Zeng, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Yadav, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Dobre, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Tsiropoulos, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  Poor, “Capacity Comparison Between MIMO-NOMA and MIMO-  OMA With Multiple Users in a Cluster,” IEEE Journal on Selected  Areas in Communications, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 35, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 2413–2424, oct 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  [8] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Islam, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Zeng, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Dobre, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Kwak, “Resource  Allocation for Downlink NOMA Systems: Key Techniques and Open  Issues,” IEEE Wireless Communications, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 25, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 40–47, apr  2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  [9] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Chen, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Gong, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Zhang, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Song, “User Pairing  and Pair Scheduling in Massive MIMO-NOMA Systems,” IEEE  Communications Letters, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 22, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 788–791, apr 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  [10] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Cheng, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Bjornson, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Larsson, “Performance Anal-  ysis of NOMA in Training-Based Multiuser MIMO Systems,” IEEE  Transactions on Wireless Communications, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 17, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 372–385, jan  2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  [11] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Dai, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Wang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Peng, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Chen, “Hybrid Precoding-  Based Millimeter-Wave Massive MIMO-NOMA With Simultaneous  Wireless Information and Power Transfer,” IEEE Journal on Selected  Areas in Communications, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 37, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 131–141, jan 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  [12] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Makki, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Chitti, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Behravan, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Alouini, “A Survey of  NOMA: Current Status and Open Research Challenges,” IEEE Open  Journal of the Communications Society, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 1, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' December 2019,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 179–189, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  [13] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Usman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Khan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Usman, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Jang, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Shin,  “On the performance of perfect and imperfect SIC in downlink non  orthogonal multiple access (NOMA),” 2016 International Conference  on Smart Green Technology in Electrical and Information Systems  (ICSGTEIS), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 102–106, oct 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  [14] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Bjornson and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Larsson, “How Energy-Efficient Can a Wireless  Communication System Become?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=',” Conference Record - Asilomar  Conference on Signals, Systems and Computers, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 2018-Octob,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 1252–1256, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  [15] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Wang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Zheng, and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Yang, “Energy-  Efficient Transmission Design in Non-orthogonal Multiple Access,”  IEEE Transactions on Vehicular Technology, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 66, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 2852–  2857, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  [16] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Mounchili and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Hamouda, “Pairing Distance Resolution and  Power Control for Massive Connectivity Improvement in NOMA  Systems,” IEEE Transactions on Vehicular Technology, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 69, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 4,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 4093–4103, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  [17] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Rezaei, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Tellambura, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Tadaion, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Heidarpour, “Rate  analysis of cell-free massive MIMO-NOMA with three linear pre-  coders,” IEEE Transactions on Communications, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 68, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 6,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 3480–3494, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  [18] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Zhang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Liu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Long, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Dong, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Leung,  “Energy Efficient User Clustering, Hybrid Precoding and Power  Optimization in Terahertz MIMO-NOMA Systems,” arXiv, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 38,  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 9, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 2074–2085, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  [19] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Maraqa, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Rajasekaran, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Al-Ahmadi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Yanikomeroglu,  and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Sait, “A Survey of Rate-Optimal Power Domain NOMA  with Enabling Technologies of Future Wireless Networks,” IEEE  Communications Surveys and Tutorials, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 22, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 2192–  2235, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  [20] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Rajoria, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Trivedi, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Godfrey, “Sum-rate optimization  for NOMA based two-tier hetnets with massive MIMO enabled  wireless backhauling,” AEU - International Journal of Electronics and  Communications, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 132, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' October 2020, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 153626, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  [21] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Kim and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Choi, “Non-Orthogonal Multiple Access in Dis-  tributed Antenna Systems for Max-Min Fairness and Max-Sum-  Rate,” IEEE Access, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 9, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 69467–69480, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  [22] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Budhiraja, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Kumar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Tyagi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Tanwar, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Han, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Suh,  and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Piran, A Systematic Review on NOMA Variants for 5G  and Beyond, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' IEEE, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  [23] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Zafari, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Gkelias, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Leung, “A Survey of Indoor Local-  ization Systems and Technologies,” IEEE Communications Surveys  and Tutorials, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 21, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 2568–2599, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  [24] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Mohamed, “Joint users selection and beamforming in downlink  milimetre-wave NOMA based on users positioning,” IET Communi-  cations, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 14, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 8, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 1234–1240, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  [25] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Marzetta, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Larsson, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Yang, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Ngo, Fundamen-  tals of Massive MIMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Cambridge University Press, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  [26] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Yang and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Marzetta, “Performance of conjugate and zero-  forcing beamforming in large-scale antenna systems,” IEEE Journal  on Selected Areas in Communications, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 31, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 172–179,  2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  [27] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  Boyd  and  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  Vandenberghe,  “Numerical  lin-  ear  algebra  background,”  [On  Line] Available:  https://web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='stanford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='edu/class/ee364a/lectures/num-lin-alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content='  [28] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Jacob and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' Abrão, “Nonorthogonal multiple access systems  optimization to ensure maximum fairness to users,” Transactions on  Emerging Telecommunications Technologies, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 31, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
+page_content=' 1–  14, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfVQTf/content/2301.03101v1.pdf'}
diff --git a/AtFQT4oBgHgl3EQf9DeI/content/tmp_files/2301.13449v1.pdf.txt b/AtFQT4oBgHgl3EQf9DeI/content/tmp_files/2301.13449v1.pdf.txt
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@@ -0,0 +1,1205 @@
+Certification Design for a Competitive Market
+Andreas Haupt
+Nicole Immorlica
+Brendan Lucier∗
+February 1, 2023
+Abstract
+Motivated by applications such as voluntary carbon markets and edu-
+cational testing, we consider a market for goods with varying but hidden
+levels of quality in the presence of a third-party certifier. The certifier
+can provide informative signals about the quality of products, and can
+charge for this service.
+Sellers choose both the quality of the product
+they produce and a certification. Prices are then determined in a compet-
+itive market. Under a single-crossing condition, we show that the levels of
+certification chosen by producers are uniquely determined at equilibrium.
+We then show how to reduce a revenue-maximizing certifier’s problem to
+a monopolistic pricing problem with non-linear valuations, and design an
+FPTAS for computing the optimal slate of certificates and their prices. In
+general, both the welfare-optimal and revenue-optimal slate of certificates
+can be arbitrarily large.
+1
+Introduction
+Many markets sell goods that have different features but appear identical to
+consumers.
+Examples include the market for carbon credits, the market for
+contract workers such as electricians or plumbers, or even some markets for
+physical goods like milk and eggs. In each of these markets, the goods have
+hidden, hard-to-verify properties that distinguish them. Nature-based carbon
+credits, for example, are sourced from a variety of forests with different pro-
+tection and longevity properties.
+Contract workers have differing skill levels
+and amount of knowledge. And physical goods are produced in a variety of
+circumstances, some more ethical and sustainable than others.
+In such settings, sellers seek to distinguish their products through costly
+certification. In the example of carbon credits, large certifiers issue certification
+standards for carbon emissions.
+Contract workers can enroll in certification
+programs or take exams to document their skill level. And farms can reach out
+to third-party organizations that certify the conditions, such as free-range or
+vegetarian-fed, under which their food is produced.
+∗We thank Robert Maddox, Yanika Meyer-Oldenburg and a seminar audience at Harvard.
+1
+arXiv:2301.13449v1  [cs.GT]  31 Jan 2023
+
+In each of these cases, a downstream market allocates goods based on cer-
+tification.
+Carbon exchanges allow for the trade of carbon certificates, and
+platforms for local services allow the trade of electrical or plumbing services.
+These markets have price formation that is independent of, and not controllable
+by, the certifier. Therefore, both welfare- and revenue-maximizing certifiers will
+need to reason about how the certification options they provide affect trade,
+and how this in turn influences the demand for certification.
+What impact does the presence of certification have on the downstream
+market? We answer this question in the context of a competitive market with
+production. In our model there is a mass of heterogeneous producers each creat-
+ing a single unit of a vertically-differentiated product. Producers can select the
+quality level of their products. The corresponding production cost is determined
+by the producer’s type and is increasing in quality. A mass of unit-demand con-
+sumers purchases these goods. Consumers also have varying types which deter-
+mine the strength of their preference for quality. Types are totally ordered and
+satisfy a single-crossing condition: higher-type producers are relatively more
+efficient at producing higher-quality goods, which higher-type consumers have
+relatively stronger preference for.
+Quality cannot be verified directly; instead, a third-party certifier offers a
+menu of certifications, each with its own requirements and certification price.
+Producers can purchase certifications of their products’ qualities from this menu.
+If their product surpasses the quality level of the certificate, it will be marketed
+as such. The certified goods, together with indistinguishable uncertified ones,
+are then sold in a competitive market.
+As in the Economics literature on certification, we assume that market par-
+ticipants are able form correct beliefs about the quality implications of a certifi-
+cate Milgrom (1981); Grossman (1981); Lizzeri (1999); DeMarzo et al. (2019). A
+big strand of empirical work considers the question of whether certifications are
+correctly interpreted in various markets (Wimmer and Chezum (2003) for race
+horses, Tadelis and Zettelmeyer (2015) on used-car auctions, Ramanarayanan
+and Snyder (2012) for dialysis screening centers, Luca (2016) for restaurant
+reviews, Elfenbein et al. (2015) for seller ratings on Ebay, Conte and Kotchen
+(2010) for voluntary carbon credits). In our model, certificates are based on ver-
+ifiable certification requirements, and at perfect Bayesian equilibrium all market
+participants hold rational beliefs about the distribution of quality implied by
+any given certificate. We note that such rational beliefs need not be aligned
+with official certificate descriptions. For example, in voluntary carbon markets,
+several empirical contributions point out that carbon certificate descriptions
+might not correspond to counterfactual mitigation outcomes, compare West
+et al. (2020, 2023); Guizar-Couti˜no et al. (2022); Clarke and Barratt ([n. d.]);
+Greenfield ([n. d.]b,n).
+The presence of certificates clearly impacts the market equilibrium because
+it influences consumers’ beliefs and hence willingness-to-pay. Our first result
+shows that, for any menu of certificates offered by the certifier, the equilibrium
+choice of certificates and the allocation outcome of the downstream market
+equilibrium is unique. Furthermore better producers (that is, those who can
+2
+
+produce higher quality at lower cost) always purchase higher (more restrictive)
+levels of certification.
+This implies that the producer-consumer matching in
+equilibrium is assortative. Better producers sell to consumers with higher value
+for quality, and at higher quality levels.
+This analysis immediately implies that the welfare-optimal menu offers every
+certificate at cost, and suggests a dynamic program for finding the welfare-
+optimal menu subject to a cardinality restriction on the number of certification
+levels that can be offered. But in many markets, certification is provided by
+third-party firms with profit-maximizing incentives.
+Our main result shows
+how to approximately construct the revenue-optimal menu in polynomial time.
+Specifically, we provide an FPTAS: we show how to compute a menu whose
+revenue for the certifier is an additive ϵ less than the optimal revenue in time
+polynomial in 1/ϵ. Our algorithm can optionally take as input a cardinality
+constraint k on the menu size (i.e., a maximum number of certification levels),
+in which case the revenue benchmark is the optimal slate of at most k certificates
+and their prices.
+Our proof contains technical insights that may be of independent interest.
+Namely, we reduce the certifier’s problem to one of a seller who wishes to sell
+a divisible good, facing buyers whose valuations are non-linear and potentially
+non-monotone in quantity. The seller corresponds to the certifier, and the buy-
+ers correspond to producer-consumer pairs. This projection of producers and
+consumers into a single economic agent is enabled by the fact that the matching
+in equilibrium is unique and assortative. The buyer valuations in this reduced
+problem are concave but not necessarily monotone as a function of quantity,
+meaning that for different buyer types the valuation may reach its maximum at
+different quantities. The valuations do satisfy a single-crossing property, which
+implies that types are totally ordered and higher-type buyers have pointwise
+higher valuations and have weakly higher preferred quantities.
+Under these
+conditions, we show that the revenue-optimal menu may be non-linear but will
+exhibit prices that are monotone in quantity. Monotonicity of prices enables
+the use of dynamic programming to construct an approximately optimal menu,
+though some care must be taken when discretizing the space of quantities and
+prices to ensure that buyer preferences do not change substantially. This non-
+linear pricing problem generalizes prior work that specifically studied the case
+of quadratic demand Bergemann et al. (2012a,b), and may be of independent
+interest.
+The menu of certifications offered by a revenue-maximizing third-party cer-
+tifier can distort welfare and result in inefficient levels of trade.
+That said,
+we show that a certifying agent entering into the market can never lead to a
+loss of welfare, no matter what menu of certification options they make avail-
+able. More specifically, we can imagine that a collection of (perhaps costly)
+certification options are available to the market as a baseline, perhaps provided
+by a public or tightly regulated agency. If a third-party certifier then enters
+the market and makes available any additional certification options into the
+market, at any price, we show that the total utility of all market participants
+(even excluding the new entrant) can only increase. A policy implication is that
+3
+
+a regulatory body concerned about inefficient certification arising from profit-
+seeking motives need not prevent certifiers from entering the market. Rather, it
+suffices to make sure that there exists some set of certifications available, either
+through third-party providers or public options, that enable an acceptable level
+of welfare from trade.
+1.1
+Related Literature
+This work contributes to the literature on certification. The early results Mil-
+grom (1981); Grossman (1981) produce unraveling type results: In equilibrium,
+the quality of a good is fully revealed. The main intuition for these results in
+models of certification is that certifying non-informatively will be interpreted
+by the market as a sign of bad quality—adverse selection is extreme. In our
+model of a revenue-maximizing certifier, there might be other reasons for certi-
+fication less informatively, as the price in the competitive market may depend
+on not only an individual seller’s quality, but all the sellers in the market. Later
+contributions Lizzeri (1999); DeMarzo et al. (2019); Acharya et al. (2011) allow
+for unsuccessful certifications, restricting the stark result obtained in Milgrom
+(1981); Grossman (1981).
+More broadly, our work is related to mechanism and information design in
+the presence of an exogenously given game played after the design. We rec-
+ommend Bergemann and Morris (2019) for a general overview of information
+design. Our work is especially related to the design of information provided to
+buyers and/or sellers of a good. Bergemann et al. (2017) considers the design of
+information for a first-price auction, where a third party can reveal a signal cor-
+related with a buyer’s valuations, and fully characterizes the achievable revenue
+and payoffs. Alijani et al. (2022) extends the analysis to a scenario with mul-
+tiple buyers. In a general mechanism design framework, Candogan and Strack
+(2021) develop an optimal disclosure policy for action recommendations in a
+game with private types and a hidden state. Dworczak (2020) designs mech-
+anisms in a setting where players participate in a finite Bayesian game after
+participating in the mechanism, so that game outcomes are impacted by infor-
+mation revealed over the course of the mechanism. He finds a cutoff structure
+of optimal mechanisms in the first stage.
+For-profit certification relates to the sale of hard information, which has
+been studied in the context of competitive markets. Ali et al. (2022) consider
+a seller holding a good of uncertain quality. The seller can purchase a quality-
+correlated signal from a revenue-maximizing intermediary before bringing the
+good to market. In general the resulting equilibria are not unique and can vary
+substantially, but by employing noisy signals the intermediary can robustly
+guarantee high revenue. Our model differs in that any certification options are
+made available to the entire market and product quality is endogenous, so the
+certifying agent can impact welfare. More generally, our work relates to the
+problem of how to sell payoff-relevant hard information to a potential buyer.
+Bergemann et al. (2018) solve for the revenue-maximizing mechanism in binary
+environments, and Bergemann et al. (2022) establish when full disclosure is
+4
+
+approximately optimal under more general spaces of actions. In contrast, our
+certifying agent is selling a signal that is valuable in that it conveys information
+to other participants in a subsequent game.
+In our mathematical analysis, we reduce the certifier’s problem to a pricing
+problem with a non-linear valuation. The non-linear pricing literature following
+Mussa and Rosen (1978) (see also treatment in Dewatripont et al. (2005) and
+(B¨orgers and Krahmer, 2015, Chapter 2.3)) studies a non-linear concave valu-
+ation with a quadratic cost. The functional form assumptions in these papers
+allow to characterize the optimal mechanism in closed form. Typically, the op-
+timal menu of offered goods is a continuum, in contrast to the linear screening
+problem first studied in the influential Myerson (1981). Our analysis will show
+that also the class of models we consider may feature infinite menus.
+In our polynomial-time approximation scheme, we use an approximation of
+a non-linear pricing model. The papers Bergemann et al. (2012a,b) consider
+approximation of non-linear single- and multi-dimensional pricing environments
+with finite menus (in the papers called “finite information”). The papers make
+functional form assumptions similar to the ones in Mussa and Rosen (1978), and
+derive rates of approximation by finite menus in these finite menus. The present
+paper allows for a general class of utility functions that satisfy a single-crossing
+condition, and, in addition to showing approximation by finite menus, shows
+that the finite-sized menu can be computed efficiently.
+Finally, our main assumption guaranteeing uniqueness of our equilibrium is
+a single-crossing condition. Single-crossing conditions are important in several
+domains, among them the interdependent private values auctions (Milgrom and
+Weber (1982)) and social choice and voting (Saporiti and Tohm´e (2006)). A re-
+cent line of work in algorithmic mechanism design has employed single-crossing
+conditions to enable approximately optimal designs in interdependent value set-
+tings Roughgarden and Talgam-Cohen (2013); Chawla et al. (2014). Closest to
+the present paper, another implication of single-crossing is adverse selection in
+markets Mirrlees (1971); Spence (1974).
+1.2
+Outline
+The rest of this article is structured as follows.
+We formalize our model in
+section 2. In section 3 we analyze the structure of equilibria given the certifier’s
+offerings. Section 4 contains our main results, a reduction of revenue-maximizing
+certification to a non-linear pricing problem, the FPTAS for its computation.
+Section 5 contains our results on welfare maximization and explores the welfare
+implications of third-party certification.
+2
+Model
+Market
+We consider a continuum market between producers and consumers.
+Producers are unit-supply and parameterized by types ψ ∈ R+ with measure G.
+Consumers are unit-demand and parameterized by types φ ∈ R+ with measure
+5
+
+F. The type measures F and G are atomless and continuous with compact
+support.
+Goods can be produced at different levels of quality, denoted q ∈ [0, 1].
+Goods of higher quality are more valuable to consumers but more costly to
+produce. We write g(q; ψ) for the cost incurred by a producer of type ψ when
+producing a good of quality q. We assume g is weakly convex and non-decreasing
+in q for every ψ and normalized so that g(0; ψ) = 0. We also write f(q; φ) for
+the value enjoyed by a consumer of type φ for a good of quality q, where f
+is weakly concave and non-decreasing in q for every φ and normalized so that
+f(0; φ) = 0. We will scale valuations so that f(q; φ) ≤ 1 for all q and φ, which
+is without loss for bounded values.
+We will assume that costs and valuations satisfy single-crossing with respect
+to the producer and consumer types, respectively. Roughly speaking, this means
+that producers (consumers) of higher types have lower marginal cost (higher
+marginal value) for producing higher-quality goods. More formally, for all φ1 <
+φ2 and q1 < q2, we have
+f(q2; φ2) − f(q1; φ2) > f(q2; φ1) − f(q1; φ1).
+Likewise, for all ψ1 < ψ2 and q1 < q2, we have
+g(q2; ψ2) − g(q1; ψ2) < g(q2; ψ1) − g(q1; ψ1).
+Transfers between producers and consumers are permitted. We assume that
+producers and consumers are risk-neutral and have quasi-linear preferences with
+respect to money. That is, if consumer φ purchases a product of quality q from
+producer ψ at a price of p, then the consumer enjoys utility
+uC((q, p); φ) = f(q; φ) − p
+and the producer’s utility is
+uP ((q, p); ψ) = p − g(q; φ).
+Certification
+Crucially, producers and consumers cannot contract on quality.
+This means that a producer cannot credibly commit to the quality of the good
+they produce, and a consumer cannot verify quality at the point of trade. But
+there is a third-party certifier who is able to determine the quality of a producer’s
+good. This verification is costly to the certifier, with cost c ≥ 0.
+After verifying the quality of a good, the certifier is able to assign to that
+good a signal (or certificate) σ ∈ Σ, where Σ is an arbitrary space of potential
+certificates.
+This certificate is visible to all producers and consumers.
+The
+certifier is permitted to collect payments from producers for this service, and
+these transfers can depend arbitrarily on the certificate produced.
+The certifier can commit to a certification menu M, which is a collection
+of certificate / transfer pairs (σ, t(σ)) along with quality requirements for each
+certificate σ. Following the literature on information design and persuasion,
+6
+
+we note that it is without loss of generality to associate each certificate signal
+σ with the set of quality levels that are eligible for that certificate. We will
+therefore assume without loss of generality that Σ = 2[0,1], the collection of all
+subsets of [0, 1], and each σ is a subset of [0, 1]. The interpretation is that a
+producer can select an item from this menu, in which case she pays the certifier
+price (or transfer) t(σ), the certifier verifies the good’s quality q, and as long
+as q ∈ σ the producer will receive certification σ. We will assume for technical
+convenience that each σ in the certifier’s menu contains a minimum element,
+meaning that inf σ ∈ σ.1
+We will write σ0 = [0, 1] for the trivial signal that conveys no additional
+information about quality. A good with certificate σ0 is functionally equivalent
+to a good that has not been certified.
+It will be notationally convenient to
+assume that the certifier always offers signal σ0 at cost 0. This allows us to
+think of every good as being certified, albeit possibly at the trivial level. If a
+producer attempts to purchase a certificate but does not satisfy that certificate’s
+requirements, they will instead be assigned certificate σ0; i.e., the certifier will
+not certify the good.
+The Competitive Market
+All certification is assumed to occur simultane-
+ously, and in advance of any trading between producers and consumers. Given
+the menu M of certificates and prices offered by the certifier, the producers’
+(production and certification) strategy is a mapping from producer type ψ to
+a choice of quality level q and certification σ. We will denote such a strategy
+Γ : ψ �→ (q, σ), and restrict attention to measurable functions Γ. In a slight
+abuse of notation, we will also use Γ to denote the measure over pairs (q, σ) of
+products with corresponding quality and certification that result when produc-
+ers apply strategy Γ.
+Goods that are assigned the same certificate are indistinguishable by the
+consumers. After all certification is complete, each producer has a single unit
+of a good marked with a certification σ. For any given certification σ, write Γσ
+for the marginal distribution over quality q of Γ restricted to certificate σ. That
+is, fixing the choices of the producers, Γσ is the distribution of levels of quality
+for a product with certification σ. Then the value of a consumer of type φ for
+a good with certificate σ can be evaluated as
+f(σ; φ) = Eq∼Γσ[f(q; φ)].
+That is, each consumer rationally evaluates the expected quality of each product
+given its certification level and the choices of the producers.
+1This excludes certificates of the form “the quality q is strictly greater than 1/2,” as
+opposed to “...at least 1/2.” Certificates of the former type are inconvenient because there is
+no single least-cost choice of quality that satisfies the certification requirement, and hence no
+utility-maximizing choice of quality for the producer. We could handle such certificates in a
+straightforward but tedious way by relaxing our equilibrium notion and assuming that each
+producer selects an ϵ-approximately utility maximizing choice of quality for some arbitrarily
+small ϵ. For the remainder of the paper we will ignore such technical issues and simply assume
+that each σ includes a minimum element.
+7
+
+Since goods with the same certification are indistinguishable to consumers,
+we can view the competitive market between producers and consumers as a
+market for certificates σ.
+A Walrasian (or Competitive) equilibrium of the
+resulting market is an allocation x(φ) of a certificates to each consumer φ, along
+with a price pσ for each certificate, such that:
+• Demand satisfaction: every consumer purchases her most-preferred good.
+That is, for every consumer type φ, f(x(φ); φ) − px(φ) ≥ f(σ; φ) − pσ for
+every σ ∈ Σ.
+• Market Clearing: every good with a positive price is sold. That is, for all
+σ, the measure of consumers φ such that x(φ) = σ is at most the measure
+of producers ψ who select level of certification σ. If pσ > 0 then these
+measures are equal.
+Since buyers (consumers) are unit-demand and hence their preferences satisfy
+the gross substitutes condition, a Walrasian equilibrium is guaranteed to ex-
+ist Gul and Stacchetti (2000).
+We will therefore assume that trade occurs
+between consumers and producers at competitive equilibrium prices given the
+choices made by the producers.
+Timeline
+To summarize, the timing of the market with certification is as
+follows:
+1. The certifier commits to a menu of certificates with corresponding prices.
+2. Each producer ψ simultaneously and privately chooses whether to produce
+a good, and if so at what level of quality.
+3. Each producer that chose to produce decides whether to certify their prod-
+uct, and at which certificate. These decisions are made simultaneously for
+all producers.
+4. The certifier verifies the products of producers who choose to certify and
+assigns certificates. Any producer who does not successfully certify re-
+ceives certificate σ0.
+5. Producers and consumers trade goods in a competitive market. I.e., trade
+occurs at market-clearing prices for the chosen levels of certification.
+Since Walrasian equilibria are not unique in general, one might wonder if
+the outcome described in the final step is well-defined. We will show in the
+next section that the competitive market equilibrium described the final step
+exists and its resulting allocation is unique for any certificate menu chosen by
+the certifier and any choice of certification levels chosen by the producers.
+8
+
+3
+Equilibrium Characterization
+In this section we describe the market outcome that will occur for any given
+menu of certificates offered by the certifier. We show that it is without loss
+of generality for the certifier to restrict to offering threshold certificates that
+guarantee that a product is at least a certain level of quality. We characterize
+the unique Walrasian market equilibrium allocation that results from any as-
+signment of such certificates to producers. We then use that characterization
+to solve for each producer’s utility-maximizing choice of certificate from the
+certifier’s menu, which will also be unique.
+3.1
+Certifications as Minimum Quality Thresholds
+A first simple observation is that since producer costs are increasing in quality
+level, and since goods at different quality levels but with the same certification
+are indistinguishable to consumers (and hence must sell at the same price),
+a producer who is assigned certificate σ will always choose to produce at the
+minimum quality level eligible for that certificate.
+Observation 3.1. Fix any certifier menu M and any production and certifica-
+tion strategy of the producers. Then for any producer ψ, selecting certificate σ
+and producing at quality q > min σ is dominated by selecting certificate σ and
+producting at quality min σ.
+Given this observation, we know that for any menu M, any equilbrium
+strategy Γ for the producers, and any certificate σ, the marginal distribution
+over quality Γσ will be a point mass at min σ. In particular, any two certificates
+with the same minimum will induce the same equilibrium beliefs over quality and
+hence have indistinguishable value to all consumers. It is therefore without loss
+of generality for the certifier to only offer certificates of the form σq = [q, 1]; i.e.,
+certificates that are differentiated only with respect to their minimum values.
+If a producer selects certificate σq, then that producer’s chosen quality level
+at equilibrium will necessarily be q. Any given certification menu M therefore
+reduces to a (possibly infinite) collection of quality levels in [0, 1] to certify.
+Motivated by this observation, we will assume for the remainder of the paper
+that all certificates are of the form [q, 1], and associate each σ = [q, 1] with its
+quality threshold q. We can then think of a certification menu M as a collection
+of pairs {(qi, ti)}, where qi is a quality threshold and ti is a corresponding price
+for certifying that quality is at least qi.
+3.2
+Uniqueness and Assortativeness of Competitive Mar-
+ket Allocations
+We now turn to an analysis of the competitive market outcome that will result
+given the strategies of the certifier and producers. Recall that we can restrict
+attention to certificates of the form σq = [q, 1] and that any good with certificate
+σq will have quality q with probability 1, so for the remainder of the section we
+9
+
+will think of a market outcome as an allocation x and prices p of quality levels.
+That is, x(φ) ∈ [0, 1] for all consumers φ, and for each q ∈ [0, 1] in menu M
+there is an associated market price pq. We emphasize that x is a mapping from
+consumers to the certified goods they buy at market, whereas Γ is a mapping
+from producers to the certificates that they choose from the certifier.
+The following lemma shows that for any choice of certification menu M and
+production and certification strategy Γ of the producers, all competitive market
+equilibria in the resulting market have the same uniquely-determined allocation.
+This allocation will be assortative, with higher-type consumers purchasing the
+higher-quality certificates.
+Lemma 3.2. Fix any certifier menu M and any strategy Γ of the produc-
+ers. Then in every competitive market outcome (x, p), the allocation x satisfies
+x(φ1) ≤ x(φ2) and px(φ1) ≤ px(φ2) for all φ1 ≤ φ2.
+Proof. Since consumers are unit-demand and each producer has a single unit of
+good, a Walrasian equilibrium (x, p) of the market is guaranteed to exist. By
+the first welfare theorem, any such equilibrium must maximize the total welfare,
+�
+φ
+f(x(φ); φ)dF(φ).
+Suppose there exist types φ1 < φ2 with x(φ1) > x(φ2). By the single-crossing
+condition, we have that
+f(x(φ1); φ2) − f(x(φ2); φ2) > f(x(φ1); φ1) − f(x(φ2); φ1)
+and hence
+f(x(φ1); φ2) + f(x(φ2); φ1) > f(x(φ1); φ1) + f(x(φ2); φ2)
+which contradicts the supposed welfare optimality of allocation x.
+We now turn to prices. Fix any consumer types φ1 < φ2, so in particular
+we know x(φ1) ≤ x(φ2), and suppose for contradiction that px(φ1) > px(φ2). By
+monotonicity of the value function f, we must have f(x(φ1); φ1) ≤ f(x(φ2); φ1).
+But this then means f(x(φ2); φ1)−px(φ2) > f(x(φ1); φ1)−px(φ1), which violates
+the competitive market condition that consumer type φ1 is choosing her most-
+preferred good. We therefore conclude that px(φ1) ≤ px(φ2), as claimed.
+3.3
+Uniqueness and Assortativeness of Certificate Selec-
+tion
+Given that market outcomes are well-defined, we next turn to the equilibrium
+choices of the producers when selecting quality levels and their corresponding
+certifications. We again show that for any menu M of certificates offered, the
+quality choices of producers are unique at equilibrium. Moreover, higher-type
+producers will always select (weakly) higher certificates. In the Lemma 3.2, we
+saw that higher-type consumers also purchase (weakly) higher certificates. As
+10
+
+we discuss later, this means the matching in any equilibrium will be assortative
+and hence constrained-efficient given the available certificates.
+Recall that a producer strategy Γ is a mapping from producer type ψ to a
+choice of certification and quality, which we know from will always coincide. We
+will therefore write Γ(ψ) = q to mean that producer ψ produces at quality level
+q and purchases certificate σq. In particular, we must have Γ(ψ) ∈ M for all ψ.
+Lemma 3.3. Fix any certification menu M offered by the certifier. Then there
+is a unique equilibrium strategy Γ for the producers, and Γ(ψ) is weakly increas-
+ing in ψ.
+Proof. Fix strategy Γ, which implies the measure of certificates chosen by the
+collection of producers. Let (x, p) denote a Walrasian equilibrium in the result-
+ing competitive market, and recall that x is uniquely determined.
+We first show that Γ is weakly increasing is ψ. Assume for contradiction
+that there exist ψ1 < ψ2 with q1 = Γ(ψ1) and q2 = Γ(ψ2) with q2 < q1. Then
+by the single-crossing condition for producers, we have g(q1; ψ1) − g(q2; ψ1) >
+g(q1; ψ2) − g(q2; ψ2).
+But then, if we let pq1 and pq2 denote the Walrasian
+equilibrium prices of q1 and q2 given Γ, we have
+(pq1 − g(q1; ψ1)) + (pq2 − g(q2; ψ2)) < (pq1 − g(q1; ψ2)) + (pq2 − g(q2; ψ1))
+which means that either
+pq1 − g(q1; ψ1) < pq2 − g(q2; ψ1)
+or
+pq2 − g(q2; ψ2) < pq1 − g(q1; ψ2).
+In other words, either producer ψ1 or ψ2 (or both) would strictly improve their
+utility by switching their choice of quality and certification. As such a swap has
+measure zero and does not influence the competitive equilibrium, this would be
+an improving deviation for the producer(s), violating the assumption that Γ is
+an equilibrium strategy for the producers.
+We have shown that Γ is weakly increasing in ψ. On the other hand, we
+know from Lemma 3.2 that the market allocation of quality levels to consumers is
+weakly increasing in φ. This means that any equilibrium outcome of production
+and trade is equivalent to one in which consumers and producers are matched
+assortatively, with higher-type consumers trading with higher-type producers.
+In other words, for any producer type ψ, there is a consumer type φ = φ(ψ) such
+that ψ always trades with φ(ψ). Specifically, φ(ψ) is such that F(φ(ψ)) = G(ψ)
+(treating F and G as cumulative distribution functions).
+Given this, we claim that Γ(ψ), the certification selected by producer ψ
+at equilibrium, will always be a certificate qi from the certifier’s menu M =
+{(qi, ti)} that maximizes f(qi; φ(ψ)) − g(qi; ψ) − ti. To see why, suppose for
+contradiction that the producer instead chooses some other certificate q′ at price
+t′ such that f(q′; φ(ψ))−g(q′; ψ)−t′ < f(qi; φ(ψ))−g(qi; ψ)−ti−ϵ for some ϵ > 0,
+11
+
+and sells to consumer φ(ψ) at an assumed market-clearing price pq′.2 Then, the
+producer ψ could instead deviate to purchasing qi at a price of ti, and offering
+it on the competitive market at a price of pq′ +g(qi; ψ)−g(q′; ψ)+(ti −t′)+ϵ/2.
+Note that if consumer φ(ψ) were to purchase from producer ψ at this price,
+then her utility would be
+f(qi; φ(ψ)) − [pq′ + g(qi; φ(ψ)) − g(q′; φ(ψ)) + (ti − t′) + ϵ/2]
+= (f(qi; φ(ψ)) − g(qi; ψ) − ti − ϵ) + (g(q′; ψ) + t′) + ϵ/2 − pq′
+> f(q′; φ(ψ)) − (g(q′; ψ) + t′) + (g(q′; ψ) + t′) − pq′ + ϵ/2
+> f(q′; φ(ψ)) − pq′.
+But since (q′, pq′) is the most-demanded offering to consumer φ(ψ) in the market
+equilibrium, this means that the offering of qi at the proposed price would be the
+most-demanded offering to consumer φ(ψ) under this deviation. In particular
+this means that some consumer would want to purchase qi at the suggested price,
+and therefore in the adjusted market equilibrium after this proposed deviation
+the price of qi must be at least this high.
+We conclude that the utility of producer ψ under this deviation is at least
+[pq′ + (g(qi; ψ) − g(q′; ψ) + (ti − t′) + ϵ/2] − g(qi; ψ) − ti = pq′ − g(q′; ψ) − t′ + ϵ/2
+> pq′ − g(q′; ψ) − t′
+and hence this deviation is strictly utility-improving for the producer, contra-
+dicting the assumption that Γ is an equilibrium.
+We therefore conclude that at equilibrium, each producer ψ chooses whichever
+certificate qi from the menu maximizes f(qi; φ(ψ)) − g(qi; ψ) − ti. The choice of
+each producer is therefore unique, up to tie-breaking on sets of measure zero.
+An immediate corollary of Lemma 3.2 and Lemma 3.3 is that the equilibrium
+outcome for a given menu M is not only essentially unique (up to the choice of
+market-clearing prices), but also has a natural assortative interpretation. Each
+producer in the market has a corresponding consumer with whom they will
+always trade. The producer will select whichever certification level maximizes
+the gains from trade between themselves and their partner consumer, less the
+price of the certification.
+Corollary 3.4. For any menu M = {(qi, ti)} of the certifier, the resulting
+equilibrium market outcome has producer ψ trade with consumer φ = φ(ψ) where
+G(ψ) = F(φ). The level of quality at which φ and ψ trade maximizes f(qi; φ) −
+g(qi; ψ) − ti, and is weakly increasing in ψ.
+2Note that as we showed Γ is weakly increasing in ψ, the deviation can not change the
+ordering of firms in terms of the certificate they purchase and hence does not change the
+consumer to which they sell.
+12
+
+4
+Revenue-Optimal Certification
+In the previous section we solved for the equilibrium outcome of the game played
+between producers and consumers given a certification menu M chosen by the
+certifier. With that characterization in hand, we now turn to the problem faced
+by a certifier who wishes to construct a revenue-maximizing slate of certificates.
+A priori, it would appear that the choice of which certificates to offer in-
+fluences the downstream competitive market allocation and prices. After all,
+producers compete for consumers and the certificates they purchase determine
+how they fare in this competition. Hence one might expect the menu of avail-
+able certificates to influence how much each producer would be willing to pay
+for any given certificate. However, as we will show, the assortative characteriza-
+tion of market outcomes means that the certifier can reason about the behavior
+of each producer type separately, without worrying about how they will jointly
+interact in the competitive market. Thus, the certifier is essentially facing a
+single buyer with an unknown type, and must simply maximize revenue subject
+to this buyer. This leads us to define a reduction from the certifier’s problem
+to that of a seller facing a (non-linear) buyer.
+4.1
+Reduction to a Non-linear Pricing Problem
+We will relate the certifier’s revenue-maximization problem to a non-linear pric-
+ing problem between a single seller and a single buyer. The buyer seeks to buy
+a perfectly divisible good. The seller may commit to a menu of quantities and
+prices. If a buyer of type θ purchases quantity q ∈ [0, 1] of the good at a total
+price of t, then the buyer utility is
+u((q, t); θ) = v(q; θ) − t,
+where v is concave in quantity q but not necessarily non-decreasing, and v(0; θ) =
+0 for all θ. The valuations v satisfy a single-crossing condition, which is that
+for θ1 < θ2 and q1 < q2, we have
+v(q2; θ2) − v(q1; θ2) > v(q2; θ1) − v(q1; θ1).
+The principal seeks to maximize revenue subject to a constant cost of production
+c > 0 for any non-zero quantity given a prior over buyer types. Given a menu
+M, we will write Rev(M) to denote the expected revenue obtained from M. We
+will also write OPT for the revenue of the revenue-maximizing choice of menu
+M.
+Proposition 4.1. The certifier’s revenue-maximization problem is equivalent
+to an instance of the non-linear pricing problem described above.
+Proof. The certifier’s problem is to design a certificate menu M = {(qi, ti)}
+that maximizes the total revenue collected, less the certification cost c paid
+to verify any non-trivial certificate qi > 0. By Corollary 3.4, given menu M,
+13
+
+each producer ψ will purchase whichever certificate qi maximizes f(qi; φ(ψ)) −
+g(qi; ψ) − ti.
+For q ∈ [0, 1], we can interpret ψ as a buyer type and define valuation
+function v(q; ψ) = f(qi; φ(ψ))−g(qi; ψ). Then since f and g both satisfy single-
+crossing with respect to their corresponding types, valuation function v does
+as well. Moreover, v is concave and v(0; ψ) = 0. By definition, the producers’
+choices of certificates from menu M corresponds precisely to the buyer’s choice
+of quantity when facing the same menu, interpreting each certificate quality
+threshold as a quantity.
+Thus the outcomes, and hence revenue, in the two
+settings are equivalent.
+Note that an immediate implication of Proposition 4.1, given Corollary 3.4,
+is that for any menu M chosen by the seller, the choice of quantity purchased
+by the buyer is weakly increasing in buyer type.3
+4.2
+Optimizing Revenue in the Non-linear Pricing Prob-
+lem
+By Proposition 4.1, to solve the certifier’s revenue maximization problem it
+suffices to optimize revenue in the non-linear pricing problem. As a first step, we
+note that in the special case where the valuations v are linear in quantity (which
+happens, for example, if the cost g and value f functions in our certification
+problem are both linear in q), this problem reduces to a standard pricing problem
+in mechanism design. A characterization due to Myerson Myerson (1981) then
+immediately establishes that it is revenue-optimal to choose a menu with only
+a single non-trivial item (q, p) with q = 1.
+Observation 4.2. If v(q; θ) is linear in q for all θ, then there is a revenue-
+optimal menu that offers only quantity q = 1 at some price p. This price p will
+be chosen to maximize p × Prθ[v(1; θ) > p].
+However, in general, non-linearity substantially changes the problem relative
+to the linear case. In particular, it is not necessarily optimal to offer a single
+menu item.
+Proposition 4.3. There are problem instances in which posting any single
+menu item is an arbitrarily poor approximation to the optimal revenue. The ap-
+proximation factor can be as large as Ω(log(H)), where H = maxθ1,θ2
+maxq v(q;θ1)
+maxq v(q;θ2)
+is the ratio between the highest and lowest maximum values across buyer types.
+Proof. Choose c = 0 and consider the valuation function v(q; θ) = q if q ≤ θ,
+and v(q; θ) = 2θ − q if q ≥ θ. That is, v is piecewise linear for each θ, with
+maximum value θ occurring at q = θ. Fix some H ≥ 1 and suppose the type
+distribution is such that Pr[θ > h] = 1/h for all h ∈ [1, H]. That is, the type
+distribution is equal-revenue on range [1, H].
+3Alternatively, this is a direct consequence of the single-crossing condition on valuation
+functions v.
+14
+
+This valuation function is concave and satisfies v(0; θ) = 0 for all θ. More-
+over, it satisfies the single-crossing condition. Indeed, for any q and any θ1 < θ2,
+we note that
+d
+dqv(q; θ1) ≤
+d
+dqv(q; θ2), since for any θ this derivative is 1 for q < θ
+and −1 for q > θ. Since v is also continuous in both q and θ, the single-crossing
+condition is implied.4 Finally, we note that this valuation function v can in-
+deed arise in our reduction from the certification problem with producers and
+consumers.5
+We now show the desired gap in approximation. Consider any menu M with
+a single non-trivial menu item (q, p). Then the revenue achieved by the seller
+is at most the welfare generated by the optimal allocation of quality level q.
+Since v(q; θ) ≤ q for all θ, this is certainly at most q times the probability that
+v(q; θ) > 0, which is q Pr[θ > q/2] ≤ q(2/q) = 2.
+On the other hand, the seller could offer a menu that includes every quality
+level q ∈ [1, H] at a price of q/2.
+A buyer of type θ would then choose to
+purchase quality level q = θ for a utility of θ/2, generating revenue θ/2.6 The
+total revenue is then E[θ/2] = O(log H).
+Posting a single menu item is therefore at best an O(log H) approximation
+to the optimal revenue, as claimed.
+We therefore know that any approximately revenue-optimal mechanism must
+sometimes have multiple non-trivial offerings on its menu. What should such a
+menu look like? Note that since buyers do not have free disposal in our setting
+(i.e., valuations are non-monotone), it is not even immediately obvious that
+higher quantities should be sold for higher prices in a revenue-optimal menu.
+We next establish that, in fact, there is always a revenue-optimal choice of menu
+for which higher quantities are sold at higher prices.
+Definition 4.4. We say a menu M = {(qi, pi)} is monotone if for all (qi, pi),
+(qj, pj) ∈ M such that qi < qj, we have pi ≤ pj.
+Lemma 4.5. For any instance of the non-linear pricing problem there is a
+revenue-optimal menu that is monotone.
+Proof. Let M be a revenue-optimal menu, and write M = {(qi, pi)}i∈Λ where
+Λ is some (possibly uncountable) index set. It is without loss to assume Λ is a
+subset of [0, 1] such that qi < qj for all i < j (e.g., by reindexing so that the
+index of qi is equal to qi). We can further assume without loss of generality that
+every item in M is purchased by some buyer type, as any element that is never
+purchased could be removed from M without impact. By Corollary 3.4, quality
+4Technically our construction only satisfies weak single-crossing since the inequality in
+derivatives is not strict. We can make the example strict by perturbing the slope of the initial
+line segment by an infinitesimal amount so that it depends on the type θ. We omit these
+details for expositional clarity.
+5In particular, take φ and ψ to be supported on [1, H], define f(q; φ) = q for all φ and
+g(q; ψ) = max{0, 2(q − ψ)} for all ψ. Then f is concave (in fact, linear), g is convex, the
+single-crossing conditions are satisfied, and v(q; θ) = f(q; φ(θ)) − g(q; θ) as required.
+6Buying a higher quality level q′ > θ is worse for the buyer because it generates less value at
+a higher price, whereas buying any quality level q′ < θ generates utility q′−q′/2 = q′/2 < θ/2.
+15
+
+levels purchased will be monotone non-decreasing in buyer type. This means
+that every item (qi, pi) is purchased by some contiguous interval of buyer types
+Ii (which may have measure zero).
+Suppose that menu M is not monotone. This means that either there is an
+element i < sup Λ such that pi > pj for all j > i, or else there exists a pair of
+menu items (qi, pi) and (qj, pj) with j > i such that (a) there exists some ℓ ∈ Λ
+with i < ℓ < j, and (b) pℓ < min{pi, pj} for all i < ℓ < j.
+Consider the former case, where there is an element i < sup Λ such that
+pi > pj for all j > i. In particular there must exist some j ∈ Λ with j > i. Let
+M ′ be the menu {(qℓ, pℓ)}ℓ∈Λ,ℓ≤i. I.e., M ′ is M with all elements with quantities
+greater than qi removed. Note that for all j ≤ i, since the types Ij preferred
+element (qj, pj) to any other element in M, they prefer element (qj, pj) to any
+other element in M ′ as well. Moreover, since purchase decisions are monotone in
+buyer type, we conclude that all types θ ∈ Iℓ with ℓ > i will purchase element
+(qi, pi) from menu M ′. But since pi > pℓ for all ℓ > i, this means that the
+revenue generated by menu M ′ is strictly greater than the revenue generated
+by menu M, contradicting the supposed optimality of menu M.
+Next consider the other case, there exists a pair of menu items (qi, pi) and
+(qj, pj) such that pℓ < min{pi, pj} for all i < ℓ < j.
+Let M ′ be the menu
+{(qℓ, pℓ)}ℓ∈Λ,ℓ≤i ∪ {(qℓ, pℓ)}ℓ∈Λ,ℓ≥j. That is, M ′ is menu M with all elements
+strictly between (qi, pi) and (qj, pj) removed. Then as in the previous case, for
+all ℓ ≤ i and ℓ ≥ j, types Iℓ all still prefer to purchase (qℓ, pℓ). In particular,
+types Ii purchase (qi, pi) and types Ij purchase (qj, pj). By monotonicity of
+purchasing decisions due to Corollary 3.4, all intermediate types θ ∈ Iℓ for
+i < ℓ < j must purchase either (qi, pi) or (qj, pj). As pi and pj are both larger
+than the prices of the elements those types were purchasing under menu M, the
+revenue of menu M ′ must be strictly larger, which is again a contradiction.
+We conclude that the prices in menu M must be monotone non-decreasing
+in quality levels, as claimed.
+4.3
+An FPTAS for Revenue
+We are now ready to consider the problem of constructing an approximately
+revenue-optimal menu for the non-linear pricing problem. For this we will make
+one further technical assumption, which is that the derivative of the valuations
+v(q; θ) with respect to q is bounded at 0. I.e., there exists some λ > 0 such
+that
+d
+dqv(0; θ) < λ for all θ. In other words, buyers are not infinitely sensitive
+to product quantity. In the context of certification, this is implied by having
+d
+dqf(0; φ) < λ for all φ, meaning that consumers are not infinitely sensitive to
+quality at 0.
+Assumption 4.6. There exists some λ > 0 such that
+d
+dqv(0; θ) < λ for all θ.
+Given this assumption, the following result provides an FPTAS for the op-
+timal menu with k items. We also show that by taking k = 1/ϵ one can obtain
+an FPTAS for the optimal (unrestricted) menu.
+16
+
+Theorem 4.7. A menu with revenue at least OPT −λϵ can be found in time
+polynomial in λ and 1/ϵ. The menu can optionally be constrained to contain at
+most k quality levels, in which case OPT is the optimal revenue achievable with
+at most k quality levels.
+Our first step to proving Theorem 4.7 is to show that we can restrict attention
+to menus with at most 1/ϵ entries at only a small loss of revenue.
+Lemma 4.8. For any monotone menu M, there exists a menu M ′ of size at
+most O(1/ϵ) such that Rev(M ′) ≥ Rev(M) − O(ϵ).
+Proof. Fix the revenue-optimal menu M = {(qi, pi)}i∈Λ. By Lemma 4.5 we can
+assume M is monotone.
+We define M ′ to be the following subset of M. Let A = {ℓϵ : 0 ≤ ℓ ≤ ⌊1/ϵ⌋}.
+Then for each a ∈ A, we add to M ′ the element (q, p) from M with smallest p
+such that p ≥ a. Then we note that M ′ contains at most ⌈1/ϵ⌉ items. Moreover,
+for each element (q, p) ∈ M there is some (q′, p′) ∈ M ′ such that p′ ≥ p − ϵ.
+Any element in M ′ will still be purchased by the types that purchased that
+element in M, as the menu of alternative options has only gotten smaller. Since
+prices are monotone in quantity, and quantities purchased are monotone in buyer
+type, we can conclude that any type that was purchasing (q, p) in the original
+menu M will then purchase an element with price at least p − ϵ in menu M ′.
+The total loss in revenue conditional on any given buyer type is therefore at
+most ϵ, and hence the total revenue loss is at most ϵ as well.
+Next, we show that we can discretize the possible (quality, price) pairs that
+appear in our menu without losing too much revenue.
+Lemma 4.9. For any monotone menu M of size k, there exists a menu M ′
+such that
+• M ′ has at most k elements,
+• for each (q, p) ∈ M ′, p is a multiple of ϵ and q = ϵ(1+ϵ)ℓ for some integer
+ℓ ≥ 0, and
+• Rev(M ′) ≥ Rev(M) − O((k + λ)ϵ).
+Proof. Let us first give the high-level idea for the construction, which is illus-
+trated in Figure 1. We will round each (quantity, price) pair on menu M in
+three steps. First, we will round each quantity down to an appropriate dis-
+cretized grid, and then also lower the corresponding price to keep constant the
+ratio of quantity to price. The concavity of the value functions will then imply
+that buyer utilities cannot be reduced by too much, multiplicatively, as a result
+of this change. This first step discretized the quantities; in the second step we
+discretize prices by rounding each price down to an appropriate grid, which can
+only increase utilities. At this point we would be almost done, except for one
+complication: we must make sure that the (small) changes in buyer utility we in-
+duce do not result in buyers switching from more expensive items to significantly
+17
+
+Figure 1: Illustration of the discretization procedure from Lemma 4.9.
+Red
+curves denote buyer valuation functions.
+Menu item (p, q) is discretized in
+quantities to (˜p, q′) by shifting along a line to the origin, then discretized in
+price to (ˆp, q′). A final discount is applied to obtain the adjusted menu item
+(p′, q′).
+cheaper items from the menu. It is here where we use the price-monotonicity of
+the menu. Since the higher-quanitity items are the more expensive ones, in our
+third step we will provide discounts for the higher-quantity menu items, which
+will offset any utility perturbations due to discretization. These discounts are
+what lead to the loss term being proportional to kϵ, rather than ϵ, in our error
+bound.
+We now move on to the formal construction. Fix menu M of size k, so that
+M = {(q1, p1), . . . , (qk, pk)} where q1 < q2 < . . . < qk. We can assume without
+loss of generality that p1 ≤ p2 ≤ . . . ≤ pk, and that each element of M is
+purchased with positive probability.
+We construct a new menu M ′ in the following sequence of steps. First, for
+each (qi, pi) with qi ≥ ϵ, let q′
+i be qi rounded down to the nearest value of ϵ(1+ϵ)ℓ
+where ℓ ≥ 0 is an integer. Then define ˜pi = pi × (q′
+i/qi), noting that we chose ˜pi
+so that ˜pi/q′
+i = pi/qi. This element (q′
+i, ˜pi) corresponds to rounding quantity but
+keeping the price to quantity ratio constant in our intuitive description above.
+For our second step, we take ˆpi to be ˜pi rounded down to the nearest multiple
+of ϵ. Finally, in our third step, we introduce our discounts. For this we define
+p′
+i = ˆpi − 3iϵ, noting that the difference between p′
+i and ˆpi is increasing in i.
+This completes our discretization procedure, so we add (q′
+i, p′
+i) to menu M ′. We
+note that menu M ′ is not necessarily monotone, may contain elements that are
+preferred by no types, and may contain elements with negative prices.
+We claim that for every type θ, if θ purchased item (qi, pi) in menu M with
+qi ≥ ϵ, then θ will purchase some (q′
+j, p′
+j) in menu M ′ such that j ≥ i. To see
+why, first consider what would happen if (q′
+i, ˜pi) were on the menu. What is
+u((q′
+i, ˜pi); θ)? Since valuation function v is concave and v(0; θ) = 0, we must
+18
+
+price
+3E
+2e:
+ (p,q)
+(p,q').
+(p,q').
+E
+quantity
+E E(1 +e) e(1+)2
+(3 + L)3
+-2E-
+(p',q').have v(q′
+i; θ) ≥ q′
+i
+qi v(qi; θ). But since q′
+i ≥
+1
+1+ϵqi, this implies
+u((q′
+i, ˜pi); θ) = v(q′
+i; θ) − ˜pi
+≥ q′
+i
+qi
+v(qi; θ) − ˜pi
+= q′
+i
+qi
+(v(qi; θ) − pi)
+≥
+1
+1 + ϵu((qi, pi); θ)
+≥ u((qi, pi); θ) − ϵ.
+Since we also know that ˜pi − ϵ ≤ ˆpi ≤ ˜pi and p′
+i = ˆpi − 3iϵ, we have
+u((q′
+i, p′
+i); θ) = u((q′
+i, ˆpi); θ) + 3iϵ
+≥ u((q′
+i, ˜pi); θ) + 3iϵ
+≥ u((qi, pi); θ) + (3i − 1)ϵ.
+On the other hand, for any j < i, the utility of purchasing (q′
+j, ˜pj) is at most
+the utility of purchasing (qj, ˜pj), which is at most ϵ more than the utility of
+purchasing (qj, pj) (since the ratio between pj and ˜pj is no greater than (1+ϵ)).
+We therefore have
+u((q′
+j, p′
+j); θ) = u((q′
+j, ˆpj); θ) + 3jϵ
+≤ u((q′
+j, ˜pj); θ) + (3j + 1)ϵ
+≤ u((qi, pi); θ) + (3j + 2)ϵ.
+Since we know that u((qi, pi); θ) ≥ u((qj, pj); θ) by assumption that θ purchases
+item (qi, pi) from menu M, we conclude that u((q′
+i, p′
+i); θ) ≥ u((q′
+j, p′
+j); θ) as well,
+since (3j + 2) ≤ (3i − 1) for i > j.
+We conclude that each type θ that purchases (qi, pi) from M with qi ≥ ϵ
+will purchase a menu item (q′
+j, p′
+j) from M ′ such that j ≥ i. Since prices are
+monotone in menu M, we conclude that the total loss in revenue can be at most
+the difference in price between pi and p′
+i for any i. This is at most O(kϵ).
+Finally, consider a type θ that purchases (qi, pi) from M with qi < ϵ. By
+Assumption 4.6, the maximum willingness to pay for any agent for quality level
+ϵ is λϵ.
+These types therefore generate revenue at most λϵ, thus regardless
+of their purchase behavior they account for a total loss in revenue of at most
+O(λϵ).
+With Lemma 4.9 in hand, we can complete the proof of Theorem 4.7 by
+employing dynamic programming to determine the revenue-optimal mechanism
+with a given maximum-quality entry. One subtlety in the construction is that
+we must be careful to account for potential cannibalization by higher-quality
+elements in the menu. We handle this by insisting that the menu we construct
+contains only elements that are selected by a non-zero measure of buyer types,
+and we check this condition when recursively applying the dynamic program.
+19
+
+Proof of Theorem 4.7. We show how to compute the optimal menu with qual-
+ities and prices chosen from a discrete indexed set of possible options, using
+dynamic programming. In general, given a quantity q and price p that lie in
+our discrete set of options, we will use Q and P to denote the integer indexing
+of q and p, respectively.
+Given any choice of Q and P and some k ≥ 1, write M[Q, P, k] for the
+optimal revenue that can be obtained using a menu with at most k elements,
+of which the one with highest quality is the one indexed by Q and P. We will
+also write L[Q, P, k] for the lowest type θ that purchases quality level Q in this
+optimal menu. We can compute M[Q, P, k] and L[Q, P, k] recursively as follows.
+If k = 1 then M[Q, P, k] is precisely p times the probability that v(q; θ) ≥ p,
+and L[Q, P, k] is precisely the infimum of types θ for which v(q; θ) ≥ p.
+For k > 1, we will consider all possible options for the next-highest quality
+level on the menu given our discretization, say (q′, p′) with Q′ < Q. For each
+choice of (Q′, P ′), we let θ(Q′, P ′) denote the type that is indifferent between
+menu items (Q′, P ′) and (Q, P), if any. Recall from the single-crossing condition
+that this choice of θ(Q′, P ′) is unique if it exists. If there is no such θ(Q′, P ′),
+then we disqualify menu item (Q′, P ′) from consideration. Otherwise, we con-
+sider L[Q′, P ′, k−1], the lowest type that purchases element (Q′, P ′) in the opti-
+mal menu with highest quality level Q′ at price P ′. If L[Q′, P ′, k−1] ≥ θ(Q′, P ′),
+then again we disqualify menu item (Q′, P ′) from consideration, as this means
+that the optimal menu containing menu item (Q, P) does not include any types
+that would purchase menu item (Q′, P ′).
+Otherwise, we have that L[Q′, P ′, k−1] < θ(Q′, P ′). We can therefore calcu-
+late the revenue from the optimal menu with highest and second-highest quality
+levels (Q, P) and (Q′, P ′) as R(Q′, P ′) = M[Q′, P ′, k − 1] + (P − P ′)Pr[θ >
+θ(Q′, P ′)]. That is, the additional revenue gain or loss due to including menu
+item (Q, P) is (P − P ′)Pr[θ > θ(Q′, P ′)], the difference due to agents with type
+greater than θ(Q′, P ′) switching from menu item (Q′, P ′) to menu item (Q, P).
+Finally, consider also the revenue that would be obtained by using only
+menu item (Q, P); call this R. If all potential choices of (Q′, P ′) were elim-
+inated or if R > R(Q′, P ′) for all potential choices of (Q′, P ′), then we set
+M[Q, P, k] = R and set L[Q, P, k] to be the infimum type θ such that v(Q; θ) ≥
+P. This corresponds to the case that the optimal menu contains only the el-
+ement (P, Q). Otherwise, let (Q′, P ′) be the choice that maximizes R(Q′, P ′),
+which by assumption is larger than R. Then we take L[Q, P, k] = θ(Q′, P ′) and
+M[Q, P, k] = R(Q′, P ′).
+We conclude that we can fill tables M and L, with each entry taking time
+proportional to ϵ−2 (the time needed to consider every possible choice (Q′, P ′)).
+As there are kϵ−2 entries in total, the total time to fill the tables is at most
+kϵ−4. The revenue-optimal mechanism with at most k menu items can then be
+obtained by taking the maximum of M[Q, P, n] over all choices of Q and P.
+Finally, by Lemma 4.8, we can take k = 1/√ϵ and our dynamic program will
+obtain a menu M such that Rev(M) is at most O(λ/√ϵ) less than the optimal
+revenue. An appropriate change of variables, taking k = 1/ϵ and discretizing to
+multiples of ϵ2, then implies that our resulting menu is at most O(λϵ) less than
+20
+
+that of the optimal menu.
+5
+Welfare Maximization
+We have studied the problem faced by a revenue-maximizing certifier. But what
+about a certifier that wishes to maximize the welfare enjoyed by the consumers
+and producers? We could think of such a certifier as a government agency who
+is offering certification services not to generate profit, but rather to maximize
+the efficiency of production and trade.
+We define the welfare of a market outcome as the sum of utilities of the
+consumers, the producers, and the certifier, taking into account all transfers
+between parties. Given a menu M of options provided by the certifier, we will
+write Wel(M) for the welfare that results in the unique market outcome resulting
+from menu M.
+An immediate consequence of Corollary 3.4 is that the welfare-optimal choice
+of menu is to offer all possible certification levels, at the cost of verification c.
+Theorem 5.1. The welfare-optimal menu of certificates offers every possible
+certification level q > 0 at a cost of c, and level 0 at a cost of 0.
+Proof. If all quality levels were visible, the welfare-maximizing outcome is would
+be for each producer ψ to trade with consumer φ(ψ) at whichever quality level
+q maximizes their gains from trade f(q; φ(ψ)) − g(q; ψ). However, since quality
+levels are hidden, producers and consumers can trade at a positive level of
+quality only if the cost of verification is paid. So if the maximum gains from
+trade is less than c, then it is preferable to trade at level 0. However, we observe
+that this is precisely the outcome implemented at equilibrium from the proposed
+certification menu, so it must be welfare-optimal over all possible menus.
+The welfare-optimal menu described above includes an arbitrarily large num-
+ber of certificates. In practice it may be helpful to find a welfare-optimal slate of
+at most k certificates. It turns out that a minor variation on the dynamic pro-
+gram described in the previous section can be used to compute an approximately
+welfare-optimizing slate, under Assumption 4.6.
+Theorem 5.2. Let M be the welfare-maximizing menu with at most k elements.
+A menu with M ′ with at most k elements and such that Wel(M ′) ≥ Wel(M) −
+O(λϵ) can be found in time polynomial in 1/ϵ.
+Proof. The proof is very similar to the one for Theorem 4.7, and strictly simpler,
+so we only briefly describe the differences here. First, it is without loss to restrict
+attention to menus that post price c for every non-trivial level of quality.
+Given any such menu M, we can discretize potential levels of quality by
+rounding down to the nearest multiple of ϵ. By Assumption 4.6 this reduces
+welfare by at most λϵ, as the welfare from each menu item is reduced by at most
+this much and each producer selects her gains-from-trade-maximizing element
+from the menu.
+21
+
+Given such a discretization, one can express the welfare-optimal menu re-
+cursively via dynamic programming as in Theorem 4.7, with the simplification
+that we need only index on quality rather than (quality, price) pairs (since all
+prices will be set to c). Rather than defining M[Q, k] (and respectively L[Q, k])
+to be the maximum revenue of a menu with k elements and maximum quality
+indexed by Q, it will be the maximum welfare of a menu with k elements and
+maximum quality indexed by Q. Our method of recursively computing M[Q, k]
+(and L[Q, k]) then remains nearly unchanged relative to Theorem 4.7.
+The
+only change to note is the actual welfare calculation, relative to the revenue
+calculation. In Theorem 4.7 we used that the revenue obtained when agents of
+type θ > θ′ purchase certificate q at price p is p times Pr[θ > θ′]. In contrast,
+the welfare obtained when producers of type ψ > ψ′ all purchase certificate q
+at price c can be calculated in closed form as
+� ψ′
+ψ (f(q; φ(η)) − g(q; η) − c)dη.
+Substituting this welfare calculation for the revenue calculation completes the
+necessary changes.
+Is the revenue-maximizing choice of menu also approximately welfare-maximizing?
+As it turns out, the welfare that results from the certifier’s reveneu-optimal menu
+can be an arbitrarily small fraction of optimal welfare. This is inherited from
+standard examples of monopolistic distortion in the linear settings, where a mo-
+nopolist might be incentivized to sell much less of their good (i.e., certification)
+than what would be efficient in order to inflate prices.
+Proposition 5.3. There is a sequence of instances for the problem where welfare
+in the revenue-optimal solution is an arbitrarily small fraction of first-best wel-
+fare. The approximation can be as bad as Ω(log H), where H = maxθ1,θ2
+maxq v(q;θ1)
+maxq v(q;θ2)
+is the ratio between the highest and lowest maximum values across buyer types.
+Proof. Take c = 0 and suppose f(q; φ) = φq and g(q; ψ) = 0 for all q and ψ. Our
+distribution over consumer types φ is an equal-revenue distribution supported
+on [1, H]; that is, F(φ) = 1 − 1
+φ for all φ ∈ [1, H]. Recalling Observation 4.2, it
+is revenue-optimal for the certifier to offer contract q = 1 at a price of H, for
+an expected welfare (and revenue) of 1. However, the optimal welfare, log(H),
+can be achieved by offering contract q = 1 at price 0.
+However, one implication of our equilibrium analysis is that adding addi-
+tional certification options to a menu of certificates cannot reduce the sum of
+utilities of the consumers and producers, regardless of the prices selected. One
+interpretation of this is that welfare cannot be harmed by a revenue-maximizing
+certifier entering a market for certification in which some certification options
+are already available.
+Proposition 5.4. Consider two certification menus M and M ′ with M ⊆ M ′.
+Then Wel(M ′) − Rev(M ′) ≥ Wel(M) − Rev(M).
+Proof. If producer ψ selects option (q, p) from certification menu M, then the
+welfare generated for the producer ψ and corresponding consumer φ, less the
+revenue raised by the certifier, is f(q; φ(ψ)) − g(q; ψ) − p. By Corollary 3.4,
+22
+
+producer ψ purchases precisely whichever menu item from M maximizes this
+quantity.
+Providing additional items can therefore only increase the welfare
+jointly enjoyed by producer type ψ and corresponding consumer φ(ψ). As this
+holds pointwise for every ψ, it holds in aggregate over all types as well.
+References
+Viral V Acharya, Peter DeMarzo, and Ilan Kremer. 2011. Endogenous informa-
+tion flows and the clustering of announcements. American Economic Review
+101, 7 (2011), 2955–79.
+S Nageeb Ali, Nima Haghpanah, Xiao Lin, and Ron Siegel. 2022. How to sell
+hard information. The Quarterly Journal of Economics 137, 1 (2022), 619–
+678.
+Reza Alijani, Siddhartha Banerjee, Kamesh Munagala, and Kangning Wang.
+2022. The limits of an information intermediary in auction design. In Proceed-
+ings of the 23rd ACM Conference on Economics and Computation. 849–868.
+Dirk Bergemann, Alessandro Bonatti, and Alex Smolin. 2018. The design and
+price of information. American economic review 108, 1 (2018), 1–48.
+Dirk Bergemann, Benjamin Brooks, and Stephen Morris. 2017. First-price auc-
+tions with general information structures: Implications for bidding and rev-
+enue. Econometrica 85, 1 (2017), 107–143.
+Dirk Bergemann, Yang Cai, Grigoris Velegkas, and Mingfei Zhao. 2022.
+Is
+Selling Complete Information (Approximately) Optimal?. In Proceedings of
+the 23rd ACM Conference on Economics and Computation. 608–663.
+Dirk Bergemann and Stephen Morris. 2019.
+Information design: A unified
+perspective. Journal of Economic Literature 57, 1 (2019), 44–95.
+Dirk Bergemann, Ji Shen, Yun Xu, and Edmund Yeh. 2012a. Multi-dimensional
+mechanism design with limited information. In Proceedings of the 13th ACM
+Conference on Electronic Commerce. 162–178.
+Dirk Bergemann, Ji Shen, Yun Xu, and Edmund M Yeh. 2012b. Mechanism
+design with limited information: the case of nonlinear pricing. In International
+Conference on Game Theory for Networks. Springer, 1–10.
+Tilman B¨orgers and Daniel Krahmer. 2015. An introduction to the theory of
+mechanism design. Oxford University Press, USA.
+Ozan Candogan and Philipp Strack. 2021. Optimal Disclosure of Information
+to a Privately Informed Receiver. In EC ’21: The 22nd ACM Conference on
+Economics and Computation, Budapest, Hungary, July 18-23, 2021, P´eter
+Bir´o, Shuchi Chawla, and Federico Echenique (Eds.). ACM, 263.
+23
+
+Shuchi Chawla, Hu Fu, and Anna Karlin. 2014. Approximate revenue maxi-
+mization in interdependent value settings. In Proceedings of the fifteenth ACM
+conference on Economics and computation. 277–294.
+Joe
+Sandler
+Clarke
+and
+Luke
+Barratt.
+[n. d.].
+Top
+airlines’
+promises
+to
+offset
+flights
+rely
+on
+‘phantom
+credits’.
+Unearthed
+([n. d.]).
+https://unearthed.greenpeace.org/2021/05/04/
+carbon-offsetting-british-airways-easyjet-verra/
+Marc N Conte and Matthew J Kotchen. 2010. Explaining the price of voluntary
+carbon offsets. Climate Change Economics 1, 02 (2010), 93–111.
+Peter M DeMarzo, Ilan Kremer, and Andrzej Skrzypacz. 2019. Test design and
+minimum standards. American Economic Review 109, 6 (2019), 2173–2207.
+Mathias Dewatripont, Patrick Bolton, et al. 2005. Contract theory. Technical
+Report. ULB–Universite Libre de Bruxelles.
+Piotr Dworczak. 2020.
+Mechanism design with aftermarkets: Cutoff mecha-
+nisms. Econometrica 88, 6 (2020), 2629–2661.
+Daniel W Elfenbein, Raymond Fisman, and Brian McManus. 2015.
+Market
+structure, reputation, and the value of quality certification. American Eco-
+nomic Journal: Microeconomics 7, 4 (2015), 83–108.
+Patrick
+Greenfield.
+[n. d.]a.
+Carbon
+offsets
+used
+by
+major
+air-
+lines
+based
+on
+flawed
+system,
+warn
+experts.
+The
+Guardian
+([n. d.]).
+https://www.theguardian.com/environment/2021/may/04/
+carbon-offsets-used-by-major-airlines-based-on-flawed-system-warn-experts
+Patrick Greenfield. [n. d.]b.
+Revealed: more than 90% of rainforest carbon
+offsets by biggest provider are worthless, analysis shows.
+The Guardian
+([n. d.]).
+https://www.theguardian.com/environment/2023/jan/18/
+revealed-forest-carbon-offsets-biggest-provider-worthless-verra-aoe
+Sanford J Grossman. 1981. The informational role of warranties and private
+disclosure about product quality. The Journal of Law and Economics 24, 3
+(1981), 461–483.
+Alejandro Guizar-Couti˜no, Julia PG Jones, Andrew Balmford, Rachel Car-
+menta, and David A Coomes. 2022. A global evaluation of the effectiveness
+of voluntary REDD+ projects at reducing deforestation and degradation in
+the moist tropics. Conservation Biology (2022), e13970.
+Faruk Gul and Ennio Stacchetti. 2000. The English auction with differentiated
+commodities. Journal of Economic theory 92, 1 (2000), 66–95.
+Alessandro Lizzeri. 1999. Information revelation and certification intermediaries.
+The RAND Journal of Economics (1999), 214–231.
+24
+
+Michael Luca. 2016. Reviews, reputation, and revenue: The case of Yelp. com.
+Com (March 15, 2016). Harvard Business School NOM Unit Working Paper
+12-016 (2016).
+Paul R Milgrom. 1981. Good news and bad news: Representation theorems and
+applications. The Bell Journal of Economics (1981), 380–391.
+Paul R Milgrom and Robert J Weber. 1982. A theory of auctions and com-
+petitive bidding. Econometrica: Journal of the Econometric Society (1982),
+1089–1122.
+James A Mirrlees. 1971.
+An exploration in the theory of optimum income
+taxation. The review of economic studies 38, 2 (1971), 175–208.
+Michael Mussa and Sherwin Rosen. 1978. Monopoly and product quality. Jour-
+nal of Economic theory 18, 2 (1978), 301–317.
+Roger B Myerson. 1981. Optimal auction design. Mathematics of operations
+research 6, 1 (1981), 58–73.
+Subramaniam Ramanarayanan and Jason Snyder. 2012. Reputations and firm
+performance: Evidence from the dialysis industry. Unpublished manuscript
+(2012).
+Tim Roughgarden and Inbal Talgam-Cohen. 2013. Optimal and near-optimal
+mechanism design with interdependent values. In Proceedings of the fourteenth
+ACM conference on Electronic commerce. 767–784.
+Alejandro Saporiti and Fernando Tohm´e. 2006. Single-crossing, strategic voting
+and the median choice rule. Social Choice and Welfare 26, 2 (2006), 363–383.
+Michael Spence. 1974. Competitive and optimal responses to signals: An anal-
+ysis of efficiency and distribution. Journal of Economic theory 7, 3 (1974),
+296–332.
+Steven Tadelis and Florian Zettelmeyer. 2015.
+Information disclosure as a
+matching mechanism: Theory and evidence from a field experiment. Ameri-
+can Economic Review 105, 2 (2015), 886–905.
+Thales AP West, Jan B¨orner, Erin O Sills, and Andreas Kontoleon. 2020. Over-
+stated carbon emission reductions from voluntary REDD+ projects in the
+Brazilian Amazon. Proceedings of the National Academy of Sciences 117, 39
+(2020), 24188–24194.
+Thales AP West, Sven Wunder, Erin O Sills, Jan B¨orner, Sami W Rifai, Alexan-
+dra N Neidermeier, and Andreas Kontoleon. 2023. Action needed to make
+carbon offsets from tropical forest conservation work for climate change mit-
+igation. arXiv preprint arXiv:2301.03354 (2023).
+Bradley S Wimmer and Brian Chezum. 2003.
+An empirical examination of
+quality certification in a “lemons market”. Economic inquiry 41, 2 (2003),
+279–291.
+25
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf,len=941
+page_content='Certification Design for a Competitive Market  Andreas Haupt  Nicole Immorlica  Brendan Lucier∗  February 1, 2023  Abstract  Motivated by applications such as voluntary carbon markets and edu-  cational testing, we consider a market for goods with varying but hidden  levels of quality in the presence of a third-party certifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' The certifier  can provide informative signals about the quality of products, and can  charge for this service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Sellers choose both the quality of the product  they produce and a certification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Prices are then determined in a compet-  itive market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Under a single-crossing condition, we show that the levels of  certification chosen by producers are uniquely determined at equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  We then show how to reduce a revenue-maximizing certifier’s problem to  a monopolistic pricing problem with non-linear valuations, and design an  FPTAS for computing the optimal slate of certificates and their prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' In  general, both the welfare-optimal and revenue-optimal slate of certificates  can be arbitrarily large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  1  Introduction  Many markets sell goods that have different features but appear identical to  consumers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Examples include the market for carbon credits, the market for  contract workers such as electricians or plumbers, or even some markets for  physical goods like milk and eggs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' In each of these markets, the goods have  hidden, hard-to-verify properties that distinguish them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Nature-based carbon  credits, for example, are sourced from a variety of forests with different pro-  tection and longevity properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Contract workers have differing skill levels  and amount of knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' And physical goods are produced in a variety of  circumstances, some more ethical and sustainable than others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  In such settings, sellers seek to distinguish their products through costly  certification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' In the example of carbon credits, large certifiers issue certification  standards for carbon emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Contract workers can enroll in certification  programs or take exams to document their skill level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' And farms can reach out  to third-party organizations that certify the conditions, such as free-range or  vegetarian-fed, under which their food is produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  ∗We thank Robert Maddox, Yanika Meyer-Oldenburg and a seminar audience at Harvard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='13449v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='GT]  31 Jan 2023  In each of these cases, a downstream market allocates goods based on cer-  tification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Carbon exchanges allow for the trade of carbon certificates, and  platforms for local services allow the trade of electrical or plumbing services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  These markets have price formation that is independent of, and not controllable  by, the certifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Therefore, both welfare- and revenue-maximizing certifiers will  need to reason about how the certification options they provide affect trade,  and how this in turn influences the demand for certification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  What impact does the presence of certification have on the downstream  market?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' We answer this question in the context of a competitive market with  production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' In our model there is a mass of heterogeneous producers each creat-  ing a single unit of a vertically-differentiated product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Producers can select the  quality level of their products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' The corresponding production cost is determined  by the producer’s type and is increasing in quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' A mass of unit-demand con-  sumers purchases these goods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Consumers also have varying types which deter-  mine the strength of their preference for quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Types are totally ordered and  satisfy a single-crossing condition: higher-type producers are relatively more  efficient at producing higher-quality goods, which higher-type consumers have  relatively stronger preference for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Quality cannot be verified directly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' instead, a third-party certifier offers a  menu of certifications, each with its own requirements and certification price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Producers can purchase certifications of their products’ qualities from this menu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  If their product surpasses the quality level of the certificate, it will be marketed  as such.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' The certified goods, together with indistinguishable uncertified ones,  are then sold in a competitive market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  As in the Economics literature on certification, we assume that market par-  ticipants are able form correct beliefs about the quality implications of a certifi-  cate Milgrom (1981);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Grossman (1981);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Lizzeri (1999);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' DeMarzo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' A  big strand of empirical work considers the question of whether certifications are  correctly interpreted in various markets (Wimmer and Chezum (2003) for race  horses, Tadelis and Zettelmeyer (2015) on used-car auctions, Ramanarayanan  and Snyder (2012) for dialysis screening centers, Luca (2016) for restaurant  reviews, Elfenbein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' (2015) for seller ratings on Ebay, Conte and Kotchen  (2010) for voluntary carbon credits).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' In our model, certificates are based on ver-  ifiable certification requirements, and at perfect Bayesian equilibrium all market  participants hold rational beliefs about the distribution of quality implied by  any given certificate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' We note that such rational beliefs need not be aligned  with official certificate descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' For example, in voluntary carbon markets,  several empirical contributions point out that carbon certificate descriptions  might not correspond to counterfactual mitigation outcomes, compare West  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' (2020, 2023);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Guizar-Couti˜no et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Clarke and Barratt ([n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=']);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Greenfield ([n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=']b,n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  The presence of certificates clearly impacts the market equilibrium because  it influences consumers’ beliefs and hence willingness-to-pay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Our first result  shows that, for any menu of certificates offered by the certifier, the equilibrium  choice of certificates and the allocation outcome of the downstream market  equilibrium is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Furthermore better producers (that is, those who can  2  produce higher quality at lower cost) always purchase higher (more restrictive)  levels of certification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  This implies that the producer-consumer matching in  equilibrium is assortative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Better producers sell to consumers with higher value  for quality, and at higher quality levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  This analysis immediately implies that the welfare-optimal menu offers every  certificate at cost, and suggests a dynamic program for finding the welfare-  optimal menu subject to a cardinality restriction on the number of certification  levels that can be offered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' But in many markets, certification is provided by  third-party firms with profit-maximizing incentives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Our main result shows  how to approximately construct the revenue-optimal menu in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Specifically, we provide an FPTAS: we show how to compute a menu whose  revenue for the certifier is an additive ϵ less than the optimal revenue in time  polynomial in 1/ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Our algorithm can optionally take as input a cardinality  constraint k on the menu size (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=', a maximum number of certification levels),  in which case the revenue benchmark is the optimal slate of at most k certificates  and their prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Our proof contains technical insights that may be of independent interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Namely, we reduce the certifier’s problem to one of a seller who wishes to sell  a divisible good, facing buyers whose valuations are non-linear and potentially  non-monotone in quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' The seller corresponds to the certifier, and the buy-  ers correspond to producer-consumer pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' This projection of producers and  consumers into a single economic agent is enabled by the fact that the matching  in equilibrium is unique and assortative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' The buyer valuations in this reduced  problem are concave but not necessarily monotone as a function of quantity,  meaning that for different buyer types the valuation may reach its maximum at  different quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' The valuations do satisfy a single-crossing property, which  implies that types are totally ordered and higher-type buyers have pointwise  higher valuations and have weakly higher preferred quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Under these  conditions, we show that the revenue-optimal menu may be non-linear but will  exhibit prices that are monotone in quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Monotonicity of prices enables  the use of dynamic programming to construct an approximately optimal menu,  though some care must be taken when discretizing the space of quantities and  prices to ensure that buyer preferences do not change substantially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' This non-  linear pricing problem generalizes prior work that specifically studied the case  of quadratic demand Bergemann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' (2012a,b), and may be of independent  interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  The menu of certifications offered by a revenue-maximizing third-party cer-  tifier can distort welfare and result in inefficient levels of trade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  That said,  we show that a certifying agent entering into the market can never lead to a  loss of welfare, no matter what menu of certification options they make avail-  able.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' More specifically, we can imagine that a collection of (perhaps costly)  certification options are available to the market as a baseline, perhaps provided  by a public or tightly regulated agency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' If a third-party certifier then enters  the market and makes available any additional certification options into the  market, at any price, we show that the total utility of all market participants  (even excluding the new entrant) can only increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' A policy implication is that  3  a regulatory body concerned about inefficient certification arising from profit-  seeking motives need not prevent certifiers from entering the market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Rather, it  suffices to make sure that there exists some set of certifications available, either  through third-party providers or public options, that enable an acceptable level  of welfare from trade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='1  Related Literature  This work contributes to the literature on certification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' The early results Mil-  grom (1981);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Grossman (1981) produce unraveling type results: In equilibrium,  the quality of a good is fully revealed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' The main intuition for these results in  models of certification is that certifying non-informatively will be interpreted  by the market as a sign of bad quality—adverse selection is extreme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' In our  model of a revenue-maximizing certifier, there might be other reasons for certi-  fication less informatively, as the price in the competitive market may depend  on not only an individual seller’s quality, but all the sellers in the market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Later  contributions Lizzeri (1999);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' DeMarzo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Acharya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' (2011) allow  for unsuccessful certifications, restricting the stark result obtained in Milgrom  (1981);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Grossman (1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  More broadly, our work is related to mechanism and information design in  the presence of an exogenously given game played after the design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' We rec-  ommend Bergemann and Morris (2019) for a general overview of information  design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Our work is especially related to the design of information provided to  buyers and/or sellers of a good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Bergemann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' (2017) considers the design of  information for a first-price auction, where a third party can reveal a signal cor-  related with a buyer’s valuations, and fully characterizes the achievable revenue  and payoffs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Alijani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' (2022) extends the analysis to a scenario with mul-  tiple buyers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' In a general mechanism design framework, Candogan and Strack  (2021) develop an optimal disclosure policy for action recommendations in a  game with private types and a hidden state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Dworczak (2020) designs mech-  anisms in a setting where players participate in a finite Bayesian game after  participating in the mechanism, so that game outcomes are impacted by infor-  mation revealed over the course of the mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' He finds a cutoff structure  of optimal mechanisms in the first stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  For-profit certification relates to the sale of hard information, which has  been studied in the context of competitive markets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Ali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' (2022) consider  a seller holding a good of uncertain quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' The seller can purchase a quality-  correlated signal from a revenue-maximizing intermediary before bringing the  good to market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' In general the resulting equilibria are not unique and can vary  substantially, but by employing noisy signals the intermediary can robustly  guarantee high revenue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Our model differs in that any certification options are  made available to the entire market and product quality is endogenous, so the  certifying agent can impact welfare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' More generally, our work relates to the  problem of how to sell payoff-relevant hard information to a potential buyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Bergemann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' (2018) solve for the revenue-maximizing mechanism in binary  environments, and Bergemann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' (2022) establish when full disclosure is  4  approximately optimal under more general spaces of actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' In contrast, our  certifying agent is selling a signal that is valuable in that it conveys information  to other participants in a subsequent game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  In our mathematical analysis, we reduce the certifier’s problem to a pricing  problem with a non-linear valuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' The non-linear pricing literature following  Mussa and Rosen (1978) (see also treatment in Dewatripont et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' (2005) and  (B¨orgers and Krahmer, 2015, Chapter 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='3)) studies a non-linear concave valu-  ation with a quadratic cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' The functional form assumptions in these papers  allow to characterize the optimal mechanism in closed form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Typically, the op-  timal menu of offered goods is a continuum, in contrast to the linear screening  problem first studied in the influential Myerson (1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Our analysis will show  that also the class of models we consider may feature infinite menus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  In our polynomial-time approximation scheme, we use an approximation of  a non-linear pricing model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' The papers Bergemann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' (2012a,b) consider  approximation of non-linear single- and multi-dimensional pricing environments  with finite menus (in the papers called “finite information”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' The papers make  functional form assumptions similar to the ones in Mussa and Rosen (1978), and  derive rates of approximation by finite menus in these finite menus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' The present  paper allows for a general class of utility functions that satisfy a single-crossing  condition, and, in addition to showing approximation by finite menus, shows  that the finite-sized menu can be computed efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Finally, our main assumption guaranteeing uniqueness of our equilibrium is  a single-crossing condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Single-crossing conditions are important in several  domains, among them the interdependent private values auctions (Milgrom and  Weber (1982)) and social choice and voting (Saporiti and Tohm´e (2006)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' A re-  cent line of work in algorithmic mechanism design has employed single-crossing  conditions to enable approximately optimal designs in interdependent value set-  tings Roughgarden and Talgam-Cohen (2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Chawla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Closest to  the present paper, another implication of single-crossing is adverse selection in  markets Mirrlees (1971);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Spence (1974).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='2  Outline  The rest of this article is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  We formalize our model in  section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' In section 3 we analyze the structure of equilibria given the certifier’s  offerings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Section 4 contains our main results, a reduction of revenue-maximizing  certification to a non-linear pricing problem, the FPTAS for its computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Section 5 contains our results on welfare maximization and explores the welfare  implications of third-party certification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  2  Model  Market  We consider a continuum market between producers and consumers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Producers are unit-supply and parameterized by types ψ ∈ R+ with measure G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Consumers are unit-demand and parameterized by types φ ∈ R+ with measure  5  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' The type measures F and G are atomless and continuous with compact  support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Goods can be produced at different levels of quality, denoted q ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Goods of higher quality are more valuable to consumers but more costly to  produce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' We write g(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' ψ) for the cost incurred by a producer of type ψ when  producing a good of quality q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' We assume g is weakly convex and non-decreasing  in q for every ψ and normalized so that g(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' ψ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' We also write f(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' φ) for  the value enjoyed by a consumer of type φ for a good of quality q, where f  is weakly concave and non-decreasing in q for every φ and normalized so that  f(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' φ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' We will scale valuations so that f(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' φ) ≤ 1 for all q and φ, which  is without loss for bounded values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  We will assume that costs and valuations satisfy single-crossing with respect  to the producer and consumer types, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Roughly speaking, this means  that producers (consumers) of higher types have lower marginal cost (higher  marginal value) for producing higher-quality goods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' More formally, for all φ1 <  φ2 and q1 < q2, we have  f(q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' φ2) − f(q1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' φ2) > f(q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' φ1) − f(q1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' φ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Likewise, for all ψ1 < ψ2 and q1 < q2, we have  g(q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' ψ2) − g(q1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' ψ2) < g(q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' ψ1) − g(q1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' ψ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Transfers between producers and consumers are permitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' We assume that  producers and consumers are risk-neutral and have quasi-linear preferences with  respect to money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' That is, if consumer φ purchases a product of quality q from  producer ψ at a price of p, then the consumer enjoys utility  uC((q, p);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' φ) = f(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' φ) − p  and the producer’s utility is  uP ((q, p);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' ψ) = p − g(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Certification  Crucially, producers and consumers cannot contract on quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  This means that a producer cannot credibly commit to the quality of the good  they produce, and a consumer cannot verify quality at the point of trade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' But  there is a third-party certifier who is able to determine the quality of a producer’s  good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' This verification is costly to the certifier, with cost c ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  After verifying the quality of a good, the certifier is able to assign to that  good a signal (or certificate) σ ∈ Σ, where Σ is an arbitrary space of potential  certificates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  This certificate is visible to all producers and consumers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  The  certifier is permitted to collect payments from producers for this service, and  these transfers can depend arbitrarily on the certificate produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  The certifier can commit to a certification menu M, which is a collection  of certificate / transfer pairs (σ, t(σ)) along with quality requirements for each  certificate σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Following the literature on information design and persuasion,  6  we note that it is without loss of generality to associate each certificate signal  σ with the set of quality levels that are eligible for that certificate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' We will  therefore assume without loss of generality that Σ = 2[0,1], the collection of all  subsets of [0, 1], and each σ is a subset of [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' The interpretation is that a  producer can select an item from this menu, in which case she pays the certifier  price (or transfer) t(σ), the certifier verifies the good’s quality q, and as long  as q ∈ σ the producer will receive certification σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' We will assume for technical  convenience that each σ in the certifier’s menu contains a minimum element,  meaning that inf σ ∈ σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='1  We will write σ0 = [0, 1] for the trivial signal that conveys no additional  information about quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' A good with certificate σ0 is functionally equivalent  to a good that has not been certified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  It will be notationally convenient to  assume that the certifier always offers signal σ0 at cost 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' This allows us to  think of every good as being certified, albeit possibly at the trivial level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' If a  producer attempts to purchase a certificate but does not satisfy that certificate’s  requirements, they will instead be assigned certificate σ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=', the certifier will  not certify the good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  The Competitive Market  All certification is assumed to occur simultane-  ously, and in advance of any trading between producers and consumers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Given  the menu M of certificates and prices offered by the certifier, the producers’  (production and certification) strategy is a mapping from producer type ψ to  a choice of quality level q and certification σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' We will denote such a strategy  Γ : ψ �→ (q, σ), and restrict attention to measurable functions Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' In a slight  abuse of notation, we will also use Γ to denote the measure over pairs (q, σ) of  products with corresponding quality and certification that result when produc-  ers apply strategy Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Goods that are assigned the same certificate are indistinguishable by the  consumers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' After all certification is complete, each producer has a single unit  of a good marked with a certification σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' For any given certification σ, write Γσ  for the marginal distribution over quality q of Γ restricted to certificate σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' That  is, fixing the choices of the producers, Γσ is the distribution of levels of quality  for a product with certification σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Then the value of a consumer of type φ for  a good with certificate σ can be evaluated as  f(σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' φ) = Eq∼Γσ[f(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' φ)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  That is, each consumer rationally evaluates the expected quality of each product  given its certification level and the choices of the producers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  1This excludes certificates of the form “the quality q is strictly greater than 1/2,” as  opposed to “.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='at least 1/2.” Certificates of the former type are inconvenient because there is  no single least-cost choice of quality that satisfies the certification requirement, and hence no  utility-maximizing choice of quality for the producer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' We could handle such certificates in a  straightforward but tedious way by relaxing our equilibrium notion and assuming that each  producer selects an ϵ-approximately utility maximizing choice of quality for some arbitrarily  small ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' For the remainder of the paper we will ignore such technical issues and simply assume  that each σ includes a minimum element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  7  Since goods with the same certification are indistinguishable to consumers,  we can view the competitive market between producers and consumers as a  market for certificates σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  A Walrasian (or Competitive) equilibrium of the  resulting market is an allocation x(φ) of a certificates to each consumer φ, along  with a price pσ for each certificate, such that:  Demand satisfaction: every consumer purchases her most-preferred good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  That is, for every consumer type φ, f(x(φ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' φ) − px(φ) ≥ f(σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' φ) − pσ for  every σ ∈ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Market Clearing: every good with a positive price is sold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' That is, for all  σ, the measure of consumers φ such that x(φ) = σ is at most the measure  of producers ψ who select level of certification σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' If pσ > 0 then these  measures are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Since buyers (consumers) are unit-demand and hence their preferences satisfy  the gross substitutes condition, a Walrasian equilibrium is guaranteed to ex-  ist Gul and Stacchetti (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  We will therefore assume that trade occurs  between consumers and producers at competitive equilibrium prices given the  choices made by the producers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Timeline  To summarize, the timing of the market with certification is as  follows:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' The certifier commits to a menu of certificates with corresponding prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Each producer ψ simultaneously and privately chooses whether to produce  a good, and if so at what level of quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Each producer that chose to produce decides whether to certify their prod-  uct, and at which certificate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' These decisions are made simultaneously for  all producers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' The certifier verifies the products of producers who choose to certify and  assigns certificates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Any producer who does not successfully certify re-  ceives certificate σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Producers and consumers trade goods in a competitive market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=', trade  occurs at market-clearing prices for the chosen levels of certification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Since Walrasian equilibria are not unique in general, one might wonder if  the outcome described in the final step is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' We will show in the  next section that the competitive market equilibrium described the final step  exists and its resulting allocation is unique for any certificate menu chosen by  the certifier and any choice of certification levels chosen by the producers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  8  3  Equilibrium Characterization  In this section we describe the market outcome that will occur for any given  menu of certificates offered by the certifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' We show that it is without loss  of generality for the certifier to restrict to offering threshold certificates that  guarantee that a product is at least a certain level of quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' We characterize  the unique Walrasian market equilibrium allocation that results from any as-  signment of such certificates to producers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' We then use that characterization  to solve for each producer’s utility-maximizing choice of certificate from the  certifier’s menu, which will also be unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='1  Certifications as Minimum Quality Thresholds  A first simple observation is that since producer costs are increasing in quality  level, and since goods at different quality levels but with the same certification  are indistinguishable to consumers (and hence must sell at the same price),  a producer who is assigned certificate σ will always choose to produce at the  minimum quality level eligible for that certificate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Observation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Fix any certifier menu M and any production and certifica-  tion strategy of the producers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Then for any producer ψ, selecting certificate σ  and producing at quality q > min σ is dominated by selecting certificate σ and  producting at quality min σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Given this observation, we know that for any menu M, any equilbrium  strategy Γ for the producers, and any certificate σ, the marginal distribution  over quality Γσ will be a point mass at min σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' In particular, any two certificates  with the same minimum will induce the same equilibrium beliefs over quality and  hence have indistinguishable value to all consumers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' It is therefore without loss  of generality for the certifier to only offer certificates of the form σq = [q, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=',  certificates that are differentiated only with respect to their minimum values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  If a producer selects certificate σq, then that producer’s chosen quality level  at equilibrium will necessarily be q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Any given certification menu M therefore  reduces to a (possibly infinite) collection of quality levels in [0, 1] to certify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Motivated by this observation, we will assume for the remainder of the paper  that all certificates are of the form [q, 1], and associate each σ = [q, 1] with its  quality threshold q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' We can then think of a certification menu M as a collection  of pairs {(qi, ti)}, where qi is a quality threshold and ti is a corresponding price  for certifying that quality is at least qi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='2  Uniqueness and Assortativeness of Competitive Mar-  ket Allocations  We now turn to an analysis of the competitive market outcome that will result  given the strategies of the certifier and producers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Recall that we can restrict  attention to certificates of the form σq = [q, 1] and that any good with certificate  σq will have quality q with probability 1, so for the remainder of the section we  9  will think of a market outcome as an allocation x and prices p of quality levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  That is, x(φ) ∈ [0, 1] for all consumers φ, and for each q ∈ [0, 1] in menu M  there is an associated market price pq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' We emphasize that x is a mapping from  consumers to the certified goods they buy at market, whereas Γ is a mapping  from producers to the certificates that they choose from the certifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  The following lemma shows that for any choice of certification menu M and  production and certification strategy Γ of the producers, all competitive market  equilibria in the resulting market have the same uniquely-determined allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  This allocation will be assortative, with higher-type consumers purchasing the  higher-quality certificates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Fix any certifier menu M and any strategy Γ of the produc-  ers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Then in every competitive market outcome (x, p), the allocation x satisfies  x(φ1) ≤ x(φ2) and px(φ1) ≤ px(φ2) for all φ1 ≤ φ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Since consumers are unit-demand and each producer has a single unit of  good, a Walrasian equilibrium (x, p) of the market is guaranteed to exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' By  the first welfare theorem, any such equilibrium must maximize the total welfare,  �  φ  f(x(φ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' φ)dF(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Suppose there exist types φ1 < φ2 with x(φ1) > x(φ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' By the single-crossing  condition, we have that  f(x(φ1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' φ2) − f(x(φ2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' φ2) > f(x(φ1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' φ1) − f(x(φ2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' φ1)  and hence  f(x(φ1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' φ2) + f(x(φ2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' φ1) > f(x(φ1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' φ1) + f(x(φ2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' φ2)  which contradicts the supposed welfare optimality of allocation x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  We now turn to prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Fix any consumer types φ1 < φ2, so in particular  we know x(φ1) ≤ x(φ2), and suppose for contradiction that px(φ1) > px(φ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' By  monotonicity of the value function f, we must have f(x(φ1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' φ1) ≤ f(x(φ2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' φ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  But this then means f(x(φ2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' φ1)−px(φ2) > f(x(φ1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' φ1)−px(φ1), which violates  the competitive market condition that consumer type φ1 is choosing her most-  preferred good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' We therefore conclude that px(φ1) ≤ px(φ2), as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='3  Uniqueness and Assortativeness of Certificate Selec-  tion  Given that market outcomes are well-defined, we next turn to the equilibrium  choices of the producers when selecting quality levels and their corresponding  certifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' We again show that for any menu M of certificates offered, the  quality choices of producers are unique at equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Moreover, higher-type  producers will always select (weakly) higher certificates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' In the Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='2, we  saw that higher-type consumers also purchase (weakly) higher certificates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' As  10  we discuss later, this means the matching in any equilibrium will be assortative  and hence constrained-efficient given the available certificates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Recall that a producer strategy Γ is a mapping from producer type ψ to a  choice of certification and quality, which we know from will always coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' We  will therefore write Γ(ψ) = q to mean that producer ψ produces at quality level  q and purchases certificate σq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' In particular, we must have Γ(ψ) ∈ M for all ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Fix any certification menu M offered by the certifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Then there  is a unique equilibrium strategy Γ for the producers, and Γ(ψ) is weakly increas-  ing in ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Fix strategy Γ, which implies the measure of certificates chosen by the  collection of producers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Let (x, p) denote a Walrasian equilibrium in the result-  ing competitive market, and recall that x is uniquely determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  We first show that Γ is weakly increasing is ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Assume for contradiction  that there exist ψ1 < ψ2 with q1 = Γ(ψ1) and q2 = Γ(ψ2) with q2 < q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Then  by the single-crossing condition for producers, we have g(q1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' ψ1) − g(q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' ψ1) >  g(q1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' ψ2) − g(q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' ψ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  But then, if we let pq1 and pq2 denote the Walrasian  equilibrium prices of q1 and q2 given Γ, we have  (pq1 − g(q1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' ψ1)) + (pq2 − g(q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' ψ2)) < (pq1 − g(q1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' ψ2)) + (pq2 − g(q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' ψ1))  which means that either  pq1 − g(q1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' ψ1) < pq2 − g(q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' ψ1)  or  pq2 − g(q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' ψ2) < pq1 − g(q1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' ψ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  In other words, either producer ψ1 or ψ2 (or both) would strictly improve their  utility by switching their choice of quality and certification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' As such a swap has  measure zero and does not influence the competitive equilibrium, this would be  an improving deviation for the producer(s), violating the assumption that Γ is  an equilibrium strategy for the producers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  We have shown that Γ is weakly increasing in ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' On the other hand, we  know from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='2 that the market allocation of quality levels to consumers is  weakly increasing in φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' This means that any equilibrium outcome of production  and trade is equivalent to one in which consumers and producers are matched  assortatively, with higher-type consumers trading with higher-type producers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  In other words, for any producer type ψ, there is a consumer type φ = φ(ψ) such  that ψ always trades with φ(ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Specifically, φ(ψ) is such that F(φ(ψ)) = G(ψ)  (treating F and G as cumulative distribution functions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Given this, we claim that Γ(ψ), the certification selected by producer ψ  at equilibrium, will always be a certificate qi from the certifier’s menu M =  {(qi, ti)} that maximizes f(qi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' φ(ψ)) − g(qi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' ψ) − ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' To see why, suppose for  contradiction that the producer instead chooses some other certificate q′ at price  t′ such that f(q′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' φ(ψ))−g(q′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' ψ)−t′ < f(qi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' φ(ψ))−g(qi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' ψ)−ti−ϵ for some ϵ > 0,  11  and sells to consumer φ(ψ) at an assumed market-clearing price pq′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='2 Then, the  producer ψ could instead deviate to purchasing qi at a price of ti, and offering  it on the competitive market at a price of pq′ +g(qi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' ψ)−g(q′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' ψ)+(ti −t′)+ϵ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Note that if consumer φ(ψ) were to purchase from producer ψ at this price,  then her utility would be  f(qi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' φ(ψ)) − [pq′ + g(qi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' φ(ψ)) − g(q′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' φ(ψ)) + (ti − t′) + ϵ/2]  = (f(qi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' φ(ψ)) − g(qi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' ψ) − ti − ϵ) + (g(q′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' ψ) + t′) + ϵ/2 − pq′  > f(q′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' φ(ψ)) − (g(q′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' ψ) + t′) + (g(q′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' ψ) + t′) − pq′ + ϵ/2  > f(q′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' φ(ψ)) − pq′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  But since (q′, pq′) is the most-demanded offering to consumer φ(ψ) in the market  equilibrium, this means that the offering of qi at the proposed price would be the  most-demanded offering to consumer φ(ψ) under this deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' In particular  this means that some consumer would want to purchase qi at the suggested price,  and therefore in the adjusted market equilibrium after this proposed deviation  the price of qi must be at least this high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  We conclude that the utility of producer ψ under this deviation is at least  [pq′ + (g(qi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' ψ) − g(q′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' ψ) + (ti − t′) + ϵ/2] − g(qi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' ψ) − ti = pq′ − g(q′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' ψ) − t′ + ϵ/2  > pq′ − g(q′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' ψ) − t′  and hence this deviation is strictly utility-improving for the producer, contra-  dicting the assumption that Γ is an equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  We therefore conclude that at equilibrium, each producer ψ chooses whichever  certificate qi from the menu maximizes f(qi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' φ(ψ)) − g(qi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' ψ) − ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' The choice of  each producer is therefore unique, up to tie-breaking on sets of measure zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  An immediate corollary of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='2 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='3 is that the equilibrium  outcome for a given menu M is not only essentially unique (up to the choice of  market-clearing prices), but also has a natural assortative interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Each  producer in the market has a corresponding consumer with whom they will  always trade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' The producer will select whichever certification level maximizes  the gains from trade between themselves and their partner consumer, less the  price of the certification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' For any menu M = {(qi, ti)} of the certifier, the resulting  equilibrium market outcome has producer ψ trade with consumer φ = φ(ψ) where  G(ψ) = F(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' The level of quality at which φ and ψ trade maximizes f(qi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' φ) −  g(qi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' ψ) − ti, and is weakly increasing in ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  2Note that as we showed Γ is weakly increasing in ψ, the deviation can not change the  ordering of firms in terms of the certificate they purchase and hence does not change the  consumer to which they sell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  12  4  Revenue-Optimal Certification  In the previous section we solved for the equilibrium outcome of the game played  between producers and consumers given a certification menu M chosen by the  certifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' With that characterization in hand, we now turn to the problem faced  by a certifier who wishes to construct a revenue-maximizing slate of certificates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  A priori, it would appear that the choice of which certificates to offer in-  fluences the downstream competitive market allocation and prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' After all,  producers compete for consumers and the certificates they purchase determine  how they fare in this competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Hence one might expect the menu of avail-  able certificates to influence how much each producer would be willing to pay  for any given certificate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' However, as we will show, the assortative characteriza-  tion of market outcomes means that the certifier can reason about the behavior  of each producer type separately, without worrying about how they will jointly  interact in the competitive market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Thus, the certifier is essentially facing a  single buyer with an unknown type, and must simply maximize revenue subject  to this buyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' This leads us to define a reduction from the certifier’s problem  to that of a seller facing a (non-linear) buyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='1  Reduction to a Non-linear Pricing Problem  We will relate the certifier’s revenue-maximization problem to a non-linear pric-  ing problem between a single seller and a single buyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' The buyer seeks to buy  a perfectly divisible good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' The seller may commit to a menu of quantities and  prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' If a buyer of type θ purchases quantity q ∈ [0, 1] of the good at a total  price of t, then the buyer utility is  u((q, t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' θ) = v(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' θ) − t,  where v is concave in quantity q but not necessarily non-decreasing, and v(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' θ) =  0 for all θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' The valuations v satisfy a single-crossing condition, which is that  for θ1 < θ2 and q1 < q2, we have  v(q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' θ2) − v(q1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' θ2) > v(q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' θ1) − v(q1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' θ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  The principal seeks to maximize revenue subject to a constant cost of production  c > 0 for any non-zero quantity given a prior over buyer types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Given a menu  M, we will write Rev(M) to denote the expected revenue obtained from M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' We  will also write OPT for the revenue of the revenue-maximizing choice of menu  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' The certifier’s revenue-maximization problem is equivalent  to an instance of the non-linear pricing problem described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' The certifier’s problem is to design a certificate menu M = {(qi, ti)}  that maximizes the total revenue collected, less the certification cost c paid  to verify any non-trivial certificate qi > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' By Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='4, given menu M,  13  each producer ψ will purchase whichever certificate qi maximizes f(qi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' φ(ψ)) −  g(qi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' ψ) − ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  For q ∈ [0, 1], we can interpret ψ as a buyer type and define valuation  function v(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' ψ) = f(qi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' φ(ψ))−g(qi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Then since f and g both satisfy single-  crossing with respect to their corresponding types, valuation function v does  as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Moreover, v is concave and v(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' ψ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' By definition, the producers’  choices of certificates from menu M corresponds precisely to the buyer’s choice  of quantity when facing the same menu, interpreting each certificate quality  threshold as a quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Thus the outcomes, and hence revenue, in the two  settings are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Note that an immediate implication of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='1, given Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='4,  is that for any menu M chosen by the seller, the choice of quantity purchased  by the buyer is weakly increasing in buyer type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='3  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='2  Optimizing Revenue in the Non-linear Pricing Prob-  lem  By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='1, to solve the certifier’s revenue maximization problem it  suffices to optimize revenue in the non-linear pricing problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' As a first step, we  note that in the special case where the valuations v are linear in quantity (which  happens, for example, if the cost g and value f functions in our certification  problem are both linear in q), this problem reduces to a standard pricing problem  in mechanism design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' A characterization due to Myerson Myerson (1981) then  immediately establishes that it is revenue-optimal to choose a menu with only  a single non-trivial item (q, p) with q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' If v(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' θ) is linear in q for all θ, then there is a revenue-  optimal menu that offers only quantity q = 1 at some price p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' This price p will  be chosen to maximize p × Prθ[v(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' θ) > p].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  However, in general, non-linearity substantially changes the problem relative  to the linear case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' In particular, it is not necessarily optimal to offer a single  menu item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' There are problem instances in which posting any single  menu item is an arbitrarily poor approximation to the optimal revenue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' The ap-  proximation factor can be as large as Ω(log(H)), where H = maxθ1,θ2  maxq v(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='θ1)  maxq v(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='θ2)  is the ratio between the highest and lowest maximum values across buyer types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Choose c = 0 and consider the valuation function v(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' θ) = q if q ≤ θ,  and v(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' θ) = 2θ − q if q ≥ θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' That is, v is piecewise linear for each θ, with  maximum value θ occurring at q = θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Fix some H ≥ 1 and suppose the type  distribution is such that Pr[θ > h] = 1/h for all h ∈ [1, H].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' That is, the type  distribution is equal-revenue on range [1, H].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  3Alternatively, this is a direct consequence of the single-crossing condition on valuation  functions v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  14  This valuation function is concave and satisfies v(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' θ) = 0 for all θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' More-  over, it satisfies the single-crossing condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Indeed, for any q and any θ1 < θ2,  we note that  d  dqv(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' θ1) ≤  d  dqv(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' θ2), since for any θ this derivative is 1 for q < θ  and −1 for q > θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Since v is also continuous in both q and θ, the single-crossing  condition is implied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='4 Finally, we note that this valuation function v can in-  deed arise in our reduction from the certification problem with producers and  consumers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='5  We now show the desired gap in approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Consider any menu M with  a single non-trivial menu item (q, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Then the revenue achieved by the seller  is at most the welfare generated by the optimal allocation of quality level q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Since v(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' θ) ≤ q for all θ, this is certainly at most q times the probability that  v(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' θ) > 0, which is q Pr[θ > q/2] ≤ q(2/q) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  On the other hand, the seller could offer a menu that includes every quality  level q ∈ [1, H] at a price of q/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  A buyer of type θ would then choose to  purchase quality level q = θ for a utility of θ/2, generating revenue θ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='6 The  total revenue is then E[θ/2] = O(log H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Posting a single menu item is therefore at best an O(log H) approximation  to the optimal revenue, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  We therefore know that any approximately revenue-optimal mechanism must  sometimes have multiple non-trivial offerings on its menu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' What should such a  menu look like?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Note that since buyers do not have free disposal in our setting  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=', valuations are non-monotone), it is not even immediately obvious that  higher quantities should be sold for higher prices in a revenue-optimal menu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  We next establish that, in fact, there is always a revenue-optimal choice of menu  for which higher quantities are sold at higher prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' We say a menu M = {(qi, pi)} is monotone if for all (qi, pi),  (qj, pj) ∈ M such that qi < qj, we have pi ≤ pj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' For any instance of the non-linear pricing problem there is a  revenue-optimal menu that is monotone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Let M be a revenue-optimal menu, and write M = {(qi, pi)}i∈Λ where  Λ is some (possibly uncountable) index set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' It is without loss to assume Λ is a  subset of [0, 1] such that qi < qj for all i < j (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=', by reindexing so that the  index of qi is equal to qi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' We can further assume without loss of generality that  every item in M is purchased by some buyer type, as any element that is never  purchased could be removed from M without impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' By Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='4, quality  4Technically our construction only satisfies weak single-crossing since the inequality in  derivatives is not strict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' We can make the example strict by perturbing the slope of the initial  line segment by an infinitesimal amount so that it depends on the type θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' We omit these  details for expositional clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  5In particular, take φ and ψ to be supported on [1, H], define f(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' φ) = q for all φ and  g(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' ψ) = max{0, 2(q − ψ)} for all ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Then f is concave (in fact, linear), g is convex, the  single-crossing conditions are satisfied, and v(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' θ) = f(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' φ(θ)) − g(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' θ) as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  6Buying a higher quality level q′ > θ is worse for the buyer because it generates less value at  a higher price, whereas buying any quality level q′ < θ generates utility q′−q′/2 = q′/2 < θ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  15  levels purchased will be monotone non-decreasing in buyer type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' This means  that every item (qi, pi) is purchased by some contiguous interval of buyer types  Ii (which may have measure zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Suppose that menu M is not monotone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' This means that either there is an  element i < sup Λ such that pi > pj for all j > i, or else there exists a pair of  menu items (qi, pi) and (qj, pj) with j > i such that (a) there exists some ℓ ∈ Λ  with i < ℓ < j, and (b) pℓ < min{pi, pj} for all i < ℓ < j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Consider the former case, where there is an element i < sup Λ such that  pi > pj for all j > i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' In particular there must exist some j ∈ Λ with j > i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Let  M ′ be the menu {(qℓ, pℓ)}ℓ∈Λ,ℓ≤i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=', M ′ is M with all elements with quantities  greater than qi removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Note that for all j ≤ i, since the types Ij preferred  element (qj, pj) to any other element in M, they prefer element (qj, pj) to any  other element in M ′ as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Moreover, since purchase decisions are monotone in  buyer type, we conclude that all types θ ∈ Iℓ with ℓ > i will purchase element  (qi, pi) from menu M ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' But since pi > pℓ for all ℓ > i, this means that the  revenue generated by menu M ′ is strictly greater than the revenue generated  by menu M, contradicting the supposed optimality of menu M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Next consider the other case, there exists a pair of menu items (qi, pi) and  (qj, pj) such that pℓ < min{pi, pj} for all i < ℓ < j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Let M ′ be the menu  {(qℓ, pℓ)}ℓ∈Λ,ℓ≤i ∪ {(qℓ, pℓ)}ℓ∈Λ,ℓ≥j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' That is, M ′ is menu M with all elements  strictly between (qi, pi) and (qj, pj) removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Then as in the previous case, for  all ℓ ≤ i and ℓ ≥ j, types Iℓ all still prefer to purchase (qℓ, pℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' In particular,  types Ii purchase (qi, pi) and types Ij purchase (qj, pj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' By monotonicity of  purchasing decisions due to Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='4, all intermediate types θ ∈ Iℓ for  i < ℓ < j must purchase either (qi, pi) or (qj, pj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' As pi and pj are both larger  than the prices of the elements those types were purchasing under menu M, the  revenue of menu M ′ must be strictly larger, which is again a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  We conclude that the prices in menu M must be monotone non-decreasing  in quality levels, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='3  An FPTAS for Revenue  We are now ready to consider the problem of constructing an approximately  revenue-optimal menu for the non-linear pricing problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' For this we will make  one further technical assumption, which is that the derivative of the valuations  v(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' θ) with respect to q is bounded at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=', there exists some λ > 0 such  that  d  dqv(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' θ) < λ for all θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' In other words, buyers are not infinitely sensitive  to product quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' In the context of certification, this is implied by having  d  dqf(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' φ) < λ for all φ, meaning that consumers are not infinitely sensitive to  quality at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' There exists some λ > 0 such that  d  dqv(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' θ) < λ for all θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Given this assumption, the following result provides an FPTAS for the op-  timal menu with k items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' We also show that by taking k = 1/ϵ one can obtain  an FPTAS for the optimal (unrestricted) menu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  16  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' A menu with revenue at least OPT −λϵ can be found in time  polynomial in λ and 1/ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' The menu can optionally be constrained to contain at  most k quality levels, in which case OPT is the optimal revenue achievable with  at most k quality levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Our first step to proving Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='7 is to show that we can restrict attention  to menus with at most 1/ϵ entries at only a small loss of revenue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' For any monotone menu M, there exists a menu M ′ of size at  most O(1/ϵ) such that Rev(M ′) ≥ Rev(M) − O(ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Fix the revenue-optimal menu M = {(qi, pi)}i∈Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='5 we can  assume M is monotone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  We define M ′ to be the following subset of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Let A = {ℓϵ : 0 ≤ ℓ ≤ ⌊1/ϵ⌋}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Then for each a ∈ A, we add to M ′ the element (q, p) from M with smallest p  such that p ≥ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Then we note that M ′ contains at most ⌈1/ϵ⌉ items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Moreover,  for each element (q, p) ∈ M there is some (q′, p′) ∈ M ′ such that p′ ≥ p − ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Any element in M ′ will still be purchased by the types that purchased that  element in M, as the menu of alternative options has only gotten smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Since  prices are monotone in quantity, and quantities purchased are monotone in buyer  type, we can conclude that any type that was purchasing (q, p) in the original  menu M will then purchase an element with price at least p − ϵ in menu M ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  The total loss in revenue conditional on any given buyer type is therefore at  most ϵ, and hence the total revenue loss is at most ϵ as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Next, we show that we can discretize the possible (quality, price) pairs that  appear in our menu without losing too much revenue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' For any monotone menu M of size k, there exists a menu M ′  such that  M ′ has at most k elements,  for each (q, p) ∈ M ′, p is a multiple of ϵ and q = ϵ(1+ϵ)ℓ for some integer  ℓ ≥ 0, and  Rev(M ′) ≥ Rev(M) − O((k + λ)ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Let us first give the high-level idea for the construction, which is illus-  trated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' We will round each (quantity, price) pair on menu M in  three steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' First, we will round each quantity down to an appropriate dis-  cretized grid, and then also lower the corresponding price to keep constant the  ratio of quantity to price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' The concavity of the value functions will then imply  that buyer utilities cannot be reduced by too much, multiplicatively, as a result  of this change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' This first step discretized the quantities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' in the second step we  discretize prices by rounding each price down to an appropriate grid, which can  only increase utilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' At this point we would be almost done, except for one  complication: we must make sure that the (small) changes in buyer utility we in-  duce do not result in buyers switching from more expensive items to significantly  17  Figure 1: Illustration of the discretization procedure from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Red  curves denote buyer valuation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Menu item (p, q) is discretized in  quantities to (˜p, q′) by shifting along a line to the origin, then discretized in  price to (ˆp, q′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' A final discount is applied to obtain the adjusted menu item  (p′, q′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  cheaper items from the menu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' It is here where we use the price-monotonicity of  the menu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Since the higher-quanitity items are the more expensive ones, in our  third step we will provide discounts for the higher-quantity menu items, which  will offset any utility perturbations due to discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' These discounts are  what lead to the loss term being proportional to kϵ, rather than ϵ, in our error  bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  We now move on to the formal construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Fix menu M of size k, so that  M = {(q1, p1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' , (qk, pk)} where q1 < q2 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' < qk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' We can assume without  loss of generality that p1 ≤ p2 ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' ≤ pk, and that each element of M is  purchased with positive probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  We construct a new menu M ′ in the following sequence of steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' First, for  each (qi, pi) with qi ≥ ϵ, let q′  i be qi rounded down to the nearest value of ϵ(1+ϵ)ℓ  where ℓ ≥ 0 is an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Then define ˜pi = pi × (q′  i/qi), noting that we chose ˜pi  so that ˜pi/q′  i = pi/qi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' This element (q′  i, ˜pi) corresponds to rounding quantity but  keeping the price to quantity ratio constant in our intuitive description above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  For our second step, we take ˆpi to be ˜pi rounded down to the nearest multiple  of ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Finally, in our third step, we introduce our discounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' For this we define  p′  i = ˆpi − 3iϵ, noting that the difference between p′  i and ˆpi is increasing in i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  This completes our discretization procedure, so we add (q′  i, p′  i) to menu M ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' We  note that menu M ′ is not necessarily monotone, may contain elements that are  preferred by no types, and may contain elements with negative prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  We claim that for every type θ, if θ purchased item (qi, pi) in menu M with  qi ≥ ϵ, then θ will purchase some (q′  j, p′  j) in menu M ′ such that j ≥ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' To see  why, first consider what would happen if (q′  i, ˜pi) were on the menu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' What is  u((q′  i, ˜pi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' θ)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Since valuation function v is concave and v(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=" θ) = 0, we must  18  price  3E  2e:  (p,q)  (p,q')." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content="  (p,q')." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content="  E  quantity  E E(1 +e) e(1+)2  (3 + L)3  2E-  (p',q')." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='have v(q′  i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' θ) ≥ q′  i  qi v(qi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' But since q′  i ≥  1  1+ϵqi, this implies  u((q′  i, ˜pi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' θ) = v(q′  i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' θ) − ˜pi  ≥ q′  i  qi  v(qi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' θ) − ˜pi  = q′  i  qi  (v(qi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' θ) − pi)  ≥  1  1 + ϵu((qi, pi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' θ)  ≥ u((qi, pi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' θ) − ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Since we also know that ˜pi − ϵ ≤ ˆpi ≤ ˜pi and p′  i = ˆpi − 3iϵ, we have  u((q′  i, p′  i);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' θ) = u((q′  i, ˆpi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' θ) + 3iϵ  ≥ u((q′  i, ˜pi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' θ) + 3iϵ  ≥ u((qi, pi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' θ) + (3i − 1)ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  On the other hand, for any j < i, the utility of purchasing (q′  j, ˜pj) is at most  the utility of purchasing (qj, ˜pj), which is at most ϵ more than the utility of  purchasing (qj, pj) (since the ratio between pj and ˜pj is no greater than (1+ϵ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  We therefore have  u((q′  j, p′  j);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' θ) = u((q′  j, ˆpj);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' θ) + 3jϵ  ≤ u((q′  j, ˜pj);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' θ) + (3j + 1)ϵ  ≤ u((qi, pi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' θ) + (3j + 2)ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Since we know that u((qi, pi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' θ) ≥ u((qj, pj);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' θ) by assumption that θ purchases  item (qi, pi) from menu M, we conclude that u((q′  i, p′  i);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' θ) ≥ u((q′  j, p′  j);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' θ) as well,  since (3j + 2) ≤ (3i − 1) for i > j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  We conclude that each type θ that purchases (qi, pi) from M with qi ≥ ϵ  will purchase a menu item (q′  j, p′  j) from M ′ such that j ≥ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Since prices are  monotone in menu M, we conclude that the total loss in revenue can be at most  the difference in price between pi and p′  i for any i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' This is at most O(kϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Finally, consider a type θ that purchases (qi, pi) from M with qi < ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' By  Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='6, the maximum willingness to pay for any agent for quality level  ϵ is λϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  These types therefore generate revenue at most λϵ, thus regardless  of their purchase behavior they account for a total loss in revenue of at most  O(λϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  With Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='9 in hand, we can complete the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='7 by  employing dynamic programming to determine the revenue-optimal mechanism  with a given maximum-quality entry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' One subtlety in the construction is that  we must be careful to account for potential cannibalization by higher-quality  elements in the menu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' We handle this by insisting that the menu we construct  contains only elements that are selected by a non-zero measure of buyer types,  and we check this condition when recursively applying the dynamic program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  19  Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' We show how to compute the optimal menu with qual-  ities and prices chosen from a discrete indexed set of possible options, using  dynamic programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' In general, given a quantity q and price p that lie in  our discrete set of options, we will use Q and P to denote the integer indexing  of q and p, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Given any choice of Q and P and some k ≥ 1, write M[Q, P, k] for the  optimal revenue that can be obtained using a menu with at most k elements,  of which the one with highest quality is the one indexed by Q and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' We will  also write L[Q, P, k] for the lowest type θ that purchases quality level Q in this  optimal menu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' We can compute M[Q, P, k] and L[Q, P, k] recursively as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  If k = 1 then M[Q, P, k] is precisely p times the probability that v(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' θ) ≥ p,  and L[Q, P, k] is precisely the infimum of types θ for which v(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' θ) ≥ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  For k > 1, we will consider all possible options for the next-highest quality  level on the menu given our discretization, say (q′, p′) with Q′ < Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' For each  choice of (Q′, P ′), we let θ(Q′, P ′) denote the type that is indifferent between  menu items (Q′, P ′) and (Q, P), if any.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Recall from the single-crossing condition  that this choice of θ(Q′, P ′) is unique if it exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' If there is no such θ(Q′, P ′),  then we disqualify menu item (Q′, P ′) from consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Otherwise, we con-  sider L[Q′, P ′, k−1], the lowest type that purchases element (Q′, P ′) in the opti-  mal menu with highest quality level Q′ at price P ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' If L[Q′, P ′, k−1] ≥ θ(Q′, P ′),  then again we disqualify menu item (Q′, P ′) from consideration, as this means  that the optimal menu containing menu item (Q, P) does not include any types  that would purchase menu item (Q′, P ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Otherwise, we have that L[Q′, P ′, k−1] < θ(Q′, P ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' We can therefore calcu-  late the revenue from the optimal menu with highest and second-highest quality  levels (Q, P) and (Q′, P ′) as R(Q′, P ′) = M[Q′, P ′, k − 1] + (P − P ′)Pr[θ >  θ(Q′, P ′)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' That is, the additional revenue gain or loss due to including menu  item (Q, P) is (P − P ′)Pr[θ > θ(Q′, P ′)], the difference due to agents with type  greater than θ(Q′, P ′) switching from menu item (Q′, P ′) to menu item (Q, P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Finally, consider also the revenue that would be obtained by using only  menu item (Q, P);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' call this R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' If all potential choices of (Q′, P ′) were elim-  inated or if R > R(Q′, P ′) for all potential choices of (Q′, P ′), then we set  M[Q, P, k] = R and set L[Q, P, k] to be the infimum type θ such that v(Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' θ) ≥  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' This corresponds to the case that the optimal menu contains only the el-  ement (P, Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Otherwise, let (Q′, P ′) be the choice that maximizes R(Q′, P ′),  which by assumption is larger than R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Then we take L[Q, P, k] = θ(Q′, P ′) and  M[Q, P, k] = R(Q′, P ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  We conclude that we can fill tables M and L, with each entry taking time  proportional to ϵ−2 (the time needed to consider every possible choice (Q′, P ′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  As there are kϵ−2 entries in total, the total time to fill the tables is at most  kϵ−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' The revenue-optimal mechanism with at most k menu items can then be  obtained by taking the maximum of M[Q, P, n] over all choices of Q and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Finally, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='8, we can take k = 1/√ϵ and our dynamic program will  obtain a menu M such that Rev(M) is at most O(λ/√ϵ) less than the optimal  revenue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' An appropriate change of variables, taking k = 1/ϵ and discretizing to  multiples of ϵ2, then implies that our resulting menu is at most O(λϵ) less than  20  that of the optimal menu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  5  Welfare Maximization  We have studied the problem faced by a revenue-maximizing certifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' But what  about a certifier that wishes to maximize the welfare enjoyed by the consumers  and producers?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' We could think of such a certifier as a government agency who  is offering certification services not to generate profit, but rather to maximize  the efficiency of production and trade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  We define the welfare of a market outcome as the sum of utilities of the  consumers, the producers, and the certifier, taking into account all transfers  between parties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Given a menu M of options provided by the certifier, we will  write Wel(M) for the welfare that results in the unique market outcome resulting  from menu M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  An immediate consequence of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='4 is that the welfare-optimal choice  of menu is to offer all possible certification levels, at the cost of verification c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' The welfare-optimal menu of certificates offers every possible  certification level q > 0 at a cost of c, and level 0 at a cost of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' If all quality levels were visible, the welfare-maximizing outcome is would  be for each producer ψ to trade with consumer φ(ψ) at whichever quality level  q maximizes their gains from trade f(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' φ(ψ)) − g(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' However, since quality  levels are hidden, producers and consumers can trade at a positive level of  quality only if the cost of verification is paid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' So if the maximum gains from  trade is less than c, then it is preferable to trade at level 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' However, we observe  that this is precisely the outcome implemented at equilibrium from the proposed  certification menu, so it must be welfare-optimal over all possible menus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  The welfare-optimal menu described above includes an arbitrarily large num-  ber of certificates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' In practice it may be helpful to find a welfare-optimal slate of  at most k certificates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' It turns out that a minor variation on the dynamic pro-  gram described in the previous section can be used to compute an approximately  welfare-optimizing slate, under Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Let M be the welfare-maximizing menu with at most k elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  A menu with M ′ with at most k elements and such that Wel(M ′) ≥ Wel(M) −  O(λϵ) can be found in time polynomial in 1/ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' The proof is very similar to the one for Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='7, and strictly simpler,  so we only briefly describe the differences here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' First, it is without loss to restrict  attention to menus that post price c for every non-trivial level of quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Given any such menu M, we can discretize potential levels of quality by  rounding down to the nearest multiple of ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' By Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='6 this reduces  welfare by at most λϵ, as the welfare from each menu item is reduced by at most  this much and each producer selects her gains-from-trade-maximizing element  from the menu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  21  Given such a discretization, one can express the welfare-optimal menu re-  cursively via dynamic programming as in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='7, with the simplification  that we need only index on quality rather than (quality, price) pairs (since all  prices will be set to c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Rather than defining M[Q, k] (and respectively L[Q, k])  to be the maximum revenue of a menu with k elements and maximum quality  indexed by Q, it will be the maximum welfare of a menu with k elements and  maximum quality indexed by Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Our method of recursively computing M[Q, k]  (and L[Q, k]) then remains nearly unchanged relative to Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  The  only change to note is the actual welfare calculation, relative to the revenue  calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' In Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='7 we used that the revenue obtained when agents of  type θ > θ′ purchase certificate q at price p is p times Pr[θ > θ′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' In contrast,  the welfare obtained when producers of type ψ > ψ′ all purchase certificate q  at price c can be calculated in closed form as  � ψ′  ψ (f(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' φ(η)) − g(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' η) − c)dη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Substituting this welfare calculation for the revenue calculation completes the  necessary changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Is the revenue-maximizing choice of menu also approximately welfare-maximizing?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  As it turns out, the welfare that results from the certifier’s reveneu-optimal menu  can be an arbitrarily small fraction of optimal welfare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' This is inherited from  standard examples of monopolistic distortion in the linear settings, where a mo-  nopolist might be incentivized to sell much less of their good (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=', certification)  than what would be efficient in order to inflate prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' There is a sequence of instances for the problem where welfare  in the revenue-optimal solution is an arbitrarily small fraction of first-best wel-  fare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' The approximation can be as bad as Ω(log H), where H = maxθ1,θ2  maxq v(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='θ1)  maxq v(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='θ2)  is the ratio between the highest and lowest maximum values across buyer types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Take c = 0 and suppose f(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' φ) = φq and g(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' ψ) = 0 for all q and ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Our  distribution over consumer types φ is an equal-revenue distribution supported  on [1, H];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' that is, F(φ) = 1 − 1  φ for all φ ∈ [1, H].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Recalling Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='2, it  is revenue-optimal for the certifier to offer contract q = 1 at a price of H, for  an expected welfare (and revenue) of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' However, the optimal welfare, log(H),  can be achieved by offering contract q = 1 at price 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  However, one implication of our equilibrium analysis is that adding addi-  tional certification options to a menu of certificates cannot reduce the sum of  utilities of the consumers and producers, regardless of the prices selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' One  interpretation of this is that welfare cannot be harmed by a revenue-maximizing  certifier entering a market for certification in which some certification options  are already available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Consider two certification menus M and M ′ with M ⊆ M ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Then Wel(M ′) − Rev(M ′) ≥ Wel(M) − Rev(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' If producer ψ selects option (q, p) from certification menu M, then the  welfare generated for the producer ψ and corresponding consumer φ, less the  revenue raised by the certifier, is f(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' φ(ψ)) − g(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' ψ) − p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' By Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='4,  22  producer ψ purchases precisely whichever menu item from M maximizes this  quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Providing additional items can therefore only increase the welfare  jointly enjoyed by producer type ψ and corresponding consumer φ(ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' As this  holds pointwise for every ψ, it holds in aggregate over all types as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  References  Viral V Acharya, Peter DeMarzo, and Ilan Kremer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Endogenous informa-  tion flows and the clustering of announcements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' American Economic Review  101, 7 (2011), 2955–79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  S Nageeb Ali, Nima Haghpanah, Xiao Lin, and Ron Siegel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' How to sell  hard information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' The Quarterly Journal of Economics 137, 1 (2022), 619–  678.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Reza Alijani, Siddhartha Banerjee, Kamesh Munagala, and Kangning Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' The limits of an information intermediary in auction design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' In Proceed-  ings of the 23rd ACM Conference on Economics and Computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' 849–868.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Dirk Bergemann, Alessandro Bonatti, and Alex Smolin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' The design and  price of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' American economic review 108, 1 (2018), 1–48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Dirk Bergemann, Benjamin Brooks, and Stephen Morris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' First-price auc-  tions with general information structures: Implications for bidding and rev-  enue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Econometrica 85, 1 (2017), 107–143.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Dirk Bergemann, Yang Cai, Grigoris Velegkas, and Mingfei Zhao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Is  Selling Complete Information (Approximately) Optimal?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='. In Proceedings of  the 23rd ACM Conference on Economics and Computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' 608–663.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Dirk Bergemann and Stephen Morris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Information design: A unified  perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Journal of Economic Literature 57, 1 (2019), 44–95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Dirk Bergemann, Ji Shen, Yun Xu, and Edmund Yeh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' 2012a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Multi-dimensional  mechanism design with limited information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' In Proceedings of the 13th ACM  Conference on Electronic Commerce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' 162–178.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Dirk Bergemann, Ji Shen, Yun Xu, and Edmund M Yeh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' 2012b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Mechanism  design with limited information: the case of nonlinear pricing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' In International  Conference on Game Theory for Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Springer, 1–10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Tilman B¨orgers and Daniel Krahmer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' An introduction to the theory of  mechanism design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Oxford University Press, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Ozan Candogan and Philipp Strack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Optimal Disclosure of Information  to a Privately Informed Receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' In EC ’21: The 22nd ACM Conference on  Economics and Computation, Budapest, Hungary, July 18-23, 2021, P´eter  Bir´o, Shuchi Chawla, and Federico Echenique (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' ACM, 263.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  23  Shuchi Chawla, Hu Fu, and Anna Karlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Approximate revenue maxi-  mization in interdependent value settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' In Proceedings of the fifteenth ACM  conference on Economics and computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' 277–294.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Joe  Sandler  Clarke  and  Luke  Barratt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  [n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Top  airlines’  promises  to  offset  flights  rely  on  ‘phantom  credits’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Unearthed  ([n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=']).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  https://unearthed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='greenpeace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='org/2021/05/04/  carbon-offsetting-british-airways-easyjet-verra/  Marc N Conte and Matthew J Kotchen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Explaining the price of voluntary  carbon offsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Climate Change Economics 1, 02 (2010), 93–111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Peter M DeMarzo, Ilan Kremer, and Andrzej Skrzypacz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Test design and  minimum standards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' American Economic Review 109, 6 (2019), 2173–2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Mathias Dewatripont, Patrick Bolton, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Contract theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Technical  Report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' ULB–Universite Libre de Bruxelles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Piotr Dworczak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Mechanism design with aftermarkets: Cutoff mecha-  nisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Econometrica 88, 6 (2020), 2629–2661.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Daniel W Elfenbein, Raymond Fisman, and Brian McManus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Market  structure, reputation, and the value of quality certification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' American Eco-  nomic Journal: Microeconomics 7, 4 (2015), 83–108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Patrick  Greenfield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  [n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=']a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Carbon  offsets  used  by  major  air-  lines  based  on  flawed  system,  warn  experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  The  Guardian  ([n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=']).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='theguardian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='com/environment/2021/may/04/  carbon-offsets-used-by-major-airlines-based-on-flawed-system-warn-experts  Patrick Greenfield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' [n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=']b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Revealed: more than 90% of rainforest carbon  offsets by biggest provider are worthless, analysis shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  The Guardian  ([n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=']).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='theguardian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='com/environment/2023/jan/18/  revealed-forest-carbon-offsets-biggest-provider-worthless-verra-aoe  Sanford J Grossman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' 1981.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' The informational role of warranties and private  disclosure about product quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' The Journal of Law and Economics 24, 3  (1981), 461–483.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Alejandro Guizar-Couti˜no, Julia PG Jones, Andrew Balmford, Rachel Car-  menta, and David A Coomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' A global evaluation of the effectiveness  of voluntary REDD+ projects at reducing deforestation and degradation in  the moist tropics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Conservation Biology (2022), e13970.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Faruk Gul and Ennio Stacchetti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' The English auction with differentiated  commodities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Journal of Economic theory 92, 1 (2000), 66–95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Alessandro Lizzeri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Information revelation and certification intermediaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  The RAND Journal of Economics (1999), 214–231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  24  Michael Luca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Reviews, reputation, and revenue: The case of Yelp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Com (March 15, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Harvard Business School NOM Unit Working Paper  12-016 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Paul R Milgrom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' 1981.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Good news and bad news: Representation theorems and  applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' The Bell Journal of Economics (1981), 380–391.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Paul R Milgrom and Robert J Weber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' 1982.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' A theory of auctions and com-  petitive bidding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Econometrica: Journal of the Econometric Society (1982),  1089–1122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  James A Mirrlees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' 1971.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  An exploration in the theory of optimum income  taxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' The review of economic studies 38, 2 (1971), 175–208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Michael Mussa and Sherwin Rosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' 1978.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Monopoly and product quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Jour-  nal of Economic theory 18, 2 (1978), 301–317.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Roger B Myerson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' 1981.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Optimal auction design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Mathematics of operations  research 6, 1 (1981), 58–73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Subramaniam Ramanarayanan and Jason Snyder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Reputations and firm  performance: Evidence from the dialysis industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Unpublished manuscript  (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Tim Roughgarden and Inbal Talgam-Cohen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Optimal and near-optimal  mechanism design with interdependent values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' In Proceedings of the fourteenth  ACM conference on Electronic commerce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' 767–784.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Alejandro Saporiti and Fernando Tohm´e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Single-crossing, strategic voting  and the median choice rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Social Choice and Welfare 26, 2 (2006), 363–383.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Michael Spence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' 1974.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Competitive and optimal responses to signals: An anal-  ysis of efficiency and distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Journal of Economic theory 7, 3 (1974),  296–332.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Steven Tadelis and Florian Zettelmeyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Information disclosure as a  matching mechanism: Theory and evidence from a field experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Ameri-  can Economic Review 105, 2 (2015), 886–905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Thales AP West, Jan B¨orner, Erin O Sills, and Andreas Kontoleon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Over-  stated carbon emission reductions from voluntary REDD+ projects in the  Brazilian Amazon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Proceedings of the National Academy of Sciences 117, 39  (2020), 24188–24194.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Thales AP West, Sven Wunder, Erin O Sills, Jan B¨orner, Sami W Rifai, Alexan-  dra N Neidermeier, and Andreas Kontoleon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Action needed to make  carbon offsets from tropical forest conservation work for climate change mit-  igation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' arXiv preprint arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='03354 (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  Bradley S Wimmer and Brian Chezum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  An empirical examination of  quality certification in a “lemons market”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content=' Economic inquiry 41, 2 (2003),  279–291.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
+page_content='  25' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf'}
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+Classical Algorithm for the Mean Value problem over
+Short-Time Hamiltonian Evolutions
+Reyhaneh Aghaei Saem1 and Ali Hamed Moosavian1
+1Phanous
+Abstract
+Simulating physical systems has been an important application of classical and quantum computers. In
+this article we present an efficient classical algorithm for simulating time-dependent quantum mechanical
+Hamiltonians over constant periods of time. The algorithm presented here computes the mean value of an
+observable over the output state of such short-time Hamiltonian evolutions. In proving the performance of
+this algorithm we use Lieb-Robinson type bounds to limit the evolution of local operators within a lightcone.
+This allows us to divide the task of simulating a large quantum system into smaller systems that can be
+handled on normal classical computers.
+1
+Introduction
+Understanding physical systems with quantum mechanical interactions have been an increasingly challenging
+and important task in the past century. Because nature is inherently quantum, many naturally occurring or
+artificially implemented phenomena cannot be explained without quantum mechanics. Some early examples
+include understanding atoms [1], diatomic molecules [2], the Meissner effect in superconductors [3] and
+black-body radiation [4]. With the advent of better theoretical and computational tools, the list of phenomena
+that have been shown to require quantum mechanics has grown enormously in the past few decades. To name
+a few, some of the more notable examples include photosynthesis [5], topological ordered phases such as
+fractional quantum Hall effect [6], isomerization of diazene [7] and non-Abelian lattice gauge theories [8].
+Indeed, because of the challenging nature of simulating quantum systems on classical computers, one of the
+earliest motivations for quantum computers has been to use them to study other quantum systems [9].
+Besides the obvious practical applications, the problem of simulating quantum systems has some critical
+complexity theory value too. On the one hand, it is known that the problem of simulating several quantum
+systems belongs to the Bounded-error Quantum Polynomial-time complete (BQP-complete) class [10, 11,
+12]. Also, there are specific problems where the complexity class of simulating them varies as the runtime
+increases, and exhibit a dynamical phase transition [13, 14].
+In this paper, we present a classical algorithm for computing the expectation value of an observable that
+can be written as a tensor product of local operators acting on each qubit. In the literature this problem is
+sometimes called the quantum mean value problem [15] (not to be confused with a similarly named open
+problem in mathematics [16]). Our algorithm evaluates the mean value of an operator where the state is
+generated by evolving a product state under a geometrically local time-dependent Hamiltonian for a short
+period of time. The setup of our problem is motivated by the state of current quantum technologies. On one
+1
+arXiv:2301.11420v1  [quant-ph]  26 Jan 2023
+
+hand, analog simulators lack fault-tolerance, which means they have a limited decoherence time, and on the
+other hand, we still do not have access to fault-tolerant universal quantum computers either.
+Suppose that we have a bounded-norm time-dependent Hamiltonian H(t) that acts on n-qubits. The
+initial state of the system is assumed to be a product state, typically |ψ(0)⟩ = |0⟩⊗n. The quantum mean value
+problem wants to compute the expectation value of an operator O with respect to |ψ(T)⟩. The expectation
+value is represented with µ:
+µ ≡ ⟨ψ(T)|O|ψ(T)⟩ .
+(1)
+We consider geometrically local time-dependent Hamiltonians that are defined on 2D or 3D lattices. 1D
+systems with gapped Hamiltonians have been thoroughly studied before [17, 18].
+The unitary evolution operator corresponding to this Hamiltonian can be written as:
+U(t) = T :
+�
+exp
+�
+− i
+¯h
+� t
+0
+H(t′)dt′
+��
+,
+(2)
+where T : [.] means the operators respect time ordering. Assuming the observable can be written as the tensor
+product of single-qubit Hermitian operators Oj acting on individual sites, O = O1 ⊗ O2 ⊗ · · · ⊗ On, we
+can write the mean value of the observable O as:
+µ = ⟨0n| U†(T)O1 ⊗ O2 ⊗ · · · ⊗ OnU(T) |0n⟩ .
+(3)
+In this paper, we introduce a classical algorithm that can approximate the mean value problem for time-
+dependent Hamiltonians with an additive error. The structure of the rest of this paper is as follows. In Sec. 2
+we give an overview of the algorithm. Section 3 explains how the dynamics of each operator is approximately
+restricted to within a lightcone. In Sec. 4 we analyze the problem of classically calculating the local unitary
+operators and compare different numerical algorithms for doing the task and in Sec. 5 we conclude.
+2
+Algorithm
+We provide a classical algorithm for approximating µ within an additive error, δ, for the special case of the
+mean value problem where the time-dependent Hamiltonian is geometrically local in 2D or 3D.
+Conceptually, the algorithm can be broken into two parts. First we use a Lieb-Robinson bound [19, 20,
+21] to limit the unitary evolution corresponding to each qubit within a lightcone. This allows us to classically
+compute the unitary operator. Then we follow the steps in [15] to divide the lattice into pseudo 1D slices, that
+can be efficiently simulated either using Matrix Product State algorithms [22, 23, 24] (for a nice review of
+these algorithms see [18]) or the algorithm ascribed to [15].
+Theorem 1 Let H(t) be a bounded-norm time-dependent lattice Hamiltonian that acts on n-qubits. Suppose
+the observable O is a tensor product of operators , O = O1 ⊗ O2 ⊗ · · · ⊗ On. For constant evolution times,
+there exists a classical algorithm that estimates µ within an additive error δ,
+|˜µ − µ| ≤ δ .
+(4)
+The error δ includes three different parts. The first contribution comes from the Lieb-Robinson bound where
+we used it to approximately limit the evolution within the lightcone of each qubit. Second, numerical methods
+such as trotterization are used to calculate the local unitary evolutions; these methods are not exact and incur
+some errors. The third part comes from the additive error in simulating a constant depth quantum circuit [15].
+In [15], they provide a classical algorithm for constant depth circuits in 2D or 3D which can approximate the
+2
+
+mean value to an additive error. In our work, the short-time evolution of a geometrically local Hamiltonian
+which is limited by the Lieb Robinson bound is comparable with the shallow quantum circuit with a constant
+depth.
+Suppose that the error of localizing the time evolution of the Hamiltonian, classical simulation of time-
+dependent unitaries and simulating shallow circuits are given by εLR, εCS and εSSC respectively. Then the
+total error is as follows:
+|˜µ − µ| ≤ εLR + εCS + εSSC = δ .
+(5)
+In Eq. (10), we will see how the first error is dictated by the simulation time, the lightcone radius and the
+Hamiltonian. One should note that εLR is bounded by εLR ≤ nε(LR)(L, T), where ε(LR)(L, T) is defined
+later in Eq. (10) as the Lieb Robinson error for a single site. Section 4 derives the dependency of the second
+error term on the simulation parameters.
+Algorithm 1: High level overview of the algorithm
+input :a √n × √n lattice of qubits, a geometrically local time-dependent Hamiltonian H(t), the
+operator O = O1 ⊗ O2 ⊗ · · · ⊗ On where each ∥Oj∥ ≤ 1, an upper bound for error δ, a
+simulation time T.
+output :an approximation for µ = ⟨ψ(T)|O|ψ(T)⟩, where |ψ(0)⟩ = |0⊗n⟩ and
+i¯h d
+dt |ψ(t)⟩ = H(t) |ψ(t)⟩
+1 Initialization:
+2 Use T and δ to calculate the lightcone radius, L, from the Lieb-Robinson bound.
+3 As in Fig. 1 partition the lattice into 4L × √n strips twice, let us call each set {Ai}i and {Bi}i. Also,
+define the sets
+�
+A0
+i
+�
+i and
+�
+B0
+i
+�
+i as the central part of the strips.
+4 Also, group sites into 2L × 2L super-sites to form a coarse-grained lattice.
+5 Calculations:
+6 for Ai ∈ {Ai}i do
+7
+initialize an MPS.
+8
+for Oj in A0
+i do
+9
+Use a classical ODE solver to calculate the local unitary operator Uj corresponding to Oj
+which includes the terms that are in the lightcone of the jth qubit.
+10
+Transform OjUj into a Matrix Product Operator.
+11
+Add the necessary sites from the current super-site to the active memory and apply the MPO
+on it.
+12
+Measure any sites that will no longer be needed.
+13
+Let us call the (not normalized) MPS outcome |ΨAi(T)⟩ .
+14 Repeat the Line 6 loop for Bjs.
+15 return ˜µ =
+��
+j ⟨ΨBj(T)|
+���
+i |ΨAi(T)⟩
+�
+The runtime of classical simulation of the time-dependent unitaries is related to the complexity of matrix
+multiplication. We can find the lightcone radius L from Section 3 and then find the number of qubits m in
+each lightcone which is O
+�
+L2�
+. For most practical cases, the fastest matrix multiplication algorithm is the
+famous Strasson algorithm with asymptotic complexity of O
+�
+(2m)log2 7�
+[25], however, the best known
+asymptotic complexity for matrix multiplication is O
+�
+(2m)2.373�
+[26]. For most physical Hamiltonians
+3
+
+Figure 1: A: It shows the relationship between simulation time and the lightcone radius. B:shows the sites
+inside the lightcone for various lightcone radii. C and D:show the set of strips {Ai}i and {Bi}i as well as
+the
+�
+A0
+i
+�
+i and
+�
+B0
+i
+�
+i sets on a 2D grid. The yellow squares represent the Oi operators that are to be applied
+on either set of strips.
+where we have translational symmetry in the bulk, we only need to calculate the unitary evolution matrix for
+each geometrical configuration of the sites once. This means that the complexity of this part of the algorithm
+would be O
+�
+22.373m�
+. But for generic Hamiltonians the number of configurations could grow linearly with
+the system size and the complexity would be O
+�
+n22.373m�
+.
+For 2D and 3D systems with n qubits, the runtime of simulating a shallow circuit is related to the last error
+term with O
+�
+nL226L2/(εSSC)2�
+and O
+�
+nL326L2n1/3/(εSSC)2�
+respectively [15]. We have provided a
+high-level overview of the 2D algorithm in Algorithm 1. Consequently the general total complexity of the algo-
+rithm for 2D and 3D systems is O
+�
+n22.373m + nL226L2/(εSSC)2�
+and O
+�
+n22.373m + nL326L2n1/3/(εSSC)2�
+respectively.
+4
+
+B.
+1.00000
+0.10000
+lerror
+0.01000
+0.00100
+Simulation time
++0.01
++ 0.02
+0.00010
++0.03
++0.04
+0.00001
+0
+2
+Lightcone radius3
+Local Unitary Operators
+According to [20], we know that a local operator OA which is defined inside a region A, remains local after a
+short-time evolution under a local Hamiltonian. Suppose that the Hamiltonian has the form
+H(t) =
+�
+e
+ue(t)he ,
+(6)
+where he acts non-trivially only on the two vertices of edge e of the graph G. Suppose that ∥ue(t)he∥ ≤ g
+for 0 ≤ t ≤ T and the maximum degree of the graph to be ∆. The Hamiltonian for terms in region A and the
+set of vertices in the L-boundary of it has the form
+HA(t) =
+�
+e⊂A∪∂L(A)
+ue(t)he .
+(7)
+The time evolution operator of this local Hamiltonian acts non-trivially only on the region A and its L-
+boundary. The time evolution operator is given by
+V(t) = T :
+�
+exp
+�
+− i
+¯h
+� t
+0
+HA(t′)dt′
+��
+.
+(8)
+This is known as Dyson series and T : [.] represents time-ordering [27, p. 551]. According to [20] the
+following Lieb-Robinson bound holds:
+||U†(T)OAU(T) − V†(T)OAV(T)|| ≤ ε(LR)(L, T) ,
+(9)
+where ε(LR)(L, T) is defined as:
+ε(LR)(L, T) =
+�
+2
+π |A| ||OA|| exp
+�
+−L(log L − log T − log(4g(δ − 1))) − 1
+2 log L
+�
+.
+(10)
+For each local operator in O = O1 ⊗ O2 ⊗ · · · ⊗ On, we only consider the Hamiltonian terms inside the
+lightcone of it. We replace the global unitary that acts on the entire system with these local unitary operators
+and apply the algorithm in Sec. 2 to approximate the mean value.
+The only remaining problem would be to find a classical algorithm for classically calculating the local
+unitary operators V(t). We analyze different approaches for doing so in Sec. 4.
+4
+Classical Simulation of the time-dependent unitaries
+4.1
+Trotterization
+Theorem 2 Let HA(t) be a bounded-norm time-dependent Hamiltonian and OA an observable inside region
+A. Let V(t) be the unitary evoluion of HA(t). We can approximate the operator V†(T)OAV(T) with
+W†(T, N)OAW(T, N) where W(t, N) is defined as
+W(t, N) =
+N= t
+δt
+�
+j=1
+exp
+�
+− i
+¯hδt HA(jδt)
+�
+,
+(11)
+5
+
+such that
+��W†(T, N)OAW(T, N) − V†(T)OAV(T)
+�� ≤ 6T 2
+N¯h ∥OA∥ ∥H′
+A(t∗)∥ = εCS,
+(12)
+where
+∥H′
+A(t∗)∥ = max
+0≤t≤T ∥H′
+A(t)∥.
+(13)
+This is a well-known textbook result that can be found in the literature. For instance see chapter IX of [28].
+4.2
+Numerical Differential Equation Solvers
+Another approach for approximating the unitary time evolution would be to derive a differential equation
+from Schrodinger’s equation and solve that numerically by using a suitable classical algorithm.
+i¯h d
+dt |ψ(t)⟩ = H(t) |ψ(t)⟩ ,
+i¯h d
+dtU(t) |ψ(0)⟩ = H(t)U(t) |ψ(0)⟩ ,
+i¯h d
+dtU(t) = H(t)U(t) .
+(14)
+If there are m qubits inside the lightcone, the matrix representation of U and H will be 2m ×2m operators
+and Eq. (14) will constitute a set of Ordinary Differential Equations (ODEs). The upside of using a classical
+ODE solver is that they can consistently attain much lower errors than what is possible from Eq. (11) or a
+higher order Suzuki-Trotter solution [30, 31] that is fine-tuned for a time-dependent problem [32, 33]. The
+downside is that they are typically orders of magnitude slower and require more memory than the straight
+forward trotterization as in Eq. (11). Assuming our lightcone is small enough, many numerical ODE solvers
+will be able to handle Eq. (14). See Fig. 2 for a comparison between multiple different ODE solvers.
+5
+Conclusions
+To conclude we have provided a classical algorithm for the mean value problem on outcomes of short time
+dependent Hamiltonian evolutions. These mean values are typically used in other algorithms such as the
+Variational Quantum Algorithm [34].
+The quantum mean value problem, almost by definition, belongs to the BQP-complete class for polynomial
+times. So, naturally we do not expect to be able to solve the polynomial time problem on a classical computer
+efficiently. Nonetheless, an important open question would be to find the minimum simulation time for getting
+a quantum speedup. The current work shows us that in order to benefit from the quantum speedup we need
+simulation times that are at least greater than constant [35]. This is in accordance with a wide variety of other
+results that target specific problems too [15, 36, 13, 37, 38, 20]. In general the problem of mapping out the
+entire dynamical complexity phase diagram is theoretically interesting.
+Another direction for future research would be to improve or generalize the current algorithm. If the
+algorithm cannot be further improved or generalized, then proving these limitations is another open question.
+6
+
+Figure 2: A comparison of different ODE solvers. The top figure shows average minimum time used by each
+ODE solver to solve Eq. (14) for different number of qubits. The bottom plot shows the average memory
+used by each solver to solve the same differential equation. The benchmark was done on a personal PC and
+each data point was repeated at least 100 times over 20 randomly generated time-dependent Hamiltonians.
+All of the methods except Trotter were picked from Julia’s DifferentialEquations.jl roster of state of the art
+ODE solvers [29], and compared to the Trotter method, they consistently had at least 10 orders of magnitude
+less error of the form Eq. (12). The Trotter solution was also implemented in Julia, and used N = 30.
+7
+
+1.0×109
+Solver
+(su)
+1.0×108
+Trotter
+runtime (
+1.0×107
+MMidpoint
+1.0×106
+MLeapfrog
+1.0×105
+1
+2
+3
+4
+5
+6
+7
+MGL4
+1.0×1010
+MNC6
+1.0×109
+(bytes)
+MGL6
+1.0×108
+memory
+1.0×107
+MNC8
+1.0×106
+MGL8
+1.0×105
+1
+2
+3
+4
+5
+6
+7
+Number of Sites6
+Acknowledgements
+We would like to thank Salman Beigi for overseeing this project and commenting on the draft. We also thank
+Leila Taghavi and Erfan Abedi for helpful discussions.
+References
+[1]
+W. Pauli. “ ¨Uber das Wasserstoffspektrum vom Standpunkt der neuen Quantenmechanik”. In: Zeitschrift
+f¨ur Physik A Hadrons and nuclei 36.5 (May 1, 1926), pp. 336–363. ISSN: 0939-7922. DOI: 10.1007/
+BF01450175. URL: https://doi.org/10.1007/BF01450175 (visited on 10/08/2022).
+[2]
+Lucy Mensing. “Die Rotations-Schwingungsbanden nach der Quantenmechanik”. In: Zeitschrift f¨ur
+Physik 36.11 (Nov. 1, 1926), pp. 814–823. ISSN: 0044-3328. DOI: 10.1007/BF01400216. URL:
+https://doi.org/10.1007/BF01400216 (visited on 10/08/2022).
+[3]
+F. London, H. London, and Frederick Alexander Lindemann. “The electromagnetic equations of the
+supraconductor”. In: Proceedings of the Royal Society of London. Series A - Mathematical and Physical
+Sciences 149.866 (Mar. 1935). Publisher: Royal Society, pp. 71–88. DOI: 10.1098/rspa.1935.
+0048. URL: https://royalsocietypublishing.org/doi/10.1098/rspa.1935.
+0048 (visited on 10/08/2022).
+[4]
+Max Planck. “Ueber das Gesetz der Energieverteilung im Normalspectrum”. In: Annalen der Physik
+309.3 (1901). eprint: https://onlinelibrary.wiley.com/doi/pdf/10.1002/andp.19013090310, pp. 553–563.
+ISSN: 1521-3889. DOI: 10.1002/andp.19013090310. URL: https://onlinelibrary.
+wiley.com/doi/abs/10.1002/andp.19013090310 (visited on 10/08/2022).
+[5]
+Dugan Hayes, Graham B. Griffin, and Gregory S. Engel. “Engineering Coherence Among Excited
+States in Synthetic Heterodimer Systems”. In: Science 340.6139 (June 21, 2013). Publisher: American
+Association for the Advancement of Science, pp. 1431–1434. DOI: 10.1126/science.1233828.
+URL: https://www.science.org/doi/10.1126/science.1233828 (visited on
+10/08/2022).
+[6]
+Michael P. Zaletel, Roger S. K. Mong, and Frank Pollmann. “Topological characterization of frac-
+tional quantum hall ground states from microscopic hamiltonians”. In: Physical Review Letters
+110.23 (June 4, 2013). Publisher: American Physical Society, p. 236801. ISSN: 00319007. DOI:
+10.1103/PhysRevLett.110.236801. URL: https://link.aps.org/doi/10.1103/
+PhysRevLett.110.236801 (visited on 10/10/2019).
+[7]
+Frank Arute et al. “Hartree-Fock on a superconducting qubit quantum computer”. In: Science 369.6507
+(Aug. 28, 2020), pp. 1084–1089. ISSN: 0036-8075. DOI: 10.1126/science.abb9811. arXiv:
+2004.04174. URL: https://www.science.org/doi/10.1126/science.abb9811.
+[8]
+D. Banerjee et al. “Atomic Quantum Simulation of U(N) and SU(N) Non-Abelian Lattice Gauge
+Theories”. In: Physical Review Letters 110.12 (Mar. 21, 2013). Publisher: American Physical Society,
+p. 125303. ISSN: 0031-9007. DOI: 10.1103/PhysRevLett.110.125303. arXiv: 1211.2242.
+URL: https://link.aps.org/doi/10.1103/PhysRevLett.110.125303 (visited on
+06/03/2019).
+[9]
+Richard P. Feynman. “Simulating physics with computers”. In: International Journal of Theoretical
+Physics 21.6 (June 1982). Publisher: Kluwer Academic Publishers-Plenum Publishers, pp. 467–488.
+ISSN: 0020-7748. DOI: 10.1007/BF02650179. URL: http://link.springer.com/10.
+1007/BF02650179 (visited on 04/14/2019).
+8
+
+[10]
+Stephen P. Jordan et al. “BQP-completeness of scattering in scalar quantum field theory”. In: Quan-
+tum 2 (Jan. 8, 2018). Publisher: Verein zur F¨orderung des Open Access Publizierens in den Quan-
+tenwissenschaften, p. 44. ISSN: 2521-327X. DOI: 10.22331/q- 2018- 01- 08- 44. arXiv:
+1703.00454. URL: https://quantum-journal.org/papers/q-2018-01-08-44/
+(visited on 04/18/2019).
+[11]
+Ning Bao et al. “Universal quantum computation by scattering in the Fermi–Hubbard model”. In:
+New Journal of Physics 17.9 (Sept. 2015). Publisher: IOP Publishing, p. 093028. ISSN: 1367-2630.
+DOI: 10.1088/1367-2630/17/9/093028. URL: https://doi.org/10.1088/1367-
+2630/17/9/093028 (visited on 10/08/2022).
+[12]
+Bei-Hua Chen, Yan Wu, and Qiong-Tao Xie. “Heun functions and quasi-exactly solvable double-well
+potentials”. In: Journal of Physics A: Mathematical and Theoretical 46.3 (Dec. 18, 2012). Publisher:
+IOP Publishing, p. 035301. ISSN: 1751-8121. DOI: 10.1088/1751-8113/46/3/035301. URL:
+https://iopscience.iop.org/article/10.1088/1751-8113/46/3/035301/
+meta (visited on 10/08/2022).
+[13]
+Abhinav Deshpande et al. “Dynamical Phase Transitions in Sampling Complexity”. In: Physical
+Review Letters 121.3 (July 19, 2018). Publisher: American Physical Society, p. 030501. ISSN: 0031-
+9007. DOI: 10.1103/PhysRevLett.121.030501. arXiv: 1703.05332. URL: https:
+//doi.org/10.1103/PhysRevLett.121.030501.
+[14]
+Adam Ehrenberg et al. “Simulation Complexity of Many-Body Localized Systems”. In: (May 25,
+2022), pp. 1–17. arXiv: 2205.12967. URL: http://arxiv.org/abs/2205.12967.
+[15]
+Sergey Bravyi, David Gosset, and Ramis Movassagh. “Classical algorithms for quantum mean values”.
+In: Nature Physics 17.3 (Mar. 2021), pp. 337–341. ISSN: 1745-2473. DOI: 10.1038/s41567-020-
+01109-8. arXiv: 1909.11485. URL: http://dx.doi.org/10.1038/s41567-020-
+01109-8.
+[16]
+Steve Smale. “The fundamental theorem of algebra and complexity theory”. In: Bulletin of the American
+Mathematical Society 4.1 (1981), pp. 1–36. ISSN: 0273-0979, 1088-9485. DOI: 10.1090/S0273-
+0979-1981-14858-8. URL: https://www.ams.org/bull/1981-04-01/S0273-
+0979-1981-14858-8/ (visited on 10/09/2022).
+[17]
+Rolando Somma. “Quantum simulations of one dimensional quantum systems”. In: 16 (Mar. 2015).
+[18]
+Ulrich Schollw¨ock. “The density-matrix renormalization group in the age of matrix product states”.
+In: Annals of Physics 326.1 (Aug. 20, 2011). ISBN: 0003-4916, pp. 96–192. ISSN: 00034916. DOI:
+10.1016/j.aop.2010.09.012. arXiv: 1008.3477. URL: http://arxiv.org/abs/
+1008.3477 (visited on 03/22/2017).
+[19]
+Bruno Nachtergaele, Yoshiko Ogata, and Robert Sims. “Propagation of Correlations in Quantum
+Lattice Systems”. In: Journal of Statistical Physics 124.1 (July 6, 2006). Publisher: Springer, pp. 1–13.
+ISSN: 0022-4715. DOI: 10.1007/s10955-006-9143-6. arXiv: math-ph/0603064. URL:
+https://link.springer.com/article/10.1007/s10955-006-9143-6.
+[20]
+Ali Hamed Moosavian, Seyed Sajad Kahani, and Salman Beigi. “Limits of Short-Time Evolution of
+Local Hamiltonians”. In: Quantum 6 (June 27, 2022), p. 744. ISSN: 2521-327X. DOI: 10.22331/
+q- 2022- 06- 27- 744. arXiv: 2104.12808. URL: https://arxiv.org/abs/2104.
+12808v3 (visited on 02/13/2022).
+9
+
+[21]
+Ryan Sweke, Jens Eisert, and Michael Kastner. “Lieb–Robinson bounds for open quantum systems
+with long-ranged interactions”. In: Journal of Physics A: Mathematical and Theoretical 52.42 (Oct. 18,
+2019). Publisher: IOP Publishing, p. 424003. ISSN: 1751-8113. DOI: 10.1088/1751-8121/
+ab3f4a. arXiv: 1906.00791. URL: http://arxiv.org/abs/1906.00791.
+[22]
+Guifr´e Vidal. “Efficient Classical Simulation of Slightly Entangled Quantum Computations”. In:
+Physical Review Letters 91.14 (Oct. 1, 2003). Publisher: American Physical Society, p. 147902. DOI:
+10.1103/PhysRevLett.91.147902. URL: https://link.aps.org/doi/10.1103/
+PhysRevLett.91.147902 (visited on 10/07/2022).
+[23]
+Nadav Yoran and Anthony J. Short. “Classical Simulation of Limited-Width Cluster-State Quantum
+Computation”. In: Physical Review Letters 96.17 (May 1, 2006). Publisher: American Physical Society,
+p. 170503. DOI: 10.1103/PhysRevLett.96.170503. URL: https://link.aps.org/
+doi/10.1103/PhysRevLett.96.170503 (visited on 10/07/2022).
+[24]
+Richard Jozsa. “On the simulation of quantum circuits”. In: arXiv:quant-ph/0603163. arXiv, Mar. 19,
+2006. DOI: 10.48550/arXiv.quant-ph/0603163. arXiv: quant-ph/0603163. URL:
+http://arxiv.org/abs/quant-ph/0603163 (visited on 10/07/2022).
+[25]
+Volker Strassen. “Gaussian elimination is not optimal”. In: Numerische Mathematik 13.4 (Aug. 1,
+1969), pp. 354–356. ISSN: 0945-3245. DOI: 10.1007/BF02165411. URL: https://doi.org/
+10.1007/BF02165411 (visited on 01/05/2023).
+[26]
+Josh Alman and Virginia Vassilevska Williams. “A Refined Laser Method and Faster Matrix Multi-
+plication”. In: CoRR abs/2010.05846 (2020). arXiv: 2010.05846. URL: https://arxiv.org/
+abs/2010.05846.
+[27]
+David J. Griffiths and Darrell F. Schroeter. Introduction to Quantum Mechanics. 3rd ed. Cambridge
+University Press, 2018. DOI: 10.1017/9781316995433.
+[28]
+Tosio Kato. Perturbation Theory for Linear Operators. 1966, pp. xix + 592.
+[29]
+Christopher Rackauckas and Qing Nie. “Differentialequations.jl–a performant and feature-rich ecosys-
+tem for solving differential equations in julia”. In: Journal of Open Research Software 5.1 (2017).
+[30]
+Masuo Suzuki. “Generalized Trotter’s formula and systematic approximants of exponential operators
+and inner derivations with applications to many-body problems”. In: Communications in Mathematical
+Physics 51.2 (June 1976). Publisher: Springer-Verlag, pp. 183–190. ISSN: 00103616. DOI: 10.1007/
+BF01609348. URL: http://link.springer.com/10.1007/BF01609348 (visited on
+11/05/2019).
+[31]
+Masuo Suzuki. “General theory of fractal path integrals with applications to many-body theories and
+statistical physics”. In: Journal of Mathematical Physics 32.2 (Feb. 4, 1991). Publisher: American
+Institute of Physics, pp. 400–407. ISSN: 0022-2488. DOI: 10.1063/1.529425. URL: http:
+//aip.scitation.org/doi/10.1063/1.529425 (visited on 11/05/2019).
+[32]
+Naomichi Hatano and Masuo Suzuki. “Finding Exponential Product Formulas of Higher Orders”.
+In: Lect. Notes Phys. 679.4 (June 2, 2005). ISBN: 3-540-27987-3, pp. 37–68. DOI: 10.1007/
+11526216_2. arXiv: math-ph/0506007. URL: http://arxiv.org/abs/math-ph/
+0506007 (visited on 08/07/2017).
+[33]
+Nathan Wiebe et al. “Higher order decompositions of ordered operator exponentials”. In: Journal of
+Physics A: Mathematical and Theoretical 43.6 (Jan. 12, 2010), p. 065203. ISSN: 1751-8113. DOI:
+10.1088/1751-8113/43/6/065203. arXiv: 0812.0562. URL: https://iopscience.
+iop.org/article/10.1088/1751-8113/43/6/065203.
+10
+
+[34]
+M. Cerezo et al. “Variational quantum algorithms”. In: Nature Reviews Physics 3.9 (Sept. 2021).
+Number: 9 Publisher: Nature Publishing Group, pp. 625–644. ISSN: 2522-5820. DOI: 10.1038/
+s42254-021-00348-9. URL: https://www.nature.com/articles/s42254-021-
+00348-9 (visited on 10/11/2022).
+[35]
+Ryan Babbush et al. “Focus beyond Quadratic Speedups for Error-Corrected Quantum Advantage”. In:
+PRX Quantum 2.1 (Mar. 29, 2021). Publisher: American Physical Society, p. 010103. DOI: 10.1103/
+PRXQuantum.2.010103. URL: https://link.aps.org/doi/10.1103/PRXQuantum.
+2.010103 (visited on 10/12/2022).
+[36]
+Sergey Bravyi et al. “Obstacles to Variational Quantum Optimization from Symmetry Protection”.
+In: Physical Review Letters 125.26 (Dec. 24, 2020), p. 260505. ISSN: 0031-9007. DOI: 10.1103/
+PhysRevLett.125.260505. arXiv: 1910.08980. URL: http://arxiv.org/abs/1910.
+08980 (visited on 05/17/2020).
+[37]
+Edward Farhi, David Gamarnik, and Sam Gutmann. “The Quantum Approximate Optimization Al-
+gorithm Needs to See the Whole Graph: Worst Case Examples”. In: (May 18, 2020). arXiv: 2005.
+08747. URL: http://arxiv.org/abs/2005.08747 (visited on 06/07/2020).
+[38]
+Edward Farhi, David Gamarnik, and Sam Gutmann. “The Quantum Approximate Optimization Al-
+gorithm Needs to See the Whole Graph: A Typical Case”. In: (Apr. 19, 2020). arXiv: 2004.09002.
+URL: http://arxiv.org/abs/2004.09002 (visited on 04/26/2020).
+11
+
diff --git a/CdFJT4oBgHgl3EQfASzG/content/tmp_files/load_file.txt b/CdFJT4oBgHgl3EQfASzG/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf,len=681
+page_content='Classical Algorithm for the Mean Value problem over  Short-Time Hamiltonian Evolutions  Reyhaneh Aghaei Saem1 and Ali Hamed Moosavian1  1Phanous  Abstract  Simulating physical systems has been an important application of classical and quantum computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' In  this article we present an efficient classical algorithm for simulating time-dependent quantum mechanical  Hamiltonians over constant periods of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' The algorithm presented here computes the mean value of an  observable over the output state of such short-time Hamiltonian evolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' In proving the performance of  this algorithm we use Lieb-Robinson type bounds to limit the evolution of local operators within a lightcone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  This allows us to divide the task of simulating a large quantum system into smaller systems that can be  handled on normal classical computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  1  Introduction  Understanding physical systems with quantum mechanical interactions have been an increasingly challenging  and important task in the past century.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' Because nature is inherently quantum, many naturally occurring or  artificially implemented phenomena cannot be explained without quantum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' Some early examples  include understanding atoms [1], diatomic molecules [2], the Meissner effect in superconductors [3] and  black-body radiation [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' With the advent of better theoretical and computational tools, the list of phenomena  that have been shown to require quantum mechanics has grown enormously in the past few decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' To name  a few, some of the more notable examples include photosynthesis [5], topological ordered phases such as  fractional quantum Hall effect [6], isomerization of diazene [7] and non-Abelian lattice gauge theories [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  Indeed, because of the challenging nature of simulating quantum systems on classical computers, one of the  earliest motivations for quantum computers has been to use them to study other quantum systems [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  Besides the obvious practical applications, the problem of simulating quantum systems has some critical  complexity theory value too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' On the one hand, it is known that the problem of simulating several quantum  systems belongs to the Bounded-error Quantum Polynomial-time complete (BQP-complete) class [10, 11,  12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' Also, there are specific problems where the complexity class of simulating them varies as the runtime  increases, and exhibit a dynamical phase transition [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  In this paper, we present a classical algorithm for computing the expectation value of an observable that  can be written as a tensor product of local operators acting on each qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' In the literature this problem is  sometimes called the quantum mean value problem [15] (not to be confused with a similarly named open  problem in mathematics [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' Our algorithm evaluates the mean value of an operator where the state is  generated by evolving a product state under a geometrically local time-dependent Hamiltonian for a short  period of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' The setup of our problem is motivated by the state of current quantum technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' On one  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='11420v1  [quant-ph]  26 Jan 2023  hand, analog simulators lack fault-tolerance, which means they have a limited decoherence time, and on the  other hand, we still do not have access to fault-tolerant universal quantum computers either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  Suppose that we have a bounded-norm time-dependent Hamiltonian H(t) that acts on n-qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' The  initial state of the system is assumed to be a product state, typically |ψ(0)⟩ = |0⟩⊗n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' The quantum mean value  problem wants to compute the expectation value of an operator O with respect to |ψ(T)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' The expectation  value is represented with µ:  µ ≡ ⟨ψ(T)|O|ψ(T)⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  (1)  We consider geometrically local time-dependent Hamiltonians that are defined on 2D or 3D lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' 1D  systems with gapped Hamiltonians have been thoroughly studied before [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  The unitary evolution operator corresponding to this Hamiltonian can be written as:  U(t) = T :  �  exp  �  − i  ¯h  � t  0  H(t′)dt′  ��  ,  (2)  where T : [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='] means the operators respect time ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' Assuming the observable can be written as the tensor  product of single-qubit Hermitian operators Oj acting on individual sites, O = O1 ⊗ O2 ⊗ · · · ⊗ On, we  can write the mean value of the observable O as:  µ = ⟨0n| U†(T)O1 ⊗ O2 ⊗ · · · ⊗ OnU(T) |0n⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  (3)  In this paper, we introduce a classical algorithm that can approximate the mean value problem for time-  dependent Hamiltonians with an additive error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' The structure of the rest of this paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' 2  we give an overview of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' Section 3 explains how the dynamics of each operator is approximately  restricted to within a lightcone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' 4 we analyze the problem of classically calculating the local unitary  operators and compare different numerical algorithms for doing the task and in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' 5 we conclude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  2  Algorithm  We provide a classical algorithm for approximating µ within an additive error, δ, for the special case of the  mean value problem where the time-dependent Hamiltonian is geometrically local in 2D or 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  Conceptually, the algorithm can be broken into two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' First we use a Lieb-Robinson bound [19, 20,  21] to limit the unitary evolution corresponding to each qubit within a lightcone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' This allows us to classically  compute the unitary operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' Then we follow the steps in [15] to divide the lattice into pseudo 1D slices, that  can be efficiently simulated either using Matrix Product State algorithms [22, 23, 24] (for a nice review of  these algorithms see [18]) or the algorithm ascribed to [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  Theorem 1 Let H(t) be a bounded-norm time-dependent lattice Hamiltonian that acts on n-qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' Suppose  the observable O is a tensor product of operators , O = O1 ⊗ O2 ⊗ · · · ⊗ On.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' For constant evolution times,  there exists a classical algorithm that estimates µ within an additive error δ,  |˜µ − µ| ≤ δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  (4)  The error δ includes three different parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' The first contribution comes from the Lieb-Robinson bound where  we used it to approximately limit the evolution within the lightcone of each qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' Second, numerical methods  such as trotterization are used to calculate the local unitary evolutions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' these methods are not exact and incur  some errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' The third part comes from the additive error in simulating a constant depth quantum circuit [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  In [15], they provide a classical algorithm for constant depth circuits in 2D or 3D which can approximate the  2  mean value to an additive error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' In our work, the short-time evolution of a geometrically local Hamiltonian  which is limited by the Lieb Robinson bound is comparable with the shallow quantum circuit with a constant  depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  Suppose that the error of localizing the time evolution of the Hamiltonian, classical simulation of time-  dependent unitaries and simulating shallow circuits are given by εLR, εCS and εSSC respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' Then the  total error is as follows:  |˜µ − µ| ≤ εLR + εCS + εSSC = δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  (5)  In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' (10), we will see how the first error is dictated by the simulation time, the lightcone radius and the  Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' One should note that εLR is bounded by εLR ≤ nε(LR)(L, T), where ε(LR)(L, T) is defined  later in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' (10) as the Lieb Robinson error for a single site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' Section 4 derives the dependency of the second  error term on the simulation parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  Algorithm 1: High level overview of the algorithm  input :a √n × √n lattice of qubits, a geometrically local time-dependent Hamiltonian H(t), the  operator O = O1 ⊗ O2 ⊗ · · · ⊗ On where each ∥Oj∥ ≤ 1, an upper bound for error δ, a  simulation time T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  output :an approximation for µ = ⟨ψ(T)|O|ψ(T)⟩, where |ψ(0)⟩ = |0⊗n⟩ and  i¯h d  dt |ψ(t)⟩ = H(t) |ψ(t)⟩  1 Initialization:  2 Use T and δ to calculate the lightcone radius, L, from the Lieb-Robinson bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  3 As in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' 1 partition the lattice into 4L × √n strips twice, let us call each set {Ai}i and {Bi}i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' Also,  define the sets  �  A0  i  �  i and  �  B0  i  �  i as the central part of the strips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  4 Also, group sites into 2L × 2L super-sites to form a coarse-grained lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  5 Calculations:  6 for Ai ∈ {Ai}i do  7  initialize an MPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  8  for Oj in A0  i do  9  Use a classical ODE solver to calculate the local unitary operator Uj corresponding to Oj  which includes the terms that are in the lightcone of the jth qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  10  Transform OjUj into a Matrix Product Operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  11  Add the necessary sites from the current super-site to the active memory and apply the MPO  on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  12  Measure any sites that will no longer be needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  13  Let us call the (not normalized) MPS outcome |ΨAi(T)⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  14 Repeat the Line 6 loop for Bjs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  15 return ˜µ =  ��  j ⟨ΨBj(T)|  ���  i |ΨAi(T)⟩  �  The runtime of classical simulation of the time-dependent unitaries is related to the complexity of matrix  multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' We can find the lightcone radius L from Section 3 and then find the number of qubits m in  each lightcone which is O  �  L2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' For most practical cases, the fastest matrix multiplication algorithm is the  famous Strasson algorithm with asymptotic complexity of O  �  (2m)log2 7�  [25], however, the best known  asymptotic complexity for matrix multiplication is O  �  (2m)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='373�  [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' For most physical Hamiltonians  3  Figure 1: A: It shows the relationship between simulation time and the lightcone radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' B:shows the sites  inside the lightcone for various lightcone radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' C and D:show the set of strips {Ai}i and {Bi}i as well as  the  �  A0  i  �  i and  �  B0  i  �  i sets on a 2D grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' The yellow squares represent the Oi operators that are to be applied  on either set of strips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  where we have translational symmetry in the bulk, we only need to calculate the unitary evolution matrix for  each geometrical configuration of the sites once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' This means that the complexity of this part of the algorithm  would be O  �  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='373m�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' But for generic Hamiltonians the number of configurations could grow linearly with  the system size and the complexity would be O  �  n22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='373m�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  For 2D and 3D systems with n qubits, the runtime of simulating a shallow circuit is related to the last error  term with O  �  nL226L2/(εSSC)2�  and O  �  nL326L2n1/3/(εSSC)2�  respectively [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' We have provided a  high-level overview of the 2D algorithm in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' Consequently the general total complexity of the algo-  rithm for 2D and 3D systems is O  �  n22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='373m + nL226L2/(εSSC)2�  and O  �  n22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='373m + nL326L2n1/3/(εSSC)2�  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  4  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='00000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='10000  lerror  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='01000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='00100  Simulation time  +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='01  + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='00010  +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='03  +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='00001  0  2  Lightcone radius3  Local Unitary Operators  According to [20], we know that a local operator OA which is defined inside a region A, remains local after a  short-time evolution under a local Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' Suppose that the Hamiltonian has the form  H(t) =  �  e  ue(t)he ,  (6)  where he acts non-trivially only on the two vertices of edge e of the graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' Suppose that ∥ue(t)he∥ ≤ g  for 0 ≤ t ≤ T and the maximum degree of the graph to be ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' The Hamiltonian for terms in region A and the  set of vertices in the L-boundary of it has the form  HA(t) =  �  e⊂A∪∂L(A)  ue(t)he .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  (7)  The time evolution operator of this local Hamiltonian acts non-trivially only on the region A and its L-  boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' The time evolution operator is given by  V(t) = T :  �  exp  �  − i  ¯h  � t  0  HA(t′)dt′  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  (8)  This is known as Dyson series and T : [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='] represents time-ordering [27, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' 551].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' According to [20] the  following Lieb-Robinson bound holds:  ||U†(T)OAU(T) − V†(T)OAV(T)|| ≤ ε(LR)(L, T) ,  (9)  where ε(LR)(L, T) is defined as:  ε(LR)(L, T) =  �  2  π |A| ||OA|| exp  �  −L(log L − log T − log(4g(δ − 1))) − 1  2 log L  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  (10)  For each local operator in O = O1 ⊗ O2 ⊗ · · · ⊗ On, we only consider the Hamiltonian terms inside the  lightcone of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' We replace the global unitary that acts on the entire system with these local unitary operators  and apply the algorithm in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' 2 to approximate the mean value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  The only remaining problem would be to find a classical algorithm for classically calculating the local  unitary operators V(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' We analyze different approaches for doing so in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  4  Classical Simulation of the time-dependent unitaries  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='1  Trotterization  Theorem 2 Let HA(t) be a bounded-norm time-dependent Hamiltonian and OA an observable inside region  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' Let V(t) be the unitary evoluion of HA(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' We can approximate the operator V†(T)OAV(T) with  W†(T, N)OAW(T, N) where W(t, N) is defined as  W(t, N) =  N= t  δt  �  j=1  exp  �  − i  ¯hδt HA(jδt)  �  ,  (11)  5  such that  ��W†(T, N)OAW(T, N) − V†(T)OAV(T)  �� ≤ 6T 2  N¯h ∥OA∥ ∥H′  A(t∗)∥ = εCS,  (12)  where  ∥H′  A(t∗)∥ = max  0≤t≤T ∥H′  A(t)∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  (13)  This is a well-known textbook result that can be found in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' For instance see chapter IX of [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='2  Numerical Differential Equation Solvers  Another approach for approximating the unitary time evolution would be to derive a differential equation  from Schrodinger’s equation and solve that numerically by using a suitable classical algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  i¯h d  dt |ψ(t)⟩ = H(t) |ψ(t)⟩ ,  i¯h d  dtU(t) |ψ(0)⟩ = H(t)U(t) |ψ(0)⟩ ,  i¯h d  dtU(t) = H(t)U(t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  (14)  If there are m qubits inside the lightcone, the matrix representation of U and H will be 2m ×2m operators  and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' (14) will constitute a set of Ordinary Differential Equations (ODEs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' The upside of using a classical  ODE solver is that they can consistently attain much lower errors than what is possible from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' (11) or a  higher order Suzuki-Trotter solution [30, 31] that is fine-tuned for a time-dependent problem [32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' The  downside is that they are typically orders of magnitude slower and require more memory than the straight  forward trotterization as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' Assuming our lightcone is small enough, many numerical ODE solvers  will be able to handle Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' 2 for a comparison between multiple different ODE solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  5  Conclusions  To conclude we have provided a classical algorithm for the mean value problem on outcomes of short time  dependent Hamiltonian evolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' These mean values are typically used in other algorithms such as the  Variational Quantum Algorithm [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  The quantum mean value problem, almost by definition, belongs to the BQP-complete class for polynomial  times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' So, naturally we do not expect to be able to solve the polynomial time problem on a classical computer  efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' Nonetheless, an important open question would be to find the minimum simulation time for getting  a quantum speedup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' The current work shows us that in order to benefit from the quantum speedup we need  simulation times that are at least greater than constant [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' This is in accordance with a wide variety of other  results that target specific problems too [15, 36, 13, 37, 38, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' In general the problem of mapping out the  entire dynamical complexity phase diagram is theoretically interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  Another direction for future research would be to improve or generalize the current algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' If the  algorithm cannot be further improved or generalized, then proving these limitations is another open question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  6  Figure 2: A comparison of different ODE solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' The top figure shows average minimum time used by each  ODE solver to solve Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' (14) for different number of qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' The bottom plot shows the average memory  used by each solver to solve the same differential equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' The benchmark was done on a personal PC and  each data point was repeated at least 100 times over 20 randomly generated time-dependent Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  All of the methods except Trotter were picked from Julia’s DifferentialEquations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='jl roster of state of the art  ODE solvers [29], and compared to the Trotter method, they consistently had at least 10 orders of magnitude  less error of the form Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' The Trotter solution was also implemented in Julia, and used N = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='0×109  Solver  (su)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='0×108  Trotter  runtime (  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='0×107  MMidpoint  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='0×106  MLeapfrog  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='0×105  1  2  3  4  5  6  7  MGL4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='0×1010  MNC6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='0×109  (bytes)  MGL6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='0×108  memory  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='0×107  MNC8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='0×106  MGL8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='0×105  1  2  3  4  5  6  7  Number of Sites6  Acknowledgements  We would like to thank Salman Beigi for overseeing this project and commenting on the draft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' We also thank  Leila Taghavi and Erfan Abedi for helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  References  [1]  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' Pauli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' “ ¨Uber das Wasserstoffspektrum vom Standpunkt der neuen Quantenmechanik”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' In: Zeitschrift  f¨ur Physik A Hadrons and nuclei 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='5 (May 1, 1926), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' 336–363.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' ISSN: 0939-7922.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='1007/  BF01450175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' URL: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='1007/BF01450175 (visited on 10/08/2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  [2]  Lucy Mensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' “Die Rotations-Schwingungsbanden nach der Quantenmechanik”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' In: Zeitschrift f¨ur  Physik 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='11 (Nov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' 1, 1926), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' 814–823.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' ISSN: 0044-3328.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
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+page_content='  [3]  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' London, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' London, and Frederick Alexander Lindemann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' “The electromagnetic equations of the  supraconductor”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' In: Proceedings of the Royal Society of London.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' Series A - Mathematical and Physical  Sciences 149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='866 (Mar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' 1935).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' Publisher: Royal Society, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' 71–88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
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+page_content=' Griffin, and Gregory S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' Engel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' “Engineering Coherence Among Excited  States in Synthetic Heterodimer Systems”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' In: Science 340.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='6139 (June 21, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' Publisher: American  Association for the Advancement of Science, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' 1431–1434.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
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+page_content='  [6]  Michael P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' Zaletel, Roger S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' Mong, and Frank Pollmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
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+page_content=' In: Physical Review Letters  110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='23 (June 4, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
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+page_content=' In: Science 369.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
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+page_content=' ISSN: 0036-8075.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
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+page_content=' arXiv:  2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='04174.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' URL: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
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+page_content='  [8]  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' Banerjee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' “Atomic Quantum Simulation of U(N) and SU(N) Non-Abelian Lattice Gauge  Theories”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' In: Physical Review Letters 110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='12 (Mar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' 21, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
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+page_content='  [9]  Richard P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' Feynman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' “Simulating physics with computers”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' In: International Journal of Theoretical  Physics 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='6 (June 1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
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+page_content=' 467–488.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
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+page_content='  8  [10]  Stephen P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' Jordan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' “BQP-completeness of scattering in scalar quantum field theory”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' In: Quan-  tum 2 (Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
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+page_content=' 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' ISSN: 2521-327X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
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+page_content='22331/q- 2018- 01- 08- 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' arXiv:  1703.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='00454.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' URL: https://quantum-journal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
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+page_content='  [11]  Ning Bao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' “Universal quantum computation by scattering in the Fermi–Hubbard model”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' In:  New Journal of Physics 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='9 (Sept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
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+page_content=' 093028.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
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+page_content='1088/1367-2630/17/9/093028.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
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+page_content='  [12]  Bei-Hua Chen, Yan Wu, and Qiong-Tao Xie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
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+page_content=' In: Journal of Physics A: Mathematical and Theoretical 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='3 (Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' 18, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
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+page_content='  [13]  Abhinav Deshpande et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' “Dynamical Phase Transitions in Sampling Complexity”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' In: Physical  Review Letters 121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='3 (July 19, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
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+page_content=' 030501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' ISSN: 0031-  9007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
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+page_content='121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
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+page_content=' arXiv: 1703.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='05332.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
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+page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='030501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  [14]  Adam Ehrenberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' “Simulation Complexity of Many-Body Localized Systems”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' In: (May 25,  2022), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' 1–17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' arXiv: 2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='12967.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' URL: http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='org/abs/2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='12967.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  [15]  Sergey Bravyi, David Gosset, and Ramis Movassagh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' “Classical algorithms for quantum mean values”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  In: Nature Physics 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='3 (Mar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' 2021), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' 337–341.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' ISSN: 1745-2473.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='1038/s41567-020-  01109-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' arXiv: 1909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='11485.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' URL: http://dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='1038/s41567-020-  01109-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  [16]  Steve Smale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' “The fundamental theorem of algebra and complexity theory”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' In: Bulletin of the American  Mathematical Society 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='1 (1981), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' 1–36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' ISSN: 0273-0979, 1088-9485.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='1090/S0273-  0979-1981-14858-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
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+page_content='ams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='org/bull/1981-04-01/S0273-  0979-1981-14858-8/ (visited on 10/09/2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  [17]  Rolando Somma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' “Quantum simulations of one dimensional quantum systems”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' In: 16 (Mar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  [18]  Ulrich Schollw¨ock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' “The density-matrix renormalization group in the age of matrix product states”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  In: Annals of Physics 326.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='1 (Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' 20, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' ISBN: 0003-4916, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' 96–192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' ISSN: 00034916.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' DOI:  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
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+page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' arXiv: 1008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='3477.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' URL: http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='org/abs/  1008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
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+page_content='  [19]  Bruno Nachtergaele, Yoshiko Ogata, and Robert Sims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' “Propagation of Correlations in Quantum  Lattice Systems”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' In: Journal of Statistical Physics 124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='1 (July 6, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' Publisher: Springer, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' 1–13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  ISSN: 0022-4715.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='1007/s10955-006-9143-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' arXiv: math-ph/0603064.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' URL:  https://link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
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+page_content='  [20]  Ali Hamed Moosavian, Seyed Sajad Kahani, and Salman Beigi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' “Limits of Short-Time Evolution of  Local Hamiltonians”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' In: Quantum 6 (June 27, 2022), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' 744.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' ISSN: 2521-327X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
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+page_content=' arXiv: 2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='12808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' URL: https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='org/abs/2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  12808v3 (visited on 02/13/2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  9  [21]  Ryan Sweke, Jens Eisert, and Michael Kastner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' “Lieb–Robinson bounds for open quantum systems  with long-ranged interactions”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' In: Journal of Physics A: Mathematical and Theoretical 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='42 (Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' 18,  2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' Publisher: IOP Publishing, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' 424003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' ISSN: 1751-8113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
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+page_content=' arXiv: 1906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='00791.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' URL: http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='org/abs/1906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='00791.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  [22]  Guifr´e Vidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' “Efficient Classical Simulation of Slightly Entangled Quantum Computations”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' In:  Physical Review Letters 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='14 (Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' 1, 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' Publisher: American Physical Society, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' 147902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' DOI:  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
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+page_content='1103/  PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='147902 (visited on 10/07/2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  [23]  Nadav Yoran and Anthony J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' Short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' “Classical Simulation of Limited-Width Cluster-State Quantum  Computation”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' In: Physical Review Letters 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='17 (May 1, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' Publisher: American Physical Society,  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' 170503.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='170503.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' URL: https://link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='aps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='org/  doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='170503 (visited on 10/07/2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  [24]  Richard Jozsa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' “On the simulation of quantum circuits”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' In: arXiv:quant-ph/0603163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' arXiv, Mar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' 19,  2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='48550/arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='quant-ph/0603163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' arXiv: quant-ph/0603163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' URL:  http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='org/abs/quant-ph/0603163 (visited on 10/07/2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  [25]  Volker Strassen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' “Gaussian elimination is not optimal”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' In: Numerische Mathematik 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='4 (Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' 1,  1969), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' 354–356.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' ISSN: 0945-3245.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
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+page_content=' URL: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='org/  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='1007/BF02165411 (visited on 01/05/2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  [26]  Josh Alman and Virginia Vassilevska Williams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' “A Refined Laser Method and Faster Matrix Multi-  plication”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' In: CoRR abs/2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='05846 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' arXiv: 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='05846.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' URL: https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='org/  abs/2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='05846.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  [27]  David J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' Griffiths and Darrell F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' Schroeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' Introduction to Quantum Mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' 3rd ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' Cambridge  University Press, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='1017/9781316995433.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  [28]  Tosio Kato.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' Perturbation Theory for Linear Operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' 1966, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' xix + 592.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  [29]  Christopher Rackauckas and Qing Nie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' “Differentialequations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='jl–a performant and feature-rich ecosys-  tem for solving differential equations in julia”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' In: Journal of Open Research Software 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='1 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  [30]  Masuo Suzuki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' “Generalized Trotter’s formula and systematic approximants of exponential operators  and inner derivations with applications to many-body problems”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' In: Communications in Mathematical  Physics 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='2 (June 1976).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' Publisher: Springer-Verlag, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' 183–190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' ISSN: 00103616.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
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+page_content='com/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
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+page_content='  [31]  Masuo Suzuki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
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+page_content=' In: Journal of Mathematical Physics 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='2 (Feb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' 4, 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' Publisher: American  Institute of Physics, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' 400–407.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' ISSN: 0022-2488.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='1063/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
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+page_content=' URL: http:  //aip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
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+page_content='529425 (visited on 11/05/2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  [32]  Naomichi Hatano and Masuo Suzuki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' “Finding Exponential Product Formulas of Higher Orders”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  In: Lect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' Notes Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' 679.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='4 (June 2, 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' ISBN: 3-540-27987-3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' 37–68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
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+page_content=' arXiv: math-ph/0506007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' URL: http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
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+page_content='  [33]  Nathan Wiebe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' “Higher order decompositions of ordered operator exponentials”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' In: Journal of  Physics A: Mathematical and Theoretical 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='6 (Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' 12, 2010), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' 065203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' ISSN: 1751-8113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
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+page_content='0562.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' URL: https://iopscience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  iop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='org/article/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
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+page_content='  10  [34]  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' Cerezo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' “Variational quantum algorithms”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' In: Nature Reviews Physics 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='9 (Sept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  Number: 9 Publisher: Nature Publishing Group, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' 625–644.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' ISSN: 2522-5820.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
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+page_content='1038/  s42254-021-00348-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
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+page_content='  [35]  Ryan Babbush et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' “Focus beyond Quadratic Speedups for Error-Corrected Quantum Advantage”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' In:  PRX Quantum 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
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+page_content=' Publisher: American Physical Society, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
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+page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='1103/  PRXQuantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='010103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' URL: https://link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='aps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='org/doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='1103/PRXQuantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='010103 (visited on 10/12/2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  [36]  Sergey Bravyi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' “Obstacles to Variational Quantum Optimization from Symmetry Protection”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  In: Physical Review Letters 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='26 (Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' 24, 2020), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' 260505.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' ISSN: 0031-9007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='1103/  PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='260505.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' arXiv: 1910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='08980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' URL: http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='org/abs/1910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  08980 (visited on 05/17/2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  [37]  Edward Farhi, David Gamarnik, and Sam Gutmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' “The Quantum Approximate Optimization Al-  gorithm Needs to See the Whole Graph: Worst Case Examples”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' In: (May 18, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' arXiv: 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  08747.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' URL: http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='org/abs/2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='08747 (visited on 06/07/2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  [38]  Edward Farhi, David Gamarnik, and Sam Gutmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' “The Quantum Approximate Optimization Al-  gorithm Needs to See the Whole Graph: A Typical Case”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' In: (Apr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' 19, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content=' arXiv: 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='09002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  URL: http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='org/abs/2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='09002 (visited on 04/26/2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
+page_content='  11' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf'}
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+arXiv:2301.01513v1  [hep-th]  4 Jan 2023
+HU-EP-23/01
+Remarks on conformal invariants for
+piecewise smooth curves and Wilson loops
+Harald Dorn 1
+Institut f¨ur Physik und IRIS Adlershof, Humboldt-Universit¨at zu Berlin,
+Zum Großen Windkanal 6, D-12489 Berlin, Germany
+Abstract
+This short note is some obvious mathematical addendum to our papers on Wilson
+loops on polygon-like contours with circular edges [1, 2].
+Using the technique of
+osculating spheres and circles we identify the conformal invariants characterising the
+kinks (cusps) of generic piecewise smooth curves in 3-dimensional space.
+1dorn@physik.hu-berlin.de
+
+1
+Introduction
+The metrical invariants of smooth curves in Euclidean D-dimensional space are length,
+curvature and (D −2) torsion parameters as functions along the curve. For piecewise
+smooth curves with cusps 2, in addition each cusp is characterised by the Euler angles
+specifying the orthogonal transformation needed to rotate the Frenet-Serret frames
+on both sides of a cusp to one another.
+What concerns the issue of conformal invariants, there is for smooth curves a
+considerable amount of mathematical papers. In analogy to the metrical invariants
+length s, curvature κ and torsion τ one finds for 3-dimensional curves, see e.g. [3],
+conformal length ω,3
+dω = √νds,
+ν =
+�
+(κ′)2 + κ2τ 2 ,
+(1)
+conformal curvature Q
+Q = 4(ν′′ − κ2ν)ν − 5(ν′)2
+8ν3
+,
+(2)
+and conformal torsion T
+T = 2(κ′)2τ + κ2τ 3 + κκ′τ ′ − κκ′′τ
+ν
+5
+2
+.
+(3)
+Use of these invariants has been made in the physical literature to characterise the
+boundary conditions for the treatment of minimal surfaces in AdS via Pohlmeyer
+reduction [4].
+But what concerns the extension to piecewise smooth curves, we did not find any
+paper in the mathematical literature. Therefore, we decided to write up what one gets
+by straightforward application of one of the various techniques used for the smooth
+case: the kinematics of osculating spheres, see e.g. [5–8] and refs. therein.
+This note is organised as follows. In the next section we find formulas expressing
+the conformal invariants for cusps, with legs made of generic smooth pieces of curves,
+in terms of their metrical invariants. Then in section 3 we consider the very special
+curves studied in our papers [1,2], i.e. polygon-like curves with circular edges. These
+curves need a separate discussion, since along the circular edges all the conformal
+invariants (1),(2),(3) are zero or ill-defined.
+2
+Conformal invariants of generic cusps
+In 3-dimensional space the osculating circle (sphere ) at a generic point x on a smooth
+curve is a circle (sphere) having contact of second (third) order with the curve at x.
+Since the order of contact is preserved under conformal transformations, osculating
+circles (spheres) at a given point of a curve are mapped to those for the image under
+2We follow the language in the Wilson loop literature and use the word cusp for a kink with
+nonzero opening angle.
+3The prime denotes derivative with respect to s.
+1
+
+a conformal map. Let us denote by ⃗t,⃗n,⃗b the unit tangent, normal and binormal
+vectors at x. Then the center a of the osculating circle S1 is given by
+a = x + 1
+κ ⃗n
+(4)
+and the center c of the osculating sphere S2 by
+c = x + 1
+κ ⃗n −
+κ′
+τκ2 ⃗b .
+(5)
+We now turn to the case where x is a point of discontinuity, i.e. the tip of a cusp.
+The cusp is then characterised by two osculating circles S1
+−, S1
++ and two osculating
+spheres S2
+−, S2
++. The index ” ± ” indicates the limits one gets by approaching x along
+the both respective legs of the cusp.
+Let us first count the number of conformal invariants we can expect for the cusp.
+There are 9 metrical invariants at hand: κ±, κ′
+±, τ± and the 3 Euler angles for the
+rotations from the Frenet frame {⃗t−,⃗n−,⃗b−} to {⃗t+,⃗n+,⃗b+}. The difference of the
+numbers of parameters of the 3-dimensional conformal and isometry group is equal
+to 4. This should result in 9 − 4 = 5 conformal parameters. Now of course two
+of them are the corresponding limits of the differential of the conformal length (1).
+Hence there remain 3 conformal parameters to be attributed genuinely to the cusp.
+To find explicit formulas for them, we start with the conformal invariants of pairs
+of spheres (Sm, Sn) of the same or of different dimension. There is a rigorous math-
+ematical treatment for arbitrary dimensions in ref. [9]. Applied to our case it means
+that to each of the pair (S1
+−, S1
++), (S1
+−, S2
++), (S2
+−, S1
++), (S2
+−, S2
++) belongs just one con-
+formal parameter. To proceed, we first study these four invariants and will show
+afterwards, that only three of them are independently.
+(S1
+−, S1
++)
+The tangents of the osculating circles agree with those of the curve.
+Therefore the related conformal invariant is
+A11 = ⃗t−⃗t+ =
+− cos α .
+(6)
+Here α is the cusp angle (understood as the opening angle, i.e. α = π in the smooth
+case).
+(S2
+−, S2
++)
+Now the conformal invariant is given by the so-called inversive product,
+see e.g. [5,9]
+A22 = R2
+− + R2
++ − (c− − c+)2
+2R−R+
+= (c− − x)(c+ − x)
+R−R+
+.
+(7)
+R− and R+ are the radii of S2
+− and S2
++. Obviously A22 is the cosine of the angle
+between the vectors pointing from x to the centers of the two osculating spheres.
+Strictly speaking, only its absolute value is invariant, since A22 changes sign under
+those special conformal transformations for which the preimage of infinity is situated
+inside just one of the spheres. The same comment applies to A12 and A21 below.
+(S2
+−, S1
++) and (S1
+−, S2
++) For a sphere and an intersecting circle the conformal invariant
+2
+
+is the scalar product of the unit vector pointing from x to the center of the sphere
+with the unit tangent of the circle
+A12 =
+⃗t−(c+ − x)
+R+
+,
+A21 = (c− − x) ⃗t+
+R−
+.
+(8)
+Using (4),(5) for both legs of the cusp we get from (7) and (8)
+A22 = κ−τ−κ+τ+ ⃗n−⃗n+ + κ′
+−κ′
++ ⃗b−⃗b+ − κ−τ−κ′
++ ⃗n−⃗b+ − κ+τ+κ′
+− ⃗b−⃗n+
+�
+κ2
+−τ 2
+− + κ′2
+−
+�
+κ2
++τ 2
++ + κ′2
++
+,
+(9)
+A12
+=
+κ+τ+ ⃗t−⃗n+ − κ′
++ ⃗t−⃗b+
+�
+κ2
++τ 2
++ + κ′2
++
+,
+(10)
+A21
+=
+κ−τ− ⃗t+⃗n− − κ′
+− ⃗t+⃗b−
+�
+κ2
+−τ 2
+− + κ′2
+−
+.
+(11)
+All the scalar products in the above formulas can be expressed in terms of three Euler
+angles ϕ, ϑ, ψ needed to rotate the Frenet frame {n−, b−, t−} to {n+, b+, t+}. Then
+we get
+A11 = ⃗t−⃗t+ = cosϑ ,
+i.e. ϑ = π − α ,
+(12)
+A22
+=
+1
+�
+κ2
+−τ 2
+− + κ′2
+−
+�
+κ2
++τ 2
++ + κ′2
++
+�
+κ−τ−κ+τ+ (cosϕcosψ − sinϕsinψcosϑ)
++κ′
+−κ′
++(cosϕcosψcosϑ − sinϕsinψ) + κ−τ−κ′
++ (sinϕcosψ + cosϕsinψcosϑ)
+−κ+τ+κ′
+− (cosϕsinψ + sinϕcosψcosϑ)
+�
+,
+(13)
+A12
+=
+sinϑ
+�
+κ2
++τ 2
++ + κ′2
++
+�
+κ+τ+ sinϕ − κ′
++ cosϕ
+�
+,
+(14)
+A21
+=
+sinϑ
+�
+κ2
+−τ 2
+− + κ′2
+−
+�
+κ−τ− sinψ + κ′
+− cosψ
+�
+.
+(15)
+The part of A22 containing the factor cosϑ is related to the product of A12 and A21
+in an obvious manner. With a bit more careful inspection one gets for the whole A22
+A22(ϕ, ϑ, ψ) =
+1
+sin2ϑ
+�
+A12(ϕ + π
+2 ) A21(ψ + π
+2 ) − cosϑ A12(ϕ)A21(ψ)
+�
+.
+(16)
+Now A12 and A21 for arguments shifted by π
+2 are not independent, but related by
+�
+A12(ϕ)
+�2 +
+�
+A12(ϕ + π
+2)
+�2 = (A21(ψ)
+�2 +
+�
+A21(ψ + π
+2 )
+�2 = sin2ϑ .
+(17)
+This means that a complete set of independent conformal invariants attributed to the
+cusp is given by the three parameters 4
+α ,
+B12 = κ+τ+ sinϕ − κ′
++ cosϕ
+�
+κ2
++τ 2
++ + κ′2
++
+,
+B21 = κ−τ− sinψ + κ′
+− cosψ
+�
+κ2
+−τ 2
+− + κ′2
+−
+.
+(18)
+4Remember ϕ, ϑ, ψ Euler angles, α = π − ϑ opening angle of the cusp.
+3
+
+Let us add a warning. Inserting by brute force ϕ = ψ = 0 into B12 and B21 one
+could reach the wrong conclusion that
+κ′
+√
+κ2τ 2+κ′2 could be an invariant for smooth
+curves. But since putting ϕ or ψ to zero is not a conformal invariant statement, this
+conclusion is not allowed and also straightforwardly proven to be wrong.
+We continue with some casual comment on the conformal invariants in 4-dimensio-
+nal space. To fix all the osculating spheres from S1 to S3 at a smooth point one needs
+κ, κ′, κ′′, τ1, τ ′
+1, τ2. This for the limits from both sides of the cusp, together with 6 Euler
+angles, needed in 4D for the rotation of the Frenet frame, gives 18 metrical parameters.
+The difference of the number of parameters oft the conformal and isometry group is
+now 5, hence we tentatively reach 13 conformal parameters. Now on both sides the
+limits of the differential of the conformal length and the first conformal torsion5 are
+not related to the cusp. Hence we can expect 13 − 4 = 9 conformal parameters to be
+attributed genuinely to the cusp.
+On the other side, among the nine pairs of osculating spheres
+(Si
+−, Sj
++), i, j = 1, 2, 3, the pairs (S2
+−, S2
++), (S2
+−, S1
++), (S1
+−, S2
++) have two invariants and
+all other only one [9]. This yields 12 conformal parameters. We keep it as an open
+question, whether there are indeed three relations of the 3D type (16),(17) to reach
+the minimal number of 9 independent parameters seen in the previous paragraph.
+We close this section with a comment on the cusp anomalous dimension for Wilson
+loops. It is generally believed to depend on the cusp angle α only, and therefore
+calculations have been done using straight edges. While in field theoretic perturbation
+theory this can be justified by power counting in the corresponding Feynman integrals,
+a rigorous proof for the generic situation at strong coupling is still lacking. We have
+presented a proof for the planar case with generically curved edges in [10]. For full
+generality in 3D, a proof of its independence from B12 and B21 is still lacking.
+3
+Comments on Wilson loops on polygon-like
+curves with circular edges
+The polygon-like curves with circular edges, whose related Wilson loops have been
+studied in our papers [1, 2], are not covered by the setting for generic curves as
+presented in the previous section. Along their edges one has constant curvature κ
+and zero torsion τ, resulting in zero conformal length (1) and undefined conformal
+curvature and torsion. In mathematical language these edges are conformal vertices.6
+Although such single edges carry no conformal data, their combination to a poly-
+gon does7 It has still no extension in the sense of conformal length, but there are of
+course the cusp angles at each cusp of the polygon and for more than 3 cusp points
+the corresponding cross ratios. In addition, like generic curves, these polygons can
+wind themselves out of a plane and exhibit torsion. In our paper [1] we have this
+issue parameterised by the introduction of torsion angles βj, defined at a given cusp
+5In 3D the conformal torsion (3) cannot be build out of the metrical parameters needed to fix all
+the osculating spheres at the cusp. But in 4D the osculating S3 inherits all the information.
+6See e.g. [3,5,6].
+7In a sense one could call it a conformal vertex with internal conformal substructure.
+4
+
+point xj by
+βj = ∡({xj, xj+1}, ccj) ,
+(19)
+where {xj, xj+1} denotes the circular edge between xj and xj+1 and ccj the circle
+fixed by the three cusp points xj−1, xj, xj+1.
+However, in contrast to B12 and B21 for cusps of generic curves, these torsion
+angles are not attributed to the local properties at the corresponding cusp. This is
+simply seen by changing in (19) the neighbouring cusp point xj+1 along the circle
+of which the edge {xj, xj+1} is a part . Then the tangent of the edge at xj remains
+the same, but the circle ccj and its tangent at xj changes. By this manipulation βj
+changes, although the local situation at the cusp at xj remains the same as before.
+We end with a remark on some setting for Wilson loops on piecewise smooth
+curves intermediate between those of full generality considered in section 2 and the
+polygons with circular edges in [1]. In conformal invariant gauge field theories as in
+e.g. N = 4 SYM there will be an anomalous conformal Ward identity of the type
+derived in [1], imposing for the Wilson loop the structure of a conformally covariant
+factor depending on the distances of the tips of the cusps times a conformally invariant
+remainder factor. In the generic case this remainder factor is a function of the cross
+ratios formed out of cusp points and the conformal cusp parameters identified in
+section 2, but in addition also a functional of the conformal invariants as functions
+along the edges.
+For the case of polygons with circular edges the remainder is a
+function of only a finite number of conformal parameters.
+To have a curve characterised by a finite number of conformal parameters but
+nevertheless having nonzero conformal length, one could consider polygons with edges
+which are pieces of curves with constant conformal curvature and torsion. Such curves
+have been classified in [11]. Among them are loxodromes on rotational surfaces.
+Acknowledgement
+I thank the Quantum Field and String Theory Group at Humboldt University for
+kind hospitality.
+5
+
+References
+[1] H.
+Dorn,
+“On
+anomalous
+conformal
+Ward
+identities
+for
+Wilson
+loops
+on
+polygon-like
+contours
+with
+circular
+edges,”
+JHEP
+03
+(2020),
+166
+doi:10.1007/JHEP03(2020)166 [arXiv:2001.03391 [hep-th]].
+[2] H. Dorn, “Wilson loops for triangular contours with circular edges,” J. Phys. A
+54 (2021) no.22, 225402 doi:10.1088/1751-8121/abe311 [arXiv:2010.14822 [hep-
+th]].
+[3] G. Cairns, R. Sharpe, and L. Webb, “Conformal Invariants for Curves and Sur-
+faces in Three Dimensional Space Forms”, Rocky Mountain J. Math. Volume 24,
+Number 3 (1994), 933-959.
+[4] Y. He, C. Huang and M. Kruczenski, “Minimal area surfaces in AdSn+1
+and Wilson loops,”
+JHEP 02 (2018),
+027 doi:10.1007/JHEP02(2018)027
+[arXiv:1712.06269 [hep-th]].
+[5] M.C. Romero-Fuster and E. Sanabria-Codesal,“Generalized evolutes, vertices
+and conformal invariants of curves in Rn+1”, Indag. Mathem., N.S., 10 (2), 297-
+305
+[6] A. Montesinos Amilibia’, M.C. Romero Fuster and E. Sanabria -Codesal, “Con-
+formal curvatures of curves in Rn+1”, lndag. Mathem., N.S., 12 (3) 369-382
+[7] R. Langevin, J. O’Hara and S. Sakata, “Space of subspheres and conformal
+invariants of curves” arXiv:1102.0344 [math.DG]
+[8] R. Langevin, J O’Hara, S Sakata, “Application of spaces of subspheres to con-
+formal invariants of curves and canal surfaces”, Annales Polonici Mathematici,
+January 2013 , doi: 10.4064/ap108-2-1
+[9] R. Sulanke, “M¨obius Invarints for Pairs of Spheres (Sm
+1 , Sl
+2) in the M¨obius Space
+Sn”, Contributions to Algebra and Geometry, Vol. 41 (2000), No.1, 233-246
+[10] H. Dorn, “Wilson loops at strong coupling for curved contours with cusps,”
+J. Phys. A 49 (2016) no.14, 145402 doi:10.1088/1751-8113/49/14/145402
+[arXiv:1509.00222 [hep-th]].
+[11] R. Sulanke, “ Submanifolds of the M¨obius space II, Frenet formulas and curves
+of constant curvatures.” Mathematische Nachrichten 100.1 (1981): 235-247.
+6
+
diff --git a/G9AzT4oBgHgl3EQfjP2K/content/tmp_files/load_file.txt b/G9AzT4oBgHgl3EQfjP2K/content/tmp_files/load_file.txt
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@@ -0,0 +1,171 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf,len=170
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='01513v1  [hep-th]  4 Jan 2023  HU-EP-23/01  Remarks on conformal invariants for  piecewise smooth curves and Wilson loops  Harald Dorn 1  Institut f¨ur Physik und IRIS Adlershof, Humboldt-Universit¨at zu Berlin,  Zum Großen Windkanal 6, D-12489 Berlin, Germany  Abstract  This short note is some obvious mathematical addendum to our papers on Wilson  loops on polygon-like contours with circular edges [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  Using the technique of  osculating spheres and circles we identify the conformal invariants characterising the  kinks (cusps) of generic piecewise smooth curves in 3-dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  1dorn@physik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='hu-berlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='de  1  Introduction  The metrical invariants of smooth curves in Euclidean D-dimensional space are length,  curvature and (D −2) torsion parameters as functions along the curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' For piecewise  smooth curves with cusps 2, in addition each cusp is characterised by the Euler angles  specifying the orthogonal transformation needed to rotate the Frenet-Serret frames  on both sides of a cusp to one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  What concerns the issue of conformal invariants, there is for smooth curves a  considerable amount of mathematical papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' In analogy to the metrical invariants  length s, curvature κ and torsion τ one finds for 3-dimensional curves, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' [3],  conformal length ω,3  dω = √νds,  ν =  �  (κ′)2 + κ2τ 2 ,  (1)  conformal curvature Q  Q = 4(ν′′ − κ2ν)ν − 5(ν′)2  8ν3  ,  (2)  and conformal torsion T  T = 2(κ′)2τ + κ2τ 3 + κκ′τ ′ − κκ′′τ  ν  5  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  (3)  Use of these invariants has been made in the physical literature to characterise the  boundary conditions for the treatment of minimal surfaces in AdS via Pohlmeyer  reduction [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  But what concerns the extension to piecewise smooth curves, we did not find any  paper in the mathematical literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' Therefore, we decided to write up what one gets  by straightforward application of one of the various techniques used for the smooth  case: the kinematics of osculating spheres, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' [5–8] and refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  This note is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' In the next section we find formulas expressing  the conformal invariants for cusps, with legs made of generic smooth pieces of curves,  in terms of their metrical invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' Then in section 3 we consider the very special  curves studied in our papers [1,2], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' polygon-like curves with circular edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' These  curves need a separate discussion, since along the circular edges all the conformal  invariants (1),(2),(3) are zero or ill-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  2  Conformal invariants of generic cusps  In 3-dimensional space the osculating circle (sphere ) at a generic point x on a smooth  curve is a circle (sphere) having contact of second (third) order with the curve at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  Since the order of contact is preserved under conformal transformations, osculating  circles (spheres) at a given point of a curve are mapped to those for the image under  2We follow the language in the Wilson loop literature and use the word cusp for a kink with  nonzero opening angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  3The prime denotes derivative with respect to s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  1  a conformal map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' Let us denote by ⃗t,⃗n,⃗b the unit tangent, normal and binormal  vectors at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' Then the center a of the osculating circle S1 is given by  a = x + 1  κ ⃗n  (4)  and the center c of the osculating sphere S2 by  c = x + 1  κ ⃗n −  κ′  τκ2 ⃗b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  (5)  We now turn to the case where x is a point of discontinuity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' the tip of a cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  The cusp is then characterised by two osculating circles S1  −, S1  + and two osculating  spheres S2  −, S2  +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' The index ” ± ” indicates the limits one gets by approaching x along  the both respective legs of the cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  Let us first count the number of conformal invariants we can expect for the cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  There are 9 metrical invariants at hand: κ±, κ′  ±, τ± and the 3 Euler angles for the  rotations from the Frenet frame {⃗t−,⃗n−,⃗b−} to {⃗t+,⃗n+,⃗b+}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' The difference of the  numbers of parameters of the 3-dimensional conformal and isometry group is equal  to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' This should result in 9 − 4 = 5 conformal parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' Now of course two  of them are the corresponding limits of the differential of the conformal length (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  Hence there remain 3 conformal parameters to be attributed genuinely to the cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  To find explicit formulas for them, we start with the conformal invariants of pairs  of spheres (Sm, Sn) of the same or of different dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' There is a rigorous math-  ematical treatment for arbitrary dimensions in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' Applied to our case it means  that to each of the pair (S1  −, S1  +), (S1  −, S2  +), (S2  −, S1  +), (S2  −, S2  +) belongs just one con-  formal parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' To proceed, we first study these four invariants and will show  afterwards, that only three of them are independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  (S1  −, S1  +)  The tangents of the osculating circles agree with those of the curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  Therefore the related conformal invariant is  A11 = ⃗t−⃗t+ =  − cos α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  (6)  Here α is the cusp angle (understood as the opening angle, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' α = π in the smooth  case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  (S2  −, S2  +)  Now the conformal invariant is given by the so-called inversive product,  see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' [5,9]  A22 = R2  − + R2  + − (c− − c+)2  2R−R+  = (c− − x)(c+ − x)  R−R+  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  (7)  R− and R+ are the radii of S2  − and S2  +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' Obviously A22 is the cosine of the angle  between the vectors pointing from x to the centers of the two osculating spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  Strictly speaking, only its absolute value is invariant, since A22 changes sign under  those special conformal transformations for which the preimage of infinity is situated  inside just one of the spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' The same comment applies to A12 and A21 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  (S2  −, S1  +) and (S1  −, S2  +) For a sphere and an intersecting circle the conformal invariant  2  is the scalar product of the unit vector pointing from x to the center of the sphere  with the unit tangent of the circle  A12 =  ⃗t−(c+ − x)  R+  ,  A21 = (c− − x) ⃗t+  R−  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  (8)  Using (4),(5) for both legs of the cusp we get from (7) and (8)  A22 = κ−τ−κ+τ+ ⃗n−⃗n+ + κ′  −κ′  + ⃗b−⃗b+ − κ−τ−κ′  + ⃗n−⃗b+ − κ+τ+κ′  − ⃗b−⃗n+  �  κ2  −τ 2  − + κ′2  −  �  κ2  +τ 2  + + κ′2  +  ,  (9)  A12  =  κ+τ+ ⃗t−⃗n+ − κ′  + ⃗t−⃗b+  �  κ2  +τ 2  + + κ′2  +  ,  (10)  A21  =  κ−τ− ⃗t+⃗n− − κ′  − ⃗t+⃗b−  �  κ2  −τ 2  − + κ′2  −  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  (11)  All the scalar products in the above formulas can be expressed in terms of three Euler  angles ϕ, ϑ, ψ needed to rotate the Frenet frame {n−, b−, t−} to {n+, b+, t+}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' Then  we get  A11 = ⃗t−⃗t+ = cosϑ ,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' ϑ = π − α ,  (12)  A22  =  1  �  κ2  −τ 2  − + κ′2  −  �  κ2  +τ 2  + + κ′2  +  �  κ−τ−κ+τ+ (cosϕcosψ − sinϕsinψcosϑ)  +κ′  −κ′  +(cosϕcosψcosϑ − sinϕsinψ) + κ−τ−κ′  + (sinϕcosψ + cosϕsinψcosϑ)  −κ+τ+κ′  − (cosϕsinψ + sinϕcosψcosϑ)  �  ,  (13)  A12  =  sinϑ  �  κ2  +τ 2  + + κ′2  +  �  κ+τ+ sinϕ − κ′  + cosϕ  �  ,  (14)  A21  =  sinϑ  �  κ2  −τ 2  − + κ′2  −  �  κ−τ− sinψ + κ′  − cosψ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  (15)  The part of A22 containing the factor cosϑ is related to the product of A12 and A21  in an obvious manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' With a bit more careful inspection one gets for the whole A22  A22(ϕ, ϑ, ψ) =  1  sin2ϑ  �  A12(ϕ + π  2 ) A21(ψ + π  2 ) − cosϑ A12(ϕ)A21(ψ)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  (16)  Now A12 and A21 for arguments shifted by π  2 are not independent, but related by  �  A12(ϕ)  �2 +  �  A12(ϕ + π  2)  �2 = (A21(ψ)  �2 +  �  A21(ψ + π  2 )  �2 = sin2ϑ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  (17)  This means that a complete set of independent conformal invariants attributed to the  cusp is given by the three parameters 4  α ,  B12 = κ+τ+ sinϕ − κ′  + cosϕ  �  κ2  +τ 2  + + κ′2  +  ,  B21 = κ−τ− sinψ + κ′  − cosψ  �  κ2  −τ 2  − + κ′2  −  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  (18)  4Remember ϕ, ϑ, ψ Euler angles, α = π − ϑ opening angle of the cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  3  Let us add a warning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' Inserting by brute force ϕ = ψ = 0 into B12 and B21 one  could reach the wrong conclusion that  κ′  √  κ2τ 2+κ′2 could be an invariant for smooth  curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' But since putting ϕ or ψ to zero is not a conformal invariant statement, this  conclusion is not allowed and also straightforwardly proven to be wrong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  We continue with some casual comment on the conformal invariants in 4-dimensio-  nal space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' To fix all the osculating spheres from S1 to S3 at a smooth point one needs  κ, κ′, κ′′, τ1, τ ′  1, τ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' This for the limits from both sides of the cusp, together with 6 Euler  angles, needed in 4D for the rotation of the Frenet frame, gives 18 metrical parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  The difference of the number of parameters oft the conformal and isometry group is  now 5, hence we tentatively reach 13 conformal parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' Now on both sides the  limits of the differential of the conformal length and the first conformal torsion5 are  not related to the cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' Hence we can expect 13 − 4 = 9 conformal parameters to be  attributed genuinely to the cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  On the other side, among the nine pairs of osculating spheres  (Si  −, Sj  +), i, j = 1, 2, 3, the pairs (S2  −, S2  +), (S2  −, S1  +), (S1  −, S2  +) have two invariants and  all other only one [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' This yields 12 conformal parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' We keep it as an open  question, whether there are indeed three relations of the 3D type (16),(17) to reach  the minimal number of 9 independent parameters seen in the previous paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  We close this section with a comment on the cusp anomalous dimension for Wilson  loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' It is generally believed to depend on the cusp angle α only, and therefore  calculations have been done using straight edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' While in field theoretic perturbation  theory this can be justified by power counting in the corresponding Feynman integrals,  a rigorous proof for the generic situation at strong coupling is still lacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' We have  presented a proof for the planar case with generically curved edges in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' For full  generality in 3D, a proof of its independence from B12 and B21 is still lacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  3  Comments on Wilson loops on polygon-like  curves with circular edges  The polygon-like curves with circular edges, whose related Wilson loops have been  studied in our papers [1, 2], are not covered by the setting for generic curves as  presented in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' Along their edges one has constant curvature κ  and zero torsion τ, resulting in zero conformal length (1) and undefined conformal  curvature and torsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' In mathematical language these edges are conformal vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='6  Although such single edges carry no conformal data, their combination to a poly-  gon does7 It has still no extension in the sense of conformal length, but there are of  course the cusp angles at each cusp of the polygon and for more than 3 cusp points  the corresponding cross ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' In addition, like generic curves, these polygons can  wind themselves out of a plane and exhibit torsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' In our paper [1] we have this  issue parameterised by the introduction of torsion angles βj, defined at a given cusp  5In 3D the conformal torsion (3) cannot be build out of the metrical parameters needed to fix all  the osculating spheres at the cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' But in 4D the osculating S3 inherits all the information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  6See e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' [3,5,6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  7In a sense one could call it a conformal vertex with internal conformal substructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  4  point xj by  βj = ∡({xj, xj+1}, ccj) ,  (19)  where {xj, xj+1} denotes the circular edge between xj and xj+1 and ccj the circle  fixed by the three cusp points xj−1, xj, xj+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  However, in contrast to B12 and B21 for cusps of generic curves, these torsion  angles are not attributed to the local properties at the corresponding cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' This is  simply seen by changing in (19) the neighbouring cusp point xj+1 along the circle  of which the edge {xj, xj+1} is a part .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' Then the tangent of the edge at xj remains  the same, but the circle ccj and its tangent at xj changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' By this manipulation βj  changes, although the local situation at the cusp at xj remains the same as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  We end with a remark on some setting for Wilson loops on piecewise smooth  curves intermediate between those of full generality considered in section 2 and the  polygons with circular edges in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' In conformal invariant gauge field theories as in  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' N = 4 SYM there will be an anomalous conformal Ward identity of the type  derived in [1], imposing for the Wilson loop the structure of a conformally covariant  factor depending on the distances of the tips of the cusps times a conformally invariant  remainder factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' In the generic case this remainder factor is a function of the cross  ratios formed out of cusp points and the conformal cusp parameters identified in  section 2, but in addition also a functional of the conformal invariants as functions  along the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  For the case of polygons with circular edges the remainder is a  function of only a finite number of conformal parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  To have a curve characterised by a finite number of conformal parameters but  nevertheless having nonzero conformal length, one could consider polygons with edges  which are pieces of curves with constant conformal curvature and torsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' Such curves  have been classified in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' Among them are loxodromes on rotational surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  Acknowledgement  I thank the Quantum Field and String Theory Group at Humboldt University for  kind hospitality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  5  References  [1] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  Dorn,  “On  anomalous  conformal  Ward  identities  for  Wilson  loops  on  polygon-like  contours  with  circular  edges,”  JHEP  03  (2020),  166  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='1007/JHEP03(2020)166 [arXiv:2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='03391 [hep-th]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  [2] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' Dorn, “Wilson loops for triangular contours with circular edges,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' A  54 (2021) no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='22, 225402 doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='1088/1751-8121/abe311 [arXiv:2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='14822 [hep-  th]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  [3] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' Cairns, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' Sharpe, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' Webb, “Conformal Invariants for Curves and Sur-  faces in Three Dimensional Space Forms”, Rocky Mountain J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' Volume 24,  Number 3 (1994), 933-959.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  [4] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' He, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' Huang and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' Kruczenski, “Minimal area surfaces in AdSn+1  and Wilson loops,”  JHEP 02 (2018),  027 doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='1007/JHEP02(2018)027  [arXiv:1712.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='06269 [hep-th]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  [5] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' Romero-Fuster and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' Sanabria-Codesal,“Generalized evolutes, vertices  and conformal invariants of curves in Rn+1”, Indag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' Mathem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=', 10 (2), 297-  305  [6] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' Montesinos Amilibia’, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' Romero Fuster and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' Sanabria -Codesal, “Con-  formal curvatures of curves in Rn+1”, lndag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' Mathem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=', 12 (3) 369-382  [7] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' Langevin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' O’Hara and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' Sakata, “Space of subspheres and conformal  invariants of curves” arXiv:1102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='0344 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='DG]  [8] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' Langevin, J O’Hara, S Sakata, “Application of spaces of subspheres to con-  formal invariants of curves and canal surfaces”, Annales Polonici Mathematici,  January 2013 , doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='4064/ap108-2-1  [9] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' Sulanke, “M¨obius Invarints for Pairs of Spheres (Sm  1 , Sl  2) in the M¨obius Space  Sn”, Contributions to Algebra and Geometry, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' 41 (2000), No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='1, 233-246  [10] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' Dorn, “Wilson loops at strong coupling for curved contours with cusps,”  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' A 49 (2016) no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='14, 145402 doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='1088/1751-8113/49/14/145402  [arXiv:1509.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='00222 [hep-th]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  [11] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' Sulanke, “ Submanifolds of the M¨obius space II, Frenet formulas and curves  of constant curvatures.” Mathematische Nachrichten 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='1 (1981): 235-247.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  6' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+Astronomy & Astrophysics manuscript no. Marchandetal2023
+©ESO 2023
+January 5, 2023
+Fast methods to track grain coagulation and ionization. III.
+Protostellar collapse with non-ideal MHD
+P. Marchand1, 2, U. Lebreuilly3, M.-M. Mac Low2, V. Guillet4, 5
+1 Institut de Recherche en Astrophysique et Planétologie, Université de Toulouse, UT3-PS, CNRS, CNES, 9 av. du Colonel Roche,
+31028 Toulouse Cedex 4, France
+2 Department of Astrophysics, American Museum of Natural History, 200 Central Park West, NY, NY, 10024, USA
+3 AIM, CEA, CNRS, Université Paris-Saclay, Université Paris Diderot, Sorbonne Paris Cité, 91191 Gif-sur-Yvette, France
+4 Université Paris-Saclay, CNRS, Institut d’astrophysique spatiale, 91405, Orsay, France
+5 Laboratoire Univers et Particules de Montpellier, Université de Montpellier, CNRS/IN2P3, CC 72, Place Eugène Bataillon, 34095
+Montpellier Cedex 5, France
+ABSTRACT
+Dust grains influence many aspects of star formation, including planet formation, opacities for radiative transfer, chemistry, and
+the magnetic field via Ohmic, Hall, and ambipolar diffusion. The size distribution of the dust grains is the primary characteristic
+influencing all these aspects. Grain size increases by coagulation throughout the star formation process. We describe here numerical
+simulations of protostellar collapse using methods described in earlier papers of this series. We compute the evolution of the grain
+size distribution from coagulation and the non-ideal magnetohydrodynamics effects self-consistently and at low numerical cost. We
+find that the coagulation efficiency is mostly affected by the time spent in high-density regions. Starting from sub-micron radii, grain
+sizes reach more than 100 µm in an inner protoplanetary disk that is only 1000 years old. We also show that the growth of grains
+significantly affects the resistivities, and indirectly the dynamics and angular momentum of the disk.
+1. Introduction
+Grains play a major role during star formation. Firstly, they
+are the seed of planet formation. While their characteristic size
+is sub-micron in the interstellar medium (ISM) (Mathis et al.
+1977), they grow by coagulation during the collapse, and can
+reach sizes larger than 10 µm in the early stages of protostel-
+lar collapse (Guillet et al. 2020; Silsbee et al. 2020; Tsukamoto
+et al. 2021; Vorobyov et al. 2022; Bate 2022). Observations also
+suggest they reach these sizes, if not larger, in the envelopes of
+Class 0-I objects (Kwon et al. 2009; Miotello et al. 2014; Le
+Gouellec et al. 2019; Galametz et al. 2019). Their growth then
+continues in protoplanetary disks until they eventually become
+planetesimals. Variations in their size also significantly impact
+non-ideal magnetohydrodynamics (MHD) effects through their
+ionization and their chemical interactions with the gas, with a
+direct feedback on the dynamics of the gas (Marchand et al.
+2016; Zhao et al. 2016, 2018; Marchand et al. 2020; Guillet et al.
+2020). Non-ideal MHD effects have been shown to be critical to
+the regulation of magnetic field and angular momentum during
+the protostellar collapse and the protoplanetary disk evolution
+(Mouschovias & Paleologou 1979; Machida et al. 2006; Duf-
+fin & Pudritz 2008; Mellon & Li 2009; Li et al. 2011; Tomida
+et al. 2015; Wurster et al. 2016; Masson et al. 2016; Vaytet et al.
+2018; Marchand et al. 2020; Lebreuilly et al. 2021). Grains are
+also the main source of opacity in protostellar environments, af-
+fecting the cooling of the gas and the observations made of those
+systems. Their high optical depth at densities ρ > 10−13 g cm−3
+leads to the formation of the first hydrostatic core (Larson 1969).
+However, numerical simulations usually do not account for
+grain coagulation self-consistently due to the great cost of com-
+puting a coagulation algorithm on-the-fly (although new meth-
+ods are being developed, see the recent work by Lombart &
+Laibe 2021). The dust evolution used to be pre-processed or
+post-processed with no self-consistent feedback on the dynam-
+ics (Rossi et al. 1991; Dullemond & Dominik 2005; Zhao et al.
+2016; Marchand et al. 2020). Recently, more and more studies
+include the growth of grains in their hydrodynamics simulations
+(Tsukamoto et al. 2021; Vericel et al. 2021; Vorobyov et al. 2022;
+Bate 2022). In Marchand et al. (2021, hereafter Paper I), we
+presented a simple and fast method to track coagulation self-
+consistently that is particularly suited for modeling star forma-
+tion. We now apply this method to non-ideal MHD protostellar
+collapse simulations. It is coupled with the second method pre-
+sented in Paper I: a fast calculation of the ionization and grain
+charge, to obtain non-ideal MHD resistivities. These 3D simula-
+tions are the first to include a self-consistent grain growth with a
+direct feedback on the dynamics through the self-consistent cal-
+culation of MHD resistivities.
+The paper is organized as follows. In Section 2, we describe
+the methods used in our study. Section 3 presents the results of
+our calculations, both analytical in Section 3.1 and numerical in
+Section 3.2. We compare our results to other works and discuss
+the caveats in Section 4, and conclude in Section 5.
+2. Methods
+We perform non-ideal MHD simulations with the RAMSES
+code (Teyssier 2002). RAMSES is an Eulerian gas dynamics
+code with adaptive mesh refinement (AMR) and self-gravity. It
+includes a monofluid treatment of non-ideal MHD effects (Mas-
+son et al. 2012; Marchand et al. 2018). We have implemented
+the methods presented in Paper I to calculate the coagulation
+and ionization of grains on-the-fly in a self-consistent manner.
+Article number, page 1 of 11
+arXiv:2301.01510v1  [astro-ph.SR]  4 Jan 2023
+
+A&A proofs: manuscript no. Marchandetal2023
+2.1. Grain coagulation and ionization
+2.1.1. Coagulation
+In Paper I, we demonstrated that the coagulation process as
+described by the Smoluchowski (1916) equation is a one-
+dimensional process with certain types of coagulation kernels.
+Consequently, the size distribution of the coagulated grains de-
+pends only on the initial size distribution and a variable χ that
+encompasses the whole history of the physical conditions seen
+by the grains. At a given χ that is integrated along the path of
+the grains, the coagulated distribution is always the same inde-
+pendently of the actual path taken. This method works for every
+coagulation kernel for which the dependence on the gas variables
+such as density and temperature can be separated from the grain
+properties such as size and mass. In Paper I and in the present
+paper, we use the turbulent kernel derived by Ormel & Cuzzi
+(2007) in the intermediate coupling regime, which is suited for
+star formation conditions. We show that in this case χ can be
+derived from integrating
+dχ = n
+3
+4
+HT − 1
+4 dt,
+(1)
+where nH is the number density of the gas, T its temperature,
+and t the time. In three-dimensional (3D) hydrodynamics simu-
+lations, only the knowledge of χ is needed to track the coagula-
+tion of grains.
+Equation (1) is a Lagrangian derivative of χ with respect to
+time along the path of the grain. We can transform it into a partial
+(Eulerian) derivative
+∂χ
+∂t + u · ∇χ = n
+3
+4
+HT − 1
+4 ,
+(2)
+where u is the velocity of the gas. We can combine equation (2)
+with the mass conservation equation
+∂ρ
+∂t + ∇ · [ρu] = 0,
+(3)
+where ρ is the gas mass density. This yields
+∂ρχ
+∂t + ∇ · (ρχu) = ρn
+3
+4
+HT − 1
+4 ,
+(4)
+which means that the quantity ρχ can be treated in an Eulerian
+framework as a passive scalar with a source term. We exploit this
+property and implement it as such in RAMSES. The value of χ
+is therefore calculated self-consistently in all cells at each time-
+step, as a mass-weighted average, ignoring any diffusion of dust
+through the gas. We used the code Ishinisan (Marchand et al.
+2021) to pre-calculate a table containing the grain size distribu-
+tion for a large number of χ values in a large relevant interval
+(between χ = 1013 and χ = 1019 in cgs units). When the size
+distribution is needed during the hydrodynamical simulation, it
+is interpolated from the table based on the value of χ.
+Our initial distribution in this paper is a Mathis et al. (1977,
+MRN) distribution. The minimum and maximum radii are amin =
+5 nm and amax = 250 nm, and the slope of the distribution is
+−3.5, so that the number density of grains n follows the variation
+dn
+da ∝ a−3.5.
+(5)
+The total quantity of grains is determined by the dust-to-gas
+mass ratio that we assume to be 0.01. In this work, we sample the
+distribution with 60 bins of size logarithmically spaced between
+10-10
+10-9
+10-8
+10-7
+10-6
+10-5
+10-4
+10-3
+10-2
+10-1
+10-3
+10-2
+10-1
+100
+101
+102
+103
+Fractional mass X (ρ/nHmp)
+Radius (µm)
+Initial MRN
+χ = 1015 cgs
+χ = 1016 cgs
+χ = 1017 cgs
+χ = 1017.5 cgs
+χ = 1018 cgs
+χ = 1018.5 cgs
+χ = 1019 cgs
+Fig. 1. State of the grain size distribution for values of χ, between 1015
+cgs and 1019 cgs. The points represent the fractional abundance of the
+size-bin as function of the effective radius of the bin.
+5 nm and 5000 µm. In Figure 1, we present the coagulated MRN
+size distribution at various χ. Below χ = 1017 cgs, the shift in
+the size distribution is negligible. For higher values, the maxi-
+mum size and the peak of the distribution are located at larger
+and larger radii, while the slope of the distribution remains sim-
+ilar. In all cases, the small grains are more abundant while the
+large grains hold more mass. The mode of the size distribution
+amax is located near the largest relevant grain size.
+As grains grow, they may experience fragmentation when the
+kinetic energy of the collision is high. Contrarily to what we de-
+rived in paper I, we find that fragmentation does not occur in
+early stages of the disk, but rather only at very high densities
+ρ > 1012 cm−3 (Lebreuilly et al. 2023) for very large grains with
+a > 0.08 cm. We demonstrate this result in Appendix C. We
+therefore neglect fragmentation in this work. We also do not ac-
+count for the grain drift with respect to the gas in this work. We
+discuss the possible consequences in Section 4.2. Drift is, how-
+ever, compatible with our coagulation model, and we detail the
+method in Appendix B. Other limitations are discussed in Sec-
+tion 4.4.
+2.1.2. Ionization and resistivities
+In Paper I, we also presented a fast method to calculate the ion-
+ization of the gas-grain mixture. For an arbitrary size distribu-
+tion, we can calculate the average electric charge of each grain
+size, the number of ions and the number of electrons, provided
+the cosmic-ray (CR) ionization rate, the density and temperature
+of the gas, the average atomic mass of ions µi and the sticking
+probability of electrons on grains se. Here, we assume µi = 28,
+which corresponds to the ion HCO+, and se = 0.6 as in Marc-
+hand et al. (2016). We also assume ζ = 5 × 10−17 s−1. The den-
+sity and the temperature, are taken from the hydrodynamic sim-
+ulation. The calculation is performed by the Newton-Raphson
+scheme described in Appendix A of Paper I. The resistivities are
+computed using a similar method to Marchand et al. (2016), with
+one difference. For each grain size, they sum over the contribu-
+tions of the whole charge distribution (between -1 to +1 in their
+case). Instead, we average the contributions using the mean elec-
+tric charge. We explicit the method in more details in Appendix
+A.
+Article number, page 2 of 11
+
+P. Marchand et al.: Fast methods to track grain coagulation and ionization. III. Protostellar collapse with non-ideal MHD
+2.2. Star formation simulations
+2.2.1. Model
+We perform four numerical simulations using the RAMSES
+code. We solve the following MHD equations
+∂ρ
+∂t + ∇ · �ρu� = 0,
+(6)
+∂ρu
+∂t + ∇ ·
+�
+ρuu +
+�
+P + B2
+2
+�
+I − BB
+�
+= −ρ∇Φ,
+(7)
+∂B
+∂t − ∇ ×
+�
+u × B − ηΩJ − ηH
+J × B
+B
++ ηAD
+(J × B) × B
+B2
+�
+= 0,
+(8)
+∇ · B = 0,
+(9)
+where u is the velocity of the gas, P its pressure, B the mag-
+netic field, J = ∇ × B the current, I the identity matrix, Φ the
+gravitational potential, and ηΩ, ηH and ηAD the ohmic, Hall and
+ambipolar resistivities. The temperature evolution is prescribed
+by the barotropic equation
+T = T0
+�������1 +
+�
+ρ
+10−13 g cm−3
+�γ−1������� ,
+(10)
+with T0 = 10 K and γ = 5/3 the adiabatic index. The initial
+condition is a sphere of gas of 1 M⊙. The radius of the sphere is
+controlled by the thermal over gravitational energy ratio α. We
+choose α = 0.3, which sets a radius of R = 2946 au. The density
+distribution has an m = 2 azimuthal perturbation
+ρ(ϕ) = ρ0(1 + δρ sin ϕ),
+(11)
+where ρ0 is the density of a uniform sphere of same mass and
+radius, and ϕ the azimuthal angle. We choose δρ = 0.05. The
+computational domain outside of the sphere is filled with gas of
+density ρ0/100. The sphere undergoes a solid rotation, with a
+ratio of rotational to gravitational energy of β = 0.02. The mag-
+netic field is initially uniform and parallel to the rotation axis.
+It is defined using the mass-to-flux ratio over the critical value
+(Mouschovias & Spitzer 1976)
+µB =
+M/ΦB
+(M/ΦB)crit
+,
+(12)
+with
+� M
+ΦB
+�
+crit
+= 0.53
+3π
+�
+5
+G.
+(13)
+Observations show that dense cores are slightly super-critical
+(Crutcher 1999), although recent numerical simulations indicate
+that observations may overestimate the actual strength of the
+magnetic field due to projection effects (Kuznetsova et al. 2020).
+We choose µB = 5 as a fiducial value. Those parameters are
+used in our reference case C-3. We change the value of α to 0.4,
+and the initial mass M to 5 M⊙ for two other cases called C-4
+and C-3-M5, respectively, to investigate the influence of the col-
+lapse time on the grain coagulation. The final run, named NC-3,
+is similar to C-3 without coagulation. The grain distribution and
+ionization evolve as described in sections 2.1, 2.1.2, and Paper I.
+The four cases are summarized in table 1.
+2.2.2. Grid and Algorithm
+The simulation box is a cube that is four times as large as the ra-
+dius of the sphere, with periodic boundary conditions. The initial
+grid is composed of 323 cells (level 5 of AMR) and is refined to
+ensure at least 10 points per Jeans length, strongly satisfying the
+Truelove et al. (1997) criterion. The maximum AMR refinement
+level is 13 for C-3 and NC-3 (resolution of 1.4 au), 14 for C-4
+(resolution of 0.96 au) and 16 for C-3-M5 (resolution of 0.90
+au).
+Simulations are performed with a 3D unsplit slope limiter
+to avoid overshooting of the magnetic field at shock boundaries,
+while keeping the second order convergence for the Hall effect.
+We use the HLLD Riemann solver (Miyoshi & Kusano 2005)
+for non-magnetic variables and the 2D HLL Riemann solver for
+the magnetic field and the Hall effect (Balsara 2012; Marchand
+et al. 2018). The Poisson equation is solved using the multigrid
+method of Guillet & Teyssier (2011) in our periodic domain.
+3. Application to star formation
+In this section, we present the results of our calculations. We first
+apply our coagulation method to an analytical one-zone model in
+Section 3.1, then in 3D MHD simulations in Section 3.2.
+3.1. Analytical collapse
+During spherical protostellar collapse, the time evolution of the
+contraction ratio of a gas cloud compared to its original radius
+x = R(t)/R0 can be described by (Flower et al. 2005):
+dx
+dt = − π
+2τff
+�
+1
+x − 1,
+(14)
+where τff is the free-fall time, given by
+τff =
+�
+3π
+32Gρ0
+,
+(15)
+where ρ0 is the initial density. Guillet et al. (2020) showed that
+assuming a uniform compression of the gas nicely reproduces
+the isothermal phase of the collapse, particularly when compar-
+ing the dust size distribution at the same gas density. In this case,
+the density of mass scales as ρ(t) = ρ0(R0/R[t])3. The gas density
+then evolves as
+1
+nH
+dnH
+dt = 1
+ρ
+dρ
+dt = −3
+x
+dx
+dt .
+(16)
+We use a second-order Runge-Kutta scheme to numerically
+integrate this equation from the beginning of the collapse at
+ρ = ρ0 until the formation of the first Larson core at ρ =
+10−13 g cm−3. The evolution of χ with ρ is plotted in Figure 2,
+assuming T = 10 K. The solid and dashed lines represent dif-
+ferent initial densities, ρ0 = 3.8 × 10−20 g cm−3 and ρ0 =
+3.8×10−18 g cm−3, respectively. At densities nH > 10−16 g cm−3,
+the value of χ is independent from ρ0 and increases as χ ∝ ρ1/4.
+This evolution is expected as χ ∼ ρ3/4t and t ∼ τff ∼ ρ−1/2. At
+ρ = 10−13 g cm−3, the coagulation variable reaches χ ≈ 5.6×1017
+cgs, which corresponds to a peak of the size distribution of ∼ 10
+µm, indicating a significant grain growth. That is consistent with
+the results of Guillet et al. (2020), who use the same collapse
+model, but solve coagulation on the fly.
+Article number, page 3 of 11
+
+A&A proofs: manuscript no. Marchandetal2023
+Table 1. Parameters of the protostellar collapse simulations: name of the simulation, thermal over gravitational energy ratio α, radius R, mass M,
+initial density ρ0 and initial magnetic field B of the initial cloud, formation time of the first hydrostatic core tfc, use of coagulation.
+Name
+α
+R (au)
+M (M⊙)
+ρ0 (g cm−3)
+B (µG)
+tfc (kyr)
+Coagulation
+C-3
+0.3
+2946
+1
+5.49 × 10−18
+133
+∼ 30
+Yes
+C-3-M5
+0.3
+14738
+5
+2.19 × 10−19
+27
+∼ 150
+Yes
+C-4
+0.4
+3930
+1
+2.31 × 10−18
+75
+∼ 47
+Yes
+NC-3
+0.3
+2946
+1
+5.49 × 10−18
+133
+∼ 30
+No
+Fig. 2. Evolution of the coagulation variable χ with increasing density
+nH during the isothermal collapse. The solid line represents an initial
+density of ρ0 = 3.8 × 10−20 g cm−3, and the dashed line ρ0 = 3.8 ×
+10−18 g cm−3.
+3.2. Numerical collapse
+The numerical simulations were run as described in Section 2
+until 1000 years after the density reaches 10−13 g cm−3, the for-
+mation of the first hydrostatic core at ∼ 30 kyr (Tab. 1). In refer-
+ence simulation C-3, a small circumstellar disk with a radius of ≈
+20 au forms and a disk wind is launched by magneto-centrifugal
+acceleration (Blandford & Payne 1982). Figure 3 shows face-on
+and edge-on slices of density at the final time-step of the simu-
+lation, with arrows indicating the gas velocity.
+3.2.1. Grain growth
+We show in Figure 4 the value of χ as a function of the density in
+simulation cells at the final time-step, for runs C-3, C-4 and C-
+3-M5. The increase is quasi unidimensional in the isothermally
+collapsing envelope for ρ < 1015 g cm−3. Beyond this density,
+there is a large spread of χ values of over an order of magnitude.
+This spread most likely occurs in gas falling in the pseudo-disk,
+then the disk and the first Larson core, or in the outflow over dif-
+ferent timescales, and spending unequal times in a given density
+range. The overall trend agrees well with the analytical calcula-
+tion.
+There is no significant difference in the χ values, and thus
+dust size distributions, between the three runs, despite run C-4
+needing 50% more time to collapse to the first Larson core stage,
+and C-3-M5 needing 400% more time. Coagulation happening
+in the isothermally collapsing envelope is therefore hardly im-
+pacted by the initial conditions, as growth accelerates with in-
+creasing density.
+-40
+-20
+ 0
+ 20
+ 40
+x (au)
+-40
+-20
+ 0
+ 20
+ 40
+y (au)
+-14
+-13
+-12
+-11
+log(ρ) (g cm-3)
+-40
+-20
+ 0
+ 20
+ 40
+x (au)
+-40
+-20
+ 0
+ 20
+ 40
+y (au)
+-40
+-20
+ 0
+ 20
+ 40
+-40
+-20
+ 0
+ 20
+ 40
+-200
+-100
+ 0
+ 100
+ 200
+x (au)
+-200
+-100
+ 0
+ 100
+ 200
+z (au)
+-17
+-16
+-15
+-14
+-13
+-12
+log(ρ) (g cm-3)
+-200
+-100
+ 0
+ 100
+ 200
+x (au)
+-200
+-100
+ 0
+ 100
+ 200
+z (au)
+-200
+-100
+ 0
+ 100
+ 200
+-200
+-100
+ 0
+ 100
+ 200
+Fig. 3. Density slices of the C-3 simulation, with grain coagulation. The
+top panel is a face-on slice of the plane z=0, and the bottom panel an
+edge-on slice of the plane y=0. White arrows represent the direction of
+the gas velocity. The snapshot is taken at the final time-step, 1000 years
+after the formation of the first Larson core.
+Figure 5 shows the mode of the size distribution as a func-
+tion of density, corresponding to the distribution of χ shown in
+Figure 4. Three regions are clearly demarcated, the first being
+the envelope, in which grain coagulation is not efficient enough
+to form large grains (ρ < 10−16 g cm−3). The second comprises
+Article number, page 4 of 11
+
+1018
+cm
+-3
+16
+X
+101s
+1014
+10-19
+10-13
+10-18
+10-17
+10~16
+10-14
+10-12
+10-11
+02-01
+p (g cmP. Marchand et al.: Fast methods to track grain coagulation and ionization. III. Protostellar collapse with non-ideal MHD
+Fig. 4. Evolution of the coagulation variable χ with increasing density
+nH in the numerical collapse models C-3 (purple), C-3-M5 (light blue),
+and C-4 (green). Each point corresponds to a simulation cell at the final
+time-step. The red line is the analytical collapse solution for ρ0 = 3.8 ×
+10−20 g cm−3.
+the pseudo-disk and the early protoplanetary disk, where grains
+grow by a factor 100 from sub-micron sizes to several tens of
+µm. The third region is the first Larson core, which has even
+larger grains that reach 400 µm within a mere 103 yr years af-
+ter its formation. That value is in line with similar recent studies
+(Kawasaki et al. 2022; Lebreuilly et al. 2023). There is little dif-
+ference between runs C-3, C-4 and C-3-M5, confirming that co-
+agulation in the envelope does not impact large grains, as found
+by previous studies (for example Silsbee et al. 2022). We discuss
+observations of large grains in the envelope in Section 4.2.
+The spatial distribution of those grain sizes for run C-3 is
+displayed in Figure 6. Size distributions shifting significantly
+from the initial MRN distribution are indeed confined to the mid-
+plane, in the disk and pseudo-disk. The bottom panel also shows
+moderately larger grains in the outflow, as they only traveled
+through the upper layers of the pseudo-disk before being ejected
+(Marchand et al. 2020). If they had been in the mid-plane of the
+disk, they would have grown much more, as coagulation is irre-
+versible in our model. The upper panel shows that grains reach
+a radius larger than 1 µm in the outskirts of the disk, within 100
+au of the center. Growth then occurs rapidly as density increases
+in the inner 15 au, which is shown by the almost overlapping
+contours delimiting amax = 5 µm and amax = 10 µm.
+3.2.2. Resistivities and gas dynamics
+We describe here the impact of grain coagulation on non-ideal
+MHD resistivities and their macroscopic effects on gas dynam-
+ics. Previous studies emulated the coagulation of grains by re-
+moving the very small grains (a < 0.1 µm) and redistributing
+their mass to the larger end of the distribution (Zhao et al. 2016,
+2018; Marchand et al. 2020). This method leads to an increase
+of resistivities, in particular the ambipolar resistivity, resulting in
+weaker coupling between the magnetic field and the gas, hence
+weaker magnetic braking and larger, more unstable disks. How-
+ever, we observe here the exact opposite behavior.
+The middle panel of figure 7 presents the volume-weighted
+average resistivities as a function of density, at the final time-
+step, for simulations C-3 and NC-3. A non-evolving size distri-
+Fig. 5. Mode of the coagulated grain size distribution as a function of
+density in simulations (purple), C-3-M5 (light blue), and C-4 (green).
+The discrete values of the sizes are due to the binning of the size distri-
+bution.
+bution produces resistivities relatively similar in the envelope,
+but two to four orders of magnitude larger at disk densities, par-
+ticularly the Ohmic and ambipolar resistivities. Consequently,
+the magnetic braking is weakened, and the gas retains more an-
+gular momentum without the grain coagulation, forming a larger
+disk, as shown in Figure 8.
+This difference in regime can be quantified by the ambipolar
+Elsasser number Am = B2/(ρηADΩ). In regions where Am < 1,
+the ambipolar diffusion has a significant impact on the dynam-
+ics of the gas. The bottom panel of Figure 7 shows the radial
+profile of the ambipolar Elsasser number in runs C-3 and NC-
+3, azimuthally-averaged in the mid-plane. The higher resistivity
+in run NC-3 results in Am < 1 in the inner ∼ 12 au, indicat-
+ing active ambipolar diffusion and magnetic field dissipation,
+while Am ≳ 104 for C-3 over the same radial range, indicat-
+ing weak ambipolar diffusion. Figure 9 compares the disk size
+and angular momentum in both simulations, further confirming
+the lower magnetic braking in run NC-3. A second effect of the
+weaker magnetic forces from the stronger ambipolar diffusion in
+run NC-3 is the absence of outflow at this stage of evolution, as
+shown in the lower panel of Figure 8.
+The discrepancy of resistivity values between actual coagu-
+lation and methods simply redistributing the mass to the large-
+mass-end of the distribution originates from the lack of very
+large grains (> 10 µm) in the latter. In both cases, removing
+the small grains decreases the electron absorption by dust, and
+therefore decreases the Ohmic resistivity. However, the ambipo-
+lar resistivity is controlled by the relative abundance of ions
+and charged grains in the gas. Although the small grain removal
+method barely changes the abundance of ions, the dominant pos-
+itive charge carriers, it reduces significantly the charged grain
+population, leading to an increase of resistivity at low density. At
+high density, both with a standard MRN and a truncated-MRN
+distribution, the grains become the dominant charge carriers and
+the abundance of ions decreases (see the dashed lines in the top
+panel of Figure 7). That does not happen with coagulation be-
+cause the number of grains decreases significantly, hence leading
+to a higher number of ions and a lower resistivity than without
+proper coagulation. This kind of method without larger grain cre-
+ation is therefore inappropriate for emulating coagulation alone
+at a low cost for non-ideal MHD calculations. At later times, at
+Article number, page 5 of 11
+
+1020
+C-3
+C-4
+C-3-M5
+1019
+Analytical collapse
+1018
+C1017
+X
+1016
+1015
+10-T8
+10-T6
+10-T4
+10-20
+10°10
+p (g cm103
+C-3
+C-4
+C-3-M5
+10°
+10-78
+10-10
+-3
+p (g cmA&A proofs: manuscript no. Marchandetal2023
+-100
+-50
+ 0
+ 50
+ 100
+x (au)
+-100
+-50
+ 0
+ 50
+ 100
+y (au)
+0.1
+1
+10
+amax (µm)
+-100
+-50
+ 0
+ 50
+ 100
+x (au)
+-100
+-50
+ 0
+ 50
+ 100
+y (au)
+-200
+-100
+ 0
+ 100
+ 200
+x (au)
+-200
+-100
+ 0
+ 100
+ 200
+z (au)
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+amax (µm)
+-200
+-100
+ 0
+ 100
+ 200
+x (au)
+-200
+-100
+ 0
+ 100
+ 200
+z (au)
+Fig. 6. Slices of the C-3 simulation with coagulation showing the mode
+of the grain size distribution amax, 1000 years after the first core forma-
+tion. The color scale is different for each panel to yield a better contrast.
+On the top panel, black contours indicate amax = 1, 5 and 10 µm, and
+amax = 0.5 and 1 µm on the bottom panel.
+densities ρ > 10−12 g cm−3, Lebreuilly et al. (2023) showed that
+the replenishment of small grains by fragmentation would lead
+to an increase in both Ohmic and ambipolar resistivities.
+4. Discussion and caveats
+4.1. Grain growth
+As explained in previous sections and displayed in Figures 4 and
+5, initial gas conditions have little to no influence on grain coag-
+ulation for later stages, and coagulation is ineffective in the en-
+velope for growing large grains. This reinforces the idea that the
+system forgets the initial conditions at the formation of the first
+hydrostatic core and the disk (Vaytet & Haugbølle 2017). Con-
+sequently, calculations such as those presented in this work may
+provide standard initial dust grain size distributions for studies of
+10-20
+10-18
+10-16
+10-14
+10-12
+10-10
+10-8
+10-20 10-19 10-18 10-17 10-16 10-15 10-14 10-13 10-12 10-11 10-10
+ne / nH, ni / nH
+ρ (g cm-3)
+Ions
+Electrons
+No coagulation
+1010
+1012
+1014
+1016
+1018
+1020
+1022
+10-20 10-19 10-18 10-17 10-16 10-15 10-14 10-13 10-12 10-11 10-10
+MHD resistivities (cm2 s-1)
+ρ (g cm-3)
+Ohm
+Ambipolar
+Hall
+No coagulation
+10-2
+10-1
+100
+101
+102
+103
+104
+105
+1
+10
+100
+Elsasser number
+r (au)
+C-3 (coagulation)
+NC-3 (No coagulation)
+Fig. 7. Top panel: Abundances of ions (purple) and electrons (green)
+as a function of density in models with coagulation (C-3; solid lines) or
+without it (NC-3; dashed lines). Middle panel: Volume averaged Ohmic
+(purple), ambipolar (green), and Hall (blue) resistivities for the same
+models. Bottom panel: Average ambipolar Elsasser number Am in the
+mid-plane as a function of radius. The thin grey line represents Am = 1.
+protoplanetary disk evolution. Although other coagulation ker-
+nels affect the size distribution in different ways (see Section
+4.4), it is certain that even young protoplanetary disks contain
+grains significantly larger than the classical MRN distribution.
+Article number, page 6 of 11
+
+P. Marchand et al.: Fast methods to track grain coagulation and ionization. III. Protostellar collapse with non-ideal MHD
+-40
+-20
+ 0
+ 20
+ 40
+x (au)
+-40
+-20
+ 0
+ 20
+ 40
+y (au)
+-14
+-13
+-12
+-11
+log(ρ) (g cm-3)
+-40
+-20
+ 0
+ 20
+ 40
+x (au)
+-40
+-20
+ 0
+ 20
+ 40
+y (au)
+-40
+-20
+ 0
+ 20
+ 40
+-40
+-20
+ 0
+ 20
+ 40
+-200
+-100
+ 0
+ 100
+ 200
+x (au)
+-200
+-100
+ 0
+ 100
+ 200
+z (au)
+-17
+-16
+-15
+-14
+-13
+-12
+log(ρ) (g cm-3)
+-200
+-100
+ 0
+ 100
+ 200
+x (au)
+-200
+-100
+ 0
+ 100
+ 200
+z (au)
+-200
+-100
+ 0
+ 100
+ 200
+-200
+-100
+ 0
+ 100
+ 200
+Fig. 8. Same as figure 3 for simulation NC-3, without grain coagulation,
+1000 years after the formation of the first Larson core.
+This has important implications for the dynamics of grains in the
+disk. Larger grains couple differently to the gas and may trigger
+the streaming instability (Youdin & Goodman 2005; Johansen
+& Youdin 2007; Yang et al. 2017), which is an early step toward
+planet formation. Although this regime is not reached in our sim-
+ulations, the fast growth of the grains in circumstellar disks could
+predict an early onset for this process.
+4.2. Large grains in envelopes and dust diffusion
+Although grain coagulation is negligible in the envelope of our
+simulations, large grain signatures in envelopes have been ob-
+served. Galametz et al. (2019), for example, report "low and
+varying dust emissivity indices" at the envelope scale for some
+Class 0 and Class I protostars. This could be due to the pres-
+ence of mm-size grains in low numbers. The time-scale to form
+such large grains in the envelope and cold ISM cores is large
+0
+5
+10
+15
+20
+25
+30
+35
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.40
+1e+51
+2e+51
+3e+51
+4e+51
+5e+51
+6e+51
+7e+51
+Disk radius (au)
+Disk angular momentum (g cm2 s-1)
+t (kyr) after First Core formation
+Disk radius
+Disk angular momentum
+No coagulation
+Fig. 9. Disk size (purple, left axis) and angular momentum (green, right
+axis) as a function of time after the formation of the first Larson core.
+Solid lines represent simulation C-3 with coagulation, and dashed lines
+are simulation NC-3 without coagulation.
+(> 100 Myr), so we exclude the possibility of early coagulation.
+Similarly, Valdivia et al. (2019) report that their synthetic polari-
+sation observations of young protostellar envelopes in RAMSES
+calculations require grains larger than 10 µm to be consistent
+with observations, which is also inconsistent with our findings
+and those of recent studies (Silsbee et al. 2022; Lebreuilly et al.
+2023, for the latest ones).
+The aerodynamic properties of the grains may cause differ-
+ential velocities between grain populations and the gas, lead-
+ing to varying dust-to-gas ratio throughout the cloud (Lebreuilly
+et al. 2020). We note however that, as they have found, dust dif-
+fusion will start to play a significant role only for very large
+grains of a few hundred microns. Generally, large grains tend
+to accumulate in higher density regions. Tsukamoto et al. (2021)
+found that this dust diffusion can lead to what they call an ash fall
+phenomenon, in which large coagulated grains (up to millimeter-
+size) from the disk are ejected by an outflow, then decouple from
+the gas and fall back in the envelope. This process may explain
+the observations of low spectral index in Class 0 envelopes, as
+the outflow fuels the envelope with large grains formed in the
+central region. Eventually, those ejected grains circle back to the
+outer edge of the disk, enriching the large-end of the size distri-
+bution. In our case, the disk wind is fueled by the upper layers of
+the pseudo-disk, in which dust only moderately grows. However,
+dust-to-gas ratio enhancement in this region due to the grain dif-
+ferential velocities may lead to an accelerated growth.
+Further studies are needed to investigate this discrepancy
+about the size of grains in envelopes between observations and
+theory.
+4.3. Coagulation in the pseudo-disk
+Grains in our simulations seem to grow faster than in Bate (2022)
+despite their use of the same turbulent kernel as in our work. The
+peaks of the distributions in that work reach a few microns at a
+maximum density of nH = 1013 cm−3 (see their Figure 7), while
+in our simulations the peak exceeds 100 µm at a lower maximum
+density of nH ≈ 3×1012 cm−3 (or ρ ≈ 10−11 g cm−3; see our Fig-
+ure 5). That discrepancy is mainly due to a different modelling
+Article number, page 7 of 11
+
+A&A proofs: manuscript no. Marchandetal2023
+of the Reynolds number to calculate the turbulent coagulation
+kernel. They assume a constant value of Re = 108, while we use
+(Ormel et al. 2009; Guillet et al. 2020)
+Re = 6.7 × 107 �
+nH
+105 cm−3
+� 1
+2 .
+(17)
+Hence, in the central regions, our Reynolds number can be larger
+by three orders of magnitude. Their grains are then stuck in the
+tightly-coupled regime where relative grain velocities are lower
+than in the intermediate coupling-regime, which is more adapted
+to this situation (as demonstrated in Section 4.1 of paper I). The
+lower relative velocities result in lower coagulation rates, and
+therefore a slower growth rate. The poor constraints on the value
+of the Reynolds number in protostellar environments therefore
+represents a source of uncertainty for grain growth by coagula-
+tion.
+An additional explanation may involve an excess of growth
+in the pseudo-disk that forms in our simulations. The pseudo-
+disk is an over-density, typically denser than ρ ≈ 10−15 g cm−3,
+created by the convergence of gas flowing along magnetic field
+lines (Galli & Shu 1993), that takes the shape of a disk perpen-
+dicular to the magnetic field, but not supported against gravity.
+It is apparent in the bottom panels of Figures 3 and 8.
+Gas in the rotationally supported disk generally comes di-
+rectly from the pseudo-disk, and the overall efficiency of coag-
+ulation is mainly affected by the time spent in high-density re-
+gions like the pseudo-disk. This is what appears in the bottom
+panel of Figure 6, in which there is a ∼ 30 au-thick layer of large
+grains in the mid-plane. A passage of grains through the pseudo-
+disk would therefore provide an acceleration of coagulation in
+the early stage of star formation, even before arrival in the disk.
+That does not happen in Bate (2022) since they do not consider
+magnetic fields, resulting in less-coagulated size distributions as
+grains enter the disk.
+4.4. Coagulation kernel
+Our coagulation methods works for every coagulation kernel
+where the environment variables and grain variables can be sep-
+arated, one example of which is the well-known turbulent ker-
+nel derived by Ormel & Cuzzi (2007) that we use. This kernel
+is appropriate to calculate the growth of large grains, but it has
+several limitations. We assume a steady-state Kolmogorov turbu-
+lence spectrum, and that the injection-scale of the turbulence is
+equal to the Jeans length corresponding to the local density (see
+section 2.2 of paper I), which may lead to an over-estimation of
+the grain collision rate. Other kernels may have different effects
+on the size distribution of grains. Guillet et al. (2020) showed
+that this turbulent kernel would increase the maximum size of
+the distribution, while a kernel derived from ambipolar diffusion,
+that creates a drift between charged and neutral grains, is effi-
+cient at removing small grains. This also happens in Bate (2022)
+and Lebreuilly et al. (2023), which combine several processes
+including Brownian motion and pressure gradients that generate
+drift velocities between grains and rapidly remove the smaller
+grains. These processes are not, however, efficient at producing
+larger grains without the help of the turbulent kernel. Contrary
+to fragmentation, the addition of those kernels would steepen the
+distribution at its lower end.
+5. Conclusions
+We present here the results of numerical simulations of protostel-
+lar collapse with dust coagulation and self-consistent calculation
+of non-ideal MHD resistivities, using the methods detailed in
+Marchand et al. (2021). We performed four simulations, includ-
+ing three with coagulation and different collapse times, and one
+without coagulation for reference. Here are our main results.
+- Coagulated size distributions retain some characteristics of
+the initial MRN distribution, in particular the dominance of
+small grains in number and large grains in mass.
+- Dust coagulation is inefficient at growing larger grains in the
+envelope, even with long free-fall times. What matters is the
+time spent in high-density regions (ρ > 10−15 g cm−3) like
+the pseudo-disk. Fragmentation can also be ignored in the
+cloud-collapse phase.
+- Grain growth is extremely rapid in the disk. The peak of the
+size distribution in mass, which is also the largest relevant
+grain size of the distribution, reaches 1 µm in the pseudo-
+disk and more than 100 µm in the inner disk only 103 yr
+after its formation.
+- Grain sizes have a significant impact on non-ideal MHD re-
+sistivities. Coagulated grains result in resistivities up to four
+orders of magnitude lower than non-coagulated grains, with
+a significant impact on the dynamics of the disk. Simple re-
+distribution approximations fail to capture this effect, as it
+occurs because of the growth of the largest grains.
+Accounting for grain coagulation is therefore necessary in
+star formation and protoplanetary disk simulations, as grains
+grow rapidly to ≥ 100 µm in radius in disks. The effects of grain
+growth on chemistry and radiative transfer will be explored in
+future papers.
+Acknowledgements. P.M. acknowledges the financial support of the Kathryn W.
+Davis Postdoctoral Fellowship of the American Museum of Natural History and
+of the European Research Council (ERC) under the European Union’s Hori-
+zon 2020 research and innovation programme (ERC Starting Grant “Chemtrip”,
+grant agreement No 949278). M.-M.M.L. was partly supported by US NSF grant
+AST18-15461. U.L. acknowledges the financial support of the European Re-
+search Council (ERC) via the ERC Synergy Grant ECOGAL (grant 855130).
+Appendix A: Calculating the resistivities
+Appendix A.1: Solving for the ionization
+Let us assume a size distribution of grains divided into N bins.
+We have the following system of equations (see Paper I)
+Zk =ψτk +
+1 − ϵ2Θ2
+1 + ϵΘαk + ϵ2Θ2 ,
+(A.1)
+⟨ ˜J(τk)⟩ =(1 − ψ) +
+2
+τk
+�
+ϵ2Θ2 + ϵΘ
+�
+1 + ϵΘαk + ϵ2Θ2 ,
+(A.2)
+ϵ =1 − ψ
+Θeψ ,
+(A.3)
+ni = −
+1
+1 − ϵ
+�
+k
+nkZk,
+(A.4)
+f(ψ) =⟨σv⟩ieϵn2
+i
+ζnH
++ nivi
+ζnH
+�
+k
+nkπa2
+k⟨ ˜J(τk)⟩ − 1 = 0.
+(A.5)
+ϵ = ne/ni < 1 is the ratio between the number of electrons and
+ions, ψ is the ratio between the electric potential of the grains
+and the kinetic energy of electrons, nk, Zk and ak represent the
+number density, the average charge and the radius of grains in
+bin k, ζ is the CR ionization rate, and vi = [8kBT/πµimH]1/2
+Article number, page 8 of 11
+
+P. Marchand et al.: Fast methods to track grain coagulation and ionization. III. Protostellar collapse with non-ideal MHD
+is the thermal velocity of ions, with T the temperature, µi the
+average atomic mass of ions and mH the mass of the pro-
+ton. We also have Θ = se[µimH/me]1/2, with se the sticking
+probability of electrons on grains and me the electron mass.
+⟨ ˜J(τk)⟩ represents the enhancement factor for ion recombina-
+tion on grains, τk = akkBT/e2 is the reduced temperature of
+grains (Draine & Sutin 1987), and αk = [8/(πτk)]1/2. Finally,
+⟨σv⟩ie = 2 × 10−7[T/300]−1/2 is the collision rate between ions
+and electrons.
+We solve the system of equations (A.1) - (A.5) for ψ and
+find ni, ne and Zk for all bins. A Newton-Raphson algorithm is
+described in details in the Appendix A of Paper I.
+Appendix A.2: The resistivities
+The collision time-scale of species s with neutral H2, the most
+abundant species in the gas, is given by
+τsH2 =
+1
+asHe
+ms + mH2
+mH2
+1
+nH2⟨σω⟩s
+.
+(A.6)
+asHe accounts for the collisions with He and is equal to 1.14
+for ions, 1.16 for electrons and 1.28 for grains (Desch &
+Mouschovias 2001). ms and mH2 are the mass of the species s
+and the H2 molecule. nH2 represents the number density of H2
+(roughly equal to the density of the gas). ⟨σω⟩s is the collision
+rate, taken from Pinto & Galli (2009) for electrons and ions
+⟨σω⟩e = 3.16 × 10−11v1.3
+rms,e,
+(A.7)
+⟨σω⟩i = 2.4 × 10−9v0.6
+rms,i,
+(A.8)
+where
+vrms,s =
+� 8kBT
+πµs,H2
+� 1
+2
+,
+(A.9)
+is in km s−1, with µs,H2 the reduced mass of species s and H2.
+These velocities have been calculated in a three-fluid formalism
+but we use them in our monofluid framework. See some related
+concerns in the appendix of Marchand et al. (2020). For grains,
+the collision rate is given by (Draine & Sutin 1987; Kunz &
+Mouschovias 2009)
+⟨σω⟩k = πa2
+k
+�8kBT
+πmH2
+� 1
+2 ��������1 +
+� π
+2τk
+� 1
+2 �������� .
+(A.10)
+We also define the conductivity σs and cyclotron frequency ωs
+of species s
+σs = nsq2
+sτsH2
+ms
+,
+(A.11)
+ωs = qsB
+cms
+,
+(A.12)
+where qs is the electric charge of species s, B the magnetic field
+strength and c the speed of light. The parallel, perpendicular and
+Hall conductivities are expressed as
+σ|| =
+�
+s
+σs,
+(A.13)
+σ⊥ =
+�
+s
+σs
+1 + (ωsτsH2)2 ,
+(A.14)
+σH =
+�
+s
+σsωsτsH2
+1 + (ωsτsH2)2 .
+(A.15)
+(A.16)
+Finally, the Ohmic, Hall and ambipolar resisitivities are defined
+as
+ηO = 1
+σ||
+,
+(A.17)
+ηH =
+σH
+σ2
+⊥ + σ2
+H
+,
+(A.18)
+ηA =
+σ⊥
+σ2
+⊥ + σ2
+H
+− 1
+σ||
+.
+(A.19)
+(A.20)
+Appendix B: Coagulation and dust-to-gas ratio
+Our coagulation model assumes that grains are perfectly cou-
+pled with the gas and well-mixed, so that the dust-to-gas mass
+ratio is constant. However, grains of different sizes have dif-
+ferent aerodynamic properties and may experience a significant
+drift through the gas. This characteristic is usually determined
+by their Stokes number, that is the ratio between the stopping
+time of the grain (Epstein 1924) and the dynamical timescale of
+the system. In particular, Lebreuilly et al. (2020) showed strong
+variations in the dust-to-gas mass ratio during the protostellar
+collapse. The disk and first core tend to be dust-enriched while
+the envelope becomes dust-depleted. This has strong implication
+for the coagulation, as a higher grain density promotes collisions
+and speeds up their growth (conversely for a lower density). In
+this section, we briefly explore a refinement to the coagulation
+model presented in Paper I.
+The general expression of the Smoluchowski (1916) equa-
+tion, from which we derive the expression of χ, is (Paper I eq. 5)
+dX(a, t)
+dt
+= CglocalnHI(a, X, t),
+(B.1)
+where C is a constant, glocal a function of the local properties of
+the gas (density, temperature...), and
+I(a, X, t) = −
+� ∞
+0
+h(m, m′)X(m, t)X(m′, t)dm′
++ 1
+2
+� m
+0
+h(m − m′, m′)X(m − m′, t)X(m′, t)dm′.
+(B.2)
+The relative number of grains of size a at a given time t is
+X(a, t) = ngrain/nH where ngrain is the number density of grains.
+We can then write X = dgx, where dg is the dust-to-gas ratio, and
+x the normalized relative number of grains. We can then rewrite
+equation (B.1)
+dx(a, t)
+dt
+= CglocalnHdgI′(a, x, t),
+(B.3)
+and include dg in the definition of our coagulation variable, giv-
+ing it the form
+dχ′ = glocalnHdgdt.
+(B.4)
+In our case, for the Ormel & Cuzzi (2007) kernel,
+dχ′ = n
+3
+4
+HT − 1
+4 dgdt.
+(B.5)
+This expression means that the dust-to-gas ratio does not change
+the evolution path of the size distribution, only the coagulation
+Article number, page 9 of 11
+
+A&A proofs: manuscript no. Marchandetal2023
+speed. A dust-to-gas ratio N times higher requires a χ value N
+times lower to reach the same coagulated state. This new defini-
+tion of χ′ can be used in an environment with varying dust-to-
+gas ratio. In hydrodynamics simulations, it is possible to couple
+it with the drift of one grain size close to the peak of the mass
+density distribution, as allowed by the method proposed by Le-
+breuilly et al. (2019).
+Although it is not yet possible to associate this coagulation
+method with the differential drift of grains of different sizes, this
+is a first step towards a self-consistent treatment of dust grains
+in hydrodynamics simulations. This method will lead to an over-
+estimate of the small-to-larger grain ratio but this is probably a
+decent approximation to get the total dust-to-gas ratio. As shown
+in Lebreuilly et al. (2020), as long as the large grains dominate
+the mass in the distribution, they also dominate the differential
+gas and dust dynamics. In other words, most of the dust enrich-
+ment comes from the dynamics of large grains.
+Appendix C: The fragmentation barrier
+Ormel et al. (2009) derived criteria for the fragmentation of dust
+aggregates. They define the rolling energy Eroll that determines
+the energy needed to restructure the grain. In Section 4.2 of pa-
+per I, we misinterpreted their results by assuming the fragmen-
+tation would occur for a kinetic energy Ekin = 5Eroll. The actual
+criterion is
+Ekin > 5NtotEbr,
+(C.1)
+where Ekin is the kinetic energy, Ebr is the breaking energy and
+Ntot is the total numbers of monomers composing the two collid-
+ing grains. Ebr is defined by (Dominik & Tielens 1997)
+Ebr = Abrγ5/3
+grain
+(a0/2)4/3
+ε2/3
+,
+(C.2)
+with Abr = 2.8 × 103, γgrain the surface energy density of the ma-
+terial, a0 the size of the monomers composing the grains, and ε
+the reduced elastic modulus. As in Ormel et al. (2009), we adopt
+a0 = 0.1 µm. Ice mantles on grains make them more resistant
+to fragmentation. Therefore, we assume bare silicates to obtain a
+lower limit for a fragmentation criterion. In this case, γgrain = 25
+erg cm−2 and ε = 2.8×1011 dyn cm−2. The kinetic energy of two
+grains of mass m and m′ reads
+Ekin = 1
+2
+mm′
+m + m′ ∆v2,
+(C.3)
+where ∆v is the relative velocity between the two grains. This
+velocity is given by the kernel of Ormel & Cuzzi (2007) that we
+use in Paper I
+∆v =
+� 3√
+8
+z0[kBG]
+1
+2 γρs
+µmH
+� 1
+2
+n
+− 1
+4
+H T − 1
+4 a
+1
+2 ,
+(C.4)
+where z0 = 2.97, kB and G are the Boltzmann and gravitational
+constants, γ = 5/3 the adiabatic index of the gas, ρs = 2.3 g cm−3
+the bulk density of the grains, µ = 2.3 the average atomic mass
+of the gas, mH the proton mass, and a the radius of the larger of
+the two grains. Assuming two identical grains and
+m = 4
+3πρsa3
+0Ntot,
+(C.5)
+equation C.1 becomes
+a > 15Ebr
+πρsa3
+0
+� 3√
+8
+z0[kBG]
+1
+2 γρs
+µmH
+�−1
+n
+1
+2
+HT
+1
+2 .
+(C.6)
+Replacing by numeric values, we find
+a > 826 µm
+�
+nH
+1010 cm−3
+� 1
+2 � T
+10 K
+� 1
+2
+.
+(C.7)
+This limit is much higher than the one derived in paper I, and we
+do not reach it in our simulations. The value of γgrain we use is
+taken from Ormel et al. (2009), but surface energies of different
+materials can reach 10 to 100 times this value, resulting in a
+much higher estimate of the fragmentation threshold (as Ebr ∼
+γ5/3
+grain). The value derived in equation (C.7) should then be taken
+as very conservative.
+References
+Balsara, D. S. 2012, Journal of Computational Physics, 231, 7476
+Bate, M. R. 2022, MNRAS, 514, 2145
+Blandford, R. D. & Payne, D. G. 1982, MNRAS, 199, 883
+Crutcher, R. M. 1999, Astrophys. J. , 520, 706
+Desch, S. J. & Mouschovias, T. C. 2001, Astrophys. J. , 550, 314
+Dominik, C. & Tielens, A. G. G. M. 1997, Astrophys. J. , 480, 647
+Draine, B. T. & Sutin, B. 1987, Astrophys. J. , 320, 803
+Duffin, D. F. & Pudritz, R. E. 2008, MNRAS, 391, 1659
+Dullemond, C. P. & Dominik, C. 2005, A&A, 434, 971
+Epstein, P. S. 1924, Physical Review, 23, 710
+Flower, D. R., Pineau Des Forêts, G., & Walmsley, C. M. 2005, A&A, 436, 933
+Galametz, M., Maury, A. J., Valdivia, V., et al. 2019, A&A, 632, A5
+Galli, D. & Shu, F. H. 1993, Astrophys. J. , 417, 220
+Guillet, T. & Teyssier, R. 2011, Journal of Computational Physics, 230, 4756
+Guillet, V., Hennebelle, P., Pineau des Forêts, G., et al. 2020, A&A, 643, A17
+Johansen, A. & Youdin, A. 2007, Astrophys. J. , 662, 627
+Kawasaki, Y., Koga, S., & Machida, M. N. 2022, MNRAS, 515, 2072
+Kunz, M. W. & Mouschovias, T. C. 2009, Astrophys. J. , 693, 1895
+Kuznetsova, A., Hartmann, L., & Heitsch, F. 2020, Astrophys. J. , 893, 73
+Kwon, W., Looney, L. W., Mundy, L. G., Chiang, H.-F., & Kemball, A. J. 2009,
+Astrophys. J. , 696, 841
+Larson, R. B. 1969, MNRAS, 145, 271
+Le Gouellec, V. J. M., Hull, C. L. H., Maury, A. J., et al. 2019, Astrophys. J. ,
+885, 106
+Lebreuilly, U., Commerçon, B., & Laibe, G. 2019, A&A, 626, A96
+Lebreuilly, U., Commerçon, B., & Laibe, G. 2020, A&A, 641, A112
+Lebreuilly, U., Hennebelle, P., Colman, T., et al. 2021, Astrophys. J., Lett. , 917,
+L10
+Lebreuilly, U., Vallucci-Goy, V., Guillet, V., Lombart, M., & Marchand, P. 2023,
+MNRAS, 518, 3326
+Li, Z.-Y., Krasnopolsky, R., & Shang, H. 2011, Astrophys. J. , 738, 180
+Lombart, M. & Laibe, G. 2021, MNRAS, 501, 4298
+Machida, M. N., Inutsuka, S., & Matsumoto, T. 2006, Astrophys. J., Lett. , 647,
+L151
+Marchand, P., Commerçon, B., & Chabrier, G. 2018, A&A, 619, A37
+Marchand, P., Guillet, V., Lebreuilly, U., & Mac Low, M. M. 2021, A&A, 649,
+A50
+Marchand, P., Masson, J., Chabrier, G., et al. 2016, A&A, 592, A18
+Marchand, P., Tomida, K., Tanaka, K. E. I., Commerçon, B., & Chabrier, G.
+2020, Astrophys. J. , 900, 180
+Masson, J., Chabrier, G., Hennebelle, P., Vaytet, N., & Commercon, B. 2016,
+A&A, 587, A32
+Masson, J., Teyssier, R., Mulet-Marquis, C., Hennebelle, P., & Chabrier, G.
+2012, Astrophys. J., Suppl. Ser. , 201, 24
+Mathis, J. S., Rumpl, W., & Nordsieck, K. H. 1977, Astrophys. J. , 217, 425
+Mellon, R. R. & Li, Z. 2009, Astrophys. J. , 698, 922
+Miotello, A., Testi, L., Lodato, G., et al. 2014, A&A, 567, A32
+Miyoshi, T. & Kusano, K. 2005, Journal of Computational Physics, 208, 315
+Mouschovias, T. C. & Paleologou, E. V. 1979, Astrophys. J. , 230, 204
+Mouschovias, T. C. & Spitzer, Jr., L. 1976, Astrophys. J. , 210, 326
+Ormel, C. W. & Cuzzi, J. N. 2007, A&A, 466, 413
+Ormel, C. W., Paszun, D., Dominik, C., & Tielens, A. G. G. M. T. 2009, A&A,
+502, 845
+Pinto, C. & Galli, D. 2009, Three-Fluid Magnetohydrodynamics in Star Forma-
+tion, ed. Tsinganos, K., Ray, T., & Stute, M., 597–599
+Rossi, S. C. F., Benevides-Soares, P., & Barbuy, B. B. 1991, A&A, 251, 587
+Silsbee, K., Akimkin, V., Ivlev, A. V., et al. 2022, Astrophys. J. , 940, 188
+Silsbee, K., Ivlev, A. V., Sipilä, O., Caselli, P., & Zhao, B. 2020, A&A, 641, A39
+Smoluchowski, M. V. 1916, Zeitschrift fur Physik, 17, 557
+Teyssier, R. 2002, A&A, 385, 337
+Tomida, K., Okuzumi, S., & Machida, M. N. 2015, Astrophys. J. , 801, 117
+Article number, page 10 of 11
+
+P. Marchand et al.: Fast methods to track grain coagulation and ionization. III. Protostellar collapse with non-ideal MHD
+Truelove, J. K., Klein, R. I., McKee, C. F., et al. 1997, Astrophys. J., Lett. , 489,
+L179
+Tsukamoto, Y., Machida, M. N., & Inutsuka, S.-i. 2021, Astrophys. J., Lett. ,
+920, L35
+Valdivia, V., Maury, A., Brauer, R., et al. 2019, MNRAS, 488, 4897
+Vaytet, N., Commerçon, B., Masson, J., González, M., & Chabrier, G. 2018,
+A&A, 615, A5
+Vaytet, N. & Haugbølle, T. 2017, A&A, 598, A116
+Vericel, A., Gonzalez, J.-F., Price, D. J., Laibe, G., & Pinte, C. 2021, MNRAS,
+507, 2318
+Vorobyov, E. I., Skliarevskii, A. M., Molyarova, T., et al. 2022, A&A, 658, A191
+Wurster, J., Price, D. J., & Bate, M. R. 2016, MNRAS, 457, 1037
+Yang, C.-C., Johansen, A., & Carrera, D. 2017, A&A, 606, A80
+Youdin, A. N. & Goodman, J. 2005, Astrophys. J. , 620, 459
+Zhao, B., Caselli, P., Li, Z.-Y., & Krasnopolsky, R. 2018, MNRAS, 473, 4868
+Zhao, B., Caselli, P., Li, Z.-Y., et al. 2016, MNRAS, 460, 2050
+Article number, page 11 of 11
+
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+page_content=' American Museum of Natural History,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
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+page_content=' Université Paris-Saclay,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Université Paris Diderot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Sorbonne Paris Cité,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 91191 Gif-sur-Yvette,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' France  4 Université Paris-Saclay,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Institut d’astrophysique spatiale,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
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+page_content=' France  5 Laboratoire Univers et Particules de Montpellier,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Université de Montpellier,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' CNRS/IN2P3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' CC 72,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Place Eugène Bataillon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 34095  Montpellier Cedex 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' France  ABSTRACT  Dust grains influence many aspects of star formation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' including planet formation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' opacities for radiative transfer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' chemistry,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' and  the magnetic field via Ohmic,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Hall,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' and ambipolar diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' The size distribution of the dust grains is the primary characteristic  influencing all these aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Grain size increases by coagulation throughout the star formation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' We describe here numerical  simulations of protostellar collapse using methods described in earlier papers of this series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' We compute the evolution of the grain  size distribution from coagulation and the non-ideal magnetohydrodynamics effects self-consistently and at low numerical cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' We  find that the coagulation efficiency is mostly affected by the time spent in high-density regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Starting from sub-micron radii, grain  sizes reach more than 100 µm in an inner protoplanetary disk that is only 1000 years old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' We also show that the growth of grains  significantly affects the resistivities, and indirectly the dynamics and angular momentum of the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Introduction  Grains play a major role during star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Firstly, they  are the seed of planet formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' While their characteristic size  is sub-micron in the interstellar medium (ISM) (Mathis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  1977), they grow by coagulation during the collapse, and can  reach sizes larger than 10 µm in the early stages of protostel-  lar collapse (Guillet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Silsbee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Tsukamoto  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Vorobyov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Bate 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Observations also  suggest they reach these sizes, if not larger, in the envelopes of  Class 0-I objects (Kwon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Miotello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Le  Gouellec et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Galametz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Their growth then  continues in protoplanetary disks until they eventually become  planetesimals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Variations in their size also significantly impact  non-ideal magnetohydrodynamics (MHD) effects through their  ionization and their chemical interactions with the gas, with a  direct feedback on the dynamics of the gas (Marchand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2016, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Marchand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Guillet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Non-ideal MHD effects have been shown to be critical to  the regulation of magnetic field and angular momentum during  the protostellar collapse and the protoplanetary disk evolution  (Mouschovias & Paleologou 1979;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Machida et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Duf-  fin & Pudritz 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Mellon & Li 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Tomida  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Wurster et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Masson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Vaytet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Marchand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Lebreuilly et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Grains are  also the main source of opacity in protostellar environments, af-  fecting the cooling of the gas and the observations made of those  systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Their high optical depth at densities ρ > 10−13 g cm−3  leads to the formation of the first hydrostatic core (Larson 1969).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  However, numerical simulations usually do not account for  grain coagulation self-consistently due to the great cost of com-  puting a coagulation algorithm on-the-fly (although new meth-  ods are being developed, see the recent work by Lombart &  Laibe 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' The dust evolution used to be pre-processed or  post-processed with no self-consistent feedback on the dynam-  ics (Rossi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Dullemond & Dominik 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Marchand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Recently, more and more studies  include the growth of grains in their hydrodynamics simulations  (Tsukamoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Vericel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Vorobyov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  Bate 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' In Marchand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' (2021, hereafter Paper I), we  presented a simple and fast method to track coagulation self-  consistently that is particularly suited for modeling star forma-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' We now apply this method to non-ideal MHD protostellar  collapse simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' It is coupled with the second method pre-  sented in Paper I: a fast calculation of the ionization and grain  charge, to obtain non-ideal MHD resistivities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' These 3D simula-  tions are the first to include a self-consistent grain growth with a  direct feedback on the dynamics through the self-consistent cal-  culation of MHD resistivities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' In Section 2, we describe  the methods used in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Section 3 presents the results of  our calculations, both analytical in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='1 and numerical in  Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' We compare our results to other works and discuss  the caveats in Section 4, and conclude in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Methods  We perform non-ideal MHD simulations with the RAMSES  code (Teyssier 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' RAMSES is an Eulerian gas dynamics  code with adaptive mesh refinement (AMR) and self-gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' It  includes a monofluid treatment of non-ideal MHD effects (Mas-  son et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Marchand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' We have implemented  the methods presented in Paper I to calculate the coagulation  and ionization of grains on-the-fly in a self-consistent manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  Article number, page 1 of 11  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='01510v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='SR]  4 Jan 2023  A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Marchandetal2023  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Grain coagulation and ionization  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Coagulation  In Paper I, we demonstrated that the coagulation process as  described by the Smoluchowski (1916) equation is a one-  dimensional process with certain types of coagulation kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  Consequently, the size distribution of the coagulated grains de-  pends only on the initial size distribution and a variable χ that  encompasses the whole history of the physical conditions seen  by the grains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' At a given χ that is integrated along the path of  the grains, the coagulated distribution is always the same inde-  pendently of the actual path taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' This method works for every  coagulation kernel for which the dependence on the gas variables  such as density and temperature can be separated from the grain  properties such as size and mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' In Paper I and in the present  paper, we use the turbulent kernel derived by Ormel & Cuzzi  (2007) in the intermediate coupling regime, which is suited for  star formation conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' We show that in this case χ can be  derived from integrating  dχ = n  3  4  HT − 1  4 dt,  (1)  where nH is the number density of the gas, T its temperature,  and t the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' In three-dimensional (3D) hydrodynamics simu-  lations, only the knowledge of χ is needed to track the coagula-  tion of grains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  Equation (1) is a Lagrangian derivative of χ with respect to  time along the path of the grain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' We can transform it into a partial  (Eulerian) derivative  ∂χ  ∂t + u · ∇χ = n  3  4  HT − 1  4 ,  (2)  where u is the velocity of the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' We can combine equation (2)  with the mass conservation equation  ∂ρ  ∂t + ∇ · [ρu] = 0,  (3)  where ρ is the gas mass density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' This yields  ∂ρχ  ∂t + ∇ · (ρχu) = ρn  3  4  HT − 1  4 ,  (4)  which means that the quantity ρχ can be treated in an Eulerian  framework as a passive scalar with a source term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' We exploit this  property and implement it as such in RAMSES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' The value of χ  is therefore calculated self-consistently in all cells at each time-  step, as a mass-weighted average, ignoring any diffusion of dust  through the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' We used the code Ishinisan (Marchand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  2021) to pre-calculate a table containing the grain size distribu-  tion for a large number of χ values in a large relevant interval  (between χ = 1013 and χ = 1019 in cgs units).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' When the size  distribution is needed during the hydrodynamical simulation, it  is interpolated from the table based on the value of χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  Our initial distribution in this paper is a Mathis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' (1977,  MRN) distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' The minimum and maximum radii are amin =  5 nm and amax = 250 nm, and the slope of the distribution is  −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='5, so that the number density of grains n follows the variation  dn  da ∝ a−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  (5)  The total quantity of grains is determined by the dust-to-gas  mass ratio that we assume to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' In this work, we sample the  distribution with 60 bins of size logarithmically spaced between  10-10  10-9  10-8  10-7  10-6  10-5  10-4  10-3  10-2  10-1  10-3  10-2  10-1  100  101  102  103  Fractional mass X (ρ/nHmp)  Radius (µm)  Initial MRN  χ = 1015 cgs  χ = 1016 cgs  χ = 1017 cgs  χ = 1017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='5 cgs  χ = 1018 cgs  χ = 1018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='5 cgs  χ = 1019 cgs  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' State of the grain size distribution for values of χ, between 1015  cgs and 1019 cgs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' The points represent the fractional abundance of the  size-bin as function of the effective radius of the bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  5 nm and 5000 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' In Figure 1, we present the coagulated MRN  size distribution at various χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Below χ = 1017 cgs, the shift in  the size distribution is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' For higher values, the maxi-  mum size and the peak of the distribution are located at larger  and larger radii, while the slope of the distribution remains sim-  ilar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' In all cases, the small grains are more abundant while the  large grains hold more mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' The mode of the size distribution  amax is located near the largest relevant grain size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  As grains grow, they may experience fragmentation when the  kinetic energy of the collision is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Contrarily to what we de-  rived in paper I, we find that fragmentation does not occur in  early stages of the disk, but rather only at very high densities  ρ > 1012 cm−3 (Lebreuilly et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2023) for very large grains with  a > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='08 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' We demonstrate this result in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' We  therefore neglect fragmentation in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' We also do not ac-  count for the grain drift with respect to the gas in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' We  discuss the possible consequences in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Drift is, how-  ever, compatible with our coagulation model, and we detail the  method in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Other limitations are discussed in Sec-  tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Ionization and resistivities  In Paper I, we also presented a fast method to calculate the ion-  ization of the gas-grain mixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' For an arbitrary size distribu-  tion, we can calculate the average electric charge of each grain  size, the number of ions and the number of electrons, provided  the cosmic-ray (CR) ionization rate, the density and temperature  of the gas, the average atomic mass of ions µi and the sticking  probability of electrons on grains se.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Here, we assume µi = 28,  which corresponds to the ion HCO+, and se = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='6 as in Marc-  hand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' We also assume ζ = 5 × 10−17 s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' The den-  sity and the temperature, are taken from the hydrodynamic sim-  ulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' The calculation is performed by the Newton-Raphson  scheme described in Appendix A of Paper I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' The resistivities are  computed using a similar method to Marchand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' (2016), with  one difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' For each grain size, they sum over the contribu-  tions of the whole charge distribution (between -1 to +1 in their  case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Instead, we average the contributions using the mean elec-  tric charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' We explicit the method in more details in Appendix  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  Article number, page 2 of 11  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Marchand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' : Fast methods to track grain coagulation and ionization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Protostellar collapse with non-ideal MHD  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Star formation simulations  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Model  We perform four numerical simulations using the RAMSES  code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' We solve the following MHD equations  ∂ρ  ∂t + ∇ · �ρu� = 0,  (6)  ∂ρu  ∂t + ∇ ·  �  ρuu +  �  P + B2  2  �  I − BB  �  = −ρ∇Φ,  (7)  ∂B  ∂t − ∇ ×  �  u × B − ηΩJ − ηH  J × B  B  + ηAD  (J × B) × B  B2  �  = 0,  (8)  ∇ · B = 0,  (9)  where u is the velocity of the gas, P its pressure, B the mag-  netic field, J = ∇ × B the current, I the identity matrix, Φ the  gravitational potential, and ηΩ, ηH and ηAD the ohmic, Hall and  ambipolar resistivities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' The temperature evolution is prescribed  by the barotropic equation  T = T0  �������1 +  �  ρ  10−13 g cm−3  �γ−1������� ,  (10)  with T0 = 10 K and γ = 5/3 the adiabatic index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' The initial  condition is a sphere of gas of 1 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' The radius of the sphere is  controlled by the thermal over gravitational energy ratio α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' We  choose α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='3, which sets a radius of R = 2946 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' The density  distribution has an m = 2 azimuthal perturbation  ρ(ϕ) = ρ0(1 + δρ sin ϕ),  (11)  where ρ0 is the density of a uniform sphere of same mass and  radius, and ϕ the azimuthal angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' We choose δρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' The  computational domain outside of the sphere is filled with gas of  density ρ0/100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' The sphere undergoes a solid rotation, with a  ratio of rotational to gravitational energy of β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' The mag-  netic field is initially uniform and parallel to the rotation axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  It is defined using the mass-to-flux ratio over the critical value  (Mouschovias & Spitzer 1976)  µB =  M/ΦB  (M/ΦB)crit  ,  (12)  with  � M  ΦB  �  crit  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='53  3π  �  5  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  (13)  Observations show that dense cores are slightly super-critical  (Crutcher 1999), although recent numerical simulations indicate  that observations may overestimate the actual strength of the  magnetic field due to projection effects (Kuznetsova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  We choose µB = 5 as a fiducial value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Those parameters are  used in our reference case C-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' We change the value of α to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='4,  and the initial mass M to 5 M⊙ for two other cases called C-4  and C-3-M5, respectively, to investigate the influence of the col-  lapse time on the grain coagulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' The final run, named NC-3,  is similar to C-3 without coagulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' The grain distribution and  ionization evolve as described in sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='2, and Paper I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  The four cases are summarized in table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Grid and Algorithm  The simulation box is a cube that is four times as large as the ra-  dius of the sphere, with periodic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' The initial  grid is composed of 323 cells (level 5 of AMR) and is refined to  ensure at least 10 points per Jeans length, strongly satisfying the  Truelove et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' (1997) criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' The maximum AMR refinement  level is 13 for C-3 and NC-3 (resolution of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='4 au), 14 for C-4  (resolution of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='96 au) and 16 for C-3-M5 (resolution of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='90  au).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  Simulations are performed with a 3D unsplit slope limiter  to avoid overshooting of the magnetic field at shock boundaries,  while keeping the second order convergence for the Hall effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  We use the HLLD Riemann solver (Miyoshi & Kusano 2005)  for non-magnetic variables and the 2D HLL Riemann solver for  the magnetic field and the Hall effect (Balsara 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Marchand  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' The Poisson equation is solved using the multigrid  method of Guillet & Teyssier (2011) in our periodic domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Application to star formation  In this section, we present the results of our calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' We first  apply our coagulation method to an analytical one-zone model in  Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='1, then in 3D MHD simulations in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Analytical collapse  During spherical protostellar collapse, the time evolution of the  contraction ratio of a gas cloud compared to its original radius  x = R(t)/R0 can be described by (Flower et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2005):  dx  dt = − π  2τff  �  1  x − 1,  (14)  where τff is the free-fall time, given by  τff =  �  3π  32Gρ0  ,  (15)  where ρ0 is the initial density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Guillet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' (2020) showed that  assuming a uniform compression of the gas nicely reproduces  the isothermal phase of the collapse, particularly when compar-  ing the dust size distribution at the same gas density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' In this case,  the density of mass scales as ρ(t) = ρ0(R0/R[t])3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' The gas density  then evolves as  1  nH  dnH  dt = 1  ρ  dρ  dt = −3  x  dx  dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  (16)  We use a second-order Runge-Kutta scheme to numerically  integrate this equation from the beginning of the collapse at  ρ = ρ0 until the formation of the first Larson core at ρ =  10−13 g cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' The evolution of χ with ρ is plotted in Figure 2,  assuming T = 10 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' The solid and dashed lines represent dif-  ferent initial densities, ρ0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='8 × 10−20 g cm−3 and ρ0 =  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='8×10−18 g cm−3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' At densities nH > 10−16 g cm−3,  the value of χ is independent from ρ0 and increases as χ ∝ ρ1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  This evolution is expected as χ ∼ ρ3/4t and t ∼ τff ∼ ρ−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' At  ρ = 10−13 g cm−3, the coagulation variable reaches χ ≈ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='6×1017  cgs, which corresponds to a peak of the size distribution of ∼ 10  µm, indicating a significant grain growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' That is consistent with  the results of Guillet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' (2020), who use the same collapse  model, but solve coagulation on the fly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  Article number, page 3 of 11  A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Marchandetal2023  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Parameters of the protostellar collapse simulations: name of the simulation, thermal over gravitational energy ratio α, radius R, mass M,  initial density ρ0 and initial magnetic field B of the initial cloud, formation time of the first hydrostatic core tfc, use of coagulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  Name  α  R (au)  M (M⊙)  ρ0 (g cm−3)  B (µG)  tfc (kyr)  Coagulation  C-3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='3  2946  1  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='49 × 10−18  133  ∼ 30  Yes  C-3-M5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='3  14738  5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='19 × 10−19  27  ∼ 150  Yes  C-4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='4  3930  1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='31 × 10−18  75  ∼ 47  Yes  NC-3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='3  2946  1  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='49 × 10−18  133  ∼ 30  No  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Evolution of the coagulation variable χ with increasing density  nH during the isothermal collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' The solid line represents an initial  density of ρ0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='8 × 10−20 g cm−3, and the dashed line ρ0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='8 ×  10−18 g cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Numerical collapse  The numerical simulations were run as described in Section 2  until 1000 years after the density reaches 10−13 g cm−3, the for-  mation of the first hydrostatic core at ∼ 30 kyr (Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' In refer-  ence simulation C-3, a small circumstellar disk with a radius of ≈  20 au forms and a disk wind is launched by magneto-centrifugal  acceleration (Blandford & Payne 1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Figure 3 shows face-on  and edge-on slices of density at the final time-step of the simu-  lation, with arrows indicating the gas velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Grain growth  We show in Figure 4 the value of χ as a function of the density in  simulation cells at the final time-step, for runs C-3, C-4 and C-  3-M5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' The increase is quasi unidimensional in the isothermally  collapsing envelope for ρ < 1015 g cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Beyond this density,  there is a large spread of χ values of over an order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  This spread most likely occurs in gas falling in the pseudo-disk,  then the disk and the first Larson core, or in the outflow over dif-  ferent timescales, and spending unequal times in a given density  range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' The overall trend agrees well with the analytical calcula-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  There is no significant difference in the χ values, and thus  dust size distributions, between the three runs, despite run C-4  needing 50% more time to collapse to the first Larson core stage,  and C-3-M5 needing 400% more time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Coagulation happening  in the isothermally collapsing envelope is therefore hardly im-  pacted by the initial conditions, as growth accelerates with in-  creasing density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  40  20  0  20  40  x (au)  40  20  0  20  40  y (au)  14  13  12  11  log(ρ) (g cm-3)  40  20  0  20  40  x (au)  40  20  0  20  40  y (au)  40  20  0  20  40  40  20  0  20  40  200  100  0  100  200  x (au)  200  100  0  100  200  z (au)  17  16  15  14  13  12  log(ρ) (g cm-3)  200  100  0  100  200  x (au)  200  100  0  100  200  z (au)  200  100  0  100  200  200  100  0  100  200  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Density slices of the C-3 simulation, with grain coagulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' The  top panel is a face-on slice of the plane z=0, and the bottom panel an  edge-on slice of the plane y=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' White arrows represent the direction of  the gas velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' The snapshot is taken at the final time-step, 1000 years  after the formation of the first Larson core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  Figure 5 shows the mode of the size distribution as a func-  tion of density, corresponding to the distribution of χ shown in  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Three regions are clearly demarcated, the first being  the envelope, in which grain coagulation is not efficient enough  to form large grains (ρ < 10−16 g cm−3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' The second comprises  Article number, page 4 of 11  1018  cm  3  16  X  101s  1014  10-19  10-13  10-18  10-17  10~16  10-14  10-12  10-11  02-01  p (g cmP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Marchand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' : Fast methods to track grain coagulation and ionization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Protostellar collapse with non-ideal MHD  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Evolution of the coagulation variable χ with increasing density  nH in the numerical collapse models C-3 (purple), C-3-M5 (light blue),  and C-4 (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Each point corresponds to a simulation cell at the final  time-step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' The red line is the analytical collapse solution for ρ0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='8 ×  10−20 g cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  the pseudo-disk and the early protoplanetary disk, where grains  grow by a factor 100 from sub-micron sizes to several tens of  µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' The third region is the first Larson core, which has even  larger grains that reach 400 µm within a mere 103 yr years af-  ter its formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' That value is in line with similar recent studies  (Kawasaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Lebreuilly et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' There is little dif-  ference between runs C-3, C-4 and C-3-M5, confirming that co-  agulation in the envelope does not impact large grains, as found  by previous studies (for example Silsbee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' We discuss  observations of large grains in the envelope in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  The spatial distribution of those grain sizes for run C-3 is  displayed in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Size distributions shifting significantly  from the initial MRN distribution are indeed confined to the mid-  plane, in the disk and pseudo-disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' The bottom panel also shows  moderately larger grains in the outflow, as they only traveled  through the upper layers of the pseudo-disk before being ejected  (Marchand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' If they had been in the mid-plane of the  disk, they would have grown much more, as coagulation is irre-  versible in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' The upper panel shows that grains reach  a radius larger than 1 µm in the outskirts of the disk, within 100  au of the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Growth then occurs rapidly as density increases  in the inner 15 au, which is shown by the almost overlapping  contours delimiting amax = 5 µm and amax = 10 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Resistivities and gas dynamics  We describe here the impact of grain coagulation on non-ideal  MHD resistivities and their macroscopic effects on gas dynam-  ics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Previous studies emulated the coagulation of grains by re-  moving the very small grains (a < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='1 µm) and redistributing  their mass to the larger end of the distribution (Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2016,  2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Marchand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' This method leads to an increase  of resistivities, in particular the ambipolar resistivity, resulting in  weaker coupling between the magnetic field and the gas, hence  weaker magnetic braking and larger, more unstable disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' How-  ever, we observe here the exact opposite behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  The middle panel of figure 7 presents the volume-weighted  average resistivities as a function of density, at the final time-  step, for simulations C-3 and NC-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' A non-evolving size distri-  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Mode of the coagulated grain size distribution as a function of  density in simulations (purple), C-3-M5 (light blue), and C-4 (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  The discrete values of the sizes are due to the binning of the size distri-  bution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  bution produces resistivities relatively similar in the envelope,  but two to four orders of magnitude larger at disk densities, par-  ticularly the Ohmic and ambipolar resistivities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Consequently,  the magnetic braking is weakened, and the gas retains more an-  gular momentum without the grain coagulation, forming a larger  disk, as shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  This difference in regime can be quantified by the ambipolar  Elsasser number Am = B2/(ρηADΩ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' In regions where Am < 1,  the ambipolar diffusion has a significant impact on the dynam-  ics of the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' The bottom panel of Figure 7 shows the radial  profile of the ambipolar Elsasser number in runs C-3 and NC-  3, azimuthally-averaged in the mid-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' The higher resistivity  in run NC-3 results in Am < 1 in the inner ∼ 12 au, indicat-  ing active ambipolar diffusion and magnetic field dissipation,  while Am ≳ 104 for C-3 over the same radial range, indicat-  ing weak ambipolar diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Figure 9 compares the disk size  and angular momentum in both simulations, further confirming  the lower magnetic braking in run NC-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' A second effect of the  weaker magnetic forces from the stronger ambipolar diffusion in  run NC-3 is the absence of outflow at this stage of evolution, as  shown in the lower panel of Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  The discrepancy of resistivity values between actual coagu-  lation and methods simply redistributing the mass to the large-  mass-end of the distribution originates from the lack of very  large grains (> 10 µm) in the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' In both cases, removing  the small grains decreases the electron absorption by dust, and  therefore decreases the Ohmic resistivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' However, the ambipo-  lar resistivity is controlled by the relative abundance of ions  and charged grains in the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Although the small grain removal  method barely changes the abundance of ions, the dominant pos-  itive charge carriers, it reduces significantly the charged grain  population, leading to an increase of resistivity at low density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' At  high density, both with a standard MRN and a truncated-MRN  distribution, the grains become the dominant charge carriers and  the abundance of ions decreases (see the dashed lines in the top  panel of Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' That does not happen with coagulation be-  cause the number of grains decreases significantly, hence leading  to a higher number of ions and a lower resistivity than without  proper coagulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' This kind of method without larger grain cre-  ation is therefore inappropriate for emulating coagulation alone  at a low cost for non-ideal MHD calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' At later times, at  Article number, page 5 of 11  1020  C-3  C-4  C-3-M5  1019  Analytical collapse  1018  C1017  X  1016  1015  10-T8  10-T6  10-T4  10-20  10°10  p (g cm103  C-3  C-4  C-3-M5  10°  10-78  10-10  3  p (g cmA&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Marchandetal2023  100  50  0  50  100  x (au)  100  50  0  50  100  y (au)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='1  1  10  amax (µm)  100  50  0  50  100  x (au)  100  50  0  50  100  y (au)  200  100  0  100  200  x (au)  200  100  0  100  200  z (au)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='9  1  amax (µm)  200  100  0  100  200  x (au)  200  100  0  100  200  z (au)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Slices of the C-3 simulation with coagulation showing the mode  of the grain size distribution amax, 1000 years after the first core forma-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' The color scale is different for each panel to yield a better contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  On the top panel, black contours indicate amax = 1, 5 and 10 µm, and  amax = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='5 and 1 µm on the bottom panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  densities ρ > 10−12 g cm−3, Lebreuilly et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' (2023) showed that  the replenishment of small grains by fragmentation would lead  to an increase in both Ohmic and ambipolar resistivities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Discussion and caveats  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Grain growth  As explained in previous sections and displayed in Figures 4 and  5, initial gas conditions have little to no influence on grain coag-  ulation for later stages, and coagulation is ineffective in the en-  velope for growing large grains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' This reinforces the idea that the  system forgets the initial conditions at the formation of the first  hydrostatic core and the disk (Vaytet & Haugbølle 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Con-  sequently,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' calculations such as those presented in this work may  provide standard initial dust grain size distributions for studies of  10-20  10-18  10-16  10-14  10-12  10-10  10-8  10-20 10-19 10-18 10-17 10-16 10-15 10-14 10-13 10-12 10-11 10-10  ne / nH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' ni / nH  ρ (g cm-3)  Ions  Electrons  No coagulation  1010  1012  1014  1016  1018  1020  1022  10-20 10-19 10-18 10-17 10-16 10-15 10-14 10-13 10-12 10-11 10-10  MHD resistivities (cm2 s-1)  ρ (g cm-3)  Ohm  Ambipolar  Hall  No coagulation  10-2  10-1  100  101  102  103  104  105  1  10  100  Elsasser number  r (au)  C-3 (coagulation)  NC-3 (No coagulation)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Top panel: Abundances of ions (purple) and electrons (green)  as a function of density in models with coagulation (C-3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' solid lines) or  without it (NC-3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' dashed lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Middle panel: Volume averaged Ohmic  (purple), ambipolar (green), and Hall (blue) resistivities for the same  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Bottom panel: Average ambipolar Elsasser number Am in the  mid-plane as a function of radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' The thin grey line represents Am = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  protoplanetary disk evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Although other coagulation ker-  nels affect the size distribution in different ways (see Section  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='4), it is certain that even young protoplanetary disks contain  grains significantly larger than the classical MRN distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  Article number, page 6 of 11  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Marchand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' : Fast methods to track grain coagulation and ionization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Protostellar collapse with non-ideal MHD  40  20  0  20  40  x (au)  40  20  0  20  40  y (au)  14  13  12  11  log(ρ) (g cm-3)  40  20  0  20  40  x (au)  40  20  0  20  40  y (au)  40  20  0  20  40  40  20  0  20  40  200  100  0  100  200  x (au)  200  100  0  100  200  z (au)  17  16  15  14  13  12  log(ρ) (g cm-3)  200  100  0  100  200  x (au)  200  100  0  100  200  z (au)  200  100  0  100  200  200  100  0  100  200  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Same as figure 3 for simulation NC-3, without grain coagulation,  1000 years after the formation of the first Larson core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  This has important implications for the dynamics of grains in the  disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Larger grains couple differently to the gas and may trigger  the streaming instability (Youdin & Goodman 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Johansen  & Youdin 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2017), which is an early step toward  planet formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Although this regime is not reached in our sim-  ulations, the fast growth of the grains in circumstellar disks could  predict an early onset for this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Large grains in envelopes and dust diffusion  Although grain coagulation is negligible in the envelope of our  simulations, large grain signatures in envelopes have been ob-  served.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Galametz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' (2019), for example, report "low and  varying dust emissivity indices" at the envelope scale for some  Class 0 and Class I protostars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' This could be due to the pres-  ence of mm-size grains in low numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' The time-scale to form  such large grains in the envelope and cold ISM cores is large  0  5  10  15  20  25  30  35  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='40  1e+51  2e+51  3e+51  4e+51  5e+51  6e+51  7e+51  Disk radius (au)  Disk angular momentum (g cm2 s-1)  t (kyr) after First Core formation  Disk radius  Disk angular momentum  No coagulation  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Disk size (purple, left axis) and angular momentum (green, right  axis) as a function of time after the formation of the first Larson core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  Solid lines represent simulation C-3 with coagulation, and dashed lines  are simulation NC-3 without coagulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  (> 100 Myr), so we exclude the possibility of early coagulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  Similarly, Valdivia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' (2019) report that their synthetic polari-  sation observations of young protostellar envelopes in RAMSES  calculations require grains larger than 10 µm to be consistent  with observations, which is also inconsistent with our findings  and those of recent studies (Silsbee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Lebreuilly et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  2023, for the latest ones).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  The aerodynamic properties of the grains may cause differ-  ential velocities between grain populations and the gas, lead-  ing to varying dust-to-gas ratio throughout the cloud (Lebreuilly  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' We note however that, as they have found, dust dif-  fusion will start to play a significant role only for very large  grains of a few hundred microns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Generally, large grains tend  to accumulate in higher density regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Tsukamoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' (2021)  found that this dust diffusion can lead to what they call an ash fall  phenomenon, in which large coagulated grains (up to millimeter-  size) from the disk are ejected by an outflow, then decouple from  the gas and fall back in the envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' This process may explain  the observations of low spectral index in Class 0 envelopes, as  the outflow fuels the envelope with large grains formed in the  central region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Eventually, those ejected grains circle back to the  outer edge of the disk, enriching the large-end of the size distri-  bution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' In our case, the disk wind is fueled by the upper layers of  the pseudo-disk, in which dust only moderately grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' However,  dust-to-gas ratio enhancement in this region due to the grain dif-  ferential velocities may lead to an accelerated growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  Further studies are needed to investigate this discrepancy  about the size of grains in envelopes between observations and  theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Coagulation in the pseudo-disk  Grains in our simulations seem to grow faster than in Bate (2022)  despite their use of the same turbulent kernel as in our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' The  peaks of the distributions in that work reach a few microns at a  maximum density of nH = 1013 cm−3 (see their Figure 7), while  in our simulations the peak exceeds 100 µm at a lower maximum  density of nH ≈ 3×1012 cm−3 (or ρ ≈ 10−11 g cm−3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' see our Fig-  ure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' That discrepancy is mainly due to a different modelling  Article number, page 7 of 11  A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Marchandetal2023  of the Reynolds number to calculate the turbulent coagulation  kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' They assume a constant value of Re = 108, while we use  (Ormel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Guillet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2020)  Re = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='7 × 107 �  nH  105 cm−3  � 1  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  (17)  Hence, in the central regions, our Reynolds number can be larger  by three orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Their grains are then stuck in the  tightly-coupled regime where relative grain velocities are lower  than in the intermediate coupling-regime, which is more adapted  to this situation (as demonstrated in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='1 of paper I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' The  lower relative velocities result in lower coagulation rates, and  therefore a slower growth rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' The poor constraints on the value  of the Reynolds number in protostellar environments therefore  represents a source of uncertainty for grain growth by coagula-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  An additional explanation may involve an excess of growth  in the pseudo-disk that forms in our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' The pseudo-  disk is an over-density, typically denser than ρ ≈ 10−15 g cm−3,  created by the convergence of gas flowing along magnetic field  lines (Galli & Shu 1993), that takes the shape of a disk perpen-  dicular to the magnetic field, but not supported against gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  It is apparent in the bottom panels of Figures 3 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  Gas in the rotationally supported disk generally comes di-  rectly from the pseudo-disk, and the overall efficiency of coag-  ulation is mainly affected by the time spent in high-density re-  gions like the pseudo-disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' This is what appears in the bottom  panel of Figure 6, in which there is a ∼ 30 au-thick layer of large  grains in the mid-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' A passage of grains through the pseudo-  disk would therefore provide an acceleration of coagulation in  the early stage of star formation, even before arrival in the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  That does not happen in Bate (2022) since they do not consider  magnetic fields, resulting in less-coagulated size distributions as  grains enter the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Coagulation kernel  Our coagulation methods works for every coagulation kernel  where the environment variables and grain variables can be sep-  arated, one example of which is the well-known turbulent ker-  nel derived by Ormel & Cuzzi (2007) that we use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' This kernel  is appropriate to calculate the growth of large grains, but it has  several limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' We assume a steady-state Kolmogorov turbu-  lence spectrum, and that the injection-scale of the turbulence is  equal to the Jeans length corresponding to the local density (see  section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='2 of paper I), which may lead to an over-estimation of  the grain collision rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Other kernels may have different effects  on the size distribution of grains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Guillet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' (2020) showed  that this turbulent kernel would increase the maximum size of  the distribution, while a kernel derived from ambipolar diffusion,  that creates a drift between charged and neutral grains, is effi-  cient at removing small grains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' This also happens in Bate (2022)  and Lebreuilly et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' (2023), which combine several processes  including Brownian motion and pressure gradients that generate  drift velocities between grains and rapidly remove the smaller  grains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' These processes are not, however, efficient at producing  larger grains without the help of the turbulent kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Contrary  to fragmentation, the addition of those kernels would steepen the  distribution at its lower end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Conclusions  We present here the results of numerical simulations of protostel-  lar collapse with dust coagulation and self-consistent calculation  of non-ideal MHD resistivities, using the methods detailed in  Marchand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' We performed four simulations, includ-  ing three with coagulation and different collapse times, and one  without coagulation for reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Here are our main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  Coagulated size distributions retain some characteristics of  the initial MRN distribution, in particular the dominance of  small grains in number and large grains in mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  Dust coagulation is inefficient at growing larger grains in the  envelope, even with long free-fall times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' What matters is the  time spent in high-density regions (ρ > 10−15 g cm−3) like  the pseudo-disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Fragmentation can also be ignored in the  cloud-collapse phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  Grain growth is extremely rapid in the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' The peak of the  size distribution in mass, which is also the largest relevant  grain size of the distribution, reaches 1 µm in the pseudo-  disk and more than 100 µm in the inner disk only 103 yr  after its formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  Grain sizes have a significant impact on non-ideal MHD re-  sistivities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Coagulated grains result in resistivities up to four  orders of magnitude lower than non-coagulated grains, with  a significant impact on the dynamics of the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Simple re-  distribution approximations fail to capture this effect, as it  occurs because of the growth of the largest grains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  Accounting for grain coagulation is therefore necessary in  star formation and protoplanetary disk simulations, as grains  grow rapidly to ≥ 100 µm in radius in disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' The effects of grain  growth on chemistry and radiative transfer will be explored in  future papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' acknowledges the financial support of the Kathryn W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  Davis Postdoctoral Fellowship of the American Museum of Natural History and  of the European Research Council (ERC) under the European Union’s Hori-  zon 2020 research and innovation programme (ERC Starting Grant “Chemtrip”,  grant agreement No 949278).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' was partly supported by US NSF grant  AST18-15461.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' acknowledges the financial support of the European Re-  search Council (ERC) via the ERC Synergy Grant ECOGAL (grant 855130).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  Appendix A: Calculating the resistivities  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='1: Solving for the ionization  Let us assume a size distribution of grains divided into N bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  We have the following system of equations (see Paper I)  Zk =ψτk +  1 − ϵ2Θ2  1 + ϵΘαk + ϵ2Θ2 ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='1)  ⟨ ˜J(τk)⟩ =(1 − ψ) +  2  τk  �  ϵ2Θ2 + ϵΘ  �  1 + ϵΘαk + ϵ2Θ2 ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='2)  ϵ =1 − ψ  Θeψ ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='3)  ni = −  1  1 − ϵ  �  k  nkZk,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='4)  f(ψ) =⟨σv⟩ieϵn2  i  ζnH  + nivi  ζnH  �  k  nkπa2  k⟨ ˜J(τk)⟩ − 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='5)  ϵ = ne/ni < 1 is the ratio between the number of electrons and  ions, ψ is the ratio between the electric potential of the grains  and the kinetic energy of electrons, nk, Zk and ak represent the  number density, the average charge and the radius of grains in  bin k, ζ is the CR ionization rate, and vi = [8kBT/πµimH]1/2  Article number, page 8 of 11  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Marchand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' : Fast methods to track grain coagulation and ionization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Protostellar collapse with non-ideal MHD  is the thermal velocity of ions, with T the temperature, µi the  average atomic mass of ions and mH the mass of the pro-  ton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' We also have Θ = se[µimH/me]1/2, with se the sticking  probability of electrons on grains and me the electron mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  ⟨ ˜J(τk)⟩ represents the enhancement factor for ion recombina-  tion on grains, τk = akkBT/e2 is the reduced temperature of  grains (Draine & Sutin 1987), and αk = [8/(πτk)]1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Finally,  ⟨σv⟩ie = 2 × 10−7[T/300]−1/2 is the collision rate between ions  and electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  We solve the system of equations (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='1) - (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='5) for ψ and  find ni, ne and Zk for all bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' A Newton-Raphson algorithm is  described in details in the Appendix A of Paper I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='2: The resistivities  The collision time-scale of species s with neutral H2, the most  abundant species in the gas, is given by  τsH2 =  1  asHe  ms + mH2  mH2  1  nH2⟨σω⟩s  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='6)  asHe accounts for the collisions with He and is equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='14  for ions, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='16 for electrons and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='28 for grains (Desch &  Mouschovias 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' ms and mH2 are the mass of the species s  and the H2 molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' nH2 represents the number density of H2  (roughly equal to the density of the gas).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' ⟨σω⟩s is the collision  rate, taken from Pinto & Galli (2009) for electrons and ions  ⟨σω⟩e = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='16 × 10−11v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='3  rms,e,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='7)  ⟨σω⟩i = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='4 × 10−9v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='6  rms,i,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='8)  where  vrms,s =  � 8kBT  πµs,H2  � 1  2  ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='9)  is in km s−1, with µs,H2 the reduced mass of species s and H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  These velocities have been calculated in a three-fluid formalism  but we use them in our monofluid framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' See some related  concerns in the appendix of Marchand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' For grains,  the collision rate is given by (Draine & Sutin 1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Kunz &  Mouschovias 2009)  ⟨σω⟩k = πa2  k  �8kBT  πmH2  � 1  2 ��������1 +  � π  2τk  � 1  2 �������� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='10)  We also define the conductivity σs and cyclotron frequency ωs  of species s  σs = nsq2  sτsH2  ms  ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='11)  ωs = qsB  cms  ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='12)  where qs is the electric charge of species s, B the magnetic field  strength and c the speed of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' The parallel, perpendicular and  Hall conductivities are expressed as  σ|| =  �  s  σs,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='13)  σ⊥ =  �  s  σs  1 + (ωsτsH2)2 ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='14)  σH =  �  s  σsωsτsH2  1 + (ωsτsH2)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='15)  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='16)  Finally, the Ohmic, Hall and ambipolar resisitivities are defined  as  ηO = 1  σ||  ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='17)  ηH =  σH  σ2  ⊥ + σ2  H  ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='18)  ηA =  σ⊥  σ2  ⊥ + σ2  H  − 1  σ||  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='19)  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='20)  Appendix B: Coagulation and dust-to-gas ratio  Our coagulation model assumes that grains are perfectly cou-  pled with the gas and well-mixed, so that the dust-to-gas mass  ratio is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' However, grains of different sizes have dif-  ferent aerodynamic properties and may experience a significant  drift through the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' This characteristic is usually determined  by their Stokes number, that is the ratio between the stopping  time of the grain (Epstein 1924) and the dynamical timescale of  the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' In particular, Lebreuilly et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' (2020) showed strong  variations in the dust-to-gas mass ratio during the protostellar  collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' The disk and first core tend to be dust-enriched while  the envelope becomes dust-depleted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' This has strong implication  for the coagulation, as a higher grain density promotes collisions  and speeds up their growth (conversely for a lower density).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' In  this section, we briefly explore a refinement to the coagulation  model presented in Paper I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  The general expression of the Smoluchowski (1916) equa-  tion, from which we derive the expression of χ, is (Paper I eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 5)  dX(a, t)  dt  = CglocalnHI(a, X, t),  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='1)  where C is a constant, glocal a function of the local properties of  the gas (density, temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='), and  I(a, X, t) = −  � ∞  0  h(m, m′)X(m, t)X(m′, t)dm′  + 1  2  � m  0  h(m − m′, m′)X(m − m′, t)X(m′, t)dm′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='2)  The relative number of grains of size a at a given time t is  X(a, t) = ngrain/nH where ngrain is the number density of grains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  We can then write X = dgx, where dg is the dust-to-gas ratio, and  x the normalized relative number of grains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' We can then rewrite  equation (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='1)  dx(a, t)  dt  = CglocalnHdgI′(a, x, t),  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='3)  and include dg in the definition of our coagulation variable, giv-  ing it the form  dχ′ = glocalnHdgdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='4)  In our case, for the Ormel & Cuzzi (2007) kernel,  dχ′ = n  3  4  HT − 1  4 dgdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='5)  This expression means that the dust-to-gas ratio does not change  the evolution path of the size distribution, only the coagulation  Article number, page 9 of 11  A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Marchandetal2023  speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' A dust-to-gas ratio N times higher requires a χ value N  times lower to reach the same coagulated state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' This new defini-  tion of χ′ can be used in an environment with varying dust-to-  gas ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' In hydrodynamics simulations, it is possible to couple  it with the drift of one grain size close to the peak of the mass  density distribution, as allowed by the method proposed by Le-  breuilly et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  Although it is not yet possible to associate this coagulation  method with the differential drift of grains of different sizes, this  is a first step towards a self-consistent treatment of dust grains  in hydrodynamics simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' This method will lead to an over-  estimate of the small-to-larger grain ratio but this is probably a  decent approximation to get the total dust-to-gas ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' As shown  in Lebreuilly et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' (2020), as long as the large grains dominate  the mass in the distribution, they also dominate the differential  gas and dust dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' In other words, most of the dust enrich-  ment comes from the dynamics of large grains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  Appendix C: The fragmentation barrier  Ormel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' (2009) derived criteria for the fragmentation of dust  aggregates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' They define the rolling energy Eroll that determines  the energy needed to restructure the grain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='2 of pa-  per I, we misinterpreted their results by assuming the fragmen-  tation would occur for a kinetic energy Ekin = 5Eroll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' The actual  criterion is  Ekin > 5NtotEbr,  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='1)  where Ekin is the kinetic energy, Ebr is the breaking energy and  Ntot is the total numbers of monomers composing the two collid-  ing grains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Ebr is defined by (Dominik & Tielens 1997)  Ebr = Abrγ5/3  grain  (a0/2)4/3  ε2/3  ,  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='2)  with Abr = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='8 × 103, γgrain the surface energy density of the ma-  terial, a0 the size of the monomers composing the grains, and ε  the reduced elastic modulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' As in Ormel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' (2009), we adopt  a0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='1 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Ice mantles on grains make them more resistant  to fragmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Therefore, we assume bare silicates to obtain a  lower limit for a fragmentation criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' In this case, γgrain = 25  erg cm−2 and ε = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='8×1011 dyn cm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' The kinetic energy of two  grains of mass m and m′ reads  Ekin = 1  2  mm′  m + m′ ∆v2,  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='3)  where ∆v is the relative velocity between the two grains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' This  velocity is given by the kernel of Ormel & Cuzzi (2007) that we  use in Paper I  ∆v =  � 3√  8  z0[kBG]  1  2 γρs  µmH  � 1  2  n  − 1  4  H T − 1  4 a  1  2 ,  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='4)  where z0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='97, kB and G are the Boltzmann and gravitational  constants, γ = 5/3 the adiabatic index of the gas, ρs = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='3 g cm−3  the bulk density of the grains, µ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='3 the average atomic mass  of the gas, mH the proton mass, and a the radius of the larger of  the two grains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Assuming two identical grains and  m = 4  3πρsa3  0Ntot,  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='5)  equation C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='1 becomes  a > 15Ebr  πρsa3  0  � 3√  8  z0[kBG]  1  2 γρs  µmH  �−1  n  1  2  HT  1  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='6)  Replacing by numeric values, we find  a > 826 µm  �  nH  1010 cm−3  � 1  2 � T  10 K  � 1  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='7)  This limit is much higher than the one derived in paper I, and we  do not reach it in our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' The value of γgrain we use is  taken from Ormel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' (2009), but surface energies of different  materials can reach 10 to 100 times this value, resulting in a  much higher estimate of the fragmentation threshold (as Ebr ∼  γ5/3  grain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' The value derived in equation (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='7) should then be taken  as very conservative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  References  Balsara, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2012, Journal of Computational Physics, 231, 7476  Bate, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2022, MNRAS, 514, 2145  Blandford, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' & Payne, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 1982, MNRAS, 199, 883  Crutcher, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 1999, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' , 520, 706  Desch, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' & Mouschovias, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2001, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' , 550, 314  Dominik, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' & Tielens, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 1997, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' , 480, 647  Draine, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' & Sutin, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 1987, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' , 320, 803  Duffin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' & Pudritz, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2008, MNRAS, 391, 1659  Dullemond, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' & Dominik, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2005, A&A, 434, 971  Epstein, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 1924, Physical Review, 23, 710  Flower, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Pineau Des Forêts, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', & Walmsley, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2005, A&A, 436, 933  Galametz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Maury, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Valdivia, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2019, A&A, 632, A5  Galli, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' & Shu, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 1993, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' , 417, 220  Guillet, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' & Teyssier, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2011, Journal of Computational Physics, 230, 4756  Guillet, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Hennebelle, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Pineau des Forêts, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2020, A&A, 643, A17  Johansen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' & Youdin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2007, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' , 662, 627  Kawasaki, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Koga, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', & Machida, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2022, MNRAS, 515, 2072  Kunz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' & Mouschovias, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2009, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' , 693, 1895  Kuznetsova, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Hartmann, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', & Heitsch, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2020, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' , 893, 73  Kwon, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Looney, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Mundy, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Chiang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', & Kemball, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2009,  Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' , 696, 841  Larson, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 1969, MNRAS, 145, 271  Le Gouellec, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Hull, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Maury, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2019, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' ,  885, 106  Lebreuilly, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Commerçon, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', & Laibe, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2019, A&A, 626, A96  Lebreuilly, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Commerçon, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', & Laibe, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2020, A&A, 641, A112  Lebreuilly, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Hennebelle, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Colman, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2021, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' , 917,  L10  Lebreuilly, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Vallucci-Goy, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Guillet, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Lombart, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', & Marchand, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2023,  MNRAS, 518, 3326  Li, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Krasnopolsky, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', & Shang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2011, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' , 738, 180  Lombart, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' & Laibe, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2021, MNRAS, 501, 4298  Machida, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Inutsuka, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', & Matsumoto, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2006, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' , 647,  L151  Marchand, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Commerçon, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', & Chabrier, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2018, A&A, 619, A37  Marchand, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Guillet, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Lebreuilly, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', & Mac Low, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2021, A&A, 649,  A50  Marchand, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Masson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Chabrier, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2016, A&A, 592, A18  Marchand, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Tomida, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Tanaka, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Commerçon, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', & Chabrier, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  2020, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' , 900, 180  Masson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Chabrier, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Hennebelle, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Vaytet, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', & Commercon, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2016,  A&A, 587, A32  Masson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Teyssier, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Mulet-Marquis, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Hennebelle, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', & Chabrier, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='  2012, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Suppl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' , 201, 24  Mathis, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Rumpl, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', & Nordsieck, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 1977, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' , 217, 425  Mellon, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' & Li, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2009, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' , 698, 922  Miotello, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Testi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Lodato, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2014, A&A, 567, A32  Miyoshi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' & Kusano, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2005, Journal of Computational Physics, 208, 315  Mouschovias, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' & Paleologou, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 1979, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' , 230, 204  Mouschovias, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' & Spitzer, Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 1976, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' , 210, 326  Ormel, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' & Cuzzi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2007, A&A, 466, 413  Ormel, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Paszun, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Dominik, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', & Tielens, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2009, A&A,  502, 845  Pinto, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' & Galli, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2009, Three-Fluid Magnetohydrodynamics in Star Forma-  tion, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Tsinganos, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Ray, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', & Stute, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', 597–599  Rossi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Benevides-Soares, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', & Barbuy, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 1991, A&A, 251, 587  Silsbee, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Akimkin, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Ivlev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2022, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' , 940, 188  Silsbee, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Ivlev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Sipilä, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Caselli, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', & Zhao, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2020, A&A, 641, A39  Smoluchowski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 1916, Zeitschrift fur Physik, 17, 557  Teyssier, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2002, A&A, 385, 337  Tomida, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Okuzumi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', & Machida, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2015, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' , 801, 117  Article number, page 10 of 11  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Marchand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' : Fast methods to track grain coagulation and ionization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' Protostellar collapse with non-ideal MHD  Truelove, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Klein, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', McKee, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 1997, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' , 489,  L179  Tsukamoto, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Machida, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', & Inutsuka, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2021, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' ,  920, L35  Valdivia, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Maury, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Brauer, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2019, MNRAS, 488, 4897  Vaytet, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Commerçon, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Masson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', González, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', & Chabrier, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2018,  A&A, 615, A5  Vaytet, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' & Haugbølle, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2017, A&A, 598, A116  Vericel, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Gonzalez, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Price, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Laibe, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', & Pinte, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2021, MNRAS,  507, 2318  Vorobyov, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Skliarevskii, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Molyarova, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2022, A&A, 658, A191  Wurster, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Price, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', & Bate, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2016, MNRAS, 457, 1037  Yang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Johansen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', & Carrera, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2017, A&A, 606, A80  Youdin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' & Goodman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2005, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' , 620, 459  Zhao, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Caselli, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Li, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', & Krasnopolsky, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2018, MNRAS, 473, 4868  Zhao, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Caselli, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', Li, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
+page_content=' 2016, MNRAS, 460, 2050  Article number, page 11 of 11' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAzT4oBgHgl3EQfi_3N/content/2301.01510v1.pdf'}
diff --git a/HtAzT4oBgHgl3EQfxv41/content/tmp_files/2301.01742v1.pdf.txt b/HtAzT4oBgHgl3EQfxv41/content/tmp_files/2301.01742v1.pdf.txt
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+arXiv:2301.01742v1  [cs.GT]  4 Jan 2023
+Tight Distortion Bounds for Distributed
+Single-Winner Metric Voting on a Line
+Alexandros A. Voudouris
+School of Computer Science and Electronic Engineering, University of Essex
+Abstract
+We consider the distributed single-winner metric voting problem on a line, where agents and
+alternative are represented by points on the line of real numbers, the agents are partitioned into
+disjoint districts, and the goal is to choose a single winning alternative in a decentralized manner. In
+particular, the choice is done by a distributed voting mechanism which first selects a representative
+alternative for each district of agents and then chooses one of these representatives as the winner.
+In this paper, we design simple distributed mechanisms that achieve distortion at most 2 +
+√
+5
+for the average-of-max and the max-of-average social cost objectives, matching the corresponding
+lower bound shown in previous work for these objectives.
+1
+Introduction and Model
+We consider the following voting problem. An instance I consists of a set N of n agents and a set A
+of m alternatives, all of whom are represented by points on the line of real numbers. For any agent
+i ∈ N and alternative x ∈ A, let δ(i, x) be the distance between i and x on the line (which is equal
+to the absolute difference between their positions). Te agents are also partitioned into a set D of
+k districts, such that each district contains at least one agent. We denote by Nd the set of agents of
+district d ∈ D, and by nd = |Nd| the size of d. Te goal is to choose an alternative with good social
+efficiency guarantees using a distributed mechanism that takes as input ordinal information about the
+instance, such as the ordering of agents and alternatives on the line, and the ordinal preferences of
+the agents over the alternatives. To be specific, the preference of an agent i over the alternatives is a
+linear ordering of the alternatives such that alternative x is ranked higher than another alternative y
+if δ(i, x) ≤ δ(i, y), breaking ties arbitrarily but consistently.
+In general, a distributed mechanism M works as follows:
+• Step 1: For each district d ∈ D, given the preferences of the agents in Nd over the alternatives
+and their relative ordering on the line, M decides a representative alternative yd ∈ A for d.
+• Step 2: Given the representatives of all districts and their relative ordering on the line, M outputs
+one of them as the overall winner M(I) ∈ �
+d∈D{yd}.
+Clearly, different distributed mechanisms can be designed by changing the method used for deciding
+the representative alternatives of the districts or the method for choosing a representative as the overall
+winner.
+Interpreting the distance δ(i, x) as the individual cost of agent i for alternative x, we measure the
+social efficiency of x by some function of the distances of all agents from x. Tere are many well-known
+social efficiency objectives that have been studied in various social choice problem, such as the social
+1
+
+cost (total or average distance of all agents) and the max cost (maximum distance among all agents). In
+the context of metric distributed voting, Anshelevich et al. [2022] considered social objectives that are
+composed by some function that is applied over the districts and some function that is applied within
+the districts. In particular, they focused on the following four objectives.
+• Te average-of-average cost of x is the average over each district of the average individual cost
+of the agents therein:
+(AVG ◦ AVG)(x) = 1
+k
+�
+d∈D
+� 1
+nd
+�
+i∈Nd
+δ(i, x)
+�
+.
+• Te max cost of x is the max individual cost over all agents:
+(MAX ◦ MAX)(x) = max
+d∈D max
+i∈Nd
+δ(i, x) = max
+i∈N δ(i, x).
+• Te average-of-max cost of x is the average over each district of the max individual cost therein:
+(AVG ◦ MAX)(x) = 1
+k
+�
+d∈D
+max
+i∈Nd
+δ(i, x).
+• Te max-of-average cost of x is the max over each district of the average individual cost therein:
+(MAX ◦ AVG)(x) = max
+d∈D
+� 1
+nd
+�
+i∈Nd
+δ(i, x)
+�
+.
+We measure the efficiency of a mechanism M with respect to a social objective f (such as the ones
+defined above) by its distortion, the worst-case ratio (over all possible instances) of the f-value of the
+alternative chosen by the mechanism over the minimum possible f-value among all alternatives:
+sup
+I=(N,A,D)
+f(M(I))
+minx∈A f(x)
+By definition, the distortion of any mechanism is at least 1. We aim to design distributed mechanisms
+with an as low distortion as possible.
+1.1
+Our Contribution
+In previous work, Anshelevich et al. [2022] considered the problem of distributed metric voting where
+agents and alternatives are in some arbitrary metric space. By carefully composing centralized voting
+mechanisms for making decisions within and over the districts, they designed distributed mechanisms
+with distortion guarantees for general metric spaces and with respect to the aforementioned objectives
+as well as more general ones. For the special case of a line metric (which is our focus here), they showed
+that the distortion of distributed mechanisms that use ordinal information is exactly 7 with respect to
+the average-of-average cost, exactly 3 with respect to the max cost, and in the interval [2+
+√
+5, 5] with
+respect to the average-of-max or the max-of-average cost objectives.
+Inspired by the recent work of Filos-Ratsikas et al. [2023] who designed distributed mechanisms
+with tight distortion bounds for the continuous distributed facility location problem (where each point
+on the line of real numbers can be considered as an alternative), we resolve the distortion of distributed
+mechanisms for voting on a line with respect to the average-of-max and the max-of-average costs. In
+2
+
+particular, we design two essentially symmetric distributed mechanism that achieve a tight distortion
+bound of 2+
+√
+5 for the average-of-max cost and the max-of-average cost. For the average-of-max cost,
+our distributed mechanism chooses as the favorite alternative of the rightmost agent in each district as
+the district representative, and then the (α · k)-the lefmost alternative as the overall winner. For the
+max-of-average cost, our distributed mechanism chooses the favorite alternative of the (α·nd)-th agent
+in each district d as the district representative, and then the rightmost alternative as the overall winner.
+We show that both mechanisms achieve distortion at most max
+�
+3−α
+1−α, 2
+α − 1
+�
+for the corresponding
+objectives; for α = 3−
+√
+5
+2
+, the two terms in the bound balance out to 2 +
+√
+5.
+1.2
+Other Related Work
+Since its definition by Procaccia and Rosenschein [2006], the distortion of voting mechanisms has
+been studied extensively for several setings under different assumptions about the preferences of the
+agents. Te most well-studied seting is that of single-winner voting that has been considered under
+the premise of normalized agent valuations [Boutilier et al., 2015, Ebadian et al., 2022, Amanatidis et al.,
+2021] as well as metric preferences [Anshelevich et al., 2018, Gkatzelis et al., 2020, Kizilkayaand Kempe,
+2022]. Several other models have also been considered, such as multi-winner voting [Caragiannis et al.,
+2017, 2022], participatory budgeting [Benad`e et al., 2021], and matching [Filos-Ratsikas et al., 2014,
+Amanatidis et al., 2022].
+Te distortion of distributed voting was first considered by [Filos-Ratsikas et al., 2020] who focused
+on bounding the distortion of max-weight mechanisms with respect to the social welfare when agents
+have normalized valuations for the alternatives. Filos-Ratsikas and Voudouris [2021] then studied the
+distortion of mechanisms for a distributed facility location seting where the agents are positioned on
+a line and the goal is to choose a single location from a set of alternative ones, which can be discrete
+(finite) or continuous (infinite). Te discrete variant was studied further and generalized to arbitrary
+metric spaces by Anshelevich et al. [2022], who also introduced and studied the average-of-max and
+max-of-average costs for the first time. Te distortion of distributed mechanisms for the continuous
+variant on a line was recently resolved by Filos-Ratsikas et al. [2023]. We refer the reader to the recent
+survey of Anshelevich et al. [2021] for more details on the distortion of voting mechanisms in different
+models, and to the survey of Chan et al. [2021] for details on related facility location models.
+2
+Average-of-Max Cost
+Let α ∈ [0, 1] be a parameter. We consider the α-Lefmost-of-Rightmost, which works as follows. For
+each district d ∈ D, the mechanism chooses the favorite alternative of the rightmost agent in d as the
+representative yd. Aferwards, it chooses the (α · k)-th lefmost alternative as the overall winner. See
+Mechanism 1.
+Mechanism 1: α-Leftmost-of-Rightmost
+for each district d ∈ D do
+yd := favorite alternative of the rightmost agent in Nd;
+return w := (α · k)-th lefmost representative;
+Teorem 2.1. For average-of-max, the distortion of the α-Leftmost-of-Rightmost mechanism is at
+most max
+�
+3−α
+1−α, 2
+α − 1
+�
+.
+3
+
+Proof. Let w be the alternative chosen by the mechanism when given as input an arbitrary instance,
+and o the optimal alternative. For each district d, let id be the most distant agent from w, and i∗
+d
+the most distant agent from o. So, (AVG ◦ MAX)(w) =
+1
+k
+�
+d∈D δ(id, w), and (AVG ◦ MAX)(o) =
+1
+k
+�
+d∈D δ(i∗
+d, o). Also, let ℓd and rd denote the lefmost and rightmost agents in d, respectively. Note
+that id, id∗ ∈ {ℓd, rd}. We consider the following two cases depending on the relative positions of w
+and o.
+Case 1: o < w.
+By the triangle inequality, and since δ(id, o) ≤ δ(i∗
+d, o), we have
+(AVG ◦ MAX)(w) = 1
+k
+�
+d∈D
+δ(id, w)
+≤ 1
+k
+�
+d∈D
+�
+δ(id, o) + δ(w, o)
+�
+≤ (AVG ◦ MAX)(o) + δ(w, o).
+(1)
+Now, let S be the set of representatives that are to the right of w. Since w is by definition the (α · k)-th
+lefmost representative, we have that |S| ≥ (1−α)·k. For every d such that yd ∈ S, since o < w ≤ yd,
+agent rd is closer to w than to o, and thus δ(rd, o) ≥ 1
+2δ(w, o). Hence,
+(AVG ◦ MAX)(o) ≥ 1
+k
+�
+d∈S
+δ(rd, o)
+≥ 1
+k · |S| · δ(w, o)
+2
+≥ 1 − α
+2
+· δ(w, o),
+or, equivalently,
+δ(w, o) ≤
+2
+1 − α · (AVG ◦ MAX)(o).
+(2)
+Hence, by (1) and (2), we obtain
+(AVG ◦ MAX)(w) ≤
+�
+1 +
+2
+1 − α
+�
+· cost(o) = 3 − α
+1 − α · (AVG ◦ MAX)(o).
+Case 2: w < o.
+Let L be a set of α·k districts from the one with the lefmost representative until the
+one with the (α · k)-th representative (which is w); denote by R the set of the remaining (1 − α) · k
+districts. Observe that:
+• For every d ∈ L, since yd is the alternative that is closest to rd and yd ≤ w < o, both rd and ℓd
+are closer to w than to o. So, δ(id, w) ≤ δ(i∗
+d, o).
+• For every d ∈ R, since δ(id, o) ≤ δ(i∗
+d, o) by the definition of i∗
+d, using the triangle inequality,
+we obtain δ(id, w) ≤ δ(id, o) + δ(w, o) ≤ δ(i∗
+d, o) + δ(w, o).
+Hence,
+(AVG ◦ MAX)(w) = 1
+k
+�
+d∈D
+δ(id, w)
+4
+
+= 1
+k
+�
+d∈L
+δ(id, w) + 1
+k
+�
+d∈R
+δ(id, w)
+≤ 1
+k
+�
+d∈L
+δ(i∗
+d, o) + 1
+k
+�
+d∈R
+�
+δ(i∗
+d, o) + δ(w, o)
+�
+= (AVG ◦ MAX)(o) + |R|
+k · δ(w, o).
+(3)
+Since rd is closer to w than to o for every d ∈ L, we also have that
+(AVG ◦ MAX)(o) ≥ 1
+k
+�
+d∈L
+δ(rd, o) ≥ |L|
+2k δ(w, o),
+or, equivalently,
+δ(w, o) ≤ 2k
+|L| · (AVG ◦ MAX)(o).
+(4)
+Terefore, by (3) and (4), we obtain
+(AVG ◦ MAX)(w) ≤ (AVG ◦ MAX)(o) + 2|R|
+|L| · (AVG ◦ MAX)(o) =
+� 2
+α − 1
+�
+· (AVG ◦ MAX)(o).
+Puting everything together, we obtain an upper bound of max
+�
+3−α
+1−α, 2
+α − 1
+�
+.
+Observe that the bound max
+�
+3−α
+1−α, 2
+α − 1
+�
+consists of two functions of α, one that is non-
+decreasing and one that is non-increasing in α. To minimize the maximum between the two, we need
+to find the value of α for which the two functions intersect. So, we need to solve the equation
+3 − α
+1 − α = 2
+α − 1 ⇔ α2 − 3α + 1 = 0.
+Since α < 1, its solution is α = 3−
+√
+5
+2
+. For this value of α, both functions have value
+2
+3−
+√
+5
+2
+−1 = 2+
+√
+5,
+and we obtain the following corollary.
+Corollary 2.2. For average-of-max, the distortion of the 3−
+√
+5
+2
+-Leftmost-of-Rightmost mechanism is
+at most 2 +
+√
+5.
+3
+Max-of-Average Cost
+Let α ∈ [0, 1] be a parameter. We consider the Rightmost-of-α-Leftmost mechanism, which works
+as follows. For each district d ∈ D, the mechanism chooses the favorite alternative of the (α · nd)-
+th agent in d as the representative yd. Aferwards, it chooses the rightmost alternative as the overall
+winner. See Mechanism 2.
+Teorem 3.1. For max-of-average, the distortion of the Rightmost-of-α-leftmost mechanism is at
+most max
+�
+3−α
+1−α, 2
+α − 1
+�
+.
+Proof. Let w be the alternative chosen be the mechanism when given as input an arbitrary instance,
+and o the optimal alternative. For any district d and alternative x, let AVGd(x) =
+1
+nd
+�
+i∈Nd δ(i, x) be
+the total average distance of the agents in d for alternative x. So, AVGd(o) ≤ (MAX◦AVG)(o) for every
+district d. Denote by d∗ a district that gives the max average cost for w, such that (MAX ◦ AVG)(w) =
+AVGd∗(w). Also, let dw be a district represented by w. In addition, let i∗ and iw be the (α · nd)-th
+lefmost agents in districts d∗ and dw, respectively. We now switch between the following two cases.
+5
+
+Mechanism 2: Rightmost-of-α-Leftmost
+for each district d ∈ D do
+yd := favorite alternative of the (α · nd)-th lefmost agent in Nd;
+return w := rightmost representative;
+Case 1: o < w.
+By the definition of d∗ and the triangle inequality, we have
+(MAX ◦ AVG)(w) = 1
+nd
+�
+i∈Nd∗
+δ(i, w)
+≤ 1
+nd
+�
+i∈Nd∗
+�
+δ(i, o) + δ(o, w)
+�
+≤ (MAX ◦ AVG)(o) + δ(o, w).
+(5)
+Denote by S the set of agents that are positioned weakly to the right of iw in dw. By the definition of
+iw, |S| ≥ (1 − α)ndw. Since o < w and w is the favorite alternative of iw, all agents in S are closer to
+w than to o, and thus δ(i, o) ≥ 1
+2δ(w, o) for any i ∈ S. Using all these, we obtain:
+AVGdw(o) =
+1
+ndw
+�
+i∈Ndw
+δ(i, o)
+≥
+1
+ndw
+�
+i∈S
+δ(i, o)
+≥
+1
+ndw
+· |S|
+2 · δ(w, o) ≥ 1 − α
+2
+· δ(w, o),
+or, equivalently,
+δ(w, o) ≤
+2
+1 − α · AVGdw(o) ≤
+2
+1 − α · (MAX ◦ AVG)(o).
+(6)
+Terefore, by (5) and (6), we obtain
+(MAX ◦ AVG)(w) ≤ (MAX ◦ AVG)(o) +
+2
+1 − α · (MAX ◦ AVG)(o) = 3 − α
+1 − α · (MAX ◦ AVG)(o).
+Case 2: w < o.
+Let L be the set of the first α · nd∗ agents of d∗ (from the lefmost agent to i∗), and
+R be the set of the remaining (1 − α)nd∗ agents. As w is the rightmost representative, yd∗ ≤ w < o.
+Since yd∗ is the favorite alternative of i∗, every agent i ∈ L prefers w over o, and thus δ(i, w) = δ(i, o).
+Using this in combination with the triangle inequality for every agent of R, we have
+(MAX ◦ AVG)(w) =
+1
+nd∗
+�
+i∈Nd∗
+δ(i, w)
+=
+1
+nd∗
+�
+i∈L
+δ(i, w) +
+1
+nd∗
+�
+i∈R
+δ(i, w)
+≤
+1
+nd∗
+�
+i∈L
+δ(i, o) + 1
+nd∗
+�
+i∈R
+�
+δ(i, o) + δ(w, o)
+�
+=
+1
+nd∗
+�
+i∈Nd∗
+δ(i, o) + |R|
+nd∗ · δ(w, o)
+6
+
+≤ (MAX ◦ AVG)(o) + |R|
+nd∗ · δ(w, o).
+(7)
+Since each agent i ∈ L prefers w over o, we also have that δ(i, o) ≥ 1
+2δ(w, o), and thus
+(MAX ◦ AVG)(o) ≥ AVGd∗(o) =
+1
+nd∗
+�
+i∈Nd∗
+δ(i, o)
+≥
+1
+nd∗
+�
+i∈Nd∗
+δ(i, o)
+≥ |L|
+2nd∗ · δ(w, o)
+or, equivalently,
+δ(w, o) ≤ 2nd∗
+|L| · (MAX ◦ AVG)(o).
+(8)
+Terefore, by (7) and (8), we obtain
+(MAX ◦ AVG)(w) ≤ (MAX ◦ AVG)(o) + 2|R|
+|L| · (MAX ◦ AVG)(o) =
+� 2
+α − 1
+�
+· (MAX ◦ AVG)(o).
+Puting everything together, we get an upper bound of max
+�
+3−α
+1−α, 2
+α − 1
+�
+.
+By optimizing over α, similarly to Section 2, we obtain the following result.
+Corollary 3.2. For max-of-average, the distortion of the Rightmost-of- 3−
+√
+5
+2
+-Leftmost mechanism is
+at most 2 +
+√
+5.
+4
+Open Qestions
+In this paper, we showed a tight distortion bound of 2 +
+√
+5 with respect to the average-of-max and
+max-of-average cost functions for the single-winner distributed single-winner voting problem, thus
+completing the distortion picture with respect to the four basic objectives considered by Anshelevich
+et al. [2022] for the line metric. Te most interestingdirection for future work is to prove tight distortion
+bounds for general metric spaces, and also consider other social objectives or setings beyond single-
+winner voting.
+References
+Georgios Amanatidis, Georgios Birmpas, Aris Filos-Ratsikas, and Alexandros A. Voudouris. Peeking
+behind the ordinal curtain: Improving distortion via cardinal queries. Artificial Intelligence, 296:
+103488, 2021.
+Georgios Amanatidis, Georgios Birmpas, Aris Filos-Ratsikas, and Alexandros A. Voudouris. A few
+queries go a long way: Information-distortion tradeoffs in matching. Journal of Artificial Intelligence
+Research, 74, 2022.
+Elliot Anshelevich, Onkar Bhardwaj, Edith Elkind, John Postl, and Piotr Skowron. Approximating
+optimal social choice under metric preferences. Artificial Intelligence, 264:27–51, 2018.
+7
+
+Elliot Anshelevich, Aris Filos-Ratsikas, Nisarg Shah, and Alexandros A. Voudouris. Distortion in so-
+cial choice problems: Te first 15 years and beyond. In Proceedings of the 30th International Joint
+Conference on Artificial Intelligence (IJCAI), pages 4294–4301, 2021.
+Elliot Anshelevich, Aris Filos-Ratsikas, and Alexandros A. Voudouris. Te distortion of distributed
+metric social choice. Artificial Intelligence, 308:103713, 2022.
+Gerdus Benad`e, Swaprava Nath, Ariel D. Procaccia, and Nisarg Shah. Preference elicitation for partic-
+ipatory budgeting. Management Science, 67(5):2813–2827, 2021.
+Craig Boutilier, Ioannis Caragiannis, Simi Haber, Tyler Lu, Ariel D. Procaccia, and Or Sheffet. Optimal
+social choice functions: A utilitarian view. Artificial Intelligence, 227:190–213, 2015.
+Ioannis Caragiannis, Swaprava Nath, Ariel D. Procaccia, and Nisarg Shah. Subset selection via implicit
+utilitarian voting. Journal of Artificial Intelligence Research, 58:123–152, 2017.
+Ioannis Caragiannis, Nisarg Shah, and Alexandros A. Voudouris. Te metric distortion of multiwinner
+voting. Artificial Intelligence, 313:103802, 2022.
+Hau Chan, Aris Filos-Ratsikas, Bo Li, Minming Li, and Chenhao Wang. Mechanism design for facility
+location problem: A survey. In Proceedings of the 30th International Joint Conference on Artificial
+Intelligence (IJCAI), pages 1–17, 2021.
+Soroush Ebadian, Anson Kahng, Dominik Peters, and Nisarg Shah. Optimized distortion and propor-
+tional fairness in voting. In Proceedings of the 23rd ACM Conference on Economics and Computation
+(EC), pages 523–600, 2022.
+Aris Filos-Ratsikas and Alexandros A. Voudouris. Approximate mechanism design for distributed facil-
+ity location. In Proceedings of the 14th International Symposium on Algorithmic Game Teory (SAGT),
+pages 49–63, 2021.
+Aris Filos-Ratsikas, Søren Kristoffer Stiil Frederiksen, and Jie Zhang. Social welfare in one-sided match-
+ings: Random priority and beyond. In Proceedings of the 7th Symposium of Algorithmic Game Teory
+(SAGT), pages 1–12, 2014.
+Aris Filos-Ratsikas, Evi Micha, and Alexandros A. Voudouris. Te distortion of distributed voting.
+Artificial Intelligence, 286:103343, 2020.
+Aris Filos-Ratsikas, Panagiotis Kanellopoulos, Alexandros A. Voudouris, and Rongsen Zhang. Setling
+the distortion of distributed facility location. In Proceedings of the 22nd International Conference on
+Autonomous Agents and Multiagent Systems (AAMAS), 2023.
+Vasilis Gkatzelis, Daniel Halpern, and Nisarg Shah. Resolving the optimal metric distortion conjecture.
+In Proceedings of the 61st Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages
+1427–1438, 2020.
+Fatih Erdem Kizilkaya and David Kempe. Plurality veto: A simple voting rule achieving optimal metric
+distortion. In Proceedings of the 31st International Joint Conference on Artificial Intelligence (IJCAI),
+pages 349–355, 2022.
+Ariel D. Procaccia and Jeffrey S. Rosenschein. Te distortion of cardinal preferences in voting. In
+International Workshop on Cooperative Information Agents (CIA), pages 317–331, 2006.
+8
+
diff --git a/HtAzT4oBgHgl3EQfxv41/content/tmp_files/load_file.txt b/HtAzT4oBgHgl3EQfxv41/content/tmp_files/load_file.txt
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@@ -0,0 +1,213 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf,len=212
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='01742v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='GT]  4 Jan 2023  Tight Distortion Bounds for Distributed  Single-Winner Metric Voting on a Line  Alexandros A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' Voudouris  School of Computer Science and Electronic Engineering, University of Essex  Abstract  We consider the distributed single-winner metric voting problem on a line, where agents and  alternative are represented by points on the line of real numbers, the agents are partitioned into  disjoint districts, and the goal is to choose a single winning alternative in a decentralized manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' In  particular, the choice is done by a distributed voting mechanism which first selects a representative  alternative for each district of agents and then chooses one of these representatives as the winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  In this paper, we design simple distributed mechanisms that achieve distortion at most 2 +  √  5  for the average-of-max and the max-of-average social cost objectives, matching the corresponding  lower bound shown in previous work for these objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  1  Introduction and Model  We consider the following voting problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' An instance I consists of a set N of n agents and a set A  of m alternatives, all of whom are represented by points on the line of real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' For any agent  i ∈ N and alternative x ∈ A, let δ(i, x) be the distance between i and x on the line (which is equal  to the absolute difference between their positions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' Te agents are also partitioned into a set D of  k districts, such that each district contains at least one agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' We denote by Nd the set of agents of  district d ∈ D, and by nd = |Nd| the size of d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' Te goal is to choose an alternative with good social  efficiency guarantees using a distributed mechanism that takes as input ordinal information about the  instance, such as the ordering of agents and alternatives on the line, and the ordinal preferences of  the agents over the alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' To be specific, the preference of an agent i over the alternatives is a  linear ordering of the alternatives such that alternative x is ranked higher than another alternative y  if δ(i, x) ≤ δ(i, y), breaking ties arbitrarily but consistently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  In general, a distributed mechanism M works as follows:  Step 1: For each district d ∈ D, given the preferences of the agents in Nd over the alternatives  and their relative ordering on the line, M decides a representative alternative yd ∈ A for d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  Step 2: Given the representatives of all districts and their relative ordering on the line, M outputs  one of them as the overall winner M(I) ∈ �  d∈D{yd}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  Clearly, different distributed mechanisms can be designed by changing the method used for deciding  the representative alternatives of the districts or the method for choosing a representative as the overall  winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  Interpreting the distance δ(i, x) as the individual cost of agent i for alternative x, we measure the  social efficiency of x by some function of the distances of all agents from x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' Tere are many well-known  social efficiency objectives that have been studied in various social choice problem, such as the social  1  cost (total or average distance of all agents) and the max cost (maximum distance among all agents).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' In  the context of metric distributed voting, Anshelevich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' [2022] considered social objectives that are  composed by some function that is applied over the districts and some function that is applied within  the districts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' In particular, they focused on the following four objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  Te average-of-average cost of x is the average over each district of the average individual cost  of the agents therein:  (AVG ◦ AVG)(x) = 1  k  �  d∈D  � 1  nd  �  i∈Nd  δ(i, x)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  Te max cost of x is the max individual cost over all agents:  (MAX ◦ MAX)(x) = max  d∈D max  i∈Nd  δ(i, x) = max  i∈N δ(i, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  Te average-of-max cost of x is the average over each district of the max individual cost therein:  (AVG ◦ MAX)(x) = 1  k  �  d∈D  max  i∈Nd  δ(i, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  Te max-of-average cost of x is the max over each district of the average individual cost therein:  (MAX ◦ AVG)(x) = max  d∈D  � 1  nd  �  i∈Nd  δ(i, x)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  We measure the efficiency of a mechanism M with respect to a social objective f (such as the ones  defined above) by its distortion, the worst-case ratio (over all possible instances) of the f-value of the  alternative chosen by the mechanism over the minimum possible f-value among all alternatives:  sup  I=(N,A,D)  f(M(I))  minx∈A f(x)  By definition, the distortion of any mechanism is at least 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' We aim to design distributed mechanisms  with an as low distortion as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='1  Our Contribution  In previous work, Anshelevich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' [2022] considered the problem of distributed metric voting where  agents and alternatives are in some arbitrary metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' By carefully composing centralized voting  mechanisms for making decisions within and over the districts, they designed distributed mechanisms  with distortion guarantees for general metric spaces and with respect to the aforementioned objectives  as well as more general ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' For the special case of a line metric (which is our focus here), they showed  that the distortion of distributed mechanisms that use ordinal information is exactly 7 with respect to  the average-of-average cost, exactly 3 with respect to the max cost, and in the interval [2+  √  5, 5] with  respect to the average-of-max or the max-of-average cost objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  Inspired by the recent work of Filos-Ratsikas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' [2023] who designed distributed mechanisms  with tight distortion bounds for the continuous distributed facility location problem (where each point  on the line of real numbers can be considered as an alternative), we resolve the distortion of distributed  mechanisms for voting on a line with respect to the average-of-max and the max-of-average costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' In  2  particular, we design two essentially symmetric distributed mechanism that achieve a tight distortion  bound of 2+  √  5 for the average-of-max cost and the max-of-average cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' For the average-of-max cost,  our distributed mechanism chooses as the favorite alternative of the rightmost agent in each district as  the district representative, and then the (α · k)-the lefmost alternative as the overall winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' For the  max-of-average cost, our distributed mechanism chooses the favorite alternative of the (α·nd)-th agent  in each district d as the district representative, and then the rightmost alternative as the overall winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  We show that both mechanisms achieve distortion at most max  �  3−α  1−α, 2  α − 1  �  for the corresponding  objectives;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' for α = 3−  √  5  2  , the two terms in the bound balance out to 2 +  √  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='2  Other Related Work  Since its definition by Procaccia and Rosenschein [2006], the distortion of voting mechanisms has  been studied extensively for several setings under different assumptions about the preferences of the  agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' Te most well-studied seting is that of single-winner voting that has been considered under  the premise of normalized agent valuations [Boutilier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=', 2015, Ebadian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=', 2022, Amanatidis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=',  2021] as well as metric preferences [Anshelevich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=', 2018, Gkatzelis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=', 2020, Kizilkayaand Kempe,  2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' Several other models have also been considered, such as multi-winner voting [Caragiannis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=',  2017, 2022], participatory budgeting [Benad`e et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=', 2021], and matching [Filos-Ratsikas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=', 2014,  Amanatidis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  Te distortion of distributed voting was first considered by [Filos-Ratsikas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=', 2020] who focused  on bounding the distortion of max-weight mechanisms with respect to the social welfare when agents  have normalized valuations for the alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' Filos-Ratsikas and Voudouris [2021] then studied the  distortion of mechanisms for a distributed facility location seting where the agents are positioned on  a line and the goal is to choose a single location from a set of alternative ones, which can be discrete  (finite) or continuous (infinite).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' Te discrete variant was studied further and generalized to arbitrary  metric spaces by Anshelevich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' [2022], who also introduced and studied the average-of-max and  max-of-average costs for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' Te distortion of distributed mechanisms for the continuous  variant on a line was recently resolved by Filos-Ratsikas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' [2023].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' We refer the reader to the recent  survey of Anshelevich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' [2021] for more details on the distortion of voting mechanisms in different  models, and to the survey of Chan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' [2021] for details on related facility location models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  2  Average-of-Max Cost  Let α ∈ [0, 1] be a parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' We consider the α-Lefmost-of-Rightmost, which works as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' For  each district d ∈ D, the mechanism chooses the favorite alternative of the rightmost agent in d as the  representative yd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' Aferwards, it chooses the (α · k)-th lefmost alternative as the overall winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' See  Mechanism 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  Mechanism 1: α-Leftmost-of-Rightmost  for each district d ∈ D do  yd := favorite alternative of the rightmost agent in Nd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  return w := (α · k)-th lefmost representative;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  Teorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' For average-of-max, the distortion of the α-Leftmost-of-Rightmost mechanism is at  most max  �  3−α  1−α, 2  α − 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  3  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' Let w be the alternative chosen by the mechanism when given as input an arbitrary instance,  and o the optimal alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' For each district d, let id be the most distant agent from w, and i∗  d  the most distant agent from o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' So, (AVG ◦ MAX)(w) =  1  k  �  d∈D δ(id, w), and (AVG ◦ MAX)(o) =  1  k  �  d∈D δ(i∗  d, o).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' Also, let ℓd and rd denote the lefmost and rightmost agents in d, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' Note  that id, id∗ ∈ {ℓd, rd}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' We consider the following two cases depending on the relative positions of w  and o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  Case 1: o < w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  By the triangle inequality, and since δ(id, o) ≤ δ(i∗  d, o), we have  (AVG ◦ MAX)(w) = 1  k  �  d∈D  δ(id, w)  ≤ 1  k  �  d∈D  �  δ(id, o) + δ(w, o)  �  ≤ (AVG ◦ MAX)(o) + δ(w, o).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  (1)  Now, let S be the set of representatives that are to the right of w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' Since w is by definition the (α · k)-th  lefmost representative, we have that |S| ≥ (1−α)·k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' For every d such that yd ∈ S, since o < w ≤ yd,  agent rd is closer to w than to o, and thus δ(rd, o) ≥ 1  2δ(w, o).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' Hence,  (AVG ◦ MAX)(o) ≥ 1  k  �  d∈S  δ(rd, o)  ≥ 1  k · |S| · δ(w, o)  2  ≥ 1 − α  2  δ(w, o),  or, equivalently,  δ(w, o) ≤  2  1 − α · (AVG ◦ MAX)(o).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  (2)  Hence, by (1) and (2), we obtain  (AVG ◦ MAX)(w) ≤  �  1 +  2  1 − α  �  cost(o) = 3 − α  1 − α · (AVG ◦ MAX)(o).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  Case 2: w < o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  Let L be a set of α·k districts from the one with the lefmost representative until the  one with the (α · k)-th representative (which is w);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' denote by R the set of the remaining (1 − α) · k  districts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' Observe that:  For every d ∈ L, since yd is the alternative that is closest to rd and yd ≤ w < o, both rd and ℓd  are closer to w than to o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' So, δ(id, w) ≤ δ(i∗  d, o).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  For every d ∈ R, since δ(id, o) ≤ δ(i∗  d, o) by the definition of i∗  d, using the triangle inequality,  we obtain δ(id, w) ≤ δ(id, o) + δ(w, o) ≤ δ(i∗  d, o) + δ(w, o).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  Hence,  (AVG ◦ MAX)(w) = 1  k  �  d∈D  δ(id, w)  4  = 1  k  �  d∈L  δ(id, w) + 1  k  �  d∈R  δ(id, w)  ≤ 1  k  �  d∈L  δ(i∗  d, o) + 1  k  �  d∈R  �  δ(i∗  d, o) + δ(w, o)  �  = (AVG ◦ MAX)(o) + |R|  k · δ(w, o).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  (3)  Since rd is closer to w than to o for every d ∈ L, we also have that  (AVG ◦ MAX)(o) ≥ 1  k  �  d∈L  δ(rd, o) ≥ |L|  2k δ(w, o),  or, equivalently,  δ(w, o) ≤ 2k  |L| · (AVG ◦ MAX)(o).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  (4)  Terefore, by (3) and (4), we obtain  (AVG ◦ MAX)(w) ≤ (AVG ◦ MAX)(o) + 2|R|  |L| · (AVG ◦ MAX)(o) =  � 2  α − 1  �  (AVG ◦ MAX)(o).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  Puting everything together, we obtain an upper bound of max  �  3−α  1−α, 2  α − 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  Observe that the bound max  �  3−α  1−α, 2  α − 1  �  consists of two functions of α, one that is non-  decreasing and one that is non-increasing in α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' To minimize the maximum between the two, we need  to find the value of α for which the two functions intersect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' So, we need to solve the equation  3 − α  1 − α = 2  α − 1 ⇔ α2 − 3α + 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  Since α < 1, its solution is α = 3−  √  5  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' For this value of α, both functions have value  2  3−  √  5  2  −1 = 2+  √  5,  and we obtain the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' For average-of-max, the distortion of the 3−  √  5  2  Leftmost-of-Rightmost mechanism is  at most 2 +  √  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  3  Max-of-Average Cost  Let α ∈ [0, 1] be a parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' We consider the Rightmost-of-α-Leftmost mechanism, which works  as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' For each district d ∈ D, the mechanism chooses the favorite alternative of the (α · nd)-  th agent in d as the representative yd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' Aferwards, it chooses the rightmost alternative as the overall  winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' See Mechanism 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  Teorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' For max-of-average, the distortion of the Rightmost-of-α-leftmost mechanism is at  most max  �  3−α  1−α, 2  α − 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' Let w be the alternative chosen be the mechanism when given as input an arbitrary instance,  and o the optimal alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' For any district d and alternative x, let AVGd(x) =  1  nd  �  i∈Nd δ(i, x) be  the total average distance of the agents in d for alternative x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' So, AVGd(o) ≤ (MAX◦AVG)(o) for every  district d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' Denote by d∗ a district that gives the max average cost for w, such that (MAX ◦ AVG)(w) =  AVGd∗(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' Also, let dw be a district represented by w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' In addition, let i∗ and iw be the (α · nd)-th  lefmost agents in districts d∗ and dw, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' We now switch between the following two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  5  Mechanism 2: Rightmost-of-α-Leftmost  for each district d ∈ D do  yd := favorite alternative of the (α · nd)-th lefmost agent in Nd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  return w := rightmost representative;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  Case 1: o < w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  By the definition of d∗ and the triangle inequality, we have  (MAX ◦ AVG)(w) = 1  nd  �  i∈Nd∗  δ(i, w)  ≤ 1  nd  �  i∈Nd∗  �  δ(i, o) + δ(o, w)  �  ≤ (MAX ◦ AVG)(o) + δ(o, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  (5)  Denote by S the set of agents that are positioned weakly to the right of iw in dw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' By the definition of  iw, |S| ≥ (1 − α)ndw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' Since o < w and w is the favorite alternative of iw, all agents in S are closer to  w than to o, and thus δ(i, o) ≥ 1  2δ(w, o) for any i ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' Using all these, we obtain:  AVGdw(o) =  1  ndw  �  i∈Ndw  δ(i, o)  ≥  1  ndw  �  i∈S  δ(i, o)  ≥  1  ndw  |S|  2 · δ(w, o) ≥ 1 − α  2  δ(w, o),  or, equivalently,  δ(w, o) ≤  2  1 − α · AVGdw(o) ≤  2  1 − α · (MAX ◦ AVG)(o).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  (6)  Terefore, by (5) and (6), we obtain  (MAX ◦ AVG)(w) ≤ (MAX ◦ AVG)(o) +  2  1 − α · (MAX ◦ AVG)(o) = 3 − α  1 − α · (MAX ◦ AVG)(o).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  Case 2: w < o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  Let L be the set of the first α · nd∗ agents of d∗ (from the lefmost agent to i∗), and  R be the set of the remaining (1 − α)nd∗ agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' As w is the rightmost representative, yd∗ ≤ w < o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  Since yd∗ is the favorite alternative of i∗, every agent i ∈ L prefers w over o, and thus δ(i, w) = δ(i, o).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  Using this in combination with the triangle inequality for every agent of R, we have  (MAX ◦ AVG)(w) =  1  nd∗  �  i∈Nd∗  δ(i, w)  =  1  nd∗  �  i∈L  δ(i, w) +  1  nd∗  �  i∈R  δ(i, w)  ≤  1  nd∗  �  i∈L  δ(i, o) + 1  nd∗  �  i∈R  �  δ(i, o) + δ(w, o)  �  =  1  nd∗  �  i∈Nd∗  δ(i, o) + |R|  nd∗ · δ(w, o)  6  ≤ (MAX ◦ AVG)(o) + |R|  nd∗ · δ(w, o).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  (7)  Since each agent i ∈ L prefers w over o, we also have that δ(i, o) ≥ 1  2δ(w, o), and thus  (MAX ◦ AVG)(o) ≥ AVGd∗(o) =  1  nd∗  �  i∈Nd∗  δ(i, o)  ≥  1  nd∗  �  i∈Nd∗  δ(i, o)  ≥ |L|  2nd∗ · δ(w, o)  or, equivalently,  δ(w, o) ≤ 2nd∗  |L| · (MAX ◦ AVG)(o).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  (8)  Terefore, by (7) and (8), we obtain  (MAX ◦ AVG)(w) ≤ (MAX ◦ AVG)(o) + 2|R|  |L| · (MAX ◦ AVG)(o) =  � 2  α − 1  �  (MAX ◦ AVG)(o).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  Puting everything together, we get an upper bound of max  �  3−α  1−α, 2  α − 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  By optimizing over α, similarly to Section 2, we obtain the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' For max-of-average, the distortion of the Rightmost-of- 3−  √  5  2  Leftmost mechanism is  at most 2 +  √  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  4  Open Qestions  In this paper, we showed a tight distortion bound of 2 +  √  5 with respect to the average-of-max and  max-of-average cost functions for the single-winner distributed single-winner voting problem, thus  completing the distortion picture with respect to the four basic objectives considered by Anshelevich  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' [2022] for the line metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' Te most interestingdirection for future work is to prove tight distortion  bounds for general metric spaces, and also consider other social objectives or setings beyond single-  winner voting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  References  Georgios Amanatidis, Georgios Birmpas, Aris Filos-Ratsikas, and Alexandros A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' Voudouris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' Peeking  behind the ordinal curtain: Improving distortion via cardinal queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' Artificial Intelligence, 296:  103488, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  Georgios Amanatidis, Georgios Birmpas, Aris Filos-Ratsikas, and Alexandros A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' Voudouris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' A few  queries go a long way: Information-distortion tradeoffs in matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' Journal of Artificial Intelligence  Research, 74, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  Elliot Anshelevich, Onkar Bhardwaj, Edith Elkind, John Postl, and Piotr Skowron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' Approximating  optimal social choice under metric preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' Artificial Intelligence, 264:27–51, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  7  Elliot Anshelevich, Aris Filos-Ratsikas, Nisarg Shah, and Alexandros A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' Voudouris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' Distortion in so-  cial choice problems: Te first 15 years and beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' In Proceedings of the 30th International Joint  Conference on Artificial Intelligence (IJCAI), pages 4294–4301, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  Elliot Anshelevich, Aris Filos-Ratsikas, and Alexandros A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' Voudouris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' Te distortion of distributed  metric social choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' Artificial Intelligence, 308:103713, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  Gerdus Benad`e, Swaprava Nath, Ariel D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' Procaccia, and Nisarg Shah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' Preference elicitation for partic-  ipatory budgeting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' Management Science, 67(5):2813–2827, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  Craig Boutilier, Ioannis Caragiannis, Simi Haber, Tyler Lu, Ariel D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' Procaccia, and Or Sheffet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' Optimal  social choice functions: A utilitarian view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' Artificial Intelligence, 227:190–213, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  Ioannis Caragiannis, Swaprava Nath, Ariel D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' Procaccia, and Nisarg Shah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' Subset selection via implicit  utilitarian voting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' Journal of Artificial Intelligence Research, 58:123–152, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  Ioannis Caragiannis, Nisarg Shah, and Alexandros A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' Voudouris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' Te metric distortion of multiwinner  voting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' Artificial Intelligence, 313:103802, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  Hau Chan, Aris Filos-Ratsikas, Bo Li, Minming Li, and Chenhao Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' Mechanism design for facility  location problem: A survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' In Proceedings of the 30th International Joint Conference on Artificial  Intelligence (IJCAI), pages 1–17, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  Soroush Ebadian, Anson Kahng, Dominik Peters, and Nisarg Shah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' Optimized distortion and propor-  tional fairness in voting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' In Proceedings of the 23rd ACM Conference on Economics and Computation  (EC), pages 523–600, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  Aris Filos-Ratsikas and Alexandros A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' Voudouris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' Approximate mechanism design for distributed facil-  ity location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' In Proceedings of the 14th International Symposium on Algorithmic Game Teory (SAGT),  pages 49–63, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  Aris Filos-Ratsikas, Søren Kristoffer Stiil Frederiksen, and Jie Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' Social welfare in one-sided match-  ings: Random priority and beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' In Proceedings of the 7th Symposium of Algorithmic Game Teory  (SAGT), pages 1–12, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  Aris Filos-Ratsikas, Evi Micha, and Alexandros A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' Voudouris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' Te distortion of distributed voting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  Artificial Intelligence, 286:103343, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  Aris Filos-Ratsikas, Panagiotis Kanellopoulos, Alexandros A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' Voudouris, and Rongsen Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' Setling  the distortion of distributed facility location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' In Proceedings of the 22nd International Conference on  Autonomous Agents and Multiagent Systems (AAMAS), 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  Vasilis Gkatzelis, Daniel Halpern, and Nisarg Shah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' Resolving the optimal metric distortion conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  In Proceedings of the 61st Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages  1427–1438, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  Fatih Erdem Kizilkaya and David Kempe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' Plurality veto: A simple voting rule achieving optimal metric  distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' In Proceedings of the 31st International Joint Conference on Artificial Intelligence (IJCAI),  pages 349–355, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  Ariel D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' Procaccia and Jeffrey S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' Rosenschein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' Te distortion of cardinal preferences in voting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content=' In  International Workshop on Cooperative Information Agents (CIA), pages 317–331, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
+page_content='  8' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfxv41/content/2301.01742v1.pdf'}
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+arXiv:2301.03365v1  [math.FA]  1 Jan 2023
+LOCALIZED BOUNDED BELOW APPROXIMATE SCHAUDER FRAMES ARE
+FINITE UNIONS OF APPROXIMATE RIESZ SEQUENCES
+K. MAHESH KRISHNA
+Post Doctoral Fellow
+Statistics and Mathematics Unit
+Indian Statistical Institute, Bangalore Centre
+Karnataka 560 059, India
+Email: kmaheshak@gmail.com
+Date: January 10, 2023
+Abstract: Based on the truth of Feichtinger conjecture by Marcus, Spielman and Srivastava [Ann. of
+Math. (2), 2015] and from the localized version by Gr¨ochenig [Adv. Comput. Math., 2003], we introduce
+the notion of localization of approximate Schauder frames (ASFs) and approximate Riesz sequences
+(ARSs). We show that localized bounded below ASFs are finite unions of ARSs.
+Keywords: Feichtinger conjecture, Frame, Riesz sequence, Localization.
+Mathematics Subject Classification (2020): 42C15, 46A45, 46B45.
+1. Introduction
+In the beginning years of 21th century, Prof. Feichtinger formulated the following conjecture based on his
+extensive work on Gabor/Weyl-Heisenberg frames (see [13] for the history and [4,14,16–19,22,25–28,36]
+for general theory).
+Conjecture 1.1. [8] (Feichtinger Conjecture/Marcus-Spielman-Srivastava Theorem) Let {τn}n
+be a frame for a Hilbert space H such that
+0 < inf
+n∈N ∥τn∥.
+Then {τn}n can be partitioned into a finite union of Riesz sequences for H.
+First breakthrough which supported Conjecture 1.1 occurred when Gr¨ochenig proved it for intrinsically
+localized frames [23]. Shortly afterwords, it has been verified for certain classes of ℓ1-self-localized frames,
+wavelet frames, Gabor frames, frames of translates, frames formed by reproducing kernels and exponential
+frames/frames of exponentials [2,3,6,29,30,35,39]. Conjecture 1.1 received great attention after establish-
+ing its equivalence with Kadison-Singer conjecture [9,10,12]. Finally, the Feichtinger conjecture has been
+solved fully by resolving Weaver’s conjecture by Marcus, Spielman, and Srivastava in 2013 [5,33,34,37,38].
+In this paper, we formulate a Banach space version of Conjecture 1.1 and prove it for bounded below
+intrinsically localized ASFs (Theorem 2.7).
+2. Localized bounded below ASFs are finite unions of ARSs
+We consider the following most general notion of approximate Schauder frames. In the entire paper, X
+is a separable Banach space and X ∗ is its dual.
+Definition 2.1.
+[7, 11, 21] Let {τn}n be a collection in X and {fn}n be a collection in X ∗. The pair
+({fn}n, {τn}n) is said to be an approximate Schauder frame (we write ASF) for X if the frame
+1
+
+K. MAHESH KRISHNA
+operator
+Sf,τ : X ∋ x �→ Sf,τx :=
+∞
+�
+n=1
+fn(x)τn ∈ X
+is a well-defined bounded linear invertible operator.
+We use the following notion of ‘bounded below’ for ASFs.
+Definition 2.2. An ASF ({fn}n, {τn}n) for X is said to be bounded below if
+inf
+n∈N |fn(τn)| > 0.
+Motivated from the definition of localization of frames [20,24], we introduce the following notion.
+Definition 2.3. An ASF ({fn}n, {τn}n) for X is said to be intrinsically/self localized if there exist
+s > 1 and A > 0 such that
+|fn(τm)| ≤
+A
+(1 + |n − m|)s ,
+∀n, m ∈ N.
+Two notions of Riesz sequences for Banach spaces exist in literature, see [1,15] and [32]. Here we define
+another.
+Definition 2.4. An ASF ({fn}n, {τn}n) for X is said to be an approximate Riesz sequence (we write
+ARS) if there exists a finite partition Q1, . . . , QN of N such that
+N =
+N
+�
+j=1
+Qj
+and for each 1 ≤ j ≤ N,
+inf
+n∈Qj
+
+|fn(τn)| −
+�
+m∈Qj,m̸=n
+|fn(τm)|
+
+ > 0.
+Note that for Hilbert spaces, if {fn}n is determined by {τn}n (Riesz representation), then Definition 2.4
+is equivalent (due to positivity) to the definition of Riesz sequence (see [23]). We now formulate the
+following conjecture (some other are formulated in [31]).
+Conjecture 2.5. Every bounded below ASF can be partitioned as a finite union of ARBs.
+We now prove Conjecture 2.5 for intrinsically localized ASFs with the help of following result.
+Theorem 2.6.
+[23]
+(i) For every s > 1,
+Ds := sup
+x∈R
+∞
+�
+n=1
+1
+(1 + |n − x|)s < ∞.
+(ii) For every s > 1, there exists a Cs > 0 (which does not depend on δ) such that
+sup
+m∈N
+�
+n∈N,n̸=m
+1
+(1 + |n − m|)s ≤ Cs
+δs ,
+whenever
+inf
+n,m∈N,n̸=m |n − m| ≥ δ.
+2
+
+LOCALIZED BOUNDED BELOW APPROXIMATE SCHAUDER FRAMES ARE
+FINITE UNIONS OF APPROXIMATE RIESZ SEQUENCES
+Theorem 2.7. Conjecture 2.5 holds for intrinsically localized bounded below ASFs.
+Proof. Our proof is highly motivated from [23]. Let ({fn}n, {τn}n) be an intrinsically localized bounded
+below ASF for X. Define
+C := inf
+n∈N |fn(τn)| > 0.
+Since ({fn}n, {τn}n) intrinsically localized there exist s > 1 and A > 0 such that
+|fn(τm)| ≤
+A
+(1 + |n − m|)s ,
+∀n, m ∈ N.
+Let Cs be the constant as in Theorem 2.6. Choose natural number M such that
+ACs
+M s ≤ C
+2 .
+We now partition N into Q1, . . . , QN such that
+N :=
+N
+�
+j=1
+Qj
+and for each 1 ≤ j ≤ N,
+inf
+n,m∈Qj,n̸=m |n − m| ≥ M.
+(Note that there are infinitely many partitions of N of this type.) Let 1 ≤ j ≤ N. Then using (ii) in
+Theorem 2.6
+sup
+n∈Qj
+�
+m∈Qj,m̸=n
+|fn(τm)| ≤ A sup
+n∈Qj
+�
+m∈Qj,m̸=n
+1
+(1 + |n − m|)s ≤ A Cs
+M s ≤ C
+2 .
+Therefore for each fixed 1 ≤ j ≤ N
+inf
+n∈Qj
+
+|fn(τn)| −
+�
+m∈Qj,m̸=n
+|fn(τm)|
+
+ = inf
+n∈Qj |fn(τn)| − sup
+n∈Qj
+�
+m∈Qj,m̸=n
+|fn(τm)|
+≥ C − C
+2 = C
+2 > 0.
+Since j was arbitrary, we get the theorem.
+□
+References
+[1] Akram Aldroubi, Qiyu Sun, and Wai-Shing Tang. p-frames and shift invariant subspaces of Lp. J. Fourier Anal. Appl.,
+7(1):1–21, 2001.
+[2] Radu Balan, Peter G. Casazza, Christopher Heil, and Zeph Landau. Density, overcompleteness, and localization of
+frames. I. Theory. J. Fourier Anal. Appl., 12(2):105–143, 2006.
+[3] Anton Baranov and Konstantin Dyakonov. The Feichtinger conjecture for reproducing kernels in model subspaces. J.
+Geom. Anal., 21(2):276–287, 2011.
+[4] John J. Benedetto and David F. Walnut. Gabor frames for L2 and related spaces. In Wavelets: mathematics and
+applications, Stud. Adv. Math., pages 97–162. CRC, Boca Raton, FL, 1994.
+[5] Marcin Bownik. The Kadison-Singer problem. In Frames and harmonic analysis, volume 706 of Contemp. Math., pages
+63–92. Amer. Math. Soc., [Providence], RI, 2018.
+[6] Marcin Bownik and Darrin Speegle. The Feichtinger conjecture for wavelet frames, Gabor frames and frames of trans-
+lates. Canad. J. Math., 58(6):1121–1143, 2006.
+[7] P. G. Casazza, S. J. Dilworth, E. Odell, Th. Schlumprecht, and A. Zs´ak. Coefficient quantization for frames in Banach
+spaces. J. Math. Anal. Appl., 348(1):66–86, 2008.
+3
+
+K. MAHESH KRISHNA
+[8] Peter G. Casazza, Ole Christensen, Alexander M. Lindner, and Roman Vershynin. Frames and the Feichtinger conjec-
+ture. Proc. Amer. Math. Soc., 133(4):1025–1033, 2005.
+[9] Peter G. Casazza and Dan Edidin. Equivalents of the Kadison-Singer problem. In Function spaces, volume 435 of
+Contemp. Math., pages 123–142. Amer. Math. Soc., Providence, RI, 2007.
+[10] Peter G. Casazza, Matthew Fickus, Janet C. Tremain, and Eric Weber. The Kadison-Singer problem in mathematics
+and engineering: a detailed account. In Operator theory, operator algebras, and applications, volume 414 of Contemp.
+Math., pages 299–355. Amer. Math. Soc., Providence, RI, 2006.
+[11] Peter G. Casazza, Deguang Han, and David R. Larson. Frames for Banach spaces. In The functional and harmonic
+analysis of wavelets and frames (San Antonio, TX, 1999), volume 247 of Contemp. Math., pages 149–182. Amer.
+Math. Soc., Providence, RI, 1999.
+[12] Peter G. Casazza, Gitta Kutyniok, Darrin Speegle, and Janet C. Tremain. A decomposition theorem for frames and
+the Feichtinger conjecture. Proc. Amer. Math. Soc., 136(6):2043–2053, 2008.
+[13] Ole Christensen. Six (seven) problems in frame theory. In New perspectives on approximation and sampling theory,
+Appl. Numer. Harmon. Anal., pages 337–358. Birkh¨auser/Springer, Cham, 2014.
+[14] Ole
+Christensen.
+An
+introduction
+to
+frames
+and
+Riesz
+bases.
+Applied
+and
+Numerical
+Harmonic
+Analysis.
+Birkh¨auser/Springer, second edition, 2016.
+[15] Ole Christensen and Diana T. Stoeva. p-frames in separable Banach spaces. Adv. Comput. Math., 18(2-4):117–126,
+2003.
+[16] R. J. Duffin and A. C. Schaeffer. A class of nonharmonic Fourier series. Trans. Amer. Math. Soc., 72:341–366, 1952.
+[17] Hans G. Feichtinger and K. Gr¨ochenig. Gabor frames and time-frequency analysis of distributions. J. Funct. Anal.,
+146(2):464–495, 1997.
+[18] Hans G. Feichtinger and Thomas Strohmer, editors. Gabor analysis and algorithms: Theory and application. Applied
+and Numerical Harmonic Analysis. Birkh¨auser Boston, Inc., Boston, MA, 1998.
+[19] Hans G. Feichtinger and Thomas Strohmer, editors. Advances in Gabor analysis. Applied and Numerical Harmonic
+Analysis. Birkh¨auser Boston, Inc., Boston, MA, 2003.
+[20] Massimo Fornasier and Karlheinz Gr¨ochenig. Intrinsic localization of frames. Constr. Approx., 22(3):395–415, 2005.
+[21] D. Freeman, E. Odell, Th. Schlumprecht, and A. Zs´ak. Unconditional structures of translates for Lp(Rd). Israel J.
+Math., 203(1):189–209, 2014.
+[22] Karlheinz Gr¨ochenig. Foundations of time-frequency analysis. Applied and Numerical Harmonic Analysis. Birkh¨auser
+Boston, Inc., Boston, MA, 2001.
+[23] Karlheinz Gr¨ochenig. Localized frames are finite unions of Riesz sequences. Adv. Comput. Math., 18(2-4):149–157,
+2003.
+[24] Karlheinz Gr¨ochenig. Localization of frames, Banach frames, and the invertibility of the frame operator. J. Fourier
+Anal. Appl., 10(2):105–132, 2004.
+[25] Karlheinz Gr¨ochenig. Gabor frames without inequalities. Int. Math. Res. Not. IMRN, (23):Art. ID rnm111, 21, 2007.
+[26] Karlheinz Gr¨ochenig. The mystery of Gabor frames. J. Fourier Anal. Appl., 20(4):865–895, 2014.
+[27] Karlheinz Gr¨ochenig, Joaquim Ortega-Cerd`a, and Jos´e Luis Romero. Deformation of Gabor systems. Adv. Math.,
+277:388–425, 2015.
+[28] Christopher Heil. History and evolution of the density theorem for Gabor frames. J. Fourier Anal. Appl., 13(2):113–166,
+2007.
+[29] Sneh Lata and Vern Paulsen. The Feichtinger conjecture and reproducing kernel Hilbert spaces. Indiana Univ. Math.
+J., 60(4):1303–1317, 2011.
+[30] Wayne Lawton. The Feichtinger conjecture for exponentials. J. Nonlinear Anal. Optim., 2(1):131–140, 2011.
+[31] K.
+Mahesh
+Krishna.
+Feichtinger
+conjectures,
+Rε-conjectures
+and
+Weaver’s
+conjectures
+for
+Banach
+spaces.
+arXiv.org/2201.00125v1 [math.FA] 1 January, 2022.
+[32] K. Mahesh Krishna and P. Sam Johnson. Dilation theorem for p-approximate Schauder frames for separable Banach
+spaces. Palest. J. Math., 11(2):384–394, 2022.
+[33] Adam Marcus and Nikhil Srivastava. The solution of the Kadison-Singer problem. In Current developments in mathe-
+matics 2016, pages 111–143. Int. Press, Somerville, MA, 2018.
+[34] Adam W. Marcus, Daniel A. Spielman, and Nikhil Srivastava. Interlacing families II: Mixed characteristic polynomials
+and the Kadison-Singer problem. Ann. of Math. (2), 182(1):327–350, 2015.
+4
+
+LOCALIZED BOUNDED BELOW APPROXIMATE SCHAUDER FRAMES ARE
+FINITE UNIONS OF APPROXIMATE RIESZ SEQUENCES
+[35] Darrin Speegle. Uniform partitions of frames of exponentials into Riesz sequences. J. Math. Anal. Appl., 348(2):739–745,
+2008.
+[36] Diana T. Stoeva. On a characterization of Riesz bases via biorthogonal sequences. J. Fourier Anal. Appl., 26(4):Paper
+No. 67, 5, 2020.
+[37] Dan Timotin. The solution of the Kadison-Singer problem. In Recent advances in operator theory and operator algebras,
+pages 117–148. CRC Press, Boca Raton, FL, 2018.
+[38] Nik Weaver. The Kadison-Singer problem in discrepancy theory. Discrete Math., 278(1-3):227–239, 2004.
+[39] Eric Weber. Algebraic aspects of the paving and Feichtinger conjectures. In Topics in operator theory. Volume 1.
+Operators, matrices and analytic functions, volume 202 of Oper. Theory Adv. Appl., pages 569–578. Birkh¨auser Verlag,
+Basel, 2010.
+5
+
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+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='03365v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='FA]  1 Jan 2023  LOCALIZED BOUNDED BELOW APPROXIMATE SCHAUDER FRAMES ARE  FINITE UNIONS OF APPROXIMATE RIESZ SEQUENCES  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' MAHESH KRISHNA  Post Doctoral Fellow  Statistics and Mathematics Unit  Indian Statistical Institute, Bangalore Centre  Karnataka 560 059, India  Email: kmaheshak@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='com  Date: January 10, 2023  Abstract: Based on the truth of Feichtinger conjecture by Marcus, Spielman and Srivastava [Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' of  Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' (2), 2015] and from the localized version by Gr¨ochenig [Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=', 2003], we introduce  the notion of localization of approximate Schauder frames (ASFs) and approximate Riesz sequences  (ARSs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' We show that localized bounded below ASFs are finite unions of ARSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  Keywords: Feichtinger conjecture, Frame, Riesz sequence, Localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  Mathematics Subject Classification (2020): 42C15, 46A45, 46B45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Introduction  In the beginning years of 21th century, Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Feichtinger formulated the following conjecture based on his  extensive work on Gabor/Weyl-Heisenberg frames (see [13] for the history and [4,14,16–19,22,25–28,36]  for general theory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' [8] (Feichtinger Conjecture/Marcus-Spielman-Srivastava Theorem) Let {τn}n  be a frame for a Hilbert space H such that  0 < inf  n∈N ∥τn∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  Then {τn}n can be partitioned into a finite union of Riesz sequences for H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  First breakthrough which supported Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='1 occurred when Gr¨ochenig proved it for intrinsically  localized frames [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Shortly afterwords, it has been verified for certain classes of ℓ1-self-localized frames,  wavelet frames, Gabor frames, frames of translates, frames formed by reproducing kernels and exponential  frames/frames of exponentials [2,3,6,29,30,35,39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='1 received great attention after establish-  ing its equivalence with Kadison-Singer conjecture [9,10,12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Finally, the Feichtinger conjecture has been  solved fully by resolving Weaver’s conjecture by Marcus, Spielman, and Srivastava in 2013 [5,33,34,37,38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  In this paper, we formulate a Banach space version of Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='1 and prove it for bounded below  intrinsically localized ASFs (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Localized bounded below ASFs are finite unions of ARSs  We consider the following most general notion of approximate Schauder frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' In the entire paper, X  is a separable Banach space and X ∗ is its dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  [7, 11, 21] Let {τn}n be a collection in X and {fn}n be a collection in X ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' The pair  ({fn}n, {τn}n) is said to be an approximate Schauder frame (we write ASF) for X if the frame  1  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' MAHESH KRISHNA  operator  Sf,τ : X ∋ x �→ Sf,τx :=  ∞  �  n=1  fn(x)τn ∈ X  is a well-defined bounded linear invertible operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  We use the following notion of ‘bounded below’ for ASFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' An ASF ({fn}n, {τn}n) for X is said to be bounded below if  inf  n∈N |fn(τn)| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  Motivated from the definition of localization of frames [20,24], we introduce the following notion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' An ASF ({fn}n, {τn}n) for X is said to be intrinsically/self localized if there exist  s > 1 and A > 0 such that  |fn(τm)| ≤  A  (1 + |n − m|)s ,  ∀n, m ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  Two notions of Riesz sequences for Banach spaces exist in literature, see [1,15] and [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Here we define  another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' An ASF ({fn}n, {τn}n) for X is said to be an approximate Riesz sequence (we write  ARS) if there exists a finite partition Q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' , QN of N such that  N =  N  �  j=1  Qj  and for each 1 ≤ j ≤ N,  inf  n∈Qj  \uf8eb  \uf8ed|fn(τn)| −  �  m∈Qj,m̸=n  |fn(τm)|  \uf8f6  \uf8f8 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  Note that for Hilbert spaces, if {fn}n is determined by {τn}n (Riesz representation), then Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='4  is equivalent (due to positivity) to the definition of Riesz sequence (see [23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' We now formulate the  following conjecture (some other are formulated in [31]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Every bounded below ASF can be partitioned as a finite union of ARBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  We now prove Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='5 for intrinsically localized ASFs with the help of following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  [23]  (i) For every s > 1,  Ds := sup  x∈R  ∞  �  n=1  1  (1 + |n − x|)s < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  (ii) For every s > 1, there exists a Cs > 0 (which does not depend on δ) such that  sup  m∈N  �  n∈N,n̸=m  1  (1 + |n − m|)s ≤ Cs  δs ,  whenever  inf  n,m∈N,n̸=m |n − m| ≥ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  2  LOCALIZED BOUNDED BELOW APPROXIMATE SCHAUDER FRAMES ARE  FINITE UNIONS OF APPROXIMATE RIESZ SEQUENCES  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='5 holds for intrinsically localized bounded below ASFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Our proof is highly motivated from [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Let ({fn}n, {τn}n) be an intrinsically localized bounded  below ASF for X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Define  C := inf  n∈N |fn(τn)| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  Since ({fn}n, {τn}n) intrinsically localized there exist s > 1 and A > 0 such that  |fn(τm)| ≤  A  (1 + |n − m|)s ,  ∀n, m ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  Let Cs be the constant as in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Choose natural number M such that  ACs  M s ≤ C  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  We now partition N into Q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' , QN such that  N :=  N  �  j=1  Qj  and for each 1 ≤ j ≤ N,  inf  n,m∈Qj,n̸=m |n − m| ≥ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  (Note that there are infinitely many partitions of N of this type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=') Let 1 ≤ j ≤ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Then using (ii) in  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='6  sup  n∈Qj  �  m∈Qj,m̸=n  |fn(τm)| ≤ A sup  n∈Qj  �  m∈Qj,m̸=n  1  (1 + |n − m|)s ≤ A Cs  M s ≤ C  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  Therefore for each fixed 1 ≤ j ≤ N  inf  n∈Qj  \uf8eb  \uf8ed|fn(τn)| −  �  m∈Qj,m̸=n  |fn(τm)|  \uf8f6  \uf8f8 = inf  n∈Qj |fn(τn)| − sup  n∈Qj  �  m∈Qj,m̸=n  |fn(τm)|  ≥ C − C  2 = C  2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  Since j was arbitrary, we get the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  □  References  [1] Akram Aldroubi, Qiyu Sun, and Wai-Shing Tang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' p-frames and shift invariant subspaces of Lp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Fourier Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=',  7(1):1–21, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  [2] Radu Balan, Peter G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Casazza, Christopher Heil, and Zeph Landau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Density, overcompleteness, and localization of  frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Fourier Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=', 12(2):105–143, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  [3] Anton Baranov and Konstantin Dyakonov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' The Feichtinger conjecture for reproducing kernels in model subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=', 21(2):276–287, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  [4] John J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Benedetto and David F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Walnut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Gabor frames for L2 and related spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' In Wavelets: mathematics and  applications, Stud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=', pages 97–162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' CRC, Boca Raton, FL, 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  [5] Marcin Bownik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' The Kadison-Singer problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' In Frames and harmonic analysis, volume 706 of Contemp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=', pages  63–92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=', [Providence], RI, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  [6] Marcin Bownik and Darrin Speegle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' The Feichtinger conjecture for wavelet frames, Gabor frames and frames of trans-  lates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
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+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=', 58(6):1121–1143, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  [7] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Casazza, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Dilworth, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Odell, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Schlumprecht, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Zs´ak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Coefficient quantization for frames in Banach  spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
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+page_content=' MAHESH KRISHNA  [8] Peter G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Casazza, Ole Christensen, Alexander M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Lindner, and Roman Vershynin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Frames and the Feichtinger conjec-  ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=', 133(4):1025–1033, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  [9] Peter G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Casazza and Dan Edidin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Equivalents of the Kadison-Singer problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' In Function spaces, volume 435 of  Contemp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=', pages 123–142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=', Providence, RI, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  [10] Peter G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Casazza, Matthew Fickus, Janet C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Tremain, and Eric Weber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' The Kadison-Singer problem in mathematics  and engineering: a detailed account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' In Operator theory, operator algebras, and applications, volume 414 of Contemp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=', pages 299–355.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=', Providence, RI, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  [11] Peter G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Casazza, Deguang Han, and David R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Larson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Frames for Banach spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' In The functional and harmonic  analysis of wavelets and frames (San Antonio, TX, 1999), volume 247 of Contemp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=', pages 149–182.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=', Providence, RI, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  [12] Peter G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Casazza, Gitta Kutyniok, Darrin Speegle, and Janet C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Tremain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' A decomposition theorem for frames and  the Feichtinger conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=', 136(6):2043–2053, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  [13] Ole Christensen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Six (seven) problems in frame theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' In New perspectives on approximation and sampling theory,  Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Numer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Harmon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=', pages 337–358.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Birkh¨auser/Springer, Cham, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  [14] Ole  Christensen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  An  introduction  to  frames  and  Riesz  bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  Applied  and  Numerical  Harmonic  Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  Birkh¨auser/Springer, second edition, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  [15] Ole Christensen and Diana T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Stoeva.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' p-frames in separable Banach spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=', 18(2-4):117–126,  2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  [16] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Duffin and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Schaeffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' A class of nonharmonic Fourier series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=', 72:341–366, 1952.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  [17] Hans G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Feichtinger and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Gr¨ochenig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Gabor frames and time-frequency analysis of distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=',  146(2):464–495, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  [18] Hans G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Feichtinger and Thomas Strohmer, editors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Gabor analysis and algorithms: Theory and application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Applied  and Numerical Harmonic Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Birkh¨auser Boston, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=', Boston, MA, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  [19] Hans G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Feichtinger and Thomas Strohmer, editors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Advances in Gabor analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Applied and Numerical Harmonic  Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Birkh¨auser Boston, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=', Boston, MA, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  [20] Massimo Fornasier and Karlheinz Gr¨ochenig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Intrinsic localization of frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Constr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=', 22(3):395–415, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  [21] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Freeman, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Odell, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Schlumprecht, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Zs´ak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Unconditional structures of translates for Lp(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Israel J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=', 203(1):189–209, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  [22] Karlheinz Gr¨ochenig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Foundations of time-frequency analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Applied and Numerical Harmonic Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Birkh¨auser  Boston, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=', Boston, MA, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  [23] Karlheinz Gr¨ochenig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Localized frames are finite unions of Riesz sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
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+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=', 18(2-4):149–157,  2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  [24] Karlheinz Gr¨ochenig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Localization of frames, Banach frames, and the invertibility of the frame operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Fourier  Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=', 10(2):105–132, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  [25] Karlheinz Gr¨ochenig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Gabor frames without inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' IMRN, (23):Art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' ID rnm111, 21, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  [26] Karlheinz Gr¨ochenig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' The mystery of Gabor frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Fourier Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=', 20(4):865–895, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  [27] Karlheinz Gr¨ochenig, Joaquim Ortega-Cerd`a, and Jos´e Luis Romero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Deformation of Gabor systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=',  277:388–425, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  [28] Christopher Heil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' History and evolution of the density theorem for Gabor frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Fourier Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=', 13(2):113–166,  2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  [29] Sneh Lata and Vern Paulsen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' The Feichtinger conjecture and reproducing kernel Hilbert spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Indiana Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=', 60(4):1303–1317, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  [30] Wayne Lawton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' The Feichtinger conjecture for exponentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Nonlinear Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Optim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=', 2(1):131–140, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  [31] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  Mahesh  Krishna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  Feichtinger  conjectures,  Rε-conjectures  and  Weaver’s  conjectures  for  Banach  spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='org/2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='00125v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='FA] 1 January, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  [32] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Mahesh Krishna and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Sam Johnson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Dilation theorem for p-approximate Schauder frames for separable Banach  spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Palest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=', 11(2):384–394, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  [33] Adam Marcus and Nikhil Srivastava.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' The solution of the Kadison-Singer problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' In Current developments in mathe-  matics 2016, pages 111–143.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Press, Somerville, MA, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  [34] Adam W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Marcus, Daniel A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Spielman, and Nikhil Srivastava.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Interlacing families II: Mixed characteristic polynomials  and the Kadison-Singer problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' (2), 182(1):327–350, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  4  LOCALIZED BOUNDED BELOW APPROXIMATE SCHAUDER FRAMES ARE  FINITE UNIONS OF APPROXIMATE RIESZ SEQUENCES  [35] Darrin Speegle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Uniform partitions of frames of exponentials into Riesz sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=', 348(2):739–745,  2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  [36] Diana T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Stoeva.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' On a characterization of Riesz bases via biorthogonal sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Fourier Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=', 26(4):Paper  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' 67, 5, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  [37] Dan Timotin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' The solution of the Kadison-Singer problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' In Recent advances in operator theory and operator algebras,  pages 117–148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' CRC Press, Boca Raton, FL, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  [38] Nik Weaver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' The Kadison-Singer problem in discrepancy theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Discrete Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=', 278(1-3):227–239, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  [39] Eric Weber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Algebraic aspects of the paving and Feichtinger conjectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' In Topics in operator theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Volume 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  Operators, matrices and analytic functions, volume 202 of Oper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Theory Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=', pages 569–578.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content=' Birkh¨auser Verlag,  Basel, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
+page_content='  5' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfsQX7/content/2301.03365v1.pdf'}
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+O R I G I N A L A R T I C L E
+Post-processing output from ensembles with and
+without parametrised convection, to create
+accurate, blended, high-fidelity rainfall forecasts
+Estíbaliz Gascón1*
+|
+Andrea Montani1
+|
+Tim D.
+Hewson1
+1European Centre for Medium-Range
+Weather Forecasts, Reading, United
+Kingdom
+Correspondence
+Estíbaliz Gascón, European Centre for
+Medium-Range Weather Forecasts,
+Reading, United Kingdom.
+Email: estibaliz.gascon@ecmwf.int
+Funding information
+The authors gratefully acknowledge
+financial support from the European Union
+Connecting Europe Facility (CEF) –
+Telecommunication Sector Programme for
+the MISTRAL – Meteo Italian
+SupercompuTing poRtAL project (Grant
+Agreement No.
+INEA/CEF/ICT/A2017/1567101).
+Flash flooding is a significant societal problem, but related
+precipitation forecasts are often poor. To address, one can
+try to use output from convection-parametrising (global)
+ensembles, post-processed to forecast at point-scale, or con-
+vection-resolving limited area ensembles. The new method-
+ology described here combines both. We apply “ecPoint-
+rainfall” post-processing to the ECMWF global ensemble.
+Alongside we use 2.2km COSMO LAM ensemble output
+(centred on Italy), and also post-process that, using a scale-
+selective neighbourhood approach to compensate for insuf-
+ficient members and to preserve consistently forecast lo-
+cal details. The two resulting scale-compatible components
+then undergo lead-time-weighted blending, to create the fi-
+nal probabilistic 6h rainfall forecasts. Product creation for
+forecasters, in this way, constituted the “Italy Flash Flood
+use case” within the EU-funded MISTRAL project; real-time
+delivery of open access products is ongoing.
+One year of verification shows that, of the five compo-
+nents (2 x raw, 2 x post-processed and blended), ecPoint
+is the most skilful. The post-processed COSMO ensem-
+ble adds most value to summer convective events in the
+evening, when the global model has an underprediction bias.
+*Corresponding author.
+1
+arXiv:2301.04485v1  [physics.ao-ph]  11 Jan 2023
+
+2
+Gascón et al.
+In two typical heavy rainfall case studies we observed un-
+derestimation of the largest point totals in the raw ECMWF
+ensemble, and overestimation in the raw COSMO ensem-
+ble. However, ecPoint elevated the ECMWF maxima and
+highlighted best the most affected areas, whilst post-processing
+of COSMO diminished extremes by eradicating unreliable
+detail, such that the final merged products looked best from
+a user perspective, and seemed to be the most skilful of
+all. Although our LAM post-processing does not implicitly
+include bias-correction (a topic for further work) our study
+nonetheless provides a unique blueprint for successfully com-
+bining ensemble rainfall forecasts from global and LAM sys-
+tems around the world. It also has important implications
+for forecast products as global ensembles move ever closer
+to having convection-permitting resolution.
+Keywords — point rainfall, flash floods, post-processing
+techniques, scale-selective neighbourhood, ensemble fore-
+cast
+1
+|
+INTRODUCTION
+Short-duration convective precipitation extremes are generally responsible for flash flooding events, which are a sig-
+nificant societal problem. This paper looks at new ways to anticipate such events by applying novel post-processing
+techniques to the output of different types of numerical model.
+Many times, numerical weather preciction (NWP) systems cannot predict precipitation with sufficient accuracy in
+terms of rainfall quantity and the particular location of the rainfall events (Silvestro et al., 2015; Gascón et al., 2016).
+This is partially because weather forecasts are often required for points and not, for example, for the large areas of
+several hundred square kilometres represented by current-generation global model grid boxes. This discrepancy can
+be addressed using high-resolution limited-area models (LAMs). Alternatively one can apply some post-processing
+methods to global forecast models, to convert from gridbox to point-forecasts (Gneiting, 2014; Hewson and Pillosu,
+2021). To better understand these model limitations, we should consider the concept of sub-grid variability, previously
+
+Gascón et al.
+3
+discussed by Hewson and Pillosu (2021). Sub-grid variability denotes the variation seen amongst point values within
+a given model gridbox. In very localised convective precipitation, which is related to flash floods occurrence, the
+sub-grid variability is often so high that forecast totals from globel models of current day resolution will ’miss’ the
+event, by maybe an order of magnitude. However, in large-scale precipitation, the sub-grid variability in the gridbox is
+usually considerably less. So a major challenge here for post-processing is dealing with variations in sub-grid variability.
+Meanwhile, although LAMs, with their much higher resolution, produce more realistic features than global models,
+small-scale predictability, if it exists, can only be maintained for a number of hours, because errors often grow rapidly
+(Melhauser and Zhang, 2012; Radhakrishna et al., 2012), particularly in convective situations.
+Another reason for the inaccuracy of heavy rainfall forecasts is the innate forecast uncertainties that affect all
+NWP, which can in principle be represented and quantified using ensemble prediction systems (Buizza, 2018). En-
+sembles have been used for several decades to predict the possible evolutions of large-scale patterns beyond the
+short-range. In recent years, their use has also become more and more widespread in the framework of LAMs, that
+have convection-permitting resolution (say <5 km in the horizontal), to facilitate also probabilistic prediction of small
+scale details that the global models innately cannot capture. The number of LAM ensemble members that are strictly
+needed to address all the convective degrees of freedom in the atmosphere exceeds what is computationally afford-
+able (usually about 20 members) by at least one order of magnitude even on day 1 (Necker et al., 2020; Kobayashi
+et al., 2020), and after that, as synoptic-scale spread among members grows, even more would be needed. Indeed
+in 2017, Raynaud and Bouttier (2017) explored how forecast skill depends on ensemble size for LAMs, which tend
+to have fewer members than global ensembles. Their study compared the benefits of increasing ensemble size from
+12 to 34 with the benefits of increasing horizontal resolution from 2.5 to 1.3 km and concluded that for lead times
+greater than 12h, increasing ensemble size provides more benefit.
+The COSMO (COnsortium for Small-scale MOdelling) ensemble over Italy is an example of a LAM ensemble.
+This is a system recently developed for Italy, that runs on an operational-experimental basis, with 2.2 km horizontal
+resolution, 65 vertical levels and 20 members (COSMO-2I-EPS, or COSMO raw, henceforth). The main features of
+the system, reported under http://cosmo-model.org/content/tasks/operational/default.htm, will be further
+discussed in the next sections. Researchers have also looked at alternative cheap approaches to solve the problem
+
+4
+Gascón et al.
+related to the lack of members, such as neighbourhood techniques and lag-based ensembles, to increase ensemble
+size without increasing computing resources (Ebert, 2008; Ben Bouallègue et al., 2013; DelSole et al., 2017; Schwartz
+and Sobash, 2017). The “neighbourhood” post-processing technique has as its central premise the notion of limited
+small-scale predictability. It acknowledges that it is unrealistic to expect high-resolution models to be accurate at
+the grid-scale. Several approaches, including the fractional coverage (Theis et al., 2005) and maximum neighborhood
+value methods (Hitchens et al., 2013), have been developed to in effect increase the number of members, and thereby
+increase also the spread, to provide a more comprehensive description of the spatial uncertainties than the raw LAM
+ensemble can provide on its own. A related challenge is how to use and evaluate the ensembles in convective situa-
+tions, where the ensemble mean is unlikely to be physically appropriate for anticipating precipitation extremes (Ancell,
+2013; Dey et al., 2016b).
+Studies have demonstrated that the skill of convective scale forecasts is very scale-dependent, and this also holds
+true for ensemble forecasts, wherein ensemble skill increases with both spatial scale and ensemble size (Roberts and
+Lean, 2008; Clark et al., 2011; Ben Bouallègue and Theis, 2013; Mittermaier, 2014).
+So, notwithstanding the above limitations, it should still be possible to give useful probabilistic forecasts of local
+precipitation by anticipating uncertainties in the precipitation location (and assuming that the ensemble is unbiased)
+(Dey et al., 2016b). One strategy to address localised precipitation uncertainties is the “scale-selective neighbourhood
+approach”. Following Dey et al. (2016a,b) and Blake et al. (2018), a similarity criterion among all possible ensemble
+forecast pairs is developed. The rationale is that we preserve only the most reliable fine-scale rainfall signals by (i)
+retaining that detail in the rainfall patterns wherever ensemble rainfall forecasts are similar to one another (e.g. where
+precipitation is strongly forced by orographic uplift) whilst (ii) ironing out the detail where there is less agreement (e.g.
+’random’ summer convection over relatively flat terrain). For each model grid point, the scale-selective neighbourhood
+approach aims first to quantify the “agreement scale” among the ensemble members. This is a metric that describes
+the spatial scale at which ensemble forecast can be deemed sufficiently similar to each other, according to a criteria
+described in the next sections. One can think of the agreement scale as being proportional to the length scale across
+which the post-processing combines together the raw forecast data; i.e. agreement scale = 2 means that data from
+two extra gridboxes in all directions around the gridbox in question is used, and so forth. So by construction, the
+
+Gascón et al.
+5
+smaller the agreement scale is (its minimum value is 0), the higher the agreement among members is.
+For global model output, as noted above, an option to address sub-grid variability issues with convective precip-
+itation is to convert gridbox forecasts to point forecasts by using a calibrated post-processing technique. The most
+common forecast post-processing techniques tend to have limitations associated, such as the large training datasets
+needed (Wilks and Hamill, 2007; Mendoza et al., 2015; Gneiting, 2014; Hamill et al., 2017; Scheuerer and Hamill,
+2015), the location dependance of the calibration method (Hamill, 2018; Taillardat and Mestre, 2020) or the difficulty
+of improving the forecasts for extremes (Scheuerer and Hamill, 2015; Whan and Schmeits, 2018; Friederichs et al.,
+2018; Taillardat et al., 2019). “ecPoint-Rainfall” is a new probabilistic non-parametric methodology developed at the
+European Centre for Medium-Range Weather Forecasts (ECMWF), which successfully addresses these challenges.
+It currently delivers, on the ECMWF ecCharts platform, experimental real-time probabilistic 12-h precipitation fore-
+casts based on output of the ensemble of the ECMWF Integrated Forecasting System (IFS ENS) (Pillosu and Hewson,
+2017; Hewson and Pillosu, 2021). Its main premise is that the forecast-versus-point-observation relationship de-
+pends not on location, but on physical considerations, allowing one to foresee different degrees of sub-grid variability
+and grid-scale bias according to the prevailing (gridbox) weather situation. It associates each gridbox from a given
+ensemble member at a given time with a specific "weather type". The output from the ecPoint calibration process is
+summarised in mapping functions of forecast errors typical for each weather type. These forecast errors are calculated
+assuming that there is a typical distribution of point rainfall outcomes in a gridbox (as would be measured by multiple
+evenly-spaced raingauges) for each specific weather type. This distribution will naturally account for the sub-grid
+variability, but innately also, via averaging, for the grid-scale NWP bias typical for that weather type. This "calibration
+by weather types" approach allows us to overcome the achilles heel (from an observation availability perspective) of
+location-dependence, and indeed create a suitably large calibration dataset for the whole world using only 1-year of
+observations and 48-hour model (Control run) forecasts.
+The upshot of using the ecPoint approach is that there are clear benefits for flash flood forecasting because
+extremes are forecast much better (e.g. more than 50 mm rainfall in 12-h). At the same time predictions of smaller
+totals are also markedly improved (Hewson and Pillosu, 2021). Ultimately, use of ecPoint ensures that the horizontal
+scales of global-ensemble-related output are much more compatible with the scales of the LAM ensemble (2.2 km)
+
+6
+Gascón et al.
+meaning that a probabilistic high-resolution blending activity can then be applied (pursuant to blending requirements
+cited by Vannitsem et al. (2021)). Here we assume that because a current-day km-scale LAM is able to explicitly
+represent at least major convective updraughts, it should be reasonably representative of point measurements (in both
+stratiform and convective situations). For striking evidence of raingauge-relative improvements in multi-threshold
+frequency biases that can be delivered by even modest model resolution enhancement, most notably in summer, see
+plots in Batignani et al. (2019) (p11 and p13, for 9km and 5km resolution models respectively).
+Ours is in fact quite a unique approach. In the literature and in operational systems, rainfall blending is much more
+commonly applied in nowcasting, that just extends a few hours ahead, using for example radar-based predictions
+alongside NWP (e.g. Lin et al. (2020)). Hamill et al. (2017) describe one multi-ensemble initiative for longer leads,
+although that provides probabilistic output only for a single class (>0.25mm/12h).
+ECMWF participated in the MISTRAL (Meteo Italian SupercompuTing PoRtAL) project (http://www.mistralportal.eu)
+with the primary goal of improving probabilistic rainfall forecast products, to help with the prediction of flash floods in
+Italy and nearby Mediterranean regions. To do that, the scale-selective neighbourhood technique developed by Dey
+et al. (2016b) was adapted and applied to the 2.2 km LAM ensemble forecasts COSMO raw. Then, ecPoint-Rainfall
+for 6-h total precipitation forecasts was also developed during the project, and computed for the global ECMWF IFS
+ENS following the technique described by Hewson and Pillosu (2021). Finally, the two post-processed outputs were
+blended together, for Italy and surrounding areas, to try to fully exploit the most skilful aspects of the two ensemble
+systems. The blending weights used were simply tapered lead-time dependant, with more weight given to ecPoint
+at longer leads as LAM details become less reliable, and also to achieve seamlessness when LAM output stops. The
+overall aim is naturally to create accurate, high fidelity forecasts, but in particular to support the compilation of reli-
+able flash floods warnings as far in advance as possible. The MISTRAL project is funded under the Connecting Europe
+Facility (CEF) - Telecommunication Sector Programme of the European Union. The overarching goal of the project was
+to facilitate and foster the re-use of datasets by weather-dependent communities, to provide added value services
+using High-Performance Computing (HPC) resources.
+The paper is divided up as follows: the methods and datasets used in the creation and validation of these new
+post-processing products are explained in section 2. Section 3 describes the verification procedures (see Table 1) and
+
+Gascón et al.
+7
+Shortname
+IFS ENS raw
+6-h ecPoint-Rainfall
+COSMO raw
+COSMO post
+merged product
+Forecast system
+ECMWF ensemble raw
+6-h Point rainfall forecast
+based on ECMWF ensemble
+COSMO ensemble for Italy
+COSMO ensemble
+post-processed
+COSMO ensemble post-processed
++ 6-h point rainfall
+Ensemble members/percentiles
+51 members
+99 percentiles
+20 members
+99 percentiles
+99 percentiles
+horizontal resolution
+∼18 km
+∼18 km
+2.2 km
+2.2 km
+2.2 km
+TAB LE 1
+Table with the 5 forecast systems evaluated in this paper and their main features
+results, and two case studies are also analysed. Finally, concluding remarks are provided in section 4.
+2
+|
+DATA AND METHODOLOGY
+2.1
+|
+Observations database
+For verification purposes, ECMWF collects non-real-time high-density observations (HDOBS) from its Member and
+Cooperating states (Haiden and Duffy, 2016), typically received at the end of each month. For Italy these are precipitation-
+only observations - there about 3500 such sites in the country. The time resolution of this Italian data goes from 6-h
+to 24-h totals, but only 6-h data has been used in this paper as the main aim was to target and capture short duration
+extreme rainfall events. In turn this is because for flash flooding incidents the rainfall responsible is typically of this
+type. The 6-h precipitation data from the HDOBS and from standard SYNOP reports is used to verify our new fore-
+cast products, using a standard "nearest neighbour" technique to match station location with model grid box (instead
+of interpolating between model gridboxes). Figure 1 shows the topography of the study area from the COSMO raw
+domain area (left) and the array of rain gauge observations used for verification, in Italy and surrounding countries
+(right). Note that observation density is equally comprehensive in mountainous areas.
+
+8
+Gascón et al.
+Lombardy
+m
+Liguria
+Corsica
+Ligurian
+Sea
+Tuscany
+Lazio
+Umbria
+GENOA
+APENNINE MOUNTAINS
+ALPS
+BOLOGNA
+Veneto
+Friuli-Veneza Giulia
+Trentino-Alto Adige
+DOLOMITES
+FIG U R E 1
+Maps showing the full domain of the COSMO-2I model, with terrain height in that model (left), and a
+sample, for one time period, of the SYNOP and HDOBS rain gauge sites used in verification for this paper (right).
+Labels show the locations of Italian regions (red boundaries), cities and geographical features referenced in the text.
+2.2
+|
+ECMWF ensemble post-processing (ecPoint)
+As mentioned before, the new Point Rainfall product aims to deliver probabilities for point measurements of rainfall
+(within a forecast model gridbox). “ecPoint” (Pillosu and Hewson, 2017; Hewson and Pillosu, 2021) is the name given
+to the post-processing philosophy, whilst the companion calibration software is called “ecPoint-calibrate”. "ecPoint-
+Rainfall 12-h" is then a name given to one output type, referring to the fact that its output is for 12-h totals.
+ecPoint is based on the concept of conditional verification, and it consists of identifying (through calibration) and
+correcting (through post-processing) errors in model rainfall forecasts according to a diagnosed gridbox weather-type.
+Weather-types regularly vary between adjacent gridboxes, between consecutive lead times, and between ensemble
+members. Each weather type used is defined by a set of value ranges for each one of the different predictors used (i.e.
+"governing variables"). We typically use 5 to 10 such predictors, and typically create 100-500 weather types. Then all
+weather types together can be represented by a single decision tree, wherein each level corresponds to a governing
+variable, each leaf denotes a weather type, and all leaves cover all possible value combinations for all governing
+
+-10
+5
+20
+100
+200
+500
+1000
+1500
+2000
+3000
+8°E
+9°E
+10°E
+11°E
+12°E
+13°E
+14°E
+15°E
+16°E
+19°E
+6°E
+17°E
+18°E
+19°E
+6°E
+7°E
+9°E
+11°E
+12°E
+13°E
+14°E
+15°E
+16°E
+17°E
+18°E
+47°N
+47°N
+47°N
+46°N
+46°N
+46°N
+45°N
+45°N
+45°N
+44°N
+44°N
+44°N
+43°N
+43°N
+43°N
+42°N
+42°N
+42°N
+41°N
+41°N
+41°N
+40°N
+40°N
+40°N
+39°N
+39°N
+39°N
+38°N
+38°N
+38°N
+37°N
+37°N
+37°N
+36°N
+36°N
+36°N
+15°E
+6°E
+7°E
+8°E
+9°E
+10°E
+11°E
+12°E
+13°E
+14°E
+16°E
+17°E
+18°E
+19°E
+6°E
+7°E
+8°E
+9°E
+10°E
+11°E
+12°E
+13°E
+14°E
+15°E
+16°E
+17°E
+18°E
+19°EGascón et al.
+9
+variables. A "mapping function" is assigned to each leaf (i.e. to each weather type). Such a mapping function defines
+the range of (rainfall) forecast errors experienced across the world whenever the said weather type occurred, in the
+calibration period, in the short range Control run forecasts used for calibration. It ultimately represents the sub-grid
+variability and the gridscale bias typically seen for those particular (weather) conditions. The final ecPoint-Rainfall
+product, for a given time interval, is created by accruing together the post-processed forecasts for each ENS member.
+It can be depicted in percentile or probability format, with the user able to define, according to purpose, the percentile
+value (e.g. 99th) or the event for the probability (e.g. >50mm/12h).
+Following the same methodology as Hewson and Pillosu (2021), and as part of MISTRAL, a global 6-h ecPoint-
+Rainfall product was developed, tested and verified, based on raw IFS ENS forecasts, and using observations from
+around the world. The raw IFS ENS comprises 50 perturbed ensemble members and 1 control member, all with a
+horizontal resolution of 18 km. The ecPoint outputs are made available for overlapping 6-h periods up to day 10,
+namely T+0-T+6 h, T+3-T+9 h,..T+234-T+240 h. The main difference compared to the ecPoint-Rainfall 12-h product
+arose during calibration, wherein the governing variables and the weather type definitions were newly customised.
+“Weather type” here refers to specific weather conditions characterising a gridbox, defined by specific threshold ranges
+of the governing parameters (e.g. > 75% of rainfall is denoted to be convective rain; mean 700hPa wind speed during
+period is in the range 5-20 m/s; ...). Similar to the calibration of ecPoint-Rainfall 12-h, control run forecast rainfall totals
+(G) at short-range (3 to 30 h lead times), covering one year (the training period), were used for calibration by comparing
+with rainfall observations (r) around the world co-located in space and time. As in Hewson and Pillosu (2021), cases
+with G<1 mm (in this case in a 6-h period) were discarded before calibration, for stability and discretisation reasons, and
+also we did not use the first three forecast hours either to avoid possible model rainfall spin-up issues. We merged
+together two sets of gauge rainfall observations (r): those sourced through international data-sharing agreements,
+and others obtained from special datasets (Haiden and Duffy, 2016; Haiden, 2018). The calibration period itself was
+1 April 2017 to 31 March 2018. The full calibration procedure is not described here but involves the segregation of all
+(G,r) pairs into sets, each of which denotes a different gridbox-weather-type. A key aim of segregation was ensuring
+that each such type has associated with it sub-grid variability levels and/or bias correction factors that differ in a
+substantive way from those of neighbouring types within the tree. In practice each gridbox-weather-type connects to
+
+10
+Gascón et al.
+a mapping function, which describes, in Probability Density Function (PDF) form, the (G,r) relationship for all occasions
+when the said type occurred during the calibration period. The PDF is recorded using a non-dimensional metric of
+"Forecast Error Ratio" (FER), defined as follows:
+F E R = (r − G)
+G
+(1)
+Later on the mapping function PDFs are used to post-process forecast rainfall totals, according to whenever and
+wherever the associated weather types re-appear in those forecasts.
+Some example mapping functions are shown in Figure 2. Green bars represent “over-prediction” when FER < 0
+whilst “under-prediction” (FER > 0) is represented in yellow and red bars. The examples in Figures 2 a and b show the
+corresponding mapping functions for total precipitation in 6-h (TP)>7 mm, with precipitation mainly convective in a,
+showing an exponential pattern but mainly stratiform in b, with a more Gaussian shape. We see a better performance
+of the model in the stratiform case because sub-grid variability is much lower. However, in convective situations, a
+"good forecast" (white bar, observations within ±25% of forecast gridbox value) is rare, and overprediction (green) is
+quite common. Also, red bars are slightly higher than they are for the stratiform case, which denotes a large underpre-
+diction, which can in turn connect, on occasion, to flash-floods events that it would have been hard to foresee using
+the raw model data alone. The other two mapping functions shown demonstrate the effect of orography. Figures 2
+c and d are for the same basic weather conditions (mainly stratiform precipitation, 4-7 mm in 6-h); but the first one
+(c) corresponds to flat areas and the second one (d) to areas with more complex (sub-grid) topography. We can see
+how the model performs less well in topographically complex regions, where the incorrect representation of those
+topographic details on the model grid (due to resolution limitations) tends to lead to overestimations (green bars) and
+significant underestimations (red bars) of rainfall occurring more often than they do in flat regions. The gridbox mean
+bias values indicate slightly greater overprediction over mountains (FER=0.85) than over flat areas (FER=0.92).
+
+Gascón et al.
+11
+BiasCorr=0.57
+BiasCorr=0.92
+BiasCorr=0.85
+BiasCorr=0.90
+FER bins
+FER bins
+FER bins
+FER bins
+a) Mainly convective, TP>7 mm/6h
+c) Mainly stratiform, 4<TP<7 mm/6h, flat areas
+d) Mainly stratiform, 4<TP<7 mm/6h, complex orography
+b) Mainly stratiform, TP>7 mm/6h
+FIG U R E 2
+Mapping function examples for specific weather conditions, shown as histograms. On (a-d) dark
+green, green, white, yellow and red bars denote, respectively, FER ranges for “mostly dry” (< -0.99), “over-prediction”
+(-0.99 to -0.25), “good forecasts” (-0.25 to 0.25), “under-prediction” (0.25 to 2) and “substantial under-prediction” (2
+to ∞). (a) is for mainly convective rainfall with large totals forecast, (b) for mainly stratiform rainfall with large totals
+forecast, (c) for mainly stratiform with moderate totals forecast over flat areas and (d) mainly stratiform with
+moderate totals forecast in "mountainous" areas. Flat and mountainous areas were defined using the anciliary model
+variable "standard deviation of filtered subgrid orography" (SDfor). “BiasCorr” values correspond to the gridbox
+mean biases (FER) for each weather type.
+.
+
+0.4
+0.4
+0.3
+0.3
+Frequencies [%]
+Frequencies [%]
+0.2
+0.2
+0.1
+0.1
+0
+0.4
+0.4
+0.3
+0.3
+Frequencies [%]
+Frequencies [%]
+0.2
+0.2
+山
+0.1
+0.112
+Gascón et al.
+FIG U R E 3
+Two small segments of the final decision tree, from the main branches (level 1, grey) of (a) CPR<0.25
+and (b) 0.25<CPR<0.5. In the full tree, there are 245 branch terminations or “leaves”, with 7 levels in total. Each leaf
+has one mapping function (FER distribution) associated (for examples see Figure 2). Note the differences between
+(a), where there is no branching for SR24h and LST but multiple branches for SDfor; and (b), where SR24h and LST
+have branching but SDfor does not.
+To arrive at governing variables for the decision tree, used to define the gridbox-weather types, the ecPoint
+philosophy combines physical reasoning, meteorological experience and the use of case studies to select variables
+that have the potential to influence sub-grid variability and/or gridscale model bias in 6-h precipitation totals. As
+in Hewson and Pillosu (2021), we tried to avoid ’double counting’ by not using two (or more) variables that define
+much the same model characteristic (e.g. two ways of calculating CAPE), and by not using variables whose influence
+on rainfall totals is believed to be well captured by the numerical model itself (e.g. humidity mixing ratio). The 6-h
+ecPoint-Rainfall calibration was informed by the pre-existing 12-h product calibration, but was otherwise developed
+independently. The 6-h weather type definitions (Figure 3) are currently based on the following seven governing
+variables (listed as used in the decision tree, first level first): convective precipitation ratio (CPR), TP, 24-h clear-sky
+direct solar radiation at surface (SR24h), local solar time (LST), standard deviation of the filtered subgrid orography
+(SDfor), 700 hPa wind speed (V700) and Convective Available Potential Energy (CAPE). Five of these had been used
+in the 12-h calibration: CPR, TP, SR24h, V700 and CAPE (Hewson and Pillosu, 2021).
+Variables SR24h, LST and SDfor are all ancillary variables that do not relate to the NWP integrations, but that
+do all have a bearing on local rainfall outcomes. SDfor provides a numerical representation of sub-grid topographic
+complexity, whilst SR24h represents the amount of direct shortwave radiation from the sun that would reach the
+
+0<SDfor<50
+SR24h<105
+LST
+SDfor
+TP<2
+SR24h
+LST
+TP<2
+12<LST<21
+SDfor
+ 50<SDfor<inf
+105<SR24h
+21<LST<33
+SDfor
+SDfor<50
+33<LST<36
+SDfor
+2<TP<4
+SR24h
+LST
+50<SDfor<100
+0.25<CPR<0.5
+SR24h<105
+12<LST<36
+SDfor
+:
+CPR<0.25
+100<SDfor
+2<TP<4 
+12<LST<21
+SDfor
+ 105<SR2h
+21<LST<33
+SDfor
+SDfor<50
+33<LST<36
+SDfor
+4<TP<7
+SR24h
+LST
+50<SDfor<100
+SR24h<105
+LST
+SDfor
+.
+4<TP<7
+100<Sdfor
+105<SR24h
+LST
+SDfor
+SDfor<50
+SR24h<105
+LST
+SDfor
+7<TP
+SR24h
+LST
+7<TP
+105<SR24h
+LST
+SDfor
+50<SdforGascón et al.
+13
+surface of the earth in 24h if skies were clear (Jm−2, horizontal plane assumed). The new variable SDfor was introduced
+in recognition of the impact that topography can have on rainfall (and also considering the particular relevance of this
+aspect to Italy’s complex topography). We see clear evidence of impact on Figure 2 c and d. Meanwhile LST was
+introduced to try to account for the known tendency of the IFS to both develop and decay diurnally-driven convection
+too early in the day (Haiden, 2018; Bechtold et al., 2014).
+In creating the decision tree, we first decide upon the breakpoints for the first variable and accordingly create
+branches to level 2. Then we consider in turn each of those branches, looking for breakpoints for the second variable.
+This process is repeated for the third, fourth, fifth, sixth and seventh variables. For breakpoint selection for each level,
+we use a semi-subjective approach (examining each variable individually). Starting near the lower end of a variable’s
+value range we invent a possible breakpoint, create two subsets, test out, increment the possible breakpoint value,
+and repeat. For each test checks are performed: the first is to ensure adequate subset sample sizes, whilst the second
+looks for significant differences between the two FER distributions using the the two-sample Kolmogorov-Smirnov
+test (Massey, 1951). A third check involves a subjective analysis of the relative proportion of over-predictions (green
+bars in Figure 2), good forecasts (white bar in Figure 2) and under-prediction (yellow and red bars in Figure 2), again
+looking for some noteworthy differences. If all checks suggest a valid breakpoint, that breakpoint is retained, leaving
+only cases with higher values to subset when searching for the next possible breakpoint. This process continues
+iteratively, working down the hierarchy and across through branches, until the whole decision tree has been grown.
+In practice the procedure involves occasional backstepping for pragmatic reasons. The total number of weather types
+obtained in this way in the calibration of 6-h ecPoint-Rainfall was 245.
+Some parameters were negated for specific tree branches, based on physical reasoning backed up by tests. For
+example for CPR<0.25, SR24h and LST are not considered key parameters, since direct insolation will not generally
+have a strong influence on stratiform cloud and rain (note the absence of sub-divisions for these variables on Figure 3
+a). However, SDfor was considered relevant for this type of precipitation, as unrepresented topographic details in the
+model can strongly affect the stratiform rainfall distribution at sub-grid level, at least when tropospheric wind speeds,
+as represented by V700, are non-negligible (note green boxes on Figure 3 a). Simply put, there can be precipitation
+underestimations in upslope areas and overestimations in downslope areas. For CPR>0.25, convective precipitation
+
+14
+Gascón et al.
+evidently plays a more important role, so the astronomical parameters can then have a more substantial influence.
+Therefore, for these decision tree “branches”, LST and SR24h are utilised (note yellow and orange boxes on Figure
+3 b) because insolation strongly influences the development and decay of land-based convection. In respect of LST,
+the aforementioned diurnal cycle biases in convection imply different FER distributions at different times of the day.
+It is important to also highlight that different branches from one variable may not each have the same number of
+branches for the next variable, due mainly to different inter-dependence although subset size can also play a limiting
+role. For example we can see how in Figure 3b larger TP values do not include any division in the LST variable. This
+was because mapping function differences seen during the breakpoint tests were negligible.
+During forecast construction we consider each gridbox around the world, for each time interval, and for each
+ensemble member, and to each of these we assign one of the 245 weather types. Then in each instance we convert
+the forecast 6-h rainfall total into 100 equi-probable point rainfall values, using the respective FER mapping function.
+We then blend together the values across the ensemble, giving us 5100 possible realisations, or “virtual ensemble
+members”, of 6-h point rainfall for each gridbox for each forecast time period. Due to storage constraints, these are
+are saved as percentiles (from 1 to 99).
+2.3
+|
+COSMO ensemble post-processing
+Our "COSMO raw" ensemble output originates from the COSMO-based Italian modelling system, referred to as
+COSMO-LAMI and operationally implemented and maintained by the HydroMeteoClimate Service of the Regional
+Agency for Prevention, Environment and Energy of Emilia-Romagna (Arpae-SIMC) in collaboration with the Regional
+Agency for Environmental Protection of Piedmont (Arpa-Piedmont) and Centro Operativo per la METeorologia, Oper-
+ational Centre for Meteorology, from the Italian Military Air Force (COMET).
+The ensemble has been running daily since the end of 2017 at the CINECA Supercomputer Centre, close to
+Bologna; it consists of 20 runs of the COSMO model at 2.2 km resolution, with 65 vertical levels, starting once a
+day at 21 UTC and with a maximum forecast lead time of 51 h. The integration domain covers Italy and part of the
+surrounding countries (bounding box latitudes: 34.5°N/48.0°N; longitudes: 4.0°E/21.2°E) with a total of 567x701x65
+
+Gascón et al.
+15
+grid points. The grid is rotated lat/lon (0.02°) with a Southern Pole of rotation (47°S/10°E). COSMO raw receives
+hourly boundary conditions from COSMO-ME-EPS, the 7 km ensemble run by COMET over the Mediterranean area.
+Perturbed Initial Conditions are provided by 20 analyses resulting from the KENDA data assimilation cycle (Schraff
+et al., 2016), which is also run operationally at CINECA. Latent heat nudging is also applied to ensemble members
+using the Italian national radar composite.
+As previously mentioned, the scale-selective neighbourhood post-processing described in Dey et al. (2016a,b),
+and Blake et al. (2018), was adapted and applied to the COSMO raw, and the final post-processed product was simply
+called "post-processed COSMO". Whilst Dey et al. (2016a,b) used instantaneous precipitation rates as the target
+metric, we used 6h precipitation totals as in Blake et al. (2018).
+Based on a neighbourhood approach, the scales over which COSMO members’ 6h totals reach a specified level of
+agreement (S, an integer) were calculated at each grid point in the domain, to give a measure of the location-dependent
+believable scales for an ensemble forecast, for each one of a set of (6h) time intervals in that forecast. In practice S
+corresponds to the finest (smallest) agreement scale at which the ensemble members become sufficiently similar to
+provide a valuable, trustworthy forecast. Here S ranges from 0 (good agreement at the model gridscale) to 5 (very
+poor agreement between ensemble members), which for COSMO respectively correspond to gridlengths of 2.2km
+to 24.2km. Here we summarize the method used to derive S, in turn, for each grid point P of the model for each 6h
+period:
+1) The difference in precipitation between the ensemble members is evaluated. The comparison involves ex-
+amining pairs of ensemble members separately, using all possible pairing combinations. For COSMO, which has 20
+members in our study, this means 190 different comparisons.
+2) If the forecasts overall are quite similar (differences are below a certain threshold), then the agreement scale at
+point P corresponds to the grid-scale of the model itself. If the fields are not that similar, then a square neighbourhood
+size = 3 x 3 grid points, centred upon the point P, is considered.
+3) Then spatial precipitation averages, across the 3x3 grid point sets are computed for for all ensemble members,
+with those values then compared in the same way, to assess similarity.
+4) If this time the forecasts are found to be quite similar then the agreement scale is set to 3x3 (i.e. S=1, to denote
+
+16
+Gascón et al.
+1 "ring" of extra gridpoints used). If the fields are not similar enough, then the scale is increased again to give a square
+5x5 grid-point neighbourhood.
+5) Until the agreement scale has been set, the code will repeat steps similar to 3) and 4), increasing the neigh-
+bourhood size each time. At some predefined level, S=5 in this case (i.e. 11x11 gridpoints), computations will stop
+even if the agreement has not been reached - this is for computational efficiency reasons. S=5 equates to a box of
+dimensions ~24x24km, which in areal terms is about 80% larger than an ECMWF ensemble gridbox.
+6) Finally, the saved agreement scale S (i.e. a neighbourhood size) defines which points’ values around grid point P
+will be used to deliver the probability distribution for rainfall at P. For example, if the agreement scale is 1, i.e. equating
+to 3x3 gridpoints, we use 20x3x3 = 180 different values from the 20 ensemble members. These values ranked define
+the rainfall CDF for point P, and we conclude by computing from this CDF percentiles 1 to 99 for precipitation at point
+P (irrespective of how many different values contributed to its CDF). We use the nearest value technique to compute
+percentiles, so that where S=0, for example, the 95th - 99th percentiles all equal the ensemble maximum. Percentile
+values are stored at the same horizontal resolution as COSMO raw.
+7) The previous steps are repeated at all grid points (and then for all of the pre-defined time intervals in the
+ensemble forecast).
+With regard to thresholds used at step (2) above, we followed closely the methodology of Dey et al. (2016b) (their
+Sections 3.1-3.2), wherein thresholds have a weak dependence on agreement scale. As in that paper we actually found,
+in some tests, that agreement scales exhibited little sensitivity to whether thresholds varied in this way or were kept
+constant.
+In the context of steps (2) to (5) above, dry weather constitutes a special case. Whilst one might imagine that all
+members showing no rain at a gridpoint constituted perfect agreement, warranting an agreement scale S=0, in practice
+in this scenario the algorithm decides that the threshold has not been met (again following Dey et al. (2016b)), and
+comparison extends to the 3 x 3 gridpoint squares. The same logic applies at each iteration, such that S=5 is always
+used if all members show dry across 11 x 11 gridpoint squares.
+Our scale-selective neighbourhood post-processing is in effect used to compensate for there being insufficient
+limited area model ensemble members. It can retain rainfall signals at the finest scales when there is good agreement
+
+Gascón et al.
+17
+between members, as might commonly happen with orographically-forced rainfall, for example. In such instances
+we are deeming the finer details in the member forecasts to be reliable. At the same time the approach can spread
+out information from nearby gridboxes when the member signals differ, and are thus deemed less reliable. Cases in
+point would be a day of scattered showers, or intra-ensemble discrepancies in time and/or space in the arrival of a
+front. It is important to appreciate that the agreement scale assigned to each gridbox is based only on the similarity
+of rainfall forecasts within the ensemble, and is assessed for each forecast period, so other factors such as orography
+are not directly used here. Equally, we do not attempt any bias corrections. The post-processed COSMO percentiles
+computed via this procedure, along with those generated by ecPoint, provide input to the subsequent blending activity
+used to create the merged product.
+
+18
+Gascón et al.
+a) COSMO-2I-EPST+24h b) 
+e) 
+h) 
+i) 
+f) 
+c) 
+T+24h
+T+24h
+T+30h
+T+30h
+T+30h
+T+36h
+T+36h
+T+36h
+d) COSMO-2Ipp
+g) Scale agreement
+mm/6h
+FIG U R E 4
+Examples of the agreement scale and its impact on post-processing of the COSMO ensemble 6h
+rainfall forecast. All panels show forecasts from a nominal data time of 00UTC on 14th November, valid for (a,d,g)
+18UTC 14th to 00UTC 15th (T+24h), (b,e,h) 00 to 06UTC 15th (T+30h) , and (c,f,i) 06 to 12UTC 15th (T+36h).
+Panels a,b,c show the raw forecast 95th percentile 6h rainfall in mm (=wettest COSMO solution at each point), d,e,f
+show the post processed 95th percentile, and g,h,i show the diagnosed agreement scale used for post-processing.
+Agreement scale values are always integers between 0 and 5. Although understanding of ’nominal data time’ is not
+critical for interpretation of this figure the concept is explained in full at the start of Section 3.2.
+
+10000
+500
+300
+200
+150
+125
+100
+80
+60
+50
+40
+30
+20
+10
+5
+2
+0.5
+5.5
+4.5
+3.5
+2.5
+1.5
+0.5
+-0.5Gascón et al.
+19
+As an example of agreement scale usage Figure 4 shows, for Northern Italy, diagnosed agreement scale values
+(g,h,i) along with the corresponding 6-h rainfall 95th percentiles for raw (a,b,c) and post-processed (d,e,f) COSMO,
+for three consecutive 6h time windows over a night and morning in November 2019. The agreement scales were
+diagnosed as described above, using all 20 members from one data time.
+The integers (S) represented in Figure 4g, h and i are fundamental because they determine how ensemble data
+is amalgamated during the COSMO post-processing. A value of zero (dark green) denotes maximal agreement in the
+scheme, and the post-processed forecast is constructed using only values for the said gridbox, across the ensemble.
+So wherever there is dark green on panels g, h or i, values on the panel immediately above mirror values on the panel
+above that, implying that local details in ensemble output are being retained because they are consistent enough to
+be believable. Where the agreement scale is 1 (or more) we use instead values from arrays of 3x3 (or more) adjacent
+gridboxes to construct the CDF. So details become less and less believable as agreement scale goes up, and are thus
+removed. In northeastern Italy for example, on panel i, where the agreement scale is mostly 2 (in the southeast corner
+of Friuli-Veneza Giulia), the post-processed product (panel f) accordingly has a smoother, more spread out look with
+smaller peaks than the raw output (panel c). On the same panels even stronger "smoothing" is apparent in the Ligurian
+Sea, where the agreement scale is 3 or 4. There are many papers on the concept of spreading out information, as we
+are doing here, and such approaches also lie at the heart of the widely used “fractions skill score” verification metric
+(Zacharov and Rezacova, 2009; Mittermaier and Roberts, 2010; Skok and Roberts, 2016).
+The apparently eastward-moving band of smaller agreement scales (greens) in this case actually relates to a front,
+and one can also identify topography-related agreement scale minima - this is all discussed in more detail in Section
+3.3 (second case, see also Figure 12). In short, poor agreement between members (large agreement scales) implies
+that at the model gridscale we have insufficient ensemble members to capture the full range of possible outcomes
+concomitant with the geographic setting and ongoing synoptic situations. By preferentially spreading information
+out in such scenarios we should be able to achieve better forecasts for users, and better skill scores in verification,
+without the large costs involved in running more members. Note however that neither approach can reduce the
+"intrinsic" predictability of a situation.
+On Figure 4a the large dry patch, remote from the front, corresponds with where S=5 (Figure 4g), in accordance
+
+20
+Gascón et al.
+with the aforementioned strategy for dealing with dry outcomes. Figure 4g then also demonstrates how this strategy
+leads to a smooth, rather than abrupt, spatial transition in agreement scale close to dry areas, with S=4 being diagnosed
+nearby. Indeed right across the domain, at each of the lead times, S varies smoothly rather than abruptly.
+Having relatively smooth spatial variations in agreement scale also tempers spatial variability in grid point value
+usage in the post-processed COSMO product, thus helping to approximately conserve rainfall "mass". In tests covering
+different synoptic types the change in domain average precipitation after post-processing was very often between -5
+and +10%, with no clear-cut lead time dependence.
+2.4
+|
+Final product: blending together post-processed ensemble outputs
+The "merged product" is the name we give to the final, blended probabilistic 6-h precipitation forecasts, delivered
+for Italy and surrounding countries. These forecasts ostensibly go from 6 h to 48 h out (from a nominal 00UTC data
+time) and have a time resolution of 3-h (i.e. overlapping 6h periods). The blending itself is applied to the ecPoint and
+COSMO post-processed forecasts (in percentile form), assigning different weights for each of the two components
+depending on the lead time (the sum of the two weights is always 1). More weight is given to COSMO at shorter
+leads, starting with 0.9 at nominal lead time 6 h, with 0.05 of weight decrease every (3 h) time step thereafter until
+42 h, after which the decrease is 0.1 at 45 h and 48 h, resulting in an actual weight of 0.1 at 48 h. The horizontal
+resolution of the merged product is kept to 2.2 km. Our “tapered blending” strategy means that across all lead times
+the forecast evolution should look relatively smooth to users, with more detail at the shortest leads when the COSMO
+output is deemed more reliable, due to less synoptic scale spread. The intention was to create a very useful, seamless
+forecast, with as accurate a representation of forecast uncertainty as the ensembles themselves and our simple "first
+principles" optimisation approach would allow. Note that Hemri et al. (2022) provides qualitative support for using
+a tapered blending approach, and indeed in that study optimal LAM ensemble weights would reach zero around
+60 h. Our application of steeper tapering to COSMO weights at long leads was a subjective compromise, to retain
+seamlessness whilst keeping COSMO weights overall a bit higher.
+Clearly our weighting strategy was not "optimized" using verification. Whilst that would on the one hand be
+
+Gascón et al.
+21
+desirable, and is in principle achievable, there would inevitably still be many subjective components, such as metric
+and threshold selection, and deciding whether or not to employ season-based or weather-type-based dependencies,
+if data availability allows. At least with our approach there is simplicity and transparency for the user regarding how
+the merged product was derived; indeed linking the merged product output to the ensemble member inputs in a
+fully explainable way is also relatively straightforward. Another downside of verification-based optimization is that
+seamlessness could be intermittently compromised if extra checks were not built in.
+In practice the lead times for our operational MISTRAL products do extend all the way to 240 h, but after 48
+hours, where the COSMO runs end, we just show ecPoint output as is (so for lead times beyond 48 h it is not strictly
+’blended’ output). In this paper output beyond the 48 h lead will not be discussed further.
+The final operational products are in fact created twice a day, blending firstly 6-h ecPoint-Rainfall from the 12
+UTC ECMWF ENS runs of the previous day (from lead time T+12 h onwards) with the 21 UTC COSMO raw outputs of
+the previous day (from lead time T+3 h onwards), and then later, when the 00 UTC ECMWF ensemble data becomes
+available for the new day, that is blended with the same 21 UTC COSMO data, to deliver an “update”. However, in this
+paper, we discuss and evaluate only the performance of the first blended output as it is the main operational product.
+Two different products are operationally produced each day: probabilities of exceeding specific 6-h total precipitation
+thresholds and 6-h total precipitation percentiles. The available thresholds in the MISTRAL platform are currently 5,
+10, 20 and 50 mm for the probability (of exceedance) product and percentiles 1, 10, 25, 50, 75, 90, 95 and 99 for
+the second product. Given the way data is stored it would however be simple to adjust these specifications if a user
+requested that.
+ecPoint was specifically designed to improve the reliability and discrimination ability of the forecast, particularly
+for large totals. By blending with the post-processed COSMO output, we aim to improve the forecasts even more,
+especially in areas of greater topographic complexity, where higher horizontal resolution and, therefore, better defini-
+tion of the topography is critical. In particular with this blended product we can, in certain situations, give increased
+specificity regarding where the most extreme totals are most likely to occur.
+
+22
+Gascón et al.
+2.5
+|
+The need for HPC infrastructure
+Within the MISTRAL project the Flash Flood Use Case was partly a demonstration of the use of supercomputer
+architecture to make real-time product delivery tractable; in this instance using the Galileo system (and since June
+2021 in Galileo100 system) in the CINECA Supercomputer Centre in Bologna (Italy). The use of HPC resources is
+crucial for this use case because:
+- Considerable computational power and high-quality I/O performance are necessary for operational production
+of the COSMO model forecasts. To provide accurate predictions, COSMO has a high-resolution grid and needs to
+run with short time steps, both of which elevate the need for computational power. The lower resolution ECMWF
+ensemble is no less demanding, due to its global coverage, but is run remotely at ECMWF as part of normal operations
+there.
+- Ensemble forecasting requires multiple runs (20 in the case of COSMO) to be performed quickly. This is ordinarily
+achieved by running members in parallel, which needs a large platform with multiple nodes.
+- We need to transform the two ensemble forecasts sets into a much more accurate form, in real-time, using
+rapid post-processing. In the ecPoint post-processing, for example, we create a new virtual 100-member ensemble
+for each of the 51 real ensemble members, for each lead time interval. This means we generate 5100 virtual ensemble
+members for each of 63 time windows, which delivers 7 Tb of output data (for just the rainfall totals) at an intermediate
+stage. Computational demands are thus very high.
+The fundamental requirement above is to deliver forecasts quickly enough for operational use, which means total
+run times of a few hours at most.
+
+Gascón et al.
+23
+3
+|
+RESULTS
+3.1
+|
+Predictable Scales
+A key user-oriented facet of the merged product is the preservation of high resolution detail (from COSMO) whenever
+and wherever it is deemed appropriate, via the agreement scales concept outlined in Section 2.3. So here we illustrate
+typical behaviour of agreement scales over a 1 year period, and highlight some implications.
+Figure 5a shows that agreement scales tend to grow with lead time in a systematic way. At a nominal lead time
+of 6h about 50% of all non-dry gridpoints have an agreement scale of 3 or less (i.e. equating to gridlengths ≤15.4km -
+Figure 5a), or in other words 50% are retaining details at a finer resolution than the current (2022) ECMWF ENS native
+resolution (18km). By 48h this has reduced to 33%. For agreement scales of 1 or less (i.e. 6.6km and 2.2km gridlengths)
+the corresponding values are ~30% and ~12%. As the length scales across which COSMO ensemble information
+is spread out start to surpass the resolution of the ECMWF ENS, ecPoint output (which delivers probabilities for
+ostensibly similar measurement scales) potentially becomes a competitive and computationally expedient alternative
+for rainfall prediction, provided its output is well-calibrated across domains with diverse topographies. These results
+also seem to lend some qualitative support to the tapered blending strategy we developed a priori, with the proviso
+that verification data needs to also support this.
+The diminishing inter-member agreement shown on Figure 5a is intuitively what one would expect, as uncertain-
+ties in the synoptic pattern (for example) become ever greater with lead time. Interestingly Dey et al. (2016b) found
+no systematic changes in agreement scales with lead time (they examined up to 36h). However we would ascribe that
+to their use of a summer period, and precipitation rate snapshots as the target variable (rather than totals); presumably
+convective cell positioning at short leads was already highly uncertain. In our case precipitation total stripes, arising
+from cell movement, would be more likely to overlap at shorter leads, giving intrinsically smaller agreement scales
+then.
+Another interpretation of agreement scale growth is that it may reflect a need for more and more members, as
+lead times extend, to usefully span all possible outcomes at fine scale, even when using a post-processing technique
+
+24
+Gascón et al.
+like this. This is also what one would intuitively expect. Such a solution would of course have cost implications.
+FIG U R E 5
+Agreement scale frequencies (in %) for post-processed COSMO during the 1 year verification period,
+for all instances with 99th percentile > 0mm. (a) Value distribution (see legend) versus nominal lead time, for
+COSMO domain land areas; km values denote effective gridbox dimensions for each agreement scale. (b) Frequency
+of agreement scale = 0 or 1, over all nominal lead time periods from 0-6h up to 18-24h (day 1). (c) as (b) but for
+24-30h up to 42-48h (day 2). Although the understanding of ’nominal lead time’ is not critical for interpretation of
+this figure the concept is explained in full at the start of sub-section 3.2.
+So where is most value likely to be added by post-processed COSMO? Figures 5b and c clearly demonstrate that
+over mountainous areas there is systematically better inter-member agreement, implying less capacity (and in principal
+less need) to adjust there to ’add value’. In other words there tends to be less lateral information spread through
+COSMO post-processing in those regions,commensurate with what one would hope and expect from orographic
+forcing arguments. On day 1, agreement scales of 0 or 1 are diagnosed as often as 50% of the time over the most
+mountainous parts of the Alps, whilst mountain chains such as the Apennines also exhibit higher frequencies than
+flatter areas nearby. Similar to this, plots in Dey et al. (2016b) show systematically better inter-member agreement
+over the more mountainous northern and western parts of the UK. In our case the frequencies diminish everywhere
+on day 2 (Figure 5c), and whilst a map of day 2 : day 1 frequency ratios is rather noisy (not shown) there is evidence
+that agreement levels hold up slightly better in mountainous areas. For example over plains north of Bologna values
+drop from about 18% on day 1 to 8% on day 2, whilst in Alpine terrain in the northeastern corner of Trentino-Alto
+
+Agreement Scale Frequencies
+Frequency of Agreement Scale = O or 1 (dry events omitted)
+during Verification Period
+10°E
+15°E
+20°E
+5°E
+10°E
+15°E
+20°E
+(land only; dry events omitted)
+%
+100%
+90%
+80%
+70%
+45
+60%
+50%
+40%
+30%
+20%
+10%
+0%
+End of Period (nominal lead time in hours)
+DAY 2
+DAY
+a)
+■012345
+10°E
+15°E
+10°E
+15°EGascón et al.
+25
+Adige they drop from 45% to just 30%.
+Even if our post-processing is much more ’active’ over flatter areas, particularly at longer leads, one should not
+forget that the targetted retention of more detail over mountains is nonetheless likely to be a key contributor to
+any general COSMO post-processing success. Indeed Blake et al. (2018) highlighted that the benefits of the vari-
+able neighbourhood approach over the USA, in terms of discrimination and reliability metrics, and relative to a fixed
+neighbourhood method, were most notable in "the West", which they attributed to that region’s greater topographic
+complexity.
+The one year skill of our post-processed COSMO output, and of the other MISTRAL forecast components, is
+explored in the next sub-section.
+3.2
+|
+Verification
+Verification of 1 year of retrospective 6-h rainfall forecasts was carried out, using as truth rain gauge observations from
+both standard SYNOP reports and specialised high-density datasets from Italy and some surrounding countries (see
+coverage example on Figure 1). Five probabilistic forecast "components" were assessed: the two raw ensembles (IFS
+ENS raw, COSMO raw), the differently post-processed versions of these (ecPoint-Rainfall, post-processed COSMO)
+and the merged version of these two post-processed products (merged product). The verification region coincides
+with COSMO raw coverage (Italy and surrounding areas, Figure 1), whilst the verification period runs from 1 February
+2019 to 31 January 2020 (in January 2019, COSMO raw was not yet fully operational which meant some days were
+missing). All verification scores are computed for 6-h non-overlapped periods (for observation times 00, 06, 12 and
+18 UTC), for (nominal) lead times T+6 h up to T+48 h. We say here "nominal" because a "T+6 h" nominal lead time (for
+example) means that for the two COSMO product sets the actual lead time is T+9 h (because the formal initialisation
+time is 21UTC the day before) whilst for the two ENS product sets it is actually T+18 h (because we are referencing
+the previous day’s 12UTC run, as indicated above). Naturally for the blended product it is a mix. This approach to
+nomenclature was taken to avoid over-complicating figures and text, but it is something that clearly needs to be borne
+in mind when interpreting results. The same nomenclature was also used on Figures 4 and 5. For the observation
+
+26
+Gascón et al.
+periods ending 03, 09, 15 and 21UTC a much reduced observational coverage prevented meaningful verification from
+being performed, even though (overlapping) products were created for those times.
+The reliability component of the Brier Score (BSrel: Brier (1950) and Wilks (2011)) was used here to evaluate the
+reliability of the forecasts. In our case the Brier Score has its binary event form, and represents the mean-squared er-
+ror of the probability forecasts, over the verification sample, for a specific threshold of 6-h accumulated precipitation.
+Meanwhile, to evaluate the capacity to discriminate events, we used Area under the Relative Operating Character-
+istic curve (AROC; Richardson (2003)). Three different precipitation thresholds are initially considered: ≥0.2, ≥10,
+and ≥30mm/6h. The latter value is quite extreme for most of Italy, and may signify a flash flood risk (depending on
+other factors). There were about 10000 reports of >30mm/6h during the year. As the COSMO raw domain is quite
+small, the sample size for thresholds larger than 30mm was too limited to provide robust verification. Following the
+same verification procedures as Hewson and Pillosu (2021) and according to Hamill and Juras (2006) recommenda-
+tions, we added AROC scores for climatological probabilities, based on all available 18-year observation series, to
+show a “zero-skill” baseline (having first discretised to probability intervals of 1% to mirror ecPoint, post-processed
+COSMO and merged product discretisation). Conditional verification scores were also computed for different sea-
+sons, and also for different SDfor ranges (to divide up according to underlying topographic complexity). For these
+classes we used an upper precipitation threshold of 20 mm/6h as case counts naturally diminish following subsetting.
+Confidence intervals associated with BSrel and AROC are obtained using 1000-block bootstrap samples, using the
+stationary bootstrap scheme with mean block length according to Politis and Romano (1994) and following the same
+approach as Hewson and Pillosu (2021). In our verification plots BSrel uses the raw ECMWF ENS as a reference point
+(i.e. we subtract component performance from that of ENS raw, replicating the reference-run plotting approach of
+Baran et al. (2019) and others), so bootstrapping is calculated from these BSrel differences instead of with absolute
+values. We also show the actual reliability of the raw ENS for reference (as dashed lines). Including this enables one
+to also see the degree by which the other systems improve (or degrade): evidently if a forecast system had perfect
+reliability its solid line would overlay the dashed line.
+
+Gascón et al.
+27
+FIG U R E 6
+Forecast verification scores for 6-h rainfall over a 1-year period, using gauge observations, for an
+extended Italian domain, for nominal lead time intervals of T+0-6h to T+42-48h; x-axis values denote interval end
+point. Lines denote ENS raw (orange), ecPoint (red), raw COSMO (light blue), post-processed COSMO (navy),
+merged product (yellow). Left column (a,c,e) displays BSrel metrics, with dashed orange line showing absolute values
+of BSrel for ENS raw, and solid lines differences relative to this for the other forecast systems (i.e. ENS raw minus
+other system). So difference>0 means the other system is more reliable, with that system rated fully reliable
+wherever its solid line reaches the "cap" provided by the dashed line. Right column (b,d,f) shows AROC as a measure
+of discrimination ability (with a climatology “baseline” shown dashed black); larger is better, upper limit is 1. Each
+row is for a different precipitation threshold: (a,b) ≥0.2 , (c,d) ≥10, (e,f) ≥30mm/6h. Error bars on all panels show
+95% confidence intervals from bootstrapping.
+Figure 6 shows BSrel differences versus ENS raw (left column), and AROC (right column) for different 6-h pre-
+cipitation thresholds. First note how for ≥0.2 mm there is a clear-cut diurnal cycle in ENS raw reliability. All other
+
+0.06
+ENS raw ref
+ENSraw
+0.05
+ecPoint6h
+Srely
+coSMOraw
+0.9
+CosMOpost
+0.04
+mergeproduct
+0.8
+AROC
+ENSraw
+ecPoint6h
+0.7
+CoOsMOraw
+@'0.02
+COSMOpost
+(BSr
+merge product
+climatology
+0.6
+0.01
+0
+0.5
+1
+6
+12
+18
+24
+30
+36
+42
+48
+6
+12
+18
+24
+30
+36
+42
+48
+Lead Times [Hours]
+Lead Times [Hours]
+×10-3
+1.5
+(BSrel)
+0.9
+0.5
+AROC
+0.8
+0.7
+0.5
+0.6
+0.5
+12
+18
+24
+30
+36
+42
+48
+6
+12
+18
+24
+30
+36
+42
+48
+Lead Times [Hours]
+Lead Times [Hours]
+×10-4
+(BSrel_IFSraw) - (BSrel)
+0.9
+0
+0.8
+.1
+0.7
+-2
+0.6
+3
+0.5
+6
+12
+18
+24
+30
+36
+42
+48
+6
+12
+18
+24
+30
+36
+42
+48
+Lead Times [Hours]
+Lead Times [Hours]28
+Gascón et al.
+forecast systems show much improved reliability at all lead times for this threshold, with any residual diurnal cycles
+in those systems being, in relative terms, negligible (Figure 6a). There is also a diurnal cycle in AROC for this 0.2mm
+threshold (Figure 6b), but this is seemingly less pronounced and is there in all components. In relative terms, ENS
+raw is least reliable by day (06-18UTC), perhaps because showery activity, which probably peaks then, tends to lead,
+in a given ENS gridbox, to some sites recording rain when others nearby do not. ecPoint and the higher resolution
+ensemble can cope with this, the raw ENS cannot.
+Diurnal cycles in reliability for the raw ENS are not so evident for the higher thresholds ≥10 and ≥30 mm (Figure
+6c, e). These spurious cycles are again addressed by ecPoint-Rainfall and by the merged product, although as the
+threshold increases the improvement imparted by the post-processing becomes less emphatic. Meanwhile COSMO
+products have their own diurnal-cycle reliability errors for the higher thresholds, which are even worse than we see
+in raw ENS. This could be in part because we do not apply any bias correction to raw COSMO or post-processed
+COSMO products, which seemingly have particular reliability issues in representing heavier evening and night-time
+rainfall (18-06UTC). One can say that ecPoint-Rainfall (red line) just about has the best overall performance in terms
+of reliability and discrimination for the 10mm and 30mm thresholds. The merged product is the best product for
+one or both metrics some of the time, notably when it can exploit better performance of post-processed COSMO.
+However, for very short lead times (06 h and 12 h), the merged product can be much worse than ecPoint-Rainfall
+for both metrics, so the high weight then afforded to COSMO raw in the merging seems inappropriate. By lowering
+this weight the merged product would have undoubtedly performed better, and this should probably be adopted in a
+future version.
+The curious short-lead behaviour in COSMO output on Figures 6c,e was briefly investigated by computing, for
+all leads, annual domain-average over-land 6h rainfall totals. These totals exhibited a decaying downward trend with
+lead time (superimposed on a diurnal cycle), with 10% more rain for nominal lead times 00-06UTC on day 1 (T+0-
+6h) than on day 2 (T+24-30h), reducing to 3% more for 18-24UTC. This suggests that COSMO forecasts may have
+had excess rainfall at short leads, degrading reliability then, although in proposing this explanation we were not able
+to directly compare with observed equivalents, due to inhomogeneous data coverage. Another curious feature of
+all the reliability plots is that ENS raw reliability (dashed lines) also tends to improve with lead time (discounting
+
+Gascón et al.
+29
+diurnal trends). This is qualitatively similar to global verification for 12h rainfall in Hewson and Pillosu (2021), in which
+reliability improves up to day 5 then plateaus, and may signify either a "slow spin up" issue and/or insufficient spread. In
+turn such trends may relate to ECMWF’s intention to optimise ENS performance, in terms of spread-skill relationships
+and other metrics, in the medium range, via tailored singular vectors (Palmer, 2019), although conversely note that
+the stochastic perturbations also used to create spread are not themselves targetting medium range performance
+(Leutbecher et al., 2017).
+COSMO post (dark blue line) shows a better resolution and reliability than raw COSMO (light blue line) in all
+instances (excepting 0.2mm reliability which looks identical). An implication of COSMO agreement scales increasing
+with lead time (Figure 5a) is that more value can in principal be added by post-processing later on, such that the rate
+of skill degradation with lead time is reduced. We see signs of this happening on Figure 6 and indeed on most other
+verification plots presented in this section (on Figures 8 and 10), most notably for AROC. Specifically the gap between
+light blue and dark blue curves tends to increase with lead time.
+A final point to note here is that, for all thresholds and all lead times, ecPoint-Rainfall has better discrimination
+ability than both the corresponding ENS raw and the post-processed COSMO forecasts.
+
+30
+Gascón et al.
+FIG U R E 7
+Forecast verification scores for 6-h rainfall for the nominal 18-24h lead = 18-24UTC, over a 1-year
+period, using gauge observations for an extended Italian domain as on Fig 4. Line colours also as on Figure 4. Rows
+signify precipitation thresholds: ≥0.2 (a,b,c), ≥10 (d,e,f) and ≥30mm/6h (g,h,i). Left column (a,d,g) shows ROC curves
+(with AROC denoted by ’AUC=’ in boxes), middle column (b,e,h) reliability diagrams and right column (c,f,i) sharpness
+(logarithmic scale on y-axis). For ROC curves and sharpness, probability discretisation reflects the finest available for
+each forecast system: 1% for ecPoint-Rainfall, COSMO post, and merged product; ~2% for ENS raw and 5% for
+COSMO raw (and 1% for climatology on ROC curves). Missing points in the profiles on (b,c,e,f,h,i) denote empty
+classes.
+Figure 7 presents ROC (Relative Operating Characteristic) curves (Figure 7a, d and g), reliability diagrams (Figure
+
+1.0
+ENS raw
+ENS raw
+ecPoint6h
+ecPoint6h
+3e+05
+COSMO raw
+Observed relative frequency
+COSMO raw
+COSMO post
+0.8
+0.8
+COSMO post
+10000 300001e+05
+ merge product
+Probability of Detection
+merge product
+Frequency (log)
+0.6
+0.6
+0.4
+0.4
+ENS raw
+AUC = 0.909
+0008
+ecPoint6h
+AUC = 0.912
+0.2
+0.2
+COSMO raw
+AUC = 0.877
+COSMO post
+AUC = 0.888
+1000
+merge product AUC = 0.918
+0.0
+AUC = 0.582
+0.0
+climatology
+0.0
+0.2 
+0.4
+0.6
+0.8
+1.0
+0
+0.10.2
+0.30.40.5
+0.60.70.80.9
+0
+0.1
+0.2
+0.3 0.4 0.5 0.6 0.7
+0.80.9
+False Alarm Rate
+Forecast probability
+Forecast probability
+1e+06
+1.0
+ENS raw
+ENS raw
+ecPoint6h
+ecPoint6h
+1e+05
+COSMO raw
+Observed relative frequency
+COSMO raw
+COSMO post
+0.8
+0.8
+COSMO post
+Probability of Detection
+ merge product
+merge product
+10000
+Frequency (log)
+0.6
+0.6
+1000
+0.4
+0.4
+100
+ENS raw
+AUC = 0.876
+ecPoint6h
+AUC = 0.936
+0.2
+0.2
+COSMO raw
+AUC = 0.893
+COSMO post
+AUC = 0.914
+merge product AUC = 0.944
+3
+0.0
+ climatology
+AUC = 0.63
+0.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0
+0.10.2
+0.40.5
+0.6
+0.7
+0.80.9
+0
+0.1
+0.2
+0.30.4
+0.5
+0.60.7
+0.8
+0.9
+False Alarm Rate
+Forecast probability
+Forecast probability
+1.0
+1.0
+1e+06
+ENS raw
+ENS raw
+ecPoint6h
+ecPoint6h
+1e+05
+COSMO raw
+Observed relative frequency
+COSMO raw
+COSMO post
+0.8
+0.8
+COSMO post
+ Probability of Detection
+ merge product
+ merge product
+10000
+Frequency (log)
+0.6
+0.6
+1000
+0.4
+0.4
+100
+ENS raw
+AUC = 0.734
+0.2 
+ecPoint6h
+AUC = 0.896
+0.2
+COSMO raw
+AUC = 0.825
+0
+COSMO post
+AUC = 0.852
+3
+merge product AUC = 0.898
+0.0
+climatology
+AUC = 0.51
+0.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0
+00.1
+0.2 0.3 0.4 0.5 0.6 0.7
+0.80.9
+False Alarm Rate
+Forecast probability
+Forecast probabilityGascón et al.
+31
+7b, e and h), and sharpness diagrams (Figure 7c, f and i) for the five forecast products at 24 h lead time. As on Figure 6
+for T+24h the merged product ROC curves exhibit better discrimination ability than any of the other 4 components,
+for all precipitation thresholds, with forecasts based on climatology performing very poorly, in relative terms. For
+discrimination for the higher precipitation thresholds (Figure 7 d, g), and relative to ENS raw, we also observe clear
+advantages for both of the COSMO systems, and in the case of COSMO raw this is in spite of its coarser discretisaton.
+We attribute this to the higher spatial resolution of 2.2 km in both COSMO products, compared to 18 km in ENS
+raw: probably many cases involve convection or orographic enhancement, both of which can deliver local peaks that
+are not reflected in the output of current generation global models. At the same time there is the caveat that due to
+the apparent closeness of the leftmost segments of the ROC curves on Figure 7 d, g it is hard to anticipate, a priori,
+what any of the systems would deliver in terms of ROC area if discretisation were increased (e.g. by running more
+members).
+Whilst the characteristic of ecPoint products having better reliability and discrimination ability than raw ENS
+for large totals, previously noted by Hewson and Pillosu (2021), is supported here, we can also note a reduction in
+the range of probabilities delivered for such events (Figures 7 f and i). No instances have been seen of probabilities
+>88% and >62% for the 10 and 30mm/6h thresholds respectively, for the 18-24h lead time depicted in the 1 year
+of data. This is not necessarily "wrong" however; note that high probabilities delivered in the two COSMO systems,
+in ENS raw, and also (to a lesser extent) in the merged product, are all over-confident - on Figures 7b, e and h the
+respective lines all fall below the diagonal. And high probability case counts are also quite large (Figures 7c, f and i)
+- e.g. ~2000 for post-processed COSMO for >60% probability of >30mm/6h. Whilst the larger ecPoint probabilities
+of >30mm/6h are, conversely, under-confident (line above diagonal), the case count here is relatively low, only ~100
+for probabilities ≥50%. Sharpness diagrams plotted in Figure 7 use the maximum probability discretisation that each
+forecast system can provide, which varies (see caption). Whilst it might look as though there are discrepancies in
+frequency of occurrence (e.g. COSMO raw seems to exhibit "more of everything") this is just due to the varying
+discretizations, and also because of relatively large absolute variations in the frequency of zero which on the log scale
+are hard to discern.
+
+32
+Gascón et al.
+FIG U R E 8
+Forecast verification scores, divided by season, for a 20mm/6h threshold, using gauge observations
+for an extended Italian domain over 1 year as on Fig 6. Line colours and styles, and x-axis meaning also as on Fig. 5.
+Left column (a,c,e,g) displays BSrel metrics. Right column (b,d,f,h) shows AROC. Seasons are DJF (a,b), MAM (c,d), JJA
+(e,f), SON (g,h). All plots include 95% confidence bars from bootstrapping.
+Figure 8 shows a seasonal breakdown of the skill metrics shown on Figure 6, for very wet episodes (here ≥20
+mm rather than ≥30 mm/6h). The majority of observations of >20mm fell in autumn (~20000 cases), followed by
+spring with ~7000 cases, whilst winter and summer saw ~5000 cases each. In summer (JJA) there were two times
+
+×10-4
+ENS raw ref
+ENS raw
+ecPoint6h
+COSMO raw
+0.9
+CoSMOpost
+mergeproduct
+0.7
+ENS raw
+ecPoint6h
+COSMO raw
+2
+0.6
+COSMOpost
+merge product
+climatology
+4
+0.5
+6
+12
+18
+24
+30
+36
+42
+48
+6
+18
+24
+30
+36
+42
+48
+Lead Times [Hours]
+Lead Times [Hours]
+×10°4
+0.9
+0.8
+AROC
+0.7
+6
+0.6
+8
+0.5
+6
+12
+18
+24
+30
+36
+42
+48
+6
+12
+18
+24
+30
+36
+42
+48
+Lead Times [Hours]
+Lead Times [Hours]
+×104
+4
+(BSrel_IFSraw) - (BSrel)
+0.9
+-2
+0.8
+AROC
+4
+6
+0.7
+8
+0.6
+-10
+-12
+0.5
+6
+12
+18
+24
+30
+36
+42
+48
+6
+12
+18
+24
+30
+36
+42
+48
+Lead Times [Hours]
+Lead Times [Hours]
+×10-3
+1.5
+(BSrel)
+0.9
+IFSraw)
+0.5
+0.8
+AROC
+0.7
+(BSrel
+0.5
+0.6
+0.5
+12
+18
+24
+30
+36
+42
+48
+6
+12
+18
+24
+30
+36
+42
+48
+Lead Times [Hours]
+Lead Times [Hours]Gascón et al.
+33
+as many cases in the evening (18-24 UTC) as in any of the other 6-h periods (not shown), which probably reflects a
+diurnally-driven convective activity peak.
+BSrel plots show some differences between seasons (Figures 8a, c, e and g), but with the highest reliability overall
+delivered by the merged product. ecPoint output is not far behind, and for T+0-6h = 00-06UTC is the clear winner
+in three seasons probably because COSMO spin down issues along with the post-processed COSMO’s high weight
+allocation at short leads are contriving to degrade the merged product, as discussed above. Whilst the raw COSMO
+outputs have relatively strong reliability in winter (DJF) post-processing is, it seems, slightly degrading this: this curious
+behaviour warrants further investigation. One hypothesis is that the true picture is not emerging because of some
+erroneous observations in Italy. A high percentage of precipitation in winter is in snow form in the Alps and the
+Apennines and there may conceivably be problems with the gauge-based heaters that should melt the snow. This
+could lead to considerable underestimation of precipitation, and after the event sometimes overestimation following
+in-funnel melting. These aspects may be erroneously nullifying some of the post-processing benefits, although it is
+also true that skill metrics for the raw ensembles would also be impacted. For ecPoint, a positive design feature of
+its calibration process is resistance to contamination from occasional erroneous extreme values (Hewson and Pillosu,
+2021), and in addition its calibration domain is global and not just Italian. Together these aspects elevate the integrity
+of the ecPoint post-processing for users, but equally limit the scope for error cancellation in the ecPoint verification
+over Italy. Thus all five products we are discussing are likely to be adversely affected by any errors in the verifying
+data, although it is difficult to pin down how much in each case.
+In summer it is striking how COSMO reliability for wet events is poor in the afternoon and evening (12-24UTC), on
+both day 1 and day 2 of the forecast. From Figure 7e,f (for just 18-24UTC) it seems this may be due to over-confidence
+throughout the probability range, and equivalently to a bias to predict too many wet events. This assertion is supported
+by reliability diagrams for a time of day when Brier Score reliability is better (06-12UTC = T+12 and T+36, not shown)
+on which the COSMO lines are closer to the diagonal. So it seems that for COSMO the handling of diurnally driven
+convection exhibits some pronounced time-of-day related reliability issues.
+In terms of discrimination ability for these very wet events, quite similar performance is observed in all the seasons
+(Figure 8b, d, f and h): ecPoint and the merged product are consistently the best, followed by post-processed COSMO,
+
+34
+Gascón et al.
+then raw COSMO, and with raw ENS exhibiting the lowest discrimination ability. It is also noticeable that the summer
+season displays a sharp AROC diurnal cycle, particularly for the raw ENS and ecPoint (Figure 8f). The ecPoint minima
+are for 18-24UTC. This is in spite of the ecpoint decision tree having incorporated local solar time dependence to
+try to mitigate known convective diurnal cycle errors in raw ENS (e.g. Figure 3b), although maybe the picture would
+have been even worse without such adjustments. Note however that we do not post-process the precipitation field
+when the model predicts no precipitation, and here this could be a key factor. One could incorporate a new type
+of dependance by, for example, creating a new ecPoint decision tree branch from the top level, for LST=18-24, that
+does not multiply forecast precipitation but instead uses additive mathematics based on other relevant model-derived
+variables, such as CAPE or lightning density, either in the said period or in the preceding 6-hours. Or alternatively one
+could even use forecast rainfall in the preceding 6 hours in a creative way.
+The ecPoint ROC curves, for >20mm/6h, for different times of day for the JJA summer period (not shown) actually
+exhibit structural differences: relative to the 00-06 and 06-12UTC periods, there is a shift to the right for 18-24UTC,
+and even more so for 12-18UTC, highlighting a lower innate ecPoint discrimination ability for those later periods. This
+is broadly consistent with the ROC areas, except that the minimum ecPoint ROC area is actually for 18-24UTC (Figure
+8f) and not for 12-18UTC, because for 12-18UTC the points ascend to reach much higher hit rates. In turn this is
+because 12-18UTC has more usable rainfall signatures in the raw ENS. The higher number of observed cases may also
+be a contributory factor (in summer the 12-18UTC observation case count for >20mm/6h is ∼2200, versus ∼1100
+for each of 00-06, 06-12 and 18-24UTC).
+During summer evening hours (18-24 UTC), post-processed COSMO is ceutbechrearly better than ecPoint. The
+much higher horizontal resolution in COSMO, and particularly a related ability to prolong daytime convection into
+the evening, which raw ENS lacks, seem to play a critical role here. This clearly benefits the merged product. In the
+32 season-lead-time combinations shown the merged product exhibits its biggest gain over ecPoint for JJA T+18-
+24 (=18-24UTC). The gain would probably also be higher for T+42-48, were it not for the fact that post-processed
+COSMO weight is then very low. Moreover, the merged product’s best season of all, considering all lead times, is
+clearly summer. Probably, if COSMO reliability were intrinsically better (Figure 8e), or could be made better with
+some bias correction procedure, the post-processed COSMO performance in summer would be even more striking.
+
+Gascón et al.
+35
+FIG U R E 9
+Observations available for each SDfor range: (a) <50 m ("plains/low hills"), (b) from 50 to 100 m ("very
+hilly"), and (c) >100 m ("mountainous")
+.
+
+*E19*E
+3s61483(3
+N.9
+4
+I2*N
+N.08
+8'E
+9"E
+10*E
+6*E
+9°E
+12*E
+13*E
+14°E
+15°E18°E
+17*E
+18°E19*E
+6*E
+1D*E
+15°E
+17*E
+1B*E19*E36
+Gascón et al.
+FIG U R E 1 0
+Forecast verification scores, for categories of topographic complexity, for a 20mm/6h threshold,
+using gauge observations for an extended Italian domain over 1 year as on Fig. 5. Line colours and styles and x-axis
+meaning also as on Fig. 5. Left column (a,c,e) displays BSrel metrics. Right column (b,d,f) shows AROC. Topographic
+complexity is represented by the sub-grid orography variable SDfor, with (a,b) for "plains/low hills" (SDfor<50m),
+(c,d) for "very hilly" (50-100m) and (e,f) for "mountainous" (>100m). All plots include 95% confidence bars from
+bootstrapping.
+The final verification plots (Figure 10) show performance of the five ensemble products for different SDfor ranges.
+As described above, SDfor is an ENS grid variable which denotes sub-grid topographic complexity, based on a 1 km
+resolution topographic map and using the standard deviation measure. Higher values naturally correspond to more
+complex terrain. Figure 9 shows the location of observations in three SDfor categories, which we can call "plains/low
+
+×104
+(BSrel_IFSraw) - (BSrel)
+0.9
+3
+0.8
+AROC
+0
+0.7
+ENS raw ref
+ENSraw
+ENS raw
+ecPoint6h
+ecPoint6h
+0.6
+COSMOraw
+COSMO raw
+COSMOpost
+2
+COSMO post
+merge product
+merge product
+climatology
+-3
+0.5
+6
+12
+18
+24
+30
+36
+42
+48
+6
+12
+18
+24
+30
+36
+42
+48
+Lead Times [Hours]
+Lead Times [Hours]
+×10°4
+5
+(BSrel_IFSraw) - (BSrel)
+0.9
+3
+0.8
+AROC
+0.7
+0.6
+0
+0.5
+6
+12
+18
+24
+30
+36
+42
+48
+6
+12
+18
+24
+30
+36
+42
+48
+Lead Times [Hours]
+Lead Times [Hours]
+×10-4
+N
+(BSrel_IFSraw) - (BSrel)
+0.9
+0
+2
+0.8
+AROC
+-4
+0.7
+-6
+0.6
+-8
+-10
+0.5
+6
+12
+18
+24
+30
+36
+42
+48
+6
+12
+18
+24
+30
+36
+42
+48
+Lead Times [Hours]
+Lead Times [Hours]Gascón et al.
+37
+hills" (a), "very hilly" (b), and "mountainous" (c). Category selection was not arbitrary but corresponds to breakpoints
+found whilst applying the decision tree calibration methodology (albeit with some rounding and set size normaliza-
+tion also applied). As such the breakpoints separate significantly different FER mapping function distributions, for 6h
+rainfall. In turn this means that we also expect some differences in independent verification for each SDfor range,
+notably for ecPoint, but also for the merged product which uses ecPoint. Indeed we saw on Figures 5b and c that
+COSMO post-processing agreement scales also depend on altitude / topographic characteristics, implying reasons to
+also expect SDfor-related differences in value added by COSMO post-processing. For the different classes the ap-
+proximate gauge numbers in our domain are 700 (plains/low hills), 900 (very hilly) and 1800 (mountainous). Although
+not homogeneous these numbers are all still large enough to give a robust annual verification picture, for thresholds
+up to 20 mm/6h (refer to error bar magnitude on Figure 10).
+Figures 10a, c and e for reliability show that ecPoint and the merged product generally perform best, implying
+some robustness across multiple topographic types, with the merged product best of all. In this regard the capacity of
+ecPoint to correct the absolute reliability errors of ENS raw (dashed) systematically reduces as topographic complexity
+increases, from >~60% improvement for plains/low hills, to <~20% for mountainous, on average. Strong performance
+of the ecPoint and merged product components holds also for discrimination ability (Figures 10b, d and f) although
+here they are on a par. As in verification by season (Figure 8), there are hints of diurnal cycles in the skill metrics,
+although some of these are hard to interpret. The most clearcut feature in this regard seems to be the AROC minimum
+for ecPoint output seen for 12 to 24UTC (and by implication a maximum for 00-12UTC). This mirrors what we saw in
+summer (Figure 8f), but is watered down by inclusion of the other seasons. Notably the signal is there for all SDfor
+classes.
+In mountainous areas meteorological models commonly have difficulty representing precipitation, and this asser-
+tion is borne out by the AROC values on Figure 10b,d,f, at least for ENS raw. Whilst the value added by post-processing
+the COSMO ensemble seems fairly consistent across terrain categories (perhaps contrary to what one might have ex-
+pected from Figures 5b and c), ecPoint seems, encouragingly, to add most value (to raw ENS) in mountainous regions.
+It probably reflects the merits of using sub-grid orography as a governing variable in the calibration for mainly large-
+scale precipitation (Figure 2c,d and Figure 3a). It may also be because there is more discrimination ability to be added
+
+38
+Gascón et al.
+in mountainous areas, and because ecPoint post-processing is targetting this effectively via its decision tree subdivi-
+sions (even if those have been less effective in improving reliability in mountainous areas). One striking result for the
+mountainous areas is that the two COSMO products are overall much less reliable than the raw ENS; indeed they
+exhibit BSrel values that are more than twice as large at some leads (Figure 10e). One might have thought that the
+COSMO model would have excelled here, especially give that steep-sided mountains and narrow mountain ranges,
+well below ENS resolution, are a common feature of the Italian landscape. But maybe there are strong biases, impact-
+ing reliability, that relate to intrinsic difficulties that even convection-resolving models have in complex terrain. The
+evening-time drop in reliability of COSMO in mountainous areas is particularly striking, mirroring what we see for
+summer performance on Figure 8e, and invites further investigation with summer time case studies. Could it be that
+summer-time evening convection over mountainous regions is a key weak point for COSMO?
+3.3
+|
+Case studies
+We discuss here two case studies of extreme rainfall in Italy, that related locally to flash floods; one was in July,
+the other in November. The aim is to complement the one-year verification figures, and to also illustrate the spatial
+characteristics of outputs of our five different forecasting "systems". It was not a given that the products themselves
+would look physically reasonable and devoid of unwanted artefacts, even if verification is positive, and "believability"
+of the merged product is of course a key requirement for users. Reassuringly, in our case studies (and others not
+shown) this aspiration is clearly met. So here we comment on how each forecast component verifies for each case,
+and to the extent possible link our remarks to verification scores in the previous sub-section.
+
+Gascón et al.
+39
+FIG U R E 1 1
+95th percentiles of 6h rainfall forecasts, for nominal lead time T+24-30 h = 00-06UTC 28th July
+2019, from nominal data time 00UTC 27th July, from (a) raw COSMO, (b) post-processed COSMO, (c) raw ENS, (d)
+ecPoint, (e) merged product. Gauge observations of verifying 6h rainfall are shown in (f); inset shows part of a UK
+Met Office synoptic analysis chart for 00UTC 28th July, with 4hPa isobar interval.
+
+0.5
+5
+10
+20
+30
+40
+50
+2
+60
+80
+100
+125
+ 150 200 300 50040
+Gascón et al.
+On 28 July 2019, a shallow cyclone centred in the Ligurian Sea (see inset on Figure 11f) delivered some extreme
+convective rainfall to Italy: in central Italy >40 mm/6h was widely reported with 80-100 mm/6h locally (Figure 11f).
+Flash flood impacts were reported and some rainfall records were broken: https://www.meteoregionelazio.it/2019/07/29/il-
+forte-maltempo-di-ieri-28-luglio-2019-datifoto-e-video-di-una-giornata-da-ricordare/.
+The 95th percentile forecast from ecPoint (Figure 11d) seems to have performed better in capturing some of the
+more extreme values than ENS raw (Figure 11c), such as in Liguria and the coastal regions of Tuscany, where ecPoint
+values are higher, implying some discrimination ability. Whilst some of the extremes appear to not be captured - the
+highest range shown over Italy is 50-60mm/6h - 99th percentile ecPoint forecast values (not shown) do reach 80-
+100mm/6h in the wettest parts of the country. In reliable forecasts the 99th percentile should naturally be exceeded
+1% of the time (once in 100 observations), and the 95th percentile 5% of the time. In fact, the proportion of observa-
+tions that exceed the 95th percentiles for raw ENS and ecPoint are 23% and 15% respectively; for the 99th percentile
+the corresponding values are 16% and 3%.
+The differences between the 95th percentiles for ecPoint (d) and raw COSMO (a) are very striking, even though
+they are considered to represent similar scales. Whilst we cannot be definitive from one case the raw COSMO rainfall
+values do look a bit excessive (values of 100-200mm cover quite large areas) compared to the range of values reported
+(all <100mm), whilst ecPoint looks more reasonable. Post-processing has a marked impact in reducing the majority of
+the raw COSMO peaks, because of large agreement scales (not shown), although totals on the resulting plot (b) still
+look too high compared to observations. In this case, the proportion of observations exceeding the 95th percentile for
+raw COSMO, post-processed COSMO and the merged product are 5%, 6.7% and 6% respectively, whilst for the 99th
+percentile, the corresponding values are 5% (same value because of ensemble size), 3.2% and 1.2%. So the merged
+product here seems to be the most useful/reliable overall; it achieves a balance between under-prediction of extremes
+in ecPoint, and over-prediction in post-processed COSMO. Meanwhile the over proliferation of extreme values at the
+99th percentile level in post-processed COSMO may well be hindering it’s discrimination ability.
+Whilst we cannot do a like-for-like comparison between the stated characteristics for this case and verification
+scores, because sample size would be too small, there are some useful and meaningful pointers pertaining to the
+merged product benefits in the data we do have: Figure 10e suggests that COSMO is particularly unreliable for large
+
+Gascón et al.
+41
+totals in mountainous areas at night, compared to raw ENS, whilst Figure 7h, together with Figure 10e, suggest this
+is because of over-prediction. Meanwhile ecPoint generally has much better reliability and discrimination in these
+circumstances (Figures 10e,f). So the merged product seems to be exploiting, but reducing, the topography-related
+enhancement delivered by COSMO, whilst benefiting all round from the higher quality but lower-resolution picture
+delivered by ecPoint.
+Success for the merged product also depends on appropriate weighting of ecPoint and post-processed COSMO;
+here it is 50-50, because of the 30h lead-time and our simple "tapered weighting" approach. The weighting versus
+lead-time profile is something that could in principal be optimised in some future version, using iterative long-period
+verification, although that would be computationally challenging.
+Over time we have examined outputs of the five different components for many cases. The characteristics we are
+seeing here are quite typical: for example, post-processed percentiles from the two ensembles tend to resemble their
+respective raw versions more than they resemble each other, whilst visually the merged product commonly looks like
+a good compromise, relative to both the input fields and to subsequent observations.
+
+42
+Gascón et al.
+FIG U R E 1 2
+Forecast probabilities for 6-h precipitation ≥50 mm, for nominal lead time T+30-36 h = 06-12UTC
+15th November 2019 (= rightmost column on Figure 4), from nominal data time 00UTC 14th November, from (a) raw
+COSMO, (b) post-processed COSMO, (c) raw ENS, (d) ecPoint, (e) merged product. Corresponding observations are
+shown in (f). Inset on (c) shows part of a UK Met Office synoptic analysis chart for 06UTC 15th November, with
+4hPa isobar interval (raw ENS probabilities beneath are all zero). Top legend applies to (a)-(e), bottom legend to (f).
+
+100
+90.5
+80.5
+70.5
+60.5
+50.5
+40.5
+35.5
+30.5
+25.5
+20.5
+16.5
+13.5
+10.5
+8.5
+6.5
+4.5
+3.5
+2.5
+1.5
+0.5
+500
+300
+200
+150
+125
+100
+90
+60
+50
+40
+30
+20
+10Gascón et al.
+43
+The second case (Figure 12) is for 15 November 2019 and corresponds to a very heavy rain event over some north-
+ern and central parts of Italy. It is the same case as shown in Figures 4 c, f and i. The event was part of a several-day pe-
+riod during which flash floods occurred quite widely: https://messaggeroveneto.gelocal.it/udine/cronaca/2019/11/16/news/nuova-
+ondata-di-maltempo-in-fvg-allerta-rossa-nel-pordenonese-1.37911793. The synoptic pattern over Italy on 15th was
+cyclonic (inset on Figure 12c), with large-scale frontal rainfall seemingly dominant. However inclusion of a warm sec-
+tor trough suggests a convective component, and the relatively strong warm sector flow also suggests orographic
+enhancement potential, notably over the Dolomites. In some mountainous areas, for example in Trentino-Alto Adige
+and northern Lombardy, the existence of large totals alongside negligible totals (Figure 12f) may not be genuine vari-
+ability, but may instead by symptomatic of higher-altitude snow not being melted in unheated gauges (the freezing
+level was of order 1500-2000m).
+Panels (a)-(e) on Figure 12 show probabilities for precipitation greater than 50mm/6h from the various MISTRAL
+ensemble components. On the two COSMO plots (a and b) one clearly sees the modelled impact on rainfall of topo-
+graphic enhancement, especially in northern Veneto, and Friuli-Venezia Giulia, but also on the southwestern flanks of
+the Apennine chain where south to southwesterly synoptic-scale flow impinges, in for example Umbria and northeast-
+ern Lazio. The high spatial variability in raw probabilities seen in all these areas seems to relate to topographic forcing.
+For fine-scale signals like these that have intra-ensemble consistency - relating to (e.g.) upslope versus downslope
+rainfall - we want to preserve rather than "smooth out" the rainfall forecasts. The post-processing algorithm aims to
+facilitate this using the agreement scale metric (Figure 4g, h and i).
+In fact, comparing Figures 4h and i with the synoptic chart (for the midpoint time of 06UTC) on Figure 12c, one can
+identify an approximately front-parallel agreement scale "trough" (with values of 0 or 1 at its centre) moving eastwards
+with the front, which in this case is actually the primary driver of ensemble consistency. However there are also lateral
+topography-related minima superimposed on this band - e.g. in the Apennines along the northern Border of Tuscany,
+and more generally in the Alps (features which are not inconsistent with the annual average geographical pattern on
+Figure 5c). Thus the COSMO post-processing picture emerging for this case, as represented on Figures 4 c and f,
+and Figures 12 a and b, is quite complex. It involves minimal spreading/smoothing being applied in the zone where
+the ensemble is confidently indicating that frontal rainfall will prevail in the given time window (e.g. over central and
+
+44
+Gascón et al.
+eastern Tuscany), with fine detail retention being further enhanced in upslope/mountainous regions in and beyond
+that zone (e.g. near the Austrian border of Friuli-Veneza Giulia).
+Meanwhile the larger agreement scales (3 or 4) diagnosed in mountainous eastern Umbria and northeastern
+Lazio exist because frontal timing differences within the ensemble are the dominant factor here. The post-processed
+COSMO probabilities (Figure 12b) are as expected smoothed out more in these particular regions (in one case with a
+peak reduced from about 15% to 7%). Given the negligible amounts of rain reported (Figure 12f), presumably due to
+slower frontal progression, such probability reductions can be considered an improvement in the precipitation forecast
+for this case.
+Large precipitation totals (>50mm/6h) were not really directly predicted by any of the ECMWF ensemble mem-
+bers (Figure 12c). Conversely, the ecPoint product (d) had finite probabilities that delineate quite well the areas af-
+fected by totals >50mm/6h. The highest probabilities are in Friulia-Venezia Giulia where the observed local frequency
+of such totals looks greatest, even if the 11-13% probability forecast (yellow) is arguably not high enough (Figure 12f)
+from a user perspective. Meanwhile post-processed COSMO (Figure 12b) generally had higher probabilities where
+observations were largest, although there were a number of "missed events" - most notably in northwestern Veneto
+where the largest total of all was reported. Thus our semi-subjective assessment concurs with the July case shown
+earlier: by blending together the two post-processed products we have created a merged product (Figure 12e) that
+highlights their combined strengths, and that therefore looks more useful than either did in isolation.
+The above results can also be cross-referenced with the most appropriate long-period verification data that we
+have (in section 3.1). Overall that data shows that for high totals, in autumn, for very hilly or mountainous areas, and
+at a nominal lead time of T+30-36, ecPoint, post-processed COSMO and the merged product are on average equally
+reliable (Figures 6e, 8g, 10c,e), whilst for discrimination ability the merged product and ecPoint are generally best,
+with post-processed COSMO significantly less good (Figures 6f, 8h, 10d,f). So the conclusions from our November
+case look to be fairly typical.
+This example also demonstrates a disadvantage of the COSMO post-processing: when (frontal) timing issues
+contaminate the picture we can end up smoothing out reliable fine-scale details (e.g. upslope and downslope rainfall),
+that are a characteristic of the wet solutions, for the wrong reasons, such that we end up mirroring a disadvantage - i.e.
+
+Gascón et al.
+45
+a lack of geographical specificity - of the lower resolution ensemble. Ultimately this is because as lead times increase
+the smaller native ensemble size becomes more and more of a limiting factor, calling for increasingly complex post-
+processing to be employed for the retention of reliable output on fine scales. In our blending system the strategy to
+counter this, in an imperfect but moderately effective way, is to reduce COSMO weight at longer leads. Perhaps there
+are lead-time limits beyond which no post-processing method for a finite size convection-resolving ensemble, however
+sophisticated, could counteract the detrimental impact, on fine-scale rainfall details, of the growth of synoptic scale
+spread. In which case the optimal low-cost solution, as the length-scale to which S corresponds approaches the
+ecPoint output grid resolution, could be to just use ecPoint output instead. In the current study and others that relate
+(Hemri et al. (2022); Harris et al. (2022); Cristina Primo-Ramos - personal communication), ecPoint becomes very
+competitive for lead times greater than 24-48 hours.
+4
+|
+DISCUSSION AND CONCLUSIONS
+This paper describes how two innovative, state-of-the-art post-processing approaches have been applied to the out-
+put of two different types of ensemble, in order to forecast rainfall in Italy and nearby regions. The final product
+is created by blending. The overall aim was to deliver much greater skill than can be achieved by using the raw en-
+sembles’ output as is, particularly for localised extreme rainfall which is a very common driver of flash flood events.
+Indeed this initiative was the ’flash flood use case’ within the EU-funded MISTRAL project. Whilst the project itself
+formally ended in January 2021, products continue to be produced in real-time and are accessible on an open-access
+web platform.
+Input data comes from one LAM ensemble (COSMO) and one global ensemble (ECMWF). These ensembles have
+horizontal resolutions of 2.2 and 18km respectively: in the LAM convection is resolved, in the global model it is
+parametrised. Post-processing of the two ensembles proceeds in different ways - employing methods commensurate
+with their resolutions, their capabilities and the lead times they afford. For the LAM, a scale-selective neighbour-
+hood post-processing technique was applied, primarily to try to compensate for there being insufficient ensemble
+
+46
+Gascón et al.
+members to capture all the convection-related degrees of freedom. Despite being scale-selective, the technique
+nonetheless preserves, probabilistically, signals at 2.2km scale. We consider this to be reasonably representative of
+point (raingauge) scale. For the global ensemble, we use instead the ecPoint downscaling technique. Its main purpose
+is to activate local-weather-dependant adjustments for gridscale bias and for sub-grid variability, to predict, proba-
+bilistically, values as measured by raingauges. Post-processing as described thus puts outputs sourced from the two
+ensembles onto compatible scales, such that a simple blending step can then be meaningfully performed, preserving
+fine-scale signals where appropriate. A much more straightforward alternative to the above would be to simply use
+conservative interpolation to upscale the LAM output to the global model scale before blending, but that would waste
+the high resolution detail (irrespective of whether it was considered reliable or not) and could not deliver forecasts of
+high impact localised extremes.
+When set alongside the fast-evolving world of artificial intelligence (AI) the scale-selective neighbourhood post-
+processing we have used could be seen as rudimentary. On the other hand its simplicity is a major strength. Rather
+than using AI we rely instead on the complexities of the ensemble integrations themselves to define what the ’believ-
+able scales’ of rainfall prediction are, ultimately for any weather scenarios that may arise. We then encapsulate those
+in a very simple and interpretable way, as an agreement scale integer between 0 and 5.
+In our system blending in the first 48h is simply based on a tapered weighting scheme, with post-processed
+COSMO output afforded highest weight early on. With this strategy we aimed to best exploit the innate resolution-
+related advantages of COSMO over ECMWF at short leads. Such advantages are seen for orographic influences, as
+demonstrated in a one year agreement scale analysis, but also for convection, as reported elsewhere, and other as-
+pects. Whilst the net reduction in discrimination ability over the 48 hours for raw COSMO may be fairly modest, for
+larger totals in summer and in autumn, when flash flood frequency over Italy tends to peak, it is more pronounced.
+And although some of this reduction can be offset with post-processing, ecPoint output is still more competitive at
+the longer leads. Meanwhile the fraction of COSMO land gridpoints inheriting probabilistic information from neigh-
+bourhoods that were smaller than 18km (=ENS and ecPoint output resolution) dropped from ~50% at T+6 to ~33%
+at T+48, with km-scale retention dropping away even more rapidly, to ~12% by T+48. Such reductions occur because
+synoptic scale spread growth is rendering the small number of COSMO members increasingly restrictive for longer
+
+Gascón et al.
+47
+leads. So for those longer leads the benefits, for rainfall prediction, of investing in costly high resolution runs, versus
+using ecPoint output alone, reduce, because the two end up largely providing forecasts on similar scales. This provides
+further support for our strategy to increase ecPoint weights for longer leads. And our approach to make this weight
+almost one by (nominal) T+48, where COSMO runs end, also ensures that products are seamless across that barrier.
+Products beyond T+48 are provided by ecPoint alone; these then end at T+240. The focal point in this paper has been
+the first 48h.
+The complex process of post-processing and blending could not have been operationally implemented without
+HPC resources. These were provided for us by CINECA in Bologna. For the users, the blended point rainfall forecasts
+are stored as percentiles 1,2,..99, for overlapping 6h periods. Whilst 6h is the shortest period for which there is
+currently sufficient global coverage of observations for ecPoint calibration, it is also a reasonable compromise from
+a computational perspective. Although periods shorter than 6h might benefit flash flood applications, that would
+increase the HPC resource requirement, compromising delivery times to the detriment of users.
+Verification for the COSMO area shows that relative to raw COSMO the scale-selective neighbourhood technique
+generally improves discrimination ability and reliability for different precipitation thresholds, although the differences
+for reliability are not that significant. Relative to raw ECMWF ENS, ecPoint delivers noteworthy gains in reliability
+and resolution across multiple thresholds, as reported also by Hewson and Pillosu (2021) in global verification for 12h
+totals. However ecPoint’s reliability gains are bordering on negative for large totals at the longer leads of 36-48h,
+reflecting a curious improvement in the reliability of the raw ENS then.
+Relative to post-processed COSMO, the ecPoint product generally provides better forecasts for all precipitation
+thresholds examined, although in some circumstances, when considering larger thresholds (e.g. 20-30mm/6h), this is
+not clearcut. Indeed for large thresholds the advantages of merging the two post-processed outputs are noteworthy
+for discrimination ability in the summer period, and reliability is generally slightly better for the merged product, for
+10 and 30mm/6h, over the year as a whole. Outside of summer time discrimination capabilities for large totals are
+generally similar for the merged product and ecPoint, except at short leads where ecPoint performs best overall,
+probably because post-processed COSMO was then afforded too much weight. Spin-down issues seem to have
+compromised the raw COSMO performance for those short lead times, and the current COSMO post-processing
+
+48
+Gascón et al.
+method cannot correct for those. Future releases could address this using e.g. lead-time dependant bias corrections,
+if appropriate, or more simply by changing the weighting function.
+A particularly strong diurnal cycle in the performance of the different ensemble products was found for summer.
+Indeed an important result from the summer verification was that the post-processed COSMO improves upon ecPoint
+in the nominal "evening" period - i.e. 18 to 24 UTC. This is a time when flash floods can often occur, and so is useful
+to highlight the benefits of blending.
+The relatively poor summer time evening performance of ecPoint shows that our attempt to correct for known
+IFS under-prediction of convective rainfall late in the day, using local solar time as a predictor, has not been that
+successful. One limiting factor in this regard is undoubtedly the current ecPoint policy to apply no adjustments to
+forecasts of zero rain. One could in principal develop another decision tree branch, at the top level, for cases of 0mm
+forecast, trained on what happened when 0mm was forecast, using for example convective rainfall predicted for the
+previous 6 hours, when local solar time indicated evening, as one predictor. This is another area for future work.
+Regarding verification by topographic complexity, ecPoint and the merged product exhibit robust performance
+for large totals (>20mm/6h) across all types of terrain (found in Italy) in our two verification metrics. Overall these
+outputs rank as the best two for all lead times, for all topography classes, with the merged product demonstrating
+better reliability than ecPoint beyond nominal T+6. As expected, our verification also confirms that both COSMO
+products have much better discrimination of large totals in areas of complex orography than the raw ENS, which is
+almost certainly due to the innately better representation of complex orography afforded by COSMO’s 2.2km spatial
+resolution.
+Another striking topographic feature is the reduced ecPoint reliability for the 06-12 UTC period in mountainous
+areas (SDfor > 100 m). The full ecPoint decision tree was designed in such a way that the impacts of both LST and
+SDfor were never considered in the same branch (for a relevant portion see Figure 3). SDfor was presumed to be
+particularly relevant for large-scale precipitation (convective component <0.25), whilst for all other classes it was not
+used, and SR24h and LST were used instead. However, we should perhaps now revisit this, to see if also including
+SDfor as a governing variable for convective events improves scores, for morning times in particular but others too.
+As with many ecPoint calibration results this could also provide useful physical pointers for IFS code development.
+
+Gascón et al.
+49
+Analysis of two case studies using plan-view charts helped us to visualise and better understand the impacts of
+both post-processing techniques, and merging, on the raw forecasts. The merged products looked plausible and were
+devoid of strange artefacts, satisfying basic user requirements. In general, for the largest (localized) rainfall totals in
+the two cases, we observed an overestimation in raw COSMO and a significant underestimation in the raw ENS. Post-
+processed COSMO and ecPoint provided more realistic-looking peaks. Whilst ecPoint also delivered useful guidance
+on where, in general, the worst affected areas would be, post-processed COSMO was sometimes able to add useful
+local detail onto that picture. Ultimately users would see a final merged product that looked more credible, more
+accurate and more useful than any of the input components on their own. Although we have not unequivocally linked
+particular rainfall measurements to flash flood events, the two episodes examined coincided with times when flash
+floods were being quite widely reported.
+Via the scale-selective neighbourhood technique for COSMO we could see in the case studies where the forecast
+fine-scale detail tends to be most reliable. As expected, topographically complex areas stand out. In one case a front-
+parallel agreement scale minimum, centred near to the most likely position of the frontal rainfall, was also very evident.
+Agreement scale increases either side were symptomatic of uncertain timing of the front. At longer leads, and as frontal
+timing concurrently becomes even more uncertain, it is logical to expect a general elevation of agreement scales, which
+is what we saw in our case. In turn this would diminish, with lead time, the innate usefulness of a high resolution ensem-
+ble (with a limited number of members) for predicting local details. Out of the LAM ensembles with resolution <3km op-
+erational in Europe in December 2021 (see http://srnwp.met.hu/C_SRNWP_project/Eumetnet_List.html), none com-
+prised more than 21 members, and only one system - a 2.2km COSMO ensemble for Switzerland - extended beyond
+54h leads. Using differently post-processed output from that ensemble blended with ecPoint output, Hemri et al.
+(2022) have recently shown (using the continuous ranked probability score on 1 year of data) that value added to
+what ecPoint alone delivers diminishes with lead time: it was minimal between 60 and 84h leads, and nil beyond 84h.
+So our inference and their results basically agree.
+Looking to the future, this study provides pointers to how real-time numerical prediction of rainfall (and other
+parameters) could evolve, as LAM ensembles become increasingly widespread, and as global models advance towards
+having convective-scale resolution. In the short term our blending approach could be adopted for other domains
+
+50
+Gascón et al.
+where other LAM ensembles run, perhaps bolstered by verification-based weighting optimisation, and some form of
+LAM bias correction where necessary. In the longer term, in what will be a new and very different era for global
+prediction, one can envisage a crunch point arising, where one can afford to run a medium range ensemble with
+convective-scale resolution, but not the hundreds or thousands of members needed to address both convective and
+synoptic scale uncertainties beyond the first few days. One could accept the jumpy medium range output that would
+arise with fewer members, though users may be dissatisfied (note that even 18km resolution ensembles can be jumpy
+for extreme events - see Figure 3 in Hewson and Pillosu (2021)). Or one could conceivably invent new post-processing
+approaches - simple or complex or a mix - to quell jumpiness and recover skill, but currently the route to that is not
+clear. On the other hand, and this is a very open question, it may be that high resolution global ensembles do advance
+the predictability frontiers for synoptic scales, by capturing important upscale effects, which would make longer lead
+usage vital even with the member number limitations. But if this proves to not be the case then another option would
+be to terminate convective scale global ensembles at some intermediate lead time, say 4 or 5 days, whilst also running
+at lower resolution to much longer leads, and use post-processing and blending techniques like those described here
+to achieve seamless jump-free output, not just locally but globally. In the end it will be a question of how to best
+utilise, for users, the expanding and evolving HPC resources, focussing not just on rainfall but also of course on other
+impact-defining parameters.
+5
+|
+ACKNOWLEDGEMENTS
+The authors gratefully acknowledge financial support from the European Union Connecting Europe Facility (CEF)
+– Telecommunication Sector Programme for the MISTRAL – Meteo Italian SupercompuTing poRtAL project (Grant
+Agreement No. INEA/CEF/ICT/A2017/1567101).
+
+Gascón et al.
+51
+references
+Ancell, B. (2013) Nonlinear Characteristics of Ensemble Perturbation Evolution and Their Application to Forecasting High-
+Impact Events. Weather Forecast., 28, 1353–1365.
+Baran, S., Leutbecher, M., Szabó, M. and Ben-Bouallegue, Z. (2019) Statistical post-processing of dual-resolution ensemble
+forecasts. Q. J. R. Meteorol. Soc., 145, 1705–1720.
+Batignani, F., Cheloni, A., Fuccello, A., Marcucci, F., Torrisi, L., Troisi, A., Zaccariello, N. and Canessa, A. (2019) Application and
+Verification of ECMWF Products 2019 - Italy. In Green Book 2019, 1–25. ECMWF.
+Bechtold, P., Semane, N., Lopez, P., Chaboureau, J.-P., Beljaars, A. and Bormann, N. (2014) Representing Equilibrium and
+Nonequilibrium Convection in Large-Scale Models. J. Atmos. Sci., 71, 734–753.
+Ben Bouallègue, Z. and Theis, S. (2013) Spatial techniques applied to precipitation ensemble forecasts: from verification
+results to probabilistic products. Meteorol. Appl., 21, 922–929.
+Ben Bouallègue, Z., Theis, S. and Gebhardt, C. (2013) Enhancing COSMO-DE ensemble forecasts by inexpensive techniques.
+Meteorol. Z., 22, 49–59.
+Blake, B. T., Carley, J. R., Alcott, T. I., Jankov, I., Pyle, M. E., Perfater, S. E. and Albright, B. (2018) An Adaptive Approach for
+the Calculation of Ensemble Gridpoint Probabilities. Weather Forecast., 33, 1063–1080.
+Brier, G. (1950) Verification of forecasts expressed in terms of probability. Mon. Weather. Rev, 78, 1–3.
+Buizza, R. (2018) Ensemble Forecasting and the Need for Calibration. In Statistical Postprocessing of Ensemble Forecasts, 15–48.
+Elsevier.
+Clark, A., Kain, J., Stensrud, D., Xue, M., Kong, F., Coniglio, M., Thomas, K., Wang, Y., Brewster, K., Gao, J., Wang, X., Weiss, S.
+and Du, J. (2011) Probabilistic Precipitation Forecast Skill as a Function of Ensemble Size and Spatial Scale in a Convection-
+Allowing Ensemble. Mon. Weather Rev., 139, 1410–1418.
+DelSole, T., Trenary, L. and Tippett, M. (2017) The weighted-average lagged ensemble. J. Adv. Model. Earth Syst., 9, 2739–2752.
+Dey, S., Plant, R., Roberts, N. and Migliorini, S. (2016a) Assessing spatial precipitation uncertainties in a convective-scale
+ensemble. Q. J. R. Meteorol. Soc, 142, 2935–2948.
+Dey, S., Roberts, N., Plant, R. and Migliorini, S. (2016b) A new method for the characterization and verification of local spatial
+predictability for convective-scale ensembles. Q. J. R. Meteorol. Soc., 142, 1982–1996.
+
+52
+Gascón et al.
+Ebert, E. E. (2008) Fuzzy verification of high-resolution gridded forecasts: a review and proposed framework. Meteorol. Appl.,
+15, 51–64.
+Friederichs, P., Wahl, S. and Buschow, S. (2018) Postprocessing for Extreme Events. In Statistical Postprocessing of Ensemble
+Forecasts, 127–154. Elsevier.
+Gascón, E., Laviola, S., Merino, A. and Miglietta, M. (2016) Analysis of a localized flash-flood event over the central Mediter-
+ranean. Atmos. Res., 182, 256–268.
+Gneiting, T. (2014) Calibration of medium-range weather forecasts. Technical Memorandum 719, ECMWF.
+Haiden, T. (2018) Use of in situ surface observations at ECMWF. ECMWF Technical Memorandum 834.
+Haiden, T. and Duffy, S. (2016) Use of high-density observations in precipitation verification. ECMWF Newsletter No. 147,
+20–25.
+Hamill, T. (2018) Practical Aspects of Statistical Postprocessing. In Statistical Postprocessing of Ensemble Forecasts, 187–217.
+Elsevier.
+Hamill, T., Engle, E., Myrick, D., Peroutka, M., Finan, C. and Scheuerer, M. (2017) The U.S. National Blend of Models for
+Statistical Postprocessing of Probability of Precipitation and Deterministic Precipitation Amount. Mon. Weather Rev., 145,
+3441–3463.
+Hamill, T. and Juras, J. (2006) Measuring forecast skill: is it real skill or is it the varying climatology? Q. J. R, Meteorol. Soc., 132,
+2905–2923.
+Harris, L., McRae, A. T. T., Chantry, M., Dueben, P. D. and Palmer, T. N. (2022) A generative deep learning approach to stochastic
+downscaling of precipitation forecasts. URL: https://arxiv.org/abs/2204.02028.
+Hemri, S., Hewson, T., Gascon, E., Rajczak, J., Bhend, J., Spririg, C., Moret, L. and Liniger, M. (2022) How do ecPoint precipita-
+tion forecasts compare with postprocessed multi-model ensemble predictions over Switzerland? Technical Memorandum
+901, ECMWF.
+Hewson, T. and Pillosu, F. (2021) A low-cost post-processing technique improves weather forecasts around the world. Com-
+mun. Earth. Environ., 2, 1–10.
+Hitchens, N. M., Brooks, H. E. and Kay, M. P. (2013) Objective Limits on Forecasting Skill of Rare Events. Weather Forecast.,
+28, 525–534.
+
+Gascón et al.
+53
+Kobayashi, K., Duc, L., Apip, Oizumi, T. and Saito, K. (2020) Ensemble flood simulation for a small dam catchment in Japan
+using nonhydrostatic model rainfalls - Part 2: Flood forecasting using 1600-member 4D-EnVar-predicted rainfalls. Nat.
+Hazards Earth Syst. Sci., 20, 755–770.
+Leutbecher, M., Lock, S.-J., Ollinaho, P., Lang, S. T. K., Balsamo, G., Bechtold, P., Bonavita, M., Christensen, H. M., Diamantakis,
+M., Dutra, E., English, S., Fisher, M., Forbes, R. M., Goddard, J., Haiden, T., Hogan, R. J., Juricke, S., Lawrence, H., MacLeod,
+D., Magnusson, L., Malardel, S., Massart, S., Sandu, I., Smolarkiewicz, P. K., Subramanian, A., Vitart, F., Wedi, N. and
+Weisheimer, A. (2017) Stochastic representations of model uncertainties at ecmwf: state of the art and future vision. Q.
+J. R. Meteorol. Soc., 143, 2315–2339.
+Lin, H. H., Tsai, C. C., Liou, J. C., Chen, Y. C., Lin, C. Y., Lin, L. Y. and Chung, K. S. (2020) Multi-weather evaluation of nowcasting
+methods including a new empirical blending scheme. Atmosphere, 11, 1–17.
+Massey, F. (1951) The Kolmogorov-Smirnov Test for Goodness of Fit. J. Am. Stat. Assoc., 46, 68–78.
+Melhauser, C. and Zhang, F. (2012) Practical and Intrinsic Predictability of Severe and Convective Weather at the Mesoscales.
+J. Atmos. Sci., 69, 3350–3371.
+Mendoza, P., Rajagopalan, B., Clark, M., Ikeda, K. and Rasmussen, R. (2015) Statistical Postprocessing of High-Resolution
+Regional Climate Model Output. Mon. Weather Rev., 143, 1533–1553.
+Mittermaier, M. (2014) A Strategy for Verifying Near-Convection-Resolving Model Forecasts at Observing Sites. Weather
+Forecast., 29, 185–204.
+Mittermaier, M. and Roberts, N. (2010) Intercomparison of Spatial Forecast Verification Methods: Identifying Skillful Spatial
+Scales Using the Fractions Skill Score. Weather Forecast., 25, 343–354.
+Necker, T., Geiss, S., Weissmann, M., Ruiz, J., Miyoshi, T. and Lien, G. Y. (2020) A convective-scale 1,000-member ensemble
+simulation and potential applications. Q. J. R. Meteorol. Soc., 146, 1423–1442.
+Palmer, T. (2019) The ecmwf ensemble prediction system: Looking back (more than) 25 years and projecting forward 25 years.
+Q. J. R. Meteorol. Soc., 145, 12–24.
+Pillosu, F. and Hewson, T. (2017) New point-rainfall forecasts for flash flood prediction. ECMWF Newsletter 153.
+Politis, D. and Romano, J. (1994) The stationary bootstrap. J. Am. Stat. Assoc., 89, 1303–1313.
+Radhakrishna, B., Zawadzki, I. and Fabry, F. (2012) Predictability of Precipitation from Continental Radar Images. Part V:
+Growth and Decay. J. Atmos. Sci, 69, 3336–3349.
+
+54
+Gascón et al.
+Raynaud, L. and Bouttier, F. (2017) The impact of horizontal resolution and ensemble size for convective-scale probabilistic
+forecasts. Q. J. R. Meteorol. Soc., 143, 3037–3047.
+Richardson, D. (2003) Economic value and skill. In Forecast Verification: A Practitioner´s Guide in Atmospheric Science, pp. 165–
+187. Wiley.
+Roberts, N. and Lean, H. (2008) Scale-Selective Verification of Rainfall Accumulations from High-Resolution Forecasts of
+Convective Events. Mon. Weather Rev., 136, 78–97.
+Scheuerer, M. and Hamill, T. (2015) Statistical Postprocessing of Ensemble Precipitation Forecasts by Fitting Censored Shifted
+Gamma Distributions. Mon. Weather Rev., 143, 4578–4596.
+Schraff, C., Reich, H., Rhodin, A., Schomburg, A., Stephan, K., Periáñez, A. and Potthast, R. (2016) Kilometre-scale ensemble
+data assimilation for the COSMO model (KENDA). Q. J. R. Meteorol. Soc., 142, 1453–1472.
+Schwartz, C. S. and Sobash, R. A. (2017) Generating Probabilistic Forecasts from Convection-Allowing Ensembles Using Neigh-
+borhood Approaches: A Review and Recommendations. Mon. Weather Rev., 145, 3397–3418.
+Silvestro, F., Rebora, N., Cummings, G. and Ferraris, L. (2015) Experiences of dealing with flash floods using an ensemble
+hydrological nowcasting chain: implications of communication accessibility and distribution of the results. J. Flood Risk
+Manag., 10, 446–462.
+Skok, G. and Roberts, N. (2016) Analysis of Fractions Skill Score properties for random precipitation fields and ECMWF fore-
+casts. Q. J. R. Meteorol. Soc., 142, 2599–2610.
+Taillardat, M., Fougères, A.-L., Naveau, F. and Mestre, O. (2019) Forest-Based and Semiparametric Methods for the Postpro-
+cessing of Rainfall Ensemble Forecasting. Weather Forecast., 34, 617–634.
+Taillardat, M. and Mestre, O. (2020) From research to applications – Examples of operational ensemble post-processing in
+France using machine learning. Nonlinear Proc. Geoph., 27, 329–347.
+Theis, S. E., Hense, A. and Damrath, U. (2005) Probabilistic precipitation forecasts from a deterministic model: a pragmatic
+approach. Meteorol. Appl., 12, 257–268.
+Vannitsem, S., Bremnes, J. B., Demaeyer, J., Evans, G. R., Flowerdew, J., Hemri, S. and ... Ylhaisi, J. (2021) Statistical post-
+processing for weather forecasts: review, challenges, and avenues in a big data world.
+Bull. Am. Meteorol. Soc., 102,
+E681–E699.
+
+Gascón et al.
+55
+Whan, K. and Schmeits, M. (2018) Comparing Area Probability Forecasts of (Extreme) Local Precipitation Using Parametric
+and Machine Learning Statistical Postprocessing Methods. Mon. Weather Rev., 146, 3651–3673.
+Wilks, D. (2011) Statistical Methods in the Atmospheric Science. In International Geophysics Series, 3rd edition, Vol. 100.
+Oxford, UK and Waltham, MA: Academic Press, pp. 704. Elsevier.
+Wilks, D. and Hamill, T. (2007) Comparison of Ensemble-MOS Methods Using GFS Reforecasts. Mon. Weather Rev., 135,
+2379–2390.
+Zacharov, P. and Rezacova, D. (2009) Using the fractions skill score to assess the relationship between an ensemble QPF
+spread and skill. Atmos. Res., 94, 684–693.
+
diff --git a/LNE3T4oBgHgl3EQfYAr5/content/tmp_files/load_file.txt b/LNE3T4oBgHgl3EQfYAr5/content/tmp_files/load_file.txt
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@@ -0,0 +1,1725 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf,len=1724
+page_content='O R I G I N A L A R T I C L E  Post-processing output from ensembles with and  without parametrised convection, to create  accurate, blended, high-fidelity rainfall forecasts  Estíbaliz Gascón1*  |  Andrea Montani1  |  Tim D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Hewson1  1European Centre for Medium-Range  Weather Forecasts, Reading, United  Kingdom  Correspondence  Estíbaliz Gascón, European Centre for  Medium-Range Weather Forecasts,  Reading, United Kingdom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Email: estibaliz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='gascon@ecmwf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='int  Funding information  The authors gratefully acknowledge  financial support from the European Union  Connecting Europe Facility (CEF) –  Telecommunication Sector Programme for  the MISTRAL – Meteo Italian  SupercompuTing poRtAL project (Grant  Agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  INEA/CEF/ICT/A2017/1567101).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Flash flooding is a significant societal problem, but related  precipitation forecasts are often poor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' To address, one can  try to use output from convection-parametrising (global)  ensembles, post-processed to forecast at point-scale, or con-  vection-resolving limited area ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The new method-  ology described here combines both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' We apply “ecPoint-  rainfall” post-processing to the ECMWF global ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Alongside we use 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='2km COSMO LAM ensemble output  (centred on Italy), and also post-process that, using a scale-  selective neighbourhood approach to compensate for insuf-  ficient members and to preserve consistently forecast lo-  cal details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The two resulting scale-compatible components  then undergo lead-time-weighted blending, to create the fi-  nal probabilistic 6h rainfall forecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Product creation for  forecasters, in this way, constituted the “Italy Flash Flood  use case” within the EU-funded MISTRAL project;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' real-time  delivery of open access products is ongoing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  One year of verification shows that, of the five compo-  nents (2 x raw, 2 x post-processed and blended), ecPoint  is the most skilful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The post-processed COSMO ensem-  ble adds most value to summer convective events in the  evening, when the global model has an underprediction bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='04485v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='ao-ph]  11 Jan 2023  2  Gascón et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  In two typical heavy rainfall case studies we observed un-  derestimation of the largest point totals in the raw ECMWF  ensemble, and overestimation in the raw COSMO ensem-  ble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' However, ecPoint elevated the ECMWF maxima and  highlighted best the most affected areas, whilst post-processing  of COSMO diminished extremes by eradicating unreliable  detail, such that the final merged products looked best from  a user perspective, and seemed to be the most skilful of  all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Although our LAM post-processing does not implicitly  include bias-correction (a topic for further work) our study  nonetheless provides a unique blueprint for successfully com-  bining ensemble rainfall forecasts from global and LAM sys-  tems around the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' It also has important implications  for forecast products as global ensembles move ever closer  to having convection-permitting resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Keywords — point rainfall, flash floods, post-processing  techniques, scale-selective neighbourhood, ensemble fore-  cast  1  |  INTRODUCTION  Short-duration convective precipitation extremes are generally responsible for flash flooding events, which are a sig-  nificant societal problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' This paper looks at new ways to anticipate such events by applying novel post-processing  techniques to the output of different types of numerical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Many times, numerical weather preciction (NWP) systems cannot predict precipitation with sufficient accuracy in  terms of rainfall quantity and the particular location of the rainfall events (Silvestro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Gascón et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  This is partially because weather forecasts are often required for points and not, for example, for the large areas of  several hundred square kilometres represented by current-generation global model grid boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' This discrepancy can  be addressed using high-resolution limited-area models (LAMs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Alternatively one can apply some post-processing  methods to global forecast models, to convert from gridbox to point-forecasts (Gneiting, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Hewson and Pillosu,  2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' To better understand these model limitations, we should consider the concept of sub-grid variability, previously  Gascón et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  3  discussed by Hewson and Pillosu (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Sub-grid variability denotes the variation seen amongst point values within  a given model gridbox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' In very localised convective precipitation, which is related to flash floods occurrence, the  sub-grid variability is often so high that forecast totals from globel models of current day resolution will ’miss’ the  event, by maybe an order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' However, in large-scale precipitation, the sub-grid variability in the gridbox is  usually considerably less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' So a major challenge here for post-processing is dealing with variations in sub-grid variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Meanwhile, although LAMs, with their much higher resolution, produce more realistic features than global models,  small-scale predictability, if it exists, can only be maintained for a number of hours, because errors often grow rapidly  (Melhauser and Zhang, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Radhakrishna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', 2012), particularly in convective situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Another reason for the inaccuracy of heavy rainfall forecasts is the innate forecast uncertainties that affect all  NWP, which can in principle be represented and quantified using ensemble prediction systems (Buizza, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' En-  sembles have been used for several decades to predict the possible evolutions of large-scale patterns beyond the  short-range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' In recent years, their use has also become more and more widespread in the framework of LAMs, that  have convection-permitting resolution (say <5 km in the horizontal), to facilitate also probabilistic prediction of small  scale details that the global models innately cannot capture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The number of LAM ensemble members that are strictly  needed to address all the convective degrees of freedom in the atmosphere exceeds what is computationally afford-  able (usually about 20 members) by at least one order of magnitude even on day 1 (Necker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Kobayashi  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', 2020), and after that, as synoptic-scale spread among members grows, even more would be needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Indeed  in 2017, Raynaud and Bouttier (2017) explored how forecast skill depends on ensemble size for LAMs, which tend  to have fewer members than global ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Their study compared the benefits of increasing ensemble size from  12 to 34 with the benefits of increasing horizontal resolution from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='5 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='3 km and concluded that for lead times  greater than 12h, increasing ensemble size provides more benefit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  The COSMO (COnsortium for Small-scale MOdelling) ensemble over Italy is an example of a LAM ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  This is a system recently developed for Italy, that runs on an operational-experimental basis, with 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='2 km horizontal  resolution, 65 vertical levels and 20 members (COSMO-2I-EPS, or COSMO raw, henceforth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The main features of  the system, reported under http://cosmo-model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='org/content/tasks/operational/default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='htm, will be further  discussed in the next sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Researchers have also looked at alternative cheap approaches to solve the problem  4  Gascón et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  related to the lack of members, such as neighbourhood techniques and lag-based ensembles, to increase ensemble  size without increasing computing resources (Ebert, 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Ben Bouallègue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' DelSole et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Schwartz  and Sobash, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The “neighbourhood” post-processing technique has as its central premise the notion of limited  small-scale predictability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' It acknowledges that it is unrealistic to expect high-resolution models to be accurate at  the grid-scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Several approaches, including the fractional coverage (Theis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', 2005) and maximum neighborhood  value methods (Hitchens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', 2013), have been developed to in effect increase the number of members, and thereby  increase also the spread, to provide a more comprehensive description of the spatial uncertainties than the raw LAM  ensemble can provide on its own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' A related challenge is how to use and evaluate the ensembles in convective situa-  tions, where the ensemble mean is unlikely to be physically appropriate for anticipating precipitation extremes (Ancell,  2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Dey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', 2016b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Studies have demonstrated that the skill of convective scale forecasts is very scale-dependent, and this also holds  true for ensemble forecasts, wherein ensemble skill increases with both spatial scale and ensemble size (Roberts and  Lean, 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Clark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Ben Bouallègue and Theis, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Mittermaier, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  So, notwithstanding the above limitations, it should still be possible to give useful probabilistic forecasts of local  precipitation by anticipating uncertainties in the precipitation location (and assuming that the ensemble is unbiased)  (Dey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', 2016b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' One strategy to address localised precipitation uncertainties is the “scale-selective neighbourhood  approach”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Following Dey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2016a,b) and Blake et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2018), a similarity criterion among all possible ensemble  forecast pairs is developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The rationale is that we preserve only the most reliable fine-scale rainfall signals by (i)  retaining that detail in the rainfall patterns wherever ensemble rainfall forecasts are similar to one another (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' where  precipitation is strongly forced by orographic uplift) whilst (ii) ironing out the detail where there is less agreement (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  ’random’ summer convection over relatively flat terrain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' For each model grid point, the scale-selective neighbourhood  approach aims first to quantify the “agreement scale” among the ensemble members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' This is a metric that describes  the spatial scale at which ensemble forecast can be deemed sufficiently similar to each other, according to a criteria  described in the next sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' One can think of the agreement scale as being proportional to the length scale across  which the post-processing combines together the raw forecast data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' agreement scale = 2 means that data from  two extra gridboxes in all directions around the gridbox in question is used, and so forth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' So by construction, the  Gascón et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  5  smaller the agreement scale is (its minimum value is 0), the higher the agreement among members is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  For global model output, as noted above, an option to address sub-grid variability issues with convective precip-  itation is to convert gridbox forecasts to point forecasts by using a calibrated post-processing technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The most  common forecast post-processing techniques tend to have limitations associated, such as the large training datasets  needed (Wilks and Hamill, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Mendoza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Gneiting, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Hamill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Scheuerer and Hamill,  2015), the location dependance of the calibration method (Hamill, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Taillardat and Mestre, 2020) or the difficulty  of improving the forecasts for extremes (Scheuerer and Hamill, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Whan and Schmeits, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Friederichs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=',  2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Taillardat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' “ecPoint-Rainfall” is a new probabilistic non-parametric methodology developed at the  European Centre for Medium-Range Weather Forecasts (ECMWF), which successfully addresses these challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  It currently delivers, on the ECMWF ecCharts platform, experimental real-time probabilistic 12-h precipitation fore-  casts based on output of the ensemble of the ECMWF Integrated Forecasting System (IFS ENS) (Pillosu and Hewson,  2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Hewson and Pillosu, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Its main premise is that the forecast-versus-point-observation relationship de-  pends not on location, but on physical considerations, allowing one to foresee different degrees of sub-grid variability  and grid-scale bias according to the prevailing (gridbox) weather situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' It associates each gridbox from a given  ensemble member at a given time with a specific "weather type".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The output from the ecPoint calibration process is  summarised in mapping functions of forecast errors typical for each weather type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' These forecast errors are calculated  assuming that there is a typical distribution of point rainfall outcomes in a gridbox (as would be measured by multiple  evenly-spaced raingauges) for each specific weather type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' This distribution will naturally account for the sub-grid  variability, but innately also, via averaging, for the grid-scale NWP bias typical for that weather type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' This "calibration  by weather types" approach allows us to overcome the achilles heel (from an observation availability perspective) of  location-dependence, and indeed create a suitably large calibration dataset for the whole world using only 1-year of  observations and 48-hour model (Control run) forecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  The upshot of using the ecPoint approach is that there are clear benefits for flash flood forecasting because  extremes are forecast much better (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' more than 50 mm rainfall in 12-h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' At the same time predictions of smaller  totals are also markedly improved (Hewson and Pillosu, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Ultimately, use of ecPoint ensures that the horizontal  scales of global-ensemble-related output are much more compatible with the scales of the LAM ensemble (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='2 km)  6  Gascón et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  meaning that a probabilistic high-resolution blending activity can then be applied (pursuant to blending requirements  cited by Vannitsem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Here we assume that because a current-day km-scale LAM is able to explicitly  represent at least major convective updraughts, it should be reasonably representative of point measurements (in both  stratiform and convective situations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' For striking evidence of raingauge-relative improvements in multi-threshold  frequency biases that can be delivered by even modest model resolution enhancement, most notably in summer, see  plots in Batignani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2019) (p11 and p13, for 9km and 5km resolution models respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Ours is in fact quite a unique approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' In the literature and in operational systems, rainfall blending is much more  commonly applied in nowcasting, that just extends a few hours ahead, using for example radar-based predictions  alongside NWP (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2020)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Hamill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2017) describe one multi-ensemble initiative for longer leads,  although that provides probabilistic output only for a single class (>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='25mm/12h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  ECMWF participated in the MISTRAL (Meteo Italian SupercompuTing PoRtAL) project (http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='mistralportal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='eu)  with the primary goal of improving probabilistic rainfall forecast products, to help with the prediction of flash floods in  Italy and nearby Mediterranean regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' To do that, the scale-selective neighbourhood technique developed by Dey  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2016b) was adapted and applied to the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='2 km LAM ensemble forecasts COSMO raw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Then, ecPoint-Rainfall  for 6-h total precipitation forecasts was also developed during the project, and computed for the global ECMWF IFS  ENS following the technique described by Hewson and Pillosu (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Finally, the two post-processed outputs were  blended together, for Italy and surrounding areas, to try to fully exploit the most skilful aspects of the two ensemble  systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The blending weights used were simply tapered lead-time dependant, with more weight given to ecPoint  at longer leads as LAM details become less reliable, and also to achieve seamlessness when LAM output stops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The  overall aim is naturally to create accurate, high fidelity forecasts, but in particular to support the compilation of reli-  able flash floods warnings as far in advance as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The MISTRAL project is funded under the Connecting Europe  Facility (CEF) - Telecommunication Sector Programme of the European Union.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The overarching goal of the project was  to facilitate and foster the re-use of datasets by weather-dependent communities, to provide added value services  using High-Performance Computing (HPC) resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  The paper is divided up as follows: the methods and datasets used in the creation and validation of these new  post-processing products are explained in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Section 3 describes the verification procedures (see Table 1) and  Gascón et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  7  Shortname  IFS ENS raw  6-h ecPoint-Rainfall  COSMO raw  COSMO post  merged product  Forecast system  ECMWF ensemble raw  6-h Point rainfall forecast  based on ECMWF ensemble  COSMO ensemble for Italy  COSMO ensemble  post-processed  COSMO ensemble post-processed  + 6-h point rainfall  Ensemble members/percentiles  51 members  99 percentiles  20 members  99 percentiles  99 percentiles  horizontal resolution  ∼18 km  ∼18 km  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='2 km  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='2 km  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='2 km  TAB LE 1  Table with the 5 forecast systems evaluated in this paper and their main features  results, and two case studies are also analysed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Finally, concluding remarks are provided in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  2  |  DATA AND METHODOLOGY  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='1  |  Observations database  For verification purposes, ECMWF collects non-real-time high-density observations (HDOBS) from its Member and  Cooperating states (Haiden and Duffy, 2016), typically received at the end of each month.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' For Italy these are precipitation-  only observations - there about 3500 such sites in the country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The time resolution of this Italian data goes from 6-h  to 24-h totals, but only 6-h data has been used in this paper as the main aim was to target and capture short duration  extreme rainfall events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' In turn this is because for flash flooding incidents the rainfall responsible is typically of this  type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The 6-h precipitation data from the HDOBS and from standard SYNOP reports is used to verify our new fore-  cast products, using a standard "nearest neighbour" technique to match station location with model grid box (instead  of interpolating between model gridboxes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Figure 1 shows the topography of the study area from the COSMO raw  domain area (left) and the array of rain gauge observations used for verification, in Italy and surrounding countries  (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Note that observation density is equally comprehensive in mountainous areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  8  Gascón et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Lombardy  m  Liguria  Corsica  Ligurian  Sea  Tuscany  Lazio  Umbria  GENOA  APENNINE MOUNTAINS  ALPS  BOLOGNA  Veneto  Friuli-Veneza Giulia  Trentino-Alto Adige  DOLOMITES  FIG U R E 1  Maps showing the full domain of the COSMO-2I model, with terrain height in that model (left), and a  sample, for one time period, of the SYNOP and HDOBS rain gauge sites used in verification for this paper (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Labels show the locations of Italian regions (red boundaries), cities and geographical features referenced in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='2  |  ECMWF ensemble post-processing (ecPoint)  As mentioned before, the new Point Rainfall product aims to deliver probabilities for point measurements of rainfall  (within a forecast model gridbox).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' “ecPoint” (Pillosu and Hewson, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Hewson and Pillosu, 2021) is the name given  to the post-processing philosophy, whilst the companion calibration software is called “ecPoint-calibrate”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' "ecPoint-  Rainfall 12-h" is then a name given to one output type, referring to the fact that its output is for 12-h totals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  ecPoint is based on the concept of conditional verification, and it consists of identifying (through calibration) and  correcting (through post-processing) errors in model rainfall forecasts according to a diagnosed gridbox weather-type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Weather-types regularly vary between adjacent gridboxes, between consecutive lead times, and between ensemble  members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Each weather type used is defined by a set of value ranges for each one of the different predictors used (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  "governing variables").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' We typically use 5 to 10 such predictors, and typically create 100-500 weather types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Then all  weather types together can be represented by a single decision tree,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' wherein each level corresponds to a governing  variable,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' each leaf denotes a weather type,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' and all leaves cover all possible value combinations for all governing  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
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+page_content='19°EGascón et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  9  variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' A "mapping function" is assigned to each leaf (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' to each weather type).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Such a mapping function defines  the range of (rainfall) forecast errors experienced across the world whenever the said weather type occurred, in the  calibration period, in the short range Control run forecasts used for calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' It ultimately represents the sub-grid  variability and the gridscale bias typically seen for those particular (weather) conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The final ecPoint-Rainfall  product, for a given time interval, is created by accruing together the post-processed forecasts for each ENS member.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  It can be depicted in percentile or probability format, with the user able to define, according to purpose, the percentile  value (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' 99th) or the event for the probability (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' >50mm/12h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Following the same methodology as Hewson and Pillosu (2021), and as part of MISTRAL, a global 6-h ecPoint-  Rainfall product was developed, tested and verified, based on raw IFS ENS forecasts, and using observations from  around the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The raw IFS ENS comprises 50 perturbed ensemble members and 1 control member, all with a  horizontal resolution of 18 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The ecPoint outputs are made available for overlapping 6-h periods up to day 10,  namely T+0-T+6 h, T+3-T+9 h,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='.T+234-T+240 h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The main difference compared to the ecPoint-Rainfall 12-h product  arose during calibration, wherein the governing variables and the weather type definitions were newly customised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  “Weather type” here refers to specific weather conditions characterising a gridbox, defined by specific threshold ranges  of the governing parameters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' > 75% of rainfall is denoted to be convective rain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' mean 700hPa wind speed during  period is in the range 5-20 m/s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Similar to the calibration of ecPoint-Rainfall 12-h, control run forecast rainfall totals  (G) at short-range (3 to 30 h lead times), covering one year (the training period), were used for calibration by comparing  with rainfall observations (r) around the world co-located in space and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' As in Hewson and Pillosu (2021), cases  with G<1 mm (in this case in a 6-h period) were discarded before calibration, for stability and discretisation reasons, and  also we did not use the first three forecast hours either to avoid possible model rainfall spin-up issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' We merged  together two sets of gauge rainfall observations (r): those sourced through international data-sharing agreements,  and others obtained from special datasets (Haiden and Duffy, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Haiden, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The calibration period itself was  1 April 2017 to 31 March 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The full calibration procedure is not described here but involves the segregation of all  (G,r) pairs into sets, each of which denotes a different gridbox-weather-type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' A key aim of segregation was ensuring  that each such type has associated with it sub-grid variability levels and/or bias correction factors that differ in a  substantive way from those of neighbouring types within the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' In practice each gridbox-weather-type connects to  10  Gascón et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  a mapping function, which describes, in Probability Density Function (PDF) form, the (G,r) relationship for all occasions  when the said type occurred during the calibration period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The PDF is recorded using a non-dimensional metric of  "Forecast Error Ratio" (FER), defined as follows:  F E R = (r − G)  G  (1)  Later on the mapping function PDFs are used to post-process forecast rainfall totals, according to whenever and  wherever the associated weather types re-appear in those forecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Some example mapping functions are shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Green bars represent “over-prediction” when FER < 0  whilst “under-prediction” (FER > 0) is represented in yellow and red bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The examples in Figures 2 a and b show the  corresponding mapping functions for total precipitation in 6-h (TP)>7 mm, with precipitation mainly convective in a,  showing an exponential pattern but mainly stratiform in b, with a more Gaussian shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' We see a better performance  of the model in the stratiform case because sub-grid variability is much lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' However, in convective situations, a  "good forecast" (white bar, observations within ±25% of forecast gridbox value) is rare, and overprediction (green) is  quite common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Also, red bars are slightly higher than they are for the stratiform case, which denotes a large underpre-  diction, which can in turn connect, on occasion, to flash-floods events that it would have been hard to foresee using  the raw model data alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The other two mapping functions shown demonstrate the effect of orography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Figures 2  c and d are for the same basic weather conditions (mainly stratiform precipitation, 4-7 mm in 6-h);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' but the first one  (c) corresponds to flat areas and the second one (d) to areas with more complex (sub-grid) topography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' We can see  how the model performs less well in topographically complex regions, where the incorrect representation of those  topographic details on the model grid (due to resolution limitations) tends to lead to overestimations (green bars) and  significant underestimations (red bars) of rainfall occurring more often than they do in flat regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The gridbox mean  bias values indicate slightly greater overprediction over mountains (FER=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='85) than over flat areas (FER=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='92).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Gascón et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  11  BiasCorr=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='57  BiasCorr=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='92  BiasCorr=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='85  BiasCorr=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='90  FER bins  FER bins  FER bins  FER bins  a) Mainly convective, TP>7 mm/6h  c) Mainly stratiform, 4<TP<7 mm/6h, flat areas  d) Mainly stratiform, 4<TP<7 mm/6h, complex orography  b) Mainly stratiform, TP>7 mm/6h  FIG U R E 2  Mapping function examples for specific weather conditions, shown as histograms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' On (a-d) dark  green, green, white, yellow and red bars denote, respectively, FER ranges for “mostly dry” (< -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='99), “over-prediction”  (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='99 to -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='25), “good forecasts” (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='25 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='25), “under-prediction” (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='25 to 2) and “substantial under-prediction” (2  to ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (a) is for mainly convective rainfall with large totals forecast, (b) for mainly stratiform rainfall with large totals  forecast, (c) for mainly stratiform with moderate totals forecast over flat areas and (d) mainly stratiform with  moderate totals forecast in "mountainous" areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Flat and mountainous areas were defined using the anciliary model  variable "standard deviation of filtered subgrid orography" (SDfor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' “BiasCorr” values correspond to the gridbox  mean biases (FER) for each weather type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='3  Frequencies [%]  Frequencies [%]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='3  Frequencies [%]  Frequencies [%]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='2  山  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='112  Gascón et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  FIG U R E 3  Two small segments of the final decision tree, from the main branches (level 1, grey) of (a) CPR<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='25  and (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='25<CPR<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' In the full tree, there are 245 branch terminations or “leaves”, with 7 levels in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Each leaf  has one mapping function (FER distribution) associated (for examples see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Note the differences between  (a), where there is no branching for SR24h and LST but multiple branches for SDfor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' and (b), where SR24h and LST  have branching but SDfor does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  To arrive at governing variables for the decision tree, used to define the gridbox-weather types, the ecPoint  philosophy combines physical reasoning, meteorological experience and the use of case studies to select variables  that have the potential to influence sub-grid variability and/or gridscale model bias in 6-h precipitation totals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' As  in Hewson and Pillosu (2021), we tried to avoid ’double counting’ by not using two (or more) variables that define  much the same model characteristic (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' two ways of calculating CAPE), and by not using variables whose influence  on rainfall totals is believed to be well captured by the numerical model itself (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' humidity mixing ratio).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The 6-h  ecPoint-Rainfall calibration was informed by the pre-existing 12-h product calibration, but was otherwise developed  independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The 6-h weather type definitions (Figure 3) are currently based on the following seven governing  variables (listed as used in the decision tree, first level first): convective precipitation ratio (CPR), TP, 24-h clear-sky  direct solar radiation at surface (SR24h), local solar time (LST), standard deviation of the filtered subgrid orography  (SDfor), 700 hPa wind speed (V700) and Convective Available Potential Energy (CAPE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Five of these had been used  in the 12-h calibration: CPR, TP, SR24h, V700 and CAPE (Hewson and Pillosu, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Variables SR24h, LST and SDfor are all ancillary variables that do not relate to the NWP integrations, but that  do all have a bearing on local rainfall outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' SDfor provides a numerical representation of sub-grid topographic  complexity, whilst SR24h represents the amount of direct shortwave radiation from the sun that would reach the  0<SDfor<50  SR24h<105  LST  SDfor  TP<2  SR24h  LST  TP<2  12<LST<21  SDfor  50<SDfor<inf  105<SR24h  21<LST<33  SDfor  SDfor<50  33<LST<36  SDfor  2<TP<4  SR24h  LST  50<SDfor<100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='25<CPR<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='5  SR24h<105  12<LST<36  SDfor  :  CPR<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='25  100<SDfor  2<TP<4  12<LST<21  SDfor  105<SR2h  21<LST<33  SDfor  SDfor<50  33<LST<36  SDfor  4<TP<7  SR24h  LST  50<SDfor<100  SR24h<105  LST  SDfor  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  4<TP<7  100<Sdfor  105<SR24h  LST  SDfor  SDfor<50  SR24h<105  LST  SDfor  7<TP  SR24h  LST  7<TP  105<SR24h  LST  SDfor  50<SdforGascón et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  13  surface of the earth in 24h if skies were clear (Jm−2, horizontal plane assumed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The new variable SDfor was introduced  in recognition of the impact that topography can have on rainfall (and also considering the particular relevance of this  aspect to Italy’s complex topography).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' We see clear evidence of impact on Figure 2 c and d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Meanwhile LST was  introduced to try to account for the known tendency of the IFS to both develop and decay diurnally-driven convection  too early in the day (Haiden, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Bechtold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  In creating the decision tree, we first decide upon the breakpoints for the first variable and accordingly create  branches to level 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Then we consider in turn each of those branches, looking for breakpoints for the second variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  This process is repeated for the third, fourth, fifth, sixth and seventh variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' For breakpoint selection for each level,  we use a semi-subjective approach (examining each variable individually).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Starting near the lower end of a variable’s  value range we invent a possible breakpoint, create two subsets, test out, increment the possible breakpoint value,  and repeat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' For each test checks are performed: the first is to ensure adequate subset sample sizes, whilst the second  looks for significant differences between the two FER distributions using the the two-sample Kolmogorov-Smirnov  test (Massey, 1951).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' A third check involves a subjective analysis of the relative proportion of over-predictions (green  bars in Figure 2), good forecasts (white bar in Figure 2) and under-prediction (yellow and red bars in Figure 2), again  looking for some noteworthy differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' If all checks suggest a valid breakpoint, that breakpoint is retained, leaving  only cases with higher values to subset when searching for the next possible breakpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' This process continues  iteratively, working down the hierarchy and across through branches, until the whole decision tree has been grown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  In practice the procedure involves occasional backstepping for pragmatic reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The total number of weather types  obtained in this way in the calibration of 6-h ecPoint-Rainfall was 245.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Some parameters were negated for specific tree branches, based on physical reasoning backed up by tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' For  example for CPR<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='25, SR24h and LST are not considered key parameters, since direct insolation will not generally  have a strong influence on stratiform cloud and rain (note the absence of sub-divisions for these variables on Figure 3  a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' However, SDfor was considered relevant for this type of precipitation, as unrepresented topographic details in the  model can strongly affect the stratiform rainfall distribution at sub-grid level, at least when tropospheric wind speeds,  as represented by V700, are non-negligible (note green boxes on Figure 3 a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Simply put, there can be precipitation  underestimations in upslope areas and overestimations in downslope areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' For CPR>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='25, convective precipitation  14  Gascón et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  evidently plays a more important role, so the astronomical parameters can then have a more substantial influence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Therefore, for these decision tree “branches”, LST and SR24h are utilised (note yellow and orange boxes on Figure  3 b) because insolation strongly influences the development and decay of land-based convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' In respect of LST,  the aforementioned diurnal cycle biases in convection imply different FER distributions at different times of the day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  It is important to also highlight that different branches from one variable may not each have the same number of  branches for the next variable, due mainly to different inter-dependence although subset size can also play a limiting  role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' For example we can see how in Figure 3b larger TP values do not include any division in the LST variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' This  was because mapping function differences seen during the breakpoint tests were negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  During forecast construction we consider each gridbox around the world, for each time interval, and for each  ensemble member, and to each of these we assign one of the 245 weather types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Then in each instance we convert  the forecast 6-h rainfall total into 100 equi-probable point rainfall values, using the respective FER mapping function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  We then blend together the values across the ensemble, giving us 5100 possible realisations, or “virtual ensemble  members”, of 6-h point rainfall for each gridbox for each forecast time period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Due to storage constraints, these are  are saved as percentiles (from 1 to 99).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='3  |  COSMO ensemble post-processing  Our "COSMO raw" ensemble output originates from the COSMO-based Italian modelling system,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' referred to as  COSMO-LAMI and operationally implemented and maintained by the HydroMeteoClimate Service of the Regional  Agency for Prevention,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Environment and Energy of Emilia-Romagna (Arpae-SIMC) in collaboration with the Regional  Agency for Environmental Protection of Piedmont (Arpa-Piedmont) and Centro Operativo per la METeorologia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Oper-  ational Centre for Meteorology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' from the Italian Military Air Force (COMET).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  The ensemble has been running daily since the end of 2017 at the CINECA Supercomputer Centre, close to  Bologna;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' it consists of 20 runs of the COSMO model at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='2 km resolution, with 65 vertical levels, starting once a  day at 21 UTC and with a maximum forecast lead time of 51 h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The integration domain covers Italy and part of the  surrounding countries (bounding box latitudes: 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='5°N/48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='0°N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' longitudes: 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='0°E/21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='2°E) with a total of 567x701x65  Gascón et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  15  grid points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The grid is rotated lat/lon (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='02°) with a Southern Pole of rotation (47°S/10°E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' COSMO raw receives  hourly boundary conditions from COSMO-ME-EPS, the 7 km ensemble run by COMET over the Mediterranean area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Perturbed Initial Conditions are provided by 20 analyses resulting from the KENDA data assimilation cycle (Schraff  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', 2016), which is also run operationally at CINECA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Latent heat nudging is also applied to ensemble members  using the Italian national radar composite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  As previously mentioned, the scale-selective neighbourhood post-processing described in Dey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2016a,b),  and Blake et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2018), was adapted and applied to the COSMO raw, and the final post-processed product was simply  called "post-processed COSMO".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Whilst Dey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2016a,b) used instantaneous precipitation rates as the target  metric, we used 6h precipitation totals as in Blake et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Based on a neighbourhood approach, the scales over which COSMO members’ 6h totals reach a specified level of  agreement (S, an integer) were calculated at each grid point in the domain, to give a measure of the location-dependent  believable scales for an ensemble forecast, for each one of a set of (6h) time intervals in that forecast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' In practice S  corresponds to the finest (smallest) agreement scale at which the ensemble members become sufficiently similar to  provide a valuable, trustworthy forecast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Here S ranges from 0 (good agreement at the model gridscale) to 5 (very  poor agreement between ensemble members), which for COSMO respectively correspond to gridlengths of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='2km  to 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='2km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Here we summarize the method used to derive S, in turn, for each grid point P of the model for each 6h  period:  1) The difference in precipitation between the ensemble members is evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The comparison involves ex-  amining pairs of ensemble members separately, using all possible pairing combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' For COSMO, which has 20  members in our study, this means 190 different comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  2) If the forecasts overall are quite similar (differences are below a certain threshold), then the agreement scale at  point P corresponds to the grid-scale of the model itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' If the fields are not that similar, then a square neighbourhood  size = 3 x 3 grid points, centred upon the point P, is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  3) Then spatial precipitation averages, across the 3x3 grid point sets are computed for for all ensemble members,  with those values then compared in the same way, to assess similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  4) If this time the forecasts are found to be quite similar then the agreement scale is set to 3x3 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' S=1, to denote  16  Gascón et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  1 "ring" of extra gridpoints used).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' If the fields are not similar enough, then the scale is increased again to give a square  5x5 grid-point neighbourhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  5) Until the agreement scale has been set, the code will repeat steps similar to 3) and 4), increasing the neigh-  bourhood size each time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' At some predefined level, S=5 in this case (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' 11x11 gridpoints), computations will stop  even if the agreement has not been reached - this is for computational efficiency reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' S=5 equates to a box of  dimensions ~24x24km, which in areal terms is about 80% larger than an ECMWF ensemble gridbox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  6) Finally, the saved agreement scale S (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' a neighbourhood size) defines which points’ values around grid point P  will be used to deliver the probability distribution for rainfall at P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' For example, if the agreement scale is 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' equating  to 3x3 gridpoints, we use 20x3x3 = 180 different values from the 20 ensemble members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' These values ranked define  the rainfall CDF for point P, and we conclude by computing from this CDF percentiles 1 to 99 for precipitation at point  P (irrespective of how many different values contributed to its CDF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' We use the nearest value technique to compute  percentiles, so that where S=0, for example, the 95th - 99th percentiles all equal the ensemble maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Percentile  values are stored at the same horizontal resolution as COSMO raw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  7) The previous steps are repeated at all grid points (and then for all of the pre-defined time intervals in the  ensemble forecast).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  With regard to thresholds used at step (2) above, we followed closely the methodology of Dey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2016b) (their  Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='1-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='2), wherein thresholds have a weak dependence on agreement scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' As in that paper we actually found,  in some tests, that agreement scales exhibited little sensitivity to whether thresholds varied in this way or were kept  constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  In the context of steps (2) to (5) above, dry weather constitutes a special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Whilst one might imagine that all  members showing no rain at a gridpoint constituted perfect agreement, warranting an agreement scale S=0, in practice  in this scenario the algorithm decides that the threshold has not been met (again following Dey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2016b)), and  comparison extends to the 3 x 3 gridpoint squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The same logic applies at each iteration, such that S=5 is always  used if all members show dry across 11 x 11 gridpoint squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Our scale-selective neighbourhood post-processing is in effect used to compensate for there being insufficient  limited area model ensemble members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' It can retain rainfall signals at the finest scales when there is good agreement  Gascón et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  17  between members, as might commonly happen with orographically-forced rainfall, for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' In such instances  we are deeming the finer details in the member forecasts to be reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' At the same time the approach can spread  out information from nearby gridboxes when the member signals differ, and are thus deemed less reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Cases in  point would be a day of scattered showers, or intra-ensemble discrepancies in time and/or space in the arrival of a  front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' It is important to appreciate that the agreement scale assigned to each gridbox is based only on the similarity  of rainfall forecasts within the ensemble, and is assessed for each forecast period, so other factors such as orography  are not directly used here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Equally, we do not attempt any bias corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The post-processed COSMO percentiles  computed via this procedure, along with those generated by ecPoint, provide input to the subsequent blending activity  used to create the merged product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  18  Gascón et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  a) COSMO-2I-EPST+24h b)  e)  h)  i)  f)  c)  T+24h  T+24h  T+30h  T+30h  T+30h  T+36h  T+36h  T+36h  d) COSMO-2Ipp  g) Scale agreement  mm/6h  FIG U R E 4  Examples of the agreement scale and its impact on post-processing of the COSMO ensemble 6h  rainfall forecast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' All panels show forecasts from a nominal data time of 00UTC on 14th November, valid for (a,d,g)  18UTC 14th to 00UTC 15th (T+24h), (b,e,h) 00 to 06UTC 15th (T+30h) , and (c,f,i) 06 to 12UTC 15th (T+36h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Panels a,b,c show the raw forecast 95th percentile 6h rainfall in mm (=wettest COSMO solution at each point), d,e,f  show the post processed 95th percentile, and g,h,i show the diagnosed agreement scale used for post-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Agreement scale values are always integers between 0 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Although understanding of ’nominal data time’ is not  critical for interpretation of this figure the concept is explained in full at the start of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  10000  500  300  200  150  125  100  80  60  50  40  30  20  10  5  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='5Gascón et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  19  As an example of agreement scale usage Figure 4 shows, for Northern Italy, diagnosed agreement scale values  (g,h,i) along with the corresponding 6-h rainfall 95th percentiles for raw (a,b,c) and post-processed (d,e,f) COSMO,  for three consecutive 6h time windows over a night and morning in November 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The agreement scales were  diagnosed as described above, using all 20 members from one data time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  The integers (S) represented in Figure 4g, h and i are fundamental because they determine how ensemble data  is amalgamated during the COSMO post-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' A value of zero (dark green) denotes maximal agreement in the  scheme, and the post-processed forecast is constructed using only values for the said gridbox, across the ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  So wherever there is dark green on panels g, h or i, values on the panel immediately above mirror values on the panel  above that, implying that local details in ensemble output are being retained because they are consistent enough to  be believable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Where the agreement scale is 1 (or more) we use instead values from arrays of 3x3 (or more) adjacent  gridboxes to construct the CDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' So details become less and less believable as agreement scale goes up, and are thus  removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' In northeastern Italy for example, on panel i, where the agreement scale is mostly 2 (in the southeast corner  of Friuli-Veneza Giulia), the post-processed product (panel f) accordingly has a smoother, more spread out look with  smaller peaks than the raw output (panel c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' On the same panels even stronger "smoothing" is apparent in the Ligurian  Sea, where the agreement scale is 3 or 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' There are many papers on the concept of spreading out information, as we  are doing here, and such approaches also lie at the heart of the widely used “fractions skill score” verification metric  (Zacharov and Rezacova, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Mittermaier and Roberts, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Skok and Roberts, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  The apparently eastward-moving band of smaller agreement scales (greens) in this case actually relates to a front,  and one can also identify topography-related agreement scale minima - this is all discussed in more detail in Section  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='3 (second case, see also Figure 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' In short, poor agreement between members (large agreement scales) implies  that at the model gridscale we have insufficient ensemble members to capture the full range of possible outcomes  concomitant with the geographic setting and ongoing synoptic situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' By preferentially spreading information  out in such scenarios we should be able to achieve better forecasts for users, and better skill scores in verification,  without the large costs involved in running more members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Note however that neither approach can reduce the  "intrinsic" predictability of a situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  On Figure 4a the large dry patch, remote from the front, corresponds with where S=5 (Figure 4g), in accordance  20  Gascón et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  with the aforementioned strategy for dealing with dry outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Figure 4g then also demonstrates how this strategy  leads to a smooth, rather than abrupt, spatial transition in agreement scale close to dry areas, with S=4 being diagnosed  nearby.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Indeed right across the domain, at each of the lead times, S varies smoothly rather than abruptly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Having relatively smooth spatial variations in agreement scale also tempers spatial variability in grid point value  usage in the post-processed COSMO product, thus helping to approximately conserve rainfall "mass".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' In tests covering  different synoptic types the change in domain average precipitation after post-processing was very often between -5  and +10%, with no clear-cut lead time dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='4  |  Final product: blending together post-processed ensemble outputs  The "merged product" is the name we give to the final, blended probabilistic 6-h precipitation forecasts, delivered  for Italy and surrounding countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' These forecasts ostensibly go from 6 h to 48 h out (from a nominal 00UTC data  time) and have a time resolution of 3-h (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' overlapping 6h periods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The blending itself is applied to the ecPoint and  COSMO post-processed forecasts (in percentile form), assigning different weights for each of the two components  depending on the lead time (the sum of the two weights is always 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' More weight is given to COSMO at shorter  leads, starting with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='9 at nominal lead time 6 h, with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='05 of weight decrease every (3 h) time step thereafter until  42 h, after which the decrease is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='1 at 45 h and 48 h, resulting in an actual weight of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='1 at 48 h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The horizontal  resolution of the merged product is kept to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='2 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Our “tapered blending” strategy means that across all lead times  the forecast evolution should look relatively smooth to users, with more detail at the shortest leads when the COSMO  output is deemed more reliable, due to less synoptic scale spread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The intention was to create a very useful, seamless  forecast, with as accurate a representation of forecast uncertainty as the ensembles themselves and our simple "first  principles" optimisation approach would allow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Note that Hemri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2022) provides qualitative support for using  a tapered blending approach, and indeed in that study optimal LAM ensemble weights would reach zero around  60 h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Our application of steeper tapering to COSMO weights at long leads was a subjective compromise, to retain  seamlessness whilst keeping COSMO weights overall a bit higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Clearly our weighting strategy was not "optimized" using verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Whilst that would on the one hand be  Gascón et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  21  desirable, and is in principle achievable, there would inevitably still be many subjective components, such as metric  and threshold selection, and deciding whether or not to employ season-based or weather-type-based dependencies,  if data availability allows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' At least with our approach there is simplicity and transparency for the user regarding how  the merged product was derived;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' indeed linking the merged product output to the ensemble member inputs in a  fully explainable way is also relatively straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Another downside of verification-based optimization is that  seamlessness could be intermittently compromised if extra checks were not built in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  In practice the lead times for our operational MISTRAL products do extend all the way to 240 h, but after 48  hours, where the COSMO runs end, we just show ecPoint output as is (so for lead times beyond 48 h it is not strictly  ’blended’ output).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' In this paper output beyond the 48 h lead will not be discussed further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  The final operational products are in fact created twice a day, blending firstly 6-h ecPoint-Rainfall from the 12  UTC ECMWF ENS runs of the previous day (from lead time T+12 h onwards) with the 21 UTC COSMO raw outputs of  the previous day (from lead time T+3 h onwards), and then later, when the 00 UTC ECMWF ensemble data becomes  available for the new day, that is blended with the same 21 UTC COSMO data, to deliver an “update”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' However, in this  paper, we discuss and evaluate only the performance of the first blended output as it is the main operational product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Two different products are operationally produced each day: probabilities of exceeding specific 6-h total precipitation  thresholds and 6-h total precipitation percentiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The available thresholds in the MISTRAL platform are currently 5,  10, 20 and 50 mm for the probability (of exceedance) product and percentiles 1, 10, 25, 50, 75, 90, 95 and 99 for  the second product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Given the way data is stored it would however be simple to adjust these specifications if a user  requested that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  ecPoint was specifically designed to improve the reliability and discrimination ability of the forecast, particularly  for large totals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' By blending with the post-processed COSMO output, we aim to improve the forecasts even more,  especially in areas of greater topographic complexity, where higher horizontal resolution and, therefore, better defini-  tion of the topography is critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' In particular with this blended product we can, in certain situations, give increased  specificity regarding where the most extreme totals are most likely to occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  22  Gascón et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='5  |  The need for HPC infrastructure  Within the MISTRAL project the Flash Flood Use Case was partly a demonstration of the use of supercomputer  architecture to make real-time product delivery tractable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' in this instance using the Galileo system (and since June  2021 in Galileo100 system) in the CINECA Supercomputer Centre in Bologna (Italy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The use of HPC resources is  crucial for this use case because:  Considerable computational power and high-quality I/O performance are necessary for operational production  of the COSMO model forecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' To provide accurate predictions, COSMO has a high-resolution grid and needs to  run with short time steps, both of which elevate the need for computational power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The lower resolution ECMWF  ensemble is no less demanding, due to its global coverage, but is run remotely at ECMWF as part of normal operations  there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Ensemble forecasting requires multiple runs (20 in the case of COSMO) to be performed quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' This is ordinarily  achieved by running members in parallel, which needs a large platform with multiple nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  We need to transform the two ensemble forecasts sets into a much more accurate form, in real-time, using  rapid post-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' In the ecPoint post-processing, for example, we create a new virtual 100-member ensemble  for each of the 51 real ensemble members, for each lead time interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' This means we generate 5100 virtual ensemble  members for each of 63 time windows, which delivers 7 Tb of output data (for just the rainfall totals) at an intermediate  stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Computational demands are thus very high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  The fundamental requirement above is to deliver forecasts quickly enough for operational use, which means total  run times of a few hours at most.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Gascón et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  23  3  |  RESULTS  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='1  |  Predictable Scales  A key user-oriented facet of the merged product is the preservation of high resolution detail (from COSMO) whenever  and wherever it is deemed appropriate, via the agreement scales concept outlined in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' So here we illustrate  typical behaviour of agreement scales over a 1 year period, and highlight some implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Figure 5a shows that agreement scales tend to grow with lead time in a systematic way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' At a nominal lead time  of 6h about 50% of all non-dry gridpoints have an agreement scale of 3 or less (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' equating to gridlengths ≤15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='4km -  Figure 5a), or in other words 50% are retaining details at a finer resolution than the current (2022) ECMWF ENS native  resolution (18km).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' By 48h this has reduced to 33%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' For agreement scales of 1 or less (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='6km and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='2km gridlengths)  the corresponding values are ~30% and ~12%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' As the length scales across which COSMO ensemble information  is spread out start to surpass the resolution of the ECMWF ENS, ecPoint output (which delivers probabilities for  ostensibly similar measurement scales) potentially becomes a competitive and computationally expedient alternative  for rainfall prediction, provided its output is well-calibrated across domains with diverse topographies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' These results  also seem to lend some qualitative support to the tapered blending strategy we developed a priori, with the proviso  that verification data needs to also support this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  The diminishing inter-member agreement shown on Figure 5a is intuitively what one would expect, as uncertain-  ties in the synoptic pattern (for example) become ever greater with lead time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Interestingly Dey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2016b) found  no systematic changes in agreement scales with lead time (they examined up to 36h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' However we would ascribe that  to their use of a summer period, and precipitation rate snapshots as the target variable (rather than totals);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' presumably  convective cell positioning at short leads was already highly uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' In our case precipitation total stripes, arising  from cell movement, would be more likely to overlap at shorter leads, giving intrinsically smaller agreement scales  then.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Another interpretation of agreement scale growth is that it may reflect a need for more and more members, as  lead times extend, to usefully span all possible outcomes at fine scale, even when using a post-processing technique  24  Gascón et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  like this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' This is also what one would intuitively expect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Such a solution would of course have cost implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  FIG U R E 5  Agreement scale frequencies (in %) for post-processed COSMO during the 1 year verification period,  for all instances with 99th percentile > 0mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (a) Value distribution (see legend) versus nominal lead time, for  COSMO domain land areas;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' km values denote effective gridbox dimensions for each agreement scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (b) Frequency  of agreement scale = 0 or 1, over all nominal lead time periods from 0-6h up to 18-24h (day 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (c) as (b) but for  24-30h up to 42-48h (day 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Although the understanding of ’nominal lead time’ is not critical for interpretation of  this figure the concept is explained in full at the start of sub-section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  So where is most value likely to be added by post-processed COSMO?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Figures 5b and c clearly demonstrate that  over mountainous areas there is systematically better inter-member agreement, implying less capacity (and in principal  less need) to adjust there to ’add value’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' In other words there tends to be less lateral information spread through  COSMO post-processing in those regions,commensurate with what one would hope and expect from orographic  forcing arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' On day 1, agreement scales of 0 or 1 are diagnosed as often as 50% of the time over the most  mountainous parts of the Alps, whilst mountain chains such as the Apennines also exhibit higher frequencies than  flatter areas nearby.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Similar to this, plots in Dey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2016b) show systematically better inter-member agreement  over the more mountainous northern and western parts of the UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' In our case the frequencies diminish everywhere  on day 2 (Figure 5c), and whilst a map of day 2 : day 1 frequency ratios is rather noisy (not shown) there is evidence  that agreement levels hold up slightly better in mountainous areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' For example over plains north of Bologna values  drop from about 18% on day 1 to 8% on day 2, whilst in Alpine terrain in the northeastern corner of Trentino-Alto  Agreement Scale Frequencies  Frequency of Agreement Scale = O or 1 (dry events omitted)  during Verification Period  10°E  15°E  20°E  5°E  10°E  15°E  20°E  (land only;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' dry events omitted)  %  100%  90%  80%  70%  45  60%  50%  40%  30%  20%  10%  0%  End of Period (nominal lead time in hours)  DAY 2  DAY  a)  ■012345  10°E  15°E  10°E  15°EGascón et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  25  Adige they drop from 45% to just 30%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Even if our post-processing is much more ’active’ over flatter areas, particularly at longer leads, one should not  forget that the targetted retention of more detail over mountains is nonetheless likely to be a key contributor to  any general COSMO post-processing success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Indeed Blake et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2018) highlighted that the benefits of the vari-  able neighbourhood approach over the USA, in terms of discrimination and reliability metrics, and relative to a fixed  neighbourhood method, were most notable in "the West", which they attributed to that region’s greater topographic  complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  The one year skill of our post-processed COSMO output, and of the other MISTRAL forecast components, is  explored in the next sub-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='2  |  Verification  Verification of 1 year of retrospective 6-h rainfall forecasts was carried out, using as truth rain gauge observations from  both standard SYNOP reports and specialised high-density datasets from Italy and some surrounding countries (see  coverage example on Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Five probabilistic forecast "components" were assessed: the two raw ensembles (IFS  ENS raw, COSMO raw), the differently post-processed versions of these (ecPoint-Rainfall, post-processed COSMO)  and the merged version of these two post-processed products (merged product).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The verification region coincides  with COSMO raw coverage (Italy and surrounding areas, Figure 1), whilst the verification period runs from 1 February  2019 to 31 January 2020 (in January 2019, COSMO raw was not yet fully operational which meant some days were  missing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' All verification scores are computed for 6-h non-overlapped periods (for observation times 00, 06, 12 and  18 UTC), for (nominal) lead times T+6 h up to T+48 h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' We say here "nominal" because a "T+6 h" nominal lead time (for  example) means that for the two COSMO product sets the actual lead time is T+9 h (because the formal initialisation  time is 21UTC the day before) whilst for the two ENS product sets it is actually T+18 h (because we are referencing  the previous day’s 12UTC run, as indicated above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Naturally for the blended product it is a mix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' This approach to  nomenclature was taken to avoid over-complicating figures and text, but it is something that clearly needs to be borne  in mind when interpreting results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The same nomenclature was also used on Figures 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' For the observation  26  Gascón et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  periods ending 03, 09, 15 and 21UTC a much reduced observational coverage prevented meaningful verification from  being performed, even though (overlapping) products were created for those times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  The reliability component of the Brier Score (BSrel: Brier (1950) and Wilks (2011)) was used here to evaluate the  reliability of the forecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' In our case the Brier Score has its binary event form, and represents the mean-squared er-  ror of the probability forecasts, over the verification sample, for a specific threshold of 6-h accumulated precipitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Meanwhile, to evaluate the capacity to discriminate events, we used Area under the Relative Operating Character-  istic curve (AROC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Richardson (2003)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Three different precipitation thresholds are initially considered: ≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='2, ≥10,  and ≥30mm/6h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The latter value is quite extreme for most of Italy, and may signify a flash flood risk (depending on  other factors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' There were about 10000 reports of >30mm/6h during the year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' As the COSMO raw domain is quite  small, the sample size for thresholds larger than 30mm was too limited to provide robust verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Following the  same verification procedures as Hewson and Pillosu (2021) and according to Hamill and Juras (2006) recommenda-  tions, we added AROC scores for climatological probabilities, based on all available 18-year observation series, to  show a “zero-skill” baseline (having first discretised to probability intervals of 1% to mirror ecPoint, post-processed  COSMO and merged product discretisation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Conditional verification scores were also computed for different sea-  sons, and also for different SDfor ranges (to divide up according to underlying topographic complexity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' For these  classes we used an upper precipitation threshold of 20 mm/6h as case counts naturally diminish following subsetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Confidence intervals associated with BSrel and AROC are obtained using 1000-block bootstrap samples, using the  stationary bootstrap scheme with mean block length according to Politis and Romano (1994) and following the same  approach as Hewson and Pillosu (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' In our verification plots BSrel uses the raw ECMWF ENS as a reference point  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' we subtract component performance from that of ENS raw, replicating the reference-run plotting approach of  Baran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2019) and others), so bootstrapping is calculated from these BSrel differences instead of with absolute  values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' We also show the actual reliability of the raw ENS for reference (as dashed lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Including this enables one  to also see the degree by which the other systems improve (or degrade): evidently if a forecast system had perfect  reliability its solid line would overlay the dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Gascón et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  27  FIG U R E 6  Forecast verification scores for 6-h rainfall over a 1-year period, using gauge observations, for an  extended Italian domain, for nominal lead time intervals of T+0-6h to T+42-48h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' x-axis values denote interval end  point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Lines denote ENS raw (orange), ecPoint (red), raw COSMO (light blue), post-processed COSMO (navy),  merged product (yellow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Left column (a,c,e) displays BSrel metrics, with dashed orange line showing absolute values  of BSrel for ENS raw, and solid lines differences relative to this for the other forecast systems (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' ENS raw minus  other system).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' So difference>0 means the other system is more reliable, with that system rated fully reliable  wherever its solid line reaches the "cap" provided by the dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Right column (b,d,f) shows AROC as a measure  of discrimination ability (with a climatology “baseline” shown dashed black);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' larger is better, upper limit is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Each  row is for a different precipitation threshold: (a,b) ≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='2 , (c,d) ≥10, (e,f) ≥30mm/6h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Error bars on all panels show  95% confidence intervals from bootstrapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Figure 6 shows BSrel differences versus ENS raw (left column), and AROC (right column) for different 6-h pre-  cipitation thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' First note how for ≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='2 mm there is a clear-cut diurnal cycle in ENS raw reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' All other  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='06  ENS raw ref  ENSraw  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='05  ecPoint6h  Srely  coSMOraw  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='9  CosMOpost  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='04  mergeproduct  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='8  AROC  ENSraw  ecPoint6h  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content="7  CoOsMOraw  @'0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='02  COSMOpost  (BSr  merge product  climatology  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='01  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='5  1  6  12  18  24  30  36  42  48  6  12  18  24  30  36  42  48  Lead Times [Hours]  Lead Times [Hours]  ×10-3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='5  (BSrel)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='5  AROC  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='5  12  18  24  30  36  42  48  6  12  18  24  30  36  42  48  Lead Times [Hours]  Lead Times [Hours]  ×10-4  (BSrel_IFSraw) - (BSrel)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='9  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='8  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='7  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='6  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='5  6  12  18  24  30  36  42  48  6  12  18  24  30  36  42  48  Lead Times [Hours]  Lead Times [Hours]28  Gascón et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  forecast systems show much improved reliability at all lead times for this threshold, with any residual diurnal cycles  in those systems being, in relative terms, negligible (Figure 6a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' There is also a diurnal cycle in AROC for this 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='2mm  threshold (Figure 6b), but this is seemingly less pronounced and is there in all components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' In relative terms, ENS  raw is least reliable by day (06-18UTC), perhaps because showery activity, which probably peaks then, tends to lead,  in a given ENS gridbox, to some sites recording rain when others nearby do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' ecPoint and the higher resolution  ensemble can cope with this, the raw ENS cannot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Diurnal cycles in reliability for the raw ENS are not so evident for the higher thresholds ≥10 and ≥30 mm (Figure  6c, e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' These spurious cycles are again addressed by ecPoint-Rainfall and by the merged product, although as the  threshold increases the improvement imparted by the post-processing becomes less emphatic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Meanwhile COSMO  products have their own diurnal-cycle reliability errors for the higher thresholds, which are even worse than we see  in raw ENS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' This could be in part because we do not apply any bias correction to raw COSMO or post-processed  COSMO products, which seemingly have particular reliability issues in representing heavier evening and night-time  rainfall (18-06UTC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' One can say that ecPoint-Rainfall (red line) just about has the best overall performance in terms  of reliability and discrimination for the 10mm and 30mm thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The merged product is the best product for  one or both metrics some of the time, notably when it can exploit better performance of post-processed COSMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  However, for very short lead times (06 h and 12 h), the merged product can be much worse than ecPoint-Rainfall  for both metrics, so the high weight then afforded to COSMO raw in the merging seems inappropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' By lowering  this weight the merged product would have undoubtedly performed better, and this should probably be adopted in a  future version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  The curious short-lead behaviour in COSMO output on Figures 6c,e was briefly investigated by computing, for  all leads, annual domain-average over-land 6h rainfall totals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' These totals exhibited a decaying downward trend with  lead time (superimposed on a diurnal cycle), with 10% more rain for nominal lead times 00-06UTC on day 1 (T+0-  6h) than on day 2 (T+24-30h), reducing to 3% more for 18-24UTC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' This suggests that COSMO forecasts may have  had excess rainfall at short leads, degrading reliability then, although in proposing this explanation we were not able  to directly compare with observed equivalents, due to inhomogeneous data coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Another curious feature of  all the reliability plots is that ENS raw reliability (dashed lines) also tends to improve with lead time (discounting  Gascón et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  29  diurnal trends).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' This is qualitatively similar to global verification for 12h rainfall in Hewson and Pillosu (2021), in which  reliability improves up to day 5 then plateaus, and may signify either a "slow spin up" issue and/or insufficient spread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' In  turn such trends may relate to ECMWF’s intention to optimise ENS performance, in terms of spread-skill relationships  and other metrics, in the medium range, via tailored singular vectors (Palmer, 2019), although conversely note that  the stochastic perturbations also used to create spread are not themselves targetting medium range performance  (Leutbecher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  COSMO post (dark blue line) shows a better resolution and reliability than raw COSMO (light blue line) in all  instances (excepting 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='2mm reliability which looks identical).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' An implication of COSMO agreement scales increasing  with lead time (Figure 5a) is that more value can in principal be added by post-processing later on, such that the rate  of skill degradation with lead time is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' We see signs of this happening on Figure 6 and indeed on most other  verification plots presented in this section (on Figures 8 and 10), most notably for AROC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Specifically the gap between  light blue and dark blue curves tends to increase with lead time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  A final point to note here is that, for all thresholds and all lead times, ecPoint-Rainfall has better discrimination  ability than both the corresponding ENS raw and the post-processed COSMO forecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  30  Gascón et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  FIG U R E 7  Forecast verification scores for 6-h rainfall for the nominal 18-24h lead = 18-24UTC, over a 1-year  period, using gauge observations for an extended Italian domain as on Fig 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Line colours also as on Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Rows  signify precipitation thresholds: ≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='2 (a,b,c), ≥10 (d,e,f) and ≥30mm/6h (g,h,i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Left column (a,d,g) shows ROC curves  (with AROC denoted by ’AUC=’ in boxes), middle column (b,e,h) reliability diagrams and right column (c,f,i) sharpness  (logarithmic scale on y-axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' For ROC curves and sharpness, probability discretisation reflects the finest available for  each forecast system: 1% for ecPoint-Rainfall, COSMO post, and merged product;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' ~2% for ENS raw and 5% for  COSMO raw (and 1% for climatology on ROC curves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Missing points in the profiles on (b,c,e,f,h,i) denote empty  classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Figure 7 presents ROC (Relative Operating Characteristic) curves (Figure 7a, d and g), reliability diagrams (Figure  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
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+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
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+page_content='9  False Alarm Rate  Forecast probability  Forecast probability  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='0  1e+06  ENS raw  ENS raw  ecPoint6h  ecPoint6h  1e+05  COSMO raw  Observed relative frequency  COSMO raw  COSMO post  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
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+page_content='8  COSMO post  Probability of Detection  merge product  merge product  10000  Frequency (log)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
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+page_content='6  1000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='4  100  ENS raw  AUC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='734  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='2  ecPoint6h  AUC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='896  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='2  COSMO raw  AUC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='825  0  COSMO post  AUC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='852  3  merge product AUC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='898  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='0  climatology  AUC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='51  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
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+page_content='0  0  00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='9  False Alarm Rate  Forecast probability  Forecast probabilityGascón et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  31  7b, e and h), and sharpness diagrams (Figure 7c, f and i) for the five forecast products at 24 h lead time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' As on Figure 6  for T+24h the merged product ROC curves exhibit better discrimination ability than any of the other 4 components,  for all precipitation thresholds, with forecasts based on climatology performing very poorly, in relative terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' For  discrimination for the higher precipitation thresholds (Figure 7 d, g), and relative to ENS raw, we also observe clear  advantages for both of the COSMO systems, and in the case of COSMO raw this is in spite of its coarser discretisaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  We attribute this to the higher spatial resolution of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='2 km in both COSMO products, compared to 18 km in ENS  raw: probably many cases involve convection or orographic enhancement, both of which can deliver local peaks that  are not reflected in the output of current generation global models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' At the same time there is the caveat that due to  the apparent closeness of the leftmost segments of the ROC curves on Figure 7 d, g it is hard to anticipate, a priori,  what any of the systems would deliver in terms of ROC area if discretisation were increased (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' by running more  members).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Whilst the characteristic of ecPoint products having better reliability and discrimination ability than raw ENS  for large totals, previously noted by Hewson and Pillosu (2021), is supported here, we can also note a reduction in  the range of probabilities delivered for such events (Figures 7 f and i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' No instances have been seen of probabilities  >88% and >62% for the 10 and 30mm/6h thresholds respectively, for the 18-24h lead time depicted in the 1 year  of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' This is not necessarily "wrong" however;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' note that high probabilities delivered in the two COSMO systems,  in ENS raw, and also (to a lesser extent) in the merged product, are all over-confident - on Figures 7b, e and h the  respective lines all fall below the diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' And high probability case counts are also quite large (Figures 7c, f and i)  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' ~2000 for post-processed COSMO for >60% probability of >30mm/6h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Whilst the larger ecPoint probabilities  of >30mm/6h are, conversely, under-confident (line above diagonal), the case count here is relatively low, only ~100  for probabilities ≥50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Sharpness diagrams plotted in Figure 7 use the maximum probability discretisation that each  forecast system can provide, which varies (see caption).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Whilst it might look as though there are discrepancies in  frequency of occurrence (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' COSMO raw seems to exhibit "more of everything") this is just due to the varying  discretizations, and also because of relatively large absolute variations in the frequency of zero which on the log scale  are hard to discern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  32  Gascón et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  FIG U R E 8  Forecast verification scores, divided by season, for a 20mm/6h threshold, using gauge observations  for an extended Italian domain over 1 year as on Fig 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Line colours and styles, and x-axis meaning also as on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Left column (a,c,e,g) displays BSrel metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Right column (b,d,f,h) shows AROC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Seasons are DJF (a,b), MAM (c,d), JJA  (e,f), SON (g,h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' All plots include 95% confidence bars from bootstrapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Figure 8 shows a seasonal breakdown of the skill metrics shown on Figure 6, for very wet episodes (here ≥20  mm rather than ≥30 mm/6h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The majority of observations of >20mm fell in autumn (~20000 cases), followed by  spring with ~7000 cases, whilst winter and summer saw ~5000 cases each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' In summer (JJA) there were two times  ×10-4  ENS raw ref  ENS raw  ecPoint6h  COSMO raw  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='9  CoSMOpost  mergeproduct  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='7  ENS raw  ecPoint6h  COSMO raw  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='6  COSMOpost  merge product  climatology  4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='5  6  12  18  24  30  36  42  48  6  18  24  30  36  42  48  Lead Times [Hours]  Lead Times [Hours]  ×10°4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='8  AROC  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='7  6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='6  8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='5  6  12  18  24  30  36  42  48  6  12  18  24  30  36  42  48  Lead Times [Hours]  Lead Times [Hours]  ×104  4  (BSrel_IFSraw) - (BSrel)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='9  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='8  AROC  4  6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='7  8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='6  10  12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='5  6  12  18  24  30  36  42  48  6  12  18  24  30  36  42  48  Lead Times [Hours]  Lead Times [Hours]  ×10-3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='5  (BSrel)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='9  IFSraw)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='8  AROC  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='7  (BSrel  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='5  12  18  24  30  36  42  48  6  12  18  24  30  36  42  48  Lead Times [Hours]  Lead Times [Hours]Gascón et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  33  as many cases in the evening (18-24 UTC) as in any of the other 6-h periods (not shown), which probably reflects a  diurnally-driven convective activity peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  BSrel plots show some differences between seasons (Figures 8a, c, e and g), but with the highest reliability overall  delivered by the merged product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' ecPoint output is not far behind, and for T+0-6h = 00-06UTC is the clear winner  in three seasons probably because COSMO spin down issues along with the post-processed COSMO’s high weight  allocation at short leads are contriving to degrade the merged product, as discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Whilst the raw COSMO  outputs have relatively strong reliability in winter (DJF) post-processing is, it seems, slightly degrading this: this curious  behaviour warrants further investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' One hypothesis is that the true picture is not emerging because of some  erroneous observations in Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' A high percentage of precipitation in winter is in snow form in the Alps and the  Apennines and there may conceivably be problems with the gauge-based heaters that should melt the snow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' This  could lead to considerable underestimation of precipitation, and after the event sometimes overestimation following  in-funnel melting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' These aspects may be erroneously nullifying some of the post-processing benefits, although it is  also true that skill metrics for the raw ensembles would also be impacted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' For ecPoint, a positive design feature of  its calibration process is resistance to contamination from occasional erroneous extreme values (Hewson and Pillosu,  2021), and in addition its calibration domain is global and not just Italian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Together these aspects elevate the integrity  of the ecPoint post-processing for users, but equally limit the scope for error cancellation in the ecPoint verification  over Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Thus all five products we are discussing are likely to be adversely affected by any errors in the verifying  data, although it is difficult to pin down how much in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  In summer it is striking how COSMO reliability for wet events is poor in the afternoon and evening (12-24UTC), on  both day 1 and day 2 of the forecast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' From Figure 7e,f (for just 18-24UTC) it seems this may be due to over-confidence  throughout the probability range, and equivalently to a bias to predict too many wet events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' This assertion is supported  by reliability diagrams for a time of day when Brier Score reliability is better (06-12UTC = T+12 and T+36, not shown)  on which the COSMO lines are closer to the diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' So it seems that for COSMO the handling of diurnally driven  convection exhibits some pronounced time-of-day related reliability issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  In terms of discrimination ability for these very wet events, quite similar performance is observed in all the seasons  (Figure 8b, d, f and h): ecPoint and the merged product are consistently the best, followed by post-processed COSMO,  34  Gascón et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  then raw COSMO, and with raw ENS exhibiting the lowest discrimination ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' It is also noticeable that the summer  season displays a sharp AROC diurnal cycle, particularly for the raw ENS and ecPoint (Figure 8f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The ecPoint minima  are for 18-24UTC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' This is in spite of the ecpoint decision tree having incorporated local solar time dependence to  try to mitigate known convective diurnal cycle errors in raw ENS (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Figure 3b), although maybe the picture would  have been even worse without such adjustments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Note however that we do not post-process the precipitation field  when the model predicts no precipitation, and here this could be a key factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' One could incorporate a new type  of dependance by, for example, creating a new ecPoint decision tree branch from the top level, for LST=18-24, that  does not multiply forecast precipitation but instead uses additive mathematics based on other relevant model-derived  variables, such as CAPE or lightning density, either in the said period or in the preceding 6-hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Or alternatively one  could even use forecast rainfall in the preceding 6 hours in a creative way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  The ecPoint ROC curves, for >20mm/6h, for different times of day for the JJA summer period (not shown) actually  exhibit structural differences: relative to the 00-06 and 06-12UTC periods, there is a shift to the right for 18-24UTC,  and even more so for 12-18UTC, highlighting a lower innate ecPoint discrimination ability for those later periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' This  is broadly consistent with the ROC areas, except that the minimum ecPoint ROC area is actually for 18-24UTC (Figure  8f) and not for 12-18UTC, because for 12-18UTC the points ascend to reach much higher hit rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' In turn this is  because 12-18UTC has more usable rainfall signatures in the raw ENS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The higher number of observed cases may also  be a contributory factor (in summer the 12-18UTC observation case count for >20mm/6h is ∼2200, versus ∼1100  for each of 00-06, 06-12 and 18-24UTC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  During summer evening hours (18-24 UTC), post-processed COSMO is ceutbechrearly better than ecPoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The  much higher horizontal resolution in COSMO, and particularly a related ability to prolong daytime convection into  the evening, which raw ENS lacks, seem to play a critical role here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' This clearly benefits the merged product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' In the  32 season-lead-time combinations shown the merged product exhibits its biggest gain over ecPoint for JJA T+18-  24 (=18-24UTC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The gain would probably also be higher for T+42-48, were it not for the fact that post-processed  COSMO weight is then very low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Moreover, the merged product’s best season of all, considering all lead times, is  clearly summer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Probably, if COSMO reliability were intrinsically better (Figure 8e), or could be made better with  some bias correction procedure, the post-processed COSMO performance in summer would be even more striking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Gascón et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  35  FIG U R E 9  Observations available for each SDfor range: (a) <50 m ("plains/low hills"), (b) from 50 to 100 m ("very  hilly"), and (c) >100 m ("mountainous")  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  E19*E  3s61483(3  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='9  4  I2*N  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='08  8\'E  9"E  10*E  6*E  9°E  12*E  13*E  14°E  15°E18°E  17*E  18°E19*E  6*E  1D*E  15°E  17*E  1B*E19*E36  Gascón et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  FIG U R E 1 0  Forecast verification scores, for categories of topographic complexity, for a 20mm/6h threshold,  using gauge observations for an extended Italian domain over 1 year as on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Line colours and styles and x-axis  meaning also as on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Left column (a,c,e) displays BSrel metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Right column (b,d,f) shows AROC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Topographic  complexity is represented by the sub-grid orography variable SDfor, with (a,b) for "plains/low hills" (SDfor<50m),  (c,d) for "very hilly" (50-100m) and (e,f) for "mountainous" (>100m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' All plots include 95% confidence bars from  bootstrapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  The final verification plots (Figure 10) show performance of the five ensemble products for different SDfor ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  As described above, SDfor is an ENS grid variable which denotes sub-grid topographic complexity, based on a 1 km  resolution topographic map and using the standard deviation measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Higher values naturally correspond to more  complex terrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Figure 9 shows the location of observations in three SDfor categories, which we can call "plains/low  ×104  (BSrel_IFSraw) - (BSrel)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='9  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='8  AROC  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='7  ENS raw ref  ENSraw  ENS raw  ecPoint6h  ecPoint6h  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='6  COSMOraw  COSMO raw  COSMOpost  2  COSMO post  merge product  merge product  climatology  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='5  6  12  18  24  30  36  42  48  6  12  18  24  30  36  42  48  Lead Times [Hours]  Lead Times [Hours]  ×10°4  5  (BSrel_IFSraw) - (BSrel)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='9  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='8  AROC  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='6  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='5  6  12  18  24  30  36  42  48  6  12  18  24  30  36  42  48  Lead Times [Hours]  Lead Times [Hours]  ×10-4  N  (BSrel_IFSraw) - (BSrel)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='9  0  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='8  AROC  4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='7  6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='6  8  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='5  6  12  18  24  30  36  42  48  6  12  18  24  30  36  42  48  Lead Times [Hours]  Lead Times [Hours]Gascón et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  37  hills" (a), "very hilly" (b), and "mountainous" (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Category selection was not arbitrary but corresponds to breakpoints  found whilst applying the decision tree calibration methodology (albeit with some rounding and set size normaliza-  tion also applied).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' As such the breakpoints separate significantly different FER mapping function distributions, for 6h  rainfall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' In turn this means that we also expect some differences in independent verification for each SDfor range,  notably for ecPoint, but also for the merged product which uses ecPoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Indeed we saw on Figures 5b and c that  COSMO post-processing agreement scales also depend on altitude / topographic characteristics, implying reasons to  also expect SDfor-related differences in value added by COSMO post-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' For the different classes the ap-  proximate gauge numbers in our domain are 700 (plains/low hills), 900 (very hilly) and 1800 (mountainous).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Although  not homogeneous these numbers are all still large enough to give a robust annual verification picture, for thresholds  up to 20 mm/6h (refer to error bar magnitude on Figure 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Figures 10a, c and e for reliability show that ecPoint and the merged product generally perform best, implying  some robustness across multiple topographic types, with the merged product best of all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' In this regard the capacity of  ecPoint to correct the absolute reliability errors of ENS raw (dashed) systematically reduces as topographic complexity  increases, from >~60% improvement for plains/low hills, to <~20% for mountainous, on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Strong performance  of the ecPoint and merged product components holds also for discrimination ability (Figures 10b, d and f) although  here they are on a par.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' As in verification by season (Figure 8), there are hints of diurnal cycles in the skill metrics,  although some of these are hard to interpret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The most clearcut feature in this regard seems to be the AROC minimum  for ecPoint output seen for 12 to 24UTC (and by implication a maximum for 00-12UTC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' This mirrors what we saw in  summer (Figure 8f), but is watered down by inclusion of the other seasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Notably the signal is there for all SDfor  classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  In mountainous areas meteorological models commonly have difficulty representing precipitation, and this asser-  tion is borne out by the AROC values on Figure 10b,d,f, at least for ENS raw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Whilst the value added by post-processing  the COSMO ensemble seems fairly consistent across terrain categories (perhaps contrary to what one might have ex-  pected from Figures 5b and c), ecPoint seems, encouragingly, to add most value (to raw ENS) in mountainous regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  It probably reflects the merits of using sub-grid orography as a governing variable in the calibration for mainly large-  scale precipitation (Figure 2c,d and Figure 3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' It may also be because there is more discrimination ability to be added  38  Gascón et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  in mountainous areas, and because ecPoint post-processing is targetting this effectively via its decision tree subdivi-  sions (even if those have been less effective in improving reliability in mountainous areas).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' One striking result for the  mountainous areas is that the two COSMO products are overall much less reliable than the raw ENS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' indeed they  exhibit BSrel values that are more than twice as large at some leads (Figure 10e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' One might have thought that the  COSMO model would have excelled here, especially give that steep-sided mountains and narrow mountain ranges,  well below ENS resolution, are a common feature of the Italian landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' But maybe there are strong biases, impact-  ing reliability, that relate to intrinsic difficulties that even convection-resolving models have in complex terrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The  evening-time drop in reliability of COSMO in mountainous areas is particularly striking, mirroring what we see for  summer performance on Figure 8e, and invites further investigation with summer time case studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Could it be that  summer-time evening convection over mountainous regions is a key weak point for COSMO?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='3  |  Case studies  We discuss here two case studies of extreme rainfall in Italy, that related locally to flash floods;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' one was in July,  the other in November.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The aim is to complement the one-year verification figures, and to also illustrate the spatial  characteristics of outputs of our five different forecasting "systems".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' It was not a given that the products themselves  would look physically reasonable and devoid of unwanted artefacts, even if verification is positive, and "believability"  of the merged product is of course a key requirement for users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Reassuringly, in our case studies (and others not  shown) this aspiration is clearly met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' So here we comment on how each forecast component verifies for each case,  and to the extent possible link our remarks to verification scores in the previous sub-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Gascón et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  39  FIG U R E 1 1  95th percentiles of 6h rainfall forecasts, for nominal lead time T+24-30 h = 00-06UTC 28th July  2019, from nominal data time 00UTC 27th July, from (a) raw COSMO, (b) post-processed COSMO, (c) raw ENS, (d)  ecPoint, (e) merged product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Gauge observations of verifying 6h rainfall are shown in (f);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' inset shows part of a UK  Met Office synoptic analysis chart for 00UTC 28th July, with 4hPa isobar interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='5  5  10  20  30  40  50  2  60  80  100  125  150 200 300 50040  Gascón et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  On 28 July 2019, a shallow cyclone centred in the Ligurian Sea (see inset on Figure 11f) delivered some extreme  convective rainfall to Italy: in central Italy >40 mm/6h was widely reported with 80-100 mm/6h locally (Figure 11f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Flash flood impacts were reported and some rainfall records were broken: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='meteoregionelazio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='it/2019/07/29/il-  forte-maltempo-di-ieri-28-luglio-2019-datifoto-e-video-di-una-giornata-da-ricordare/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  The 95th percentile forecast from ecPoint (Figure 11d) seems to have performed better in capturing some of the  more extreme values than ENS raw (Figure 11c), such as in Liguria and the coastal regions of Tuscany, where ecPoint  values are higher, implying some discrimination ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Whilst some of the extremes appear to not be captured - the  highest range shown over Italy is 50-60mm/6h - 99th percentile ecPoint forecast values (not shown) do reach 80-  100mm/6h in the wettest parts of the country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' In reliable forecasts the 99th percentile should naturally be exceeded  1% of the time (once in 100 observations), and the 95th percentile 5% of the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' In fact, the proportion of observa-  tions that exceed the 95th percentiles for raw ENS and ecPoint are 23% and 15% respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' for the 99th percentile  the corresponding values are 16% and 3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  The differences between the 95th percentiles for ecPoint (d) and raw COSMO (a) are very striking, even though  they are considered to represent similar scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Whilst we cannot be definitive from one case the raw COSMO rainfall  values do look a bit excessive (values of 100-200mm cover quite large areas) compared to the range of values reported  (all <100mm), whilst ecPoint looks more reasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Post-processing has a marked impact in reducing the majority of  the raw COSMO peaks, because of large agreement scales (not shown), although totals on the resulting plot (b) still  look too high compared to observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' In this case, the proportion of observations exceeding the 95th percentile for  raw COSMO, post-processed COSMO and the merged product are 5%, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='7% and 6% respectively, whilst for the 99th  percentile, the corresponding values are 5% (same value because of ensemble size), 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='2% and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' So the merged  product here seems to be the most useful/reliable overall;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' it achieves a balance between under-prediction of extremes  in ecPoint, and over-prediction in post-processed COSMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Meanwhile the over proliferation of extreme values at the  99th percentile level in post-processed COSMO may well be hindering it’s discrimination ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Whilst we cannot do a like-for-like comparison between the stated characteristics for this case and verification  scores, because sample size would be too small, there are some useful and meaningful pointers pertaining to the  merged product benefits in the data we do have: Figure 10e suggests that COSMO is particularly unreliable for large  Gascón et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  41  totals in mountainous areas at night, compared to raw ENS, whilst Figure 7h, together with Figure 10e, suggest this  is because of over-prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Meanwhile ecPoint generally has much better reliability and discrimination in these  circumstances (Figures 10e,f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' So the merged product seems to be exploiting, but reducing, the topography-related  enhancement delivered by COSMO, whilst benefiting all round from the higher quality but lower-resolution picture  delivered by ecPoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Success for the merged product also depends on appropriate weighting of ecPoint and post-processed COSMO;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  here it is 50-50, because of the 30h lead-time and our simple "tapered weighting" approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The weighting versus  lead-time profile is something that could in principal be optimised in some future version, using iterative long-period  verification, although that would be computationally challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Over time we have examined outputs of the five different components for many cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The characteristics we are  seeing here are quite typical: for example, post-processed percentiles from the two ensembles tend to resemble their  respective raw versions more than they resemble each other, whilst visually the merged product commonly looks like  a good compromise, relative to both the input fields and to subsequent observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  42  Gascón et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  FIG U R E 1 2  Forecast probabilities for 6-h precipitation ≥50 mm, for nominal lead time T+30-36 h = 06-12UTC  15th November 2019 (= rightmost column on Figure 4), from nominal data time 00UTC 14th November, from (a) raw  COSMO, (b) post-processed COSMO, (c) raw ENS, (d) ecPoint, (e) merged product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Corresponding observations are  shown in (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Inset on (c) shows part of a UK Met Office synoptic analysis chart for 06UTC 15th November, with  4hPa isobar interval (raw ENS probabilities beneath are all zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Top legend applies to (a)-(e), bottom legend to (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  100  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='5  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='5  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='5  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='5  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='5  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='5  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='5  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='5  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='5  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='5  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='5  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='5  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='5  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='5  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='5  500  300  200  150  125  100  90  60  50  40  30  20  10Gascón et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  43  The second case (Figure 12) is for 15 November 2019 and corresponds to a very heavy rain event over some north-  ern and central parts of Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' It is the same case as shown in Figures 4 c, f and i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The event was part of a several-day pe-  riod during which flash floods occurred quite widely: https://messaggeroveneto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='gelocal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='it/udine/cronaca/2019/11/16/news/nuova-  ondata-di-maltempo-in-fvg-allerta-rossa-nel-pordenonese-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='37911793.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The synoptic pattern over Italy on 15th was  cyclonic (inset on Figure 12c), with large-scale frontal rainfall seemingly dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' However inclusion of a warm sec-  tor trough suggests a convective component, and the relatively strong warm sector flow also suggests orographic  enhancement potential, notably over the Dolomites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' In some mountainous areas, for example in Trentino-Alto Adige  and northern Lombardy, the existence of large totals alongside negligible totals (Figure 12f) may not be genuine vari-  ability, but may instead by symptomatic of higher-altitude snow not being melted in unheated gauges (the freezing  level was of order 1500-2000m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Panels (a)-(e) on Figure 12 show probabilities for precipitation greater than 50mm/6h from the various MISTRAL  ensemble components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' On the two COSMO plots (a and b) one clearly sees the modelled impact on rainfall of topo-  graphic enhancement, especially in northern Veneto, and Friuli-Venezia Giulia, but also on the southwestern flanks of  the Apennine chain where south to southwesterly synoptic-scale flow impinges, in for example Umbria and northeast-  ern Lazio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The high spatial variability in raw probabilities seen in all these areas seems to relate to topographic forcing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  For fine-scale signals like these that have intra-ensemble consistency - relating to (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=') upslope versus downslope  rainfall - we want to preserve rather than "smooth out" the rainfall forecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The post-processing algorithm aims to  facilitate this using the agreement scale metric (Figure 4g, h and i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  In fact, comparing Figures 4h and i with the synoptic chart (for the midpoint time of 06UTC) on Figure 12c, one can  identify an approximately front-parallel agreement scale "trough" (with values of 0 or 1 at its centre) moving eastwards  with the front, which in this case is actually the primary driver of ensemble consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' However there are also lateral  topography-related minima superimposed on this band - e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' in the Apennines along the northern Border of Tuscany,  and more generally in the Alps (features which are not inconsistent with the annual average geographical pattern on  Figure 5c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Thus the COSMO post-processing picture emerging for this case, as represented on Figures 4 c and f,  and Figures 12 a and b, is quite complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' It involves minimal spreading/smoothing being applied in the zone where  the ensemble is confidently indicating that frontal rainfall will prevail in the given time window (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' over central and  44  Gascón et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  eastern Tuscany), with fine detail retention being further enhanced in upslope/mountainous regions in and beyond  that zone (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' near the Austrian border of Friuli-Veneza Giulia).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Meanwhile the larger agreement scales (3 or 4) diagnosed in mountainous eastern Umbria and northeastern  Lazio exist because frontal timing differences within the ensemble are the dominant factor here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The post-processed  COSMO probabilities (Figure 12b) are as expected smoothed out more in these particular regions (in one case with a  peak reduced from about 15% to 7%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Given the negligible amounts of rain reported (Figure 12f), presumably due to  slower frontal progression, such probability reductions can be considered an improvement in the precipitation forecast  for this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Large precipitation totals (>50mm/6h) were not really directly predicted by any of the ECMWF ensemble mem-  bers (Figure 12c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Conversely, the ecPoint product (d) had finite probabilities that delineate quite well the areas af-  fected by totals >50mm/6h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The highest probabilities are in Friulia-Venezia Giulia where the observed local frequency  of such totals looks greatest, even if the 11-13% probability forecast (yellow) is arguably not high enough (Figure 12f)  from a user perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Meanwhile post-processed COSMO (Figure 12b) generally had higher probabilities where  observations were largest, although there were a number of "missed events" - most notably in northwestern Veneto  where the largest total of all was reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Thus our semi-subjective assessment concurs with the July case shown  earlier: by blending together the two post-processed products we have created a merged product (Figure 12e) that  highlights their combined strengths, and that therefore looks more useful than either did in isolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  The above results can also be cross-referenced with the most appropriate long-period verification data that we  have (in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Overall that data shows that for high totals, in autumn, for very hilly or mountainous areas, and  at a nominal lead time of T+30-36, ecPoint, post-processed COSMO and the merged product are on average equally  reliable (Figures 6e, 8g, 10c,e), whilst for discrimination ability the merged product and ecPoint are generally best,  with post-processed COSMO significantly less good (Figures 6f, 8h, 10d,f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' So the conclusions from our November  case look to be fairly typical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  This example also demonstrates a disadvantage of the COSMO post-processing: when (frontal) timing issues  contaminate the picture we can end up smoothing out reliable fine-scale details (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' upslope and downslope rainfall),  that are a characteristic of the wet solutions, for the wrong reasons, such that we end up mirroring a disadvantage - i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Gascón et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  45  a lack of geographical specificity - of the lower resolution ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Ultimately this is because as lead times increase  the smaller native ensemble size becomes more and more of a limiting factor, calling for increasingly complex post-  processing to be employed for the retention of reliable output on fine scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' In our blending system the strategy to  counter this, in an imperfect but moderately effective way, is to reduce COSMO weight at longer leads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Perhaps there  are lead-time limits beyond which no post-processing method for a finite size convection-resolving ensemble, however  sophisticated, could counteract the detrimental impact, on fine-scale rainfall details, of the growth of synoptic scale  spread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' In which case the optimal low-cost solution, as the length-scale to which S corresponds approaches the  ecPoint output grid resolution, could be to just use ecPoint output instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' In the current study and others that relate  (Hemri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Harris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Cristina Primo-Ramos - personal communication), ecPoint becomes very  competitive for lead times greater than 24-48 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  4  |  DISCUSSION AND CONCLUSIONS  This paper describes how two innovative, state-of-the-art post-processing approaches have been applied to the out-  put of two different types of ensemble, in order to forecast rainfall in Italy and nearby regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The final product  is created by blending.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The overall aim was to deliver much greater skill than can be achieved by using the raw en-  sembles’ output as is, particularly for localised extreme rainfall which is a very common driver of flash flood events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Indeed this initiative was the ’flash flood use case’ within the EU-funded MISTRAL project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Whilst the project itself  formally ended in January 2021, products continue to be produced in real-time and are accessible on an open-access  web platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Input data comes from one LAM ensemble (COSMO) and one global ensemble (ECMWF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' These ensembles have  horizontal resolutions of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='2 and 18km respectively: in the LAM convection is resolved, in the global model it is  parametrised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Post-processing of the two ensembles proceeds in different ways - employing methods commensurate  with their resolutions, their capabilities and the lead times they afford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' For the LAM, a scale-selective neighbour-  hood post-processing technique was applied, primarily to try to compensate for there being insufficient ensemble  46  Gascón et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  members to capture all the convection-related degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Despite being scale-selective, the technique  nonetheless preserves, probabilistically, signals at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='2km scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' We consider this to be reasonably representative of  point (raingauge) scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' For the global ensemble, we use instead the ecPoint downscaling technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Its main purpose  is to activate local-weather-dependant adjustments for gridscale bias and for sub-grid variability, to predict, proba-  bilistically, values as measured by raingauges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Post-processing as described thus puts outputs sourced from the two  ensembles onto compatible scales, such that a simple blending step can then be meaningfully performed, preserving  fine-scale signals where appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' A much more straightforward alternative to the above would be to simply use  conservative interpolation to upscale the LAM output to the global model scale before blending, but that would waste  the high resolution detail (irrespective of whether it was considered reliable or not) and could not deliver forecasts of  high impact localised extremes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  When set alongside the fast-evolving world of artificial intelligence (AI) the scale-selective neighbourhood post-  processing we have used could be seen as rudimentary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' On the other hand its simplicity is a major strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Rather  than using AI we rely instead on the complexities of the ensemble integrations themselves to define what the ’believ-  able scales’ of rainfall prediction are, ultimately for any weather scenarios that may arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' We then encapsulate those  in a very simple and interpretable way, as an agreement scale integer between 0 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  In our system blending in the first 48h is simply based on a tapered weighting scheme, with post-processed  COSMO output afforded highest weight early on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' With this strategy we aimed to best exploit the innate resolution-  related advantages of COSMO over ECMWF at short leads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Such advantages are seen for orographic influences, as  demonstrated in a one year agreement scale analysis, but also for convection, as reported elsewhere, and other as-  pects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Whilst the net reduction in discrimination ability over the 48 hours for raw COSMO may be fairly modest, for  larger totals in summer and in autumn, when flash flood frequency over Italy tends to peak, it is more pronounced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  And although some of this reduction can be offset with post-processing, ecPoint output is still more competitive at  the longer leads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Meanwhile the fraction of COSMO land gridpoints inheriting probabilistic information from neigh-  bourhoods that were smaller than 18km (=ENS and ecPoint output resolution) dropped from ~50% at T+6 to ~33%  at T+48, with km-scale retention dropping away even more rapidly, to ~12% by T+48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Such reductions occur because  synoptic scale spread growth is rendering the small number of COSMO members increasingly restrictive for longer  Gascón et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  47  leads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' So for those longer leads the benefits, for rainfall prediction, of investing in costly high resolution runs, versus  using ecPoint output alone, reduce, because the two end up largely providing forecasts on similar scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' This provides  further support for our strategy to increase ecPoint weights for longer leads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' And our approach to make this weight  almost one by (nominal) T+48, where COSMO runs end, also ensures that products are seamless across that barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Products beyond T+48 are provided by ecPoint alone;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' these then end at T+240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The focal point in this paper has been  the first 48h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  The complex process of post-processing and blending could not have been operationally implemented without  HPC resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' These were provided for us by CINECA in Bologna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' For the users, the blended point rainfall forecasts  are stored as percentiles 1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='.99, for overlapping 6h periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Whilst 6h is the shortest period for which there is  currently sufficient global coverage of observations for ecPoint calibration, it is also a reasonable compromise from  a computational perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Although periods shorter than 6h might benefit flash flood applications, that would  increase the HPC resource requirement, compromising delivery times to the detriment of users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Verification for the COSMO area shows that relative to raw COSMO the scale-selective neighbourhood technique  generally improves discrimination ability and reliability for different precipitation thresholds, although the differences  for reliability are not that significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Relative to raw ECMWF ENS, ecPoint delivers noteworthy gains in reliability  and resolution across multiple thresholds, as reported also by Hewson and Pillosu (2021) in global verification for 12h  totals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' However ecPoint’s reliability gains are bordering on negative for large totals at the longer leads of 36-48h,  reflecting a curious improvement in the reliability of the raw ENS then.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Relative to post-processed COSMO, the ecPoint product generally provides better forecasts for all precipitation  thresholds examined, although in some circumstances, when considering larger thresholds (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' 20-30mm/6h), this is  not clearcut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Indeed for large thresholds the advantages of merging the two post-processed outputs are noteworthy  for discrimination ability in the summer period, and reliability is generally slightly better for the merged product, for  10 and 30mm/6h, over the year as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Outside of summer time discrimination capabilities for large totals are  generally similar for the merged product and ecPoint, except at short leads where ecPoint performs best overall,  probably because post-processed COSMO was then afforded too much weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Spin-down issues seem to have  compromised the raw COSMO performance for those short lead times, and the current COSMO post-processing  48  Gascón et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  method cannot correct for those.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Future releases could address this using e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' lead-time dependant bias corrections,  if appropriate, or more simply by changing the weighting function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  A particularly strong diurnal cycle in the performance of the different ensemble products was found for summer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Indeed an important result from the summer verification was that the post-processed COSMO improves upon ecPoint  in the nominal "evening" period - i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' 18 to 24 UTC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' This is a time when flash floods can often occur, and so is useful  to highlight the benefits of blending.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  The relatively poor summer time evening performance of ecPoint shows that our attempt to correct for known  IFS under-prediction of convective rainfall late in the day, using local solar time as a predictor, has not been that  successful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' One limiting factor in this regard is undoubtedly the current ecPoint policy to apply no adjustments to  forecasts of zero rain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' One could in principal develop another decision tree branch, at the top level, for cases of 0mm  forecast, trained on what happened when 0mm was forecast, using for example convective rainfall predicted for the  previous 6 hours, when local solar time indicated evening, as one predictor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' This is another area for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Regarding verification by topographic complexity, ecPoint and the merged product exhibit robust performance  for large totals (>20mm/6h) across all types of terrain (found in Italy) in our two verification metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Overall these  outputs rank as the best two for all lead times, for all topography classes, with the merged product demonstrating  better reliability than ecPoint beyond nominal T+6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' As expected, our verification also confirms that both COSMO  products have much better discrimination of large totals in areas of complex orography than the raw ENS, which is  almost certainly due to the innately better representation of complex orography afforded by COSMO’s 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='2km spatial  resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Another striking topographic feature is the reduced ecPoint reliability for the 06-12 UTC period in mountainous  areas (SDfor > 100 m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The full ecPoint decision tree was designed in such a way that the impacts of both LST and  SDfor were never considered in the same branch (for a relevant portion see Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' SDfor was presumed to be  particularly relevant for large-scale precipitation (convective component <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='25), whilst for all other classes it was not  used, and SR24h and LST were used instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' However, we should perhaps now revisit this, to see if also including  SDfor as a governing variable for convective events improves scores, for morning times in particular but others too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  As with many ecPoint calibration results this could also provide useful physical pointers for IFS code development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Gascón et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  49  Analysis of two case studies using plan-view charts helped us to visualise and better understand the impacts of  both post-processing techniques, and merging, on the raw forecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' The merged products looked plausible and were  devoid of strange artefacts, satisfying basic user requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' In general, for the largest (localized) rainfall totals in  the two cases, we observed an overestimation in raw COSMO and a significant underestimation in the raw ENS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Post-  processed COSMO and ecPoint provided more realistic-looking peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Whilst ecPoint also delivered useful guidance  on where, in general, the worst affected areas would be, post-processed COSMO was sometimes able to add useful  local detail onto that picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Ultimately users would see a final merged product that looked more credible, more  accurate and more useful than any of the input components on their own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Although we have not unequivocally linked  particular rainfall measurements to flash flood events, the two episodes examined coincided with times when flash  floods were being quite widely reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Via the scale-selective neighbourhood technique for COSMO we could see in the case studies where the forecast  fine-scale detail tends to be most reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' As expected, topographically complex areas stand out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' In one case a front-  parallel agreement scale minimum, centred near to the most likely position of the frontal rainfall, was also very evident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Agreement scale increases either side were symptomatic of uncertain timing of the front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' At longer leads, and as frontal  timing concurrently becomes even more uncertain, it is logical to expect a general elevation of agreement scales, which  is what we saw in our case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' In turn this would diminish, with lead time, the innate usefulness of a high resolution ensem-  ble (with a limited number of members) for predicting local details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Out of the LAM ensembles with resolution <3km op-  erational in Europe in December 2021 (see http://srnwp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='hu/C_SRNWP_project/Eumetnet_List.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='html), none com-  prised more than 21 members, and only one system - a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='2km COSMO ensemble for Switzerland - extended beyond  54h leads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Using differently post-processed output from that ensemble blended with ecPoint output, Hemri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  (2022) have recently shown (using the continuous ranked probability score on 1 year of data) that value added to  what ecPoint alone delivers diminishes with lead time: it was minimal between 60 and 84h leads, and nil beyond 84h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  So our inference and their results basically agree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Looking to the future, this study provides pointers to how real-time numerical prediction of rainfall (and other  parameters) could evolve, as LAM ensembles become increasingly widespread, and as global models advance towards  having convective-scale resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' In the short term our blending approach could be adopted for other domains  50  Gascón et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  where other LAM ensembles run, perhaps bolstered by verification-based weighting optimisation, and some form of  LAM bias correction where necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' In the longer term, in what will be a new and very different era for global  prediction, one can envisage a crunch point arising, where one can afford to run a medium range ensemble with  convective-scale resolution, but not the hundreds or thousands of members needed to address both convective and  synoptic scale uncertainties beyond the first few days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' One could accept the jumpy medium range output that would  arise with fewer members, though users may be dissatisfied (note that even 18km resolution ensembles can be jumpy  for extreme events - see Figure 3 in Hewson and Pillosu (2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Or one could conceivably invent new post-processing  approaches - simple or complex or a mix - to quell jumpiness and recover skill, but currently the route to that is not  clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' On the other hand, and this is a very open question, it may be that high resolution global ensembles do advance  the predictability frontiers for synoptic scales, by capturing important upscale effects, which would make longer lead  usage vital even with the member number limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' But if this proves to not be the case then another option would  be to terminate convective scale global ensembles at some intermediate lead time, say 4 or 5 days, whilst also running  at lower resolution to much longer leads, and use post-processing and blending techniques like those described here  to achieve seamless jump-free output, not just locally but globally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' In the end it will be a question of how to best  utilise, for users, the expanding and evolving HPC resources, focussing not just on rainfall but also of course on other  impact-defining parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  5  |  ACKNOWLEDGEMENTS  The authors gratefully acknowledge financial support from the European Union Connecting Europe Facility (CEF)  – Telecommunication Sector Programme for the MISTRAL – Meteo Italian SupercompuTing poRtAL project (Grant  Agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' INEA/CEF/ICT/A2017/1567101).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Gascón et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  51  references  Ancell, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2013) Nonlinear Characteristics of Ensemble Perturbation Evolution and Their Application to Forecasting High-  Impact Events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Weather Forecast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', 28, 1353–1365.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Baran, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Leutbecher, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Szabó, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' and Ben-Bouallegue, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2019) Statistical post-processing of dual-resolution ensemble  forecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Meteorol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', 145, 1705–1720.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Batignani, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Cheloni, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Fuccello, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Marcucci, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Torrisi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Troisi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Zaccariello, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' and Canessa, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2019) Application and  Verification of ECMWF Products 2019 - Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' In Green Book 2019, 1–25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' ECMWF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Bechtold, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Semane, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Lopez, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Chaboureau, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Beljaars, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' and Bormann, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2014) Representing Equilibrium and  Nonequilibrium Convection in Large-Scale Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Atmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', 71, 734–753.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Ben Bouallègue, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' and Theis, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2013) Spatial techniques applied to precipitation ensemble forecasts: from verification  results to probabilistic products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Meteorol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', 21, 922–929.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Ben Bouallègue, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Theis, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' and Gebhardt, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2013) Enhancing COSMO-DE ensemble forecasts by inexpensive techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Meteorol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', 22, 49–59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Blake, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Carley, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Alcott, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Jankov, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Pyle, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Perfater, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' and Albright, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2018) An Adaptive Approach for  the Calculation of Ensemble Gridpoint Probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Weather Forecast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', 33, 1063–1080.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Brier, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (1950) Verification of forecasts expressed in terms of probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Weather.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Rev, 78, 1–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Buizza, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2018) Ensemble Forecasting and the Need for Calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' In Statistical Postprocessing of Ensemble Forecasts, 15–48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Elsevier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Clark, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Kain, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Stensrud, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Xue, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Kong, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Coniglio, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Thomas, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Brewster, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Gao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Weiss, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  and Du, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2011) Probabilistic Precipitation Forecast Skill as a Function of Ensemble Size and Spatial Scale in a Convection-  Allowing Ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Weather Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', 139, 1410–1418.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  DelSole, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Trenary, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' and Tippett, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2017) The weighted-average lagged ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Earth Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', 9, 2739–2752.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Dey, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Plant, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Roberts, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' and Migliorini, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2016a) Assessing spatial precipitation uncertainties in a convective-scale  ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Meteorol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Soc, 142, 2935–2948.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Dey, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Roberts, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Plant, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' and Migliorini, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2016b) A new method for the characterization and verification of local spatial  predictability for convective-scale ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Meteorol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', 142, 1982–1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  52  Gascón et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Ebert, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2008) Fuzzy verification of high-resolution gridded forecasts: a review and proposed framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Meteorol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=',  15, 51–64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Friederichs, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Wahl, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' and Buschow, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2018) Postprocessing for Extreme Events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' In Statistical Postprocessing of Ensemble  Forecasts, 127–154.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Elsevier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Gascón, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Laviola, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Merino, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' and Miglietta, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2016) Analysis of a localized flash-flood event over the central Mediter-  ranean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Atmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', 182, 256–268.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Gneiting, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2014) Calibration of medium-range weather forecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Technical Memorandum 719, ECMWF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Haiden, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2018) Use of in situ surface observations at ECMWF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' ECMWF Technical Memorandum 834.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Haiden, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' and Duffy, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2016) Use of high-density observations in precipitation verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' ECMWF Newsletter No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' 147,  20–25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Hamill, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2018) Practical Aspects of Statistical Postprocessing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' In Statistical Postprocessing of Ensemble Forecasts, 187–217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Elsevier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Hamill, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Engle, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Myrick, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Peroutka, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Finan, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' and Scheuerer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2017) The U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' National Blend of Models for  Statistical Postprocessing of Probability of Precipitation and Deterministic Precipitation Amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Weather Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', 145,  3441–3463.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Hamill, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' and Juras, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2006) Measuring forecast skill: is it real skill or is it the varying climatology?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' R, Meteorol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', 132,  2905–2923.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Harris, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
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+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
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+page_content=', Dueben, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' and Palmer, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
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+page_content=', Hewson, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
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+page_content=', Spririg, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Moret, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' and Liniger, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
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+page_content=' Technical Memorandum  901, ECMWF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Hewson, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' and Pillosu, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
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+page_content=', 2, 1–10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Hitchens, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Brooks, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' and Kay, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2013) Objective Limits on Forecasting Skill of Rare Events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Weather Forecast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=',  28, 525–534.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Gascón et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  53  Kobayashi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Duc, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Apip, Oizumi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' and Saito, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
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+page_content='  Hazards Earth Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', 20, 755–770.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Leutbecher, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Lock, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Ollinaho, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Lang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Balsamo, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Bechtold, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Bonavita, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Christensen, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Diamantakis,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
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+page_content=', Fisher, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Forbes, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Goddard, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Haiden, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Hogan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Juricke, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Lawrence, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', MacLeod,  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Magnusson, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Malardel, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Massart, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Sandu, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Smolarkiewicz, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Subramanian, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Vitart, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Wedi, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' and  Weisheimer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2017) Stochastic representations of model uncertainties at ecmwf: state of the art and future vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Meteorol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', 143, 2315–2339.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Lin, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Tsai, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Liou, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Lin, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Lin, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' and Chung, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2020) Multi-weather evaluation of nowcasting  methods including a new empirical blending scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Atmosphere, 11, 1–17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Massey, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (1951) The Kolmogorov-Smirnov Test for Goodness of Fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Assoc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', 46, 68–78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Melhauser, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' and Zhang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2012) Practical and Intrinsic Predictability of Severe and Convective Weather at the Mesoscales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Atmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', 69, 3350–3371.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Mendoza, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Rajagopalan, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Clark, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Ikeda, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' and Rasmussen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2015) Statistical Postprocessing of High-Resolution  Regional Climate Model Output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Weather Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', 143, 1533–1553.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Mittermaier, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2014) A Strategy for Verifying Near-Convection-Resolving Model Forecasts at Observing Sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Weather  Forecast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', 29, 185–204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Mittermaier, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' and Roberts, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2010) Intercomparison of Spatial Forecast Verification Methods: Identifying Skillful Spatial  Scales Using the Fractions Skill Score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Weather Forecast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', 25, 343–354.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Necker, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Geiss, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Weissmann, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Ruiz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Miyoshi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' and Lien, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2020) A convective-scale 1,000-member ensemble  simulation and potential applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Meteorol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', 146, 1423–1442.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Palmer, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2019) The ecmwf ensemble prediction system: Looking back (more than) 25 years and projecting forward 25 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Meteorol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', 145, 12–24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Pillosu, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' and Hewson, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2017) New point-rainfall forecasts for flash flood prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' ECMWF Newsletter 153.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Politis, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' and Romano, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (1994) The stationary bootstrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Assoc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', 89, 1303–1313.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Radhakrishna, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Zawadzki, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' and Fabry, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2012) Predictability of Precipitation from Continental Radar Images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Part V:  Growth and Decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Atmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Sci, 69, 3336–3349.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  54  Gascón et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Raynaud, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' and Bouttier, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2017) The impact of horizontal resolution and ensemble size for convective-scale probabilistic  forecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Meteorol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', 143, 3037–3047.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Richardson, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2003) Economic value and skill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' In Forecast Verification: A Practitioner´s Guide in Atmospheric Science, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' 165–  187.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Wiley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Roberts, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' and Lean, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2008) Scale-Selective Verification of Rainfall Accumulations from High-Resolution Forecasts of  Convective Events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Weather Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', 136, 78–97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Scheuerer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' and Hamill, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2015) Statistical Postprocessing of Ensemble Precipitation Forecasts by Fitting Censored Shifted  Gamma Distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Weather Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', 143, 4578–4596.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Schraff, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Reich, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Rhodin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Schomburg, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Stephan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Periáñez, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' and Potthast, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2016) Kilometre-scale ensemble  data assimilation for the COSMO model (KENDA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Meteorol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', 142, 1453–1472.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Schwartz, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' and Sobash, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2017) Generating Probabilistic Forecasts from Convection-Allowing Ensembles Using Neigh-  borhood Approaches: A Review and Recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Weather Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', 145, 3397–3418.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Silvestro, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Rebora, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Cummings, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' and Ferraris, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2015) Experiences of dealing with flash floods using an ensemble  hydrological nowcasting chain: implications of communication accessibility and distribution of the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Flood Risk  Manag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', 10, 446–462.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Skok, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' and Roberts, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2016) Analysis of Fractions Skill Score properties for random precipitation fields and ECMWF fore-  casts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Meteorol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', 142, 2599–2610.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Taillardat, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Fougères, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Naveau, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' and Mestre, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2019) Forest-Based and Semiparametric Methods for the Postpro-  cessing of Rainfall Ensemble Forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Weather Forecast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', 34, 617–634.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Taillardat, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' and Mestre, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2020) From research to applications – Examples of operational ensemble post-processing in  France using machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Nonlinear Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Geoph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', 27, 329–347.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Theis, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Hense, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' and Damrath, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2005) Probabilistic precipitation forecasts from a deterministic model: a pragmatic  approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Meteorol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', 12, 257–268.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Vannitsem, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Bremnes, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Demaeyer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Evans, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Flowerdew, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', Hemri, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' and .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Ylhaisi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2021) Statistical post-  processing for weather forecasts: review, challenges, and avenues in a big data world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Meteorol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', 102,  E681–E699.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Gascón et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  55  Whan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' and Schmeits, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2018) Comparing Area Probability Forecasts of (Extreme) Local Precipitation Using Parametric  and Machine Learning Statistical Postprocessing Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Weather Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', 146, 3651–3673.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Wilks, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2011) Statistical Methods in the Atmospheric Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' In International Geophysics Series, 3rd edition, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Oxford, UK and Waltham, MA: Academic Press, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' 704.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Elsevier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Wilks, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' and Hamill, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2007) Comparison of Ensemble-MOS Methods Using GFS Reforecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Weather Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', 135,  2379–2390.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content='  Zacharov, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' and Rezacova, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' (2009) Using the fractions skill score to assess the relationship between an ensemble QPF  spread and skill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Atmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
+page_content=', 94, 684–693.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfYAr5/content/2301.04485v1.pdf'}
diff --git a/M9FIT4oBgHgl3EQfcitV/content/tmp_files/2301.11266v1.pdf.txt b/M9FIT4oBgHgl3EQfcitV/content/tmp_files/2301.11266v1.pdf.txt
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@@ -0,0 +1,1080 @@
+ 
+ 
+ 
+ 
+1 
+ 
+Production and measurement of stellar neutron spectrum at 30 keV  
+ 
+ Javier Praena1, Guido Martín-Hernández2, Javier García López3,4 
+ 
+1 Dpto. Física Atómica, Molecular y Nuclear, Universidad de Granada, Granada, Spain.  
+2 Centro de Aplicaciones Tecnológicas y Desarrollo Nuclear, 5ta y 30, Playa, La Habana, Cuba. 
+3 Dpto. Física Atómica, Molecular y Nuclear, Universidad de Sevilla, Sevilla, Spain. 
+3 Centro Nacional de Aceleradores (Universidad de Sevilla, Junta de Andalucía, CSIC), Sevilla, Spain 
+ 
+E-mail: jpraena@ugr.es 
+ 
+Abstract. In the present experiment we measure the forward-emitted angle-integrated neutron 
+spectrum of the 7Li(p,n)7Be reaction via time-of-flight neutron spectrometry at Joint Research Center. 
+The proton beam is shaped to a Gaussian-like distribution with energies close to the reaction threshold 
+upstream the thick lithium target. The angle-integrated neutron spectrum resembles a Maxwellian or 
+stellar neutron spectrum at kT=30 keV.  
+ 
+1. INTRODUCTION
+One of the most interesting topics in nuclear 
+astrophysics is related to the nucleosynthesis of the 
+elements beyond iron, which is mainly produced 
+via successive neutron-capture reactions and beta 
+decays [1,2]. The neutron capture cross section 
+data are fundamental ingredients for the calculation 
+of the stellar reaction rates and for reproducing the 
+observed abundance of the elements in the 
+Universe [3]. Since neutron velocities follow a 
+Maxwell-Boltzmann probability distribution, the 
+neutron capture cross section in a stellar site is 
+known as Maxwellian-Averaged Cross Section 
+(MACS), defined as 
+ 
+  
+     
+⟨  ⟩
+  
+  
+√  
+∫
+ ( )     
+  
+    
+ 
+ 
+∫
+   
+  
+    
+ 
+ 
+           ( ) 
+ 
+Where ⟨  ⟩ is the reaction rate per particle 
+pair;    is the particle most probable thermal 
+velocity; and  ( ) is the differential neutron 
+capture cross section of the element involved as a 
+function of the neutron energy. One of the most 
+prolific sites for nucleosynthesis is the He-
+burning in red giants stars, where the typical 
+temperature is 0.348 GK and corresponds to 
+kT=30 keV. In consequence, the MACS at 
+kT=30 keV (MACS30) are key data for stellar 
+nucleosynthesis [4,5].  
+The experimental determination of the MACS 
+can be carried out via time-of-flight (TOF) 
+technique or via activation technique. The 
+TOF (or differential measurement) provides 
+the cross section as a function of the neutron 
+energy,  ( ), which allows calculating the 
+MACS at several stellar temperatures; if 
+possible, is the best option. However, if the 
+mass sample is significantly lower than 
+milligrams, the TOF measurement lacks of 
+enough neutron fluence, and the MACS can 
+only be measured by activation (or integral 
+measurement) 
+with 
+high 
+flux. 
+Integral 
+measurements will provide the MACS but the 
+neutron spectrum irradiating the sample must 
+be corrected to a stellar or maxwellian 
+spectrum. This correction depends on the 
+differential cross section of the isotope for 
+which the MACS is being measuring by 
+activation. Many isotopes lack of differential 
+cross sections in consequence the spectrum 
+correction cannot be properly performed in 
+integral measurements.  Thus, the best option 
+in integral measurements is to generate an 
+exact stellar neutron spectrum at a certain 
+stellar energy (kT) to avoid this correction.  
+The production of stellar neutron beams 
+started several decades ago. In 1966 Pönitz 
+showed that an absolute integral cross section 
+using proton energy near the reaction threshold for 
+thick Li target could be determined [6]. Near the 
+7Li(p,n)7Be reaction threshold, 1881 keV, the 
+neutron beam is collimated in a forward cone and 
+
+ 
+ 
+ 
+ 
+2 
+ 
+completely covered with a sample, thus, the 
+neutron fluence could be determined with the 7Be 
+activity, an absolutely measurement with no 
+reference. Later, Beer & Käppeler showed that the 
+angle-integrated distribution of the produced 
+neutrons could be close to a Maxwellian at kT=25 
+keV for proton energy Ep=1912 keV [7]. In 1988, 
+Ratynski & Käppeler remeasured the neutron 
+spectrum with improved sensitivity and over a 
+larger energy range [8]. They demonstrated that the 
+integration over the cone of 70º of aperture yields 
+to quasi Maxwellian neutron at kT=25 keV 
+(QMNS-25) with a lack of neutrons above 110 
+keV. They showed that the MACS30 could be 
+experimentally 
+obtained 
+in 
+an 
+integral 
+measurement with the QMNS-25. However, small 
+corrections due the difference between true 
+Maxwellian at kT=25 keV and the QMNS-25 were 
+needed. Then, an extrapolation from 25 to 30 keV 
+was also performed. Since then, this method has 
+been used for measuring MACS30 of many 
+isotopes relevant in stellar evolution and 
+nucleosynthesis of elements.  
+The correction from QMNS-25 to a Maxwellian 
+spectrum at 30 keV (MNS-30) depends on the 
+differential cross section; therefore, it becomes 
+a problem for poorly known or unknown cross 
+sections as the case of short-life isotopes. For 
+this reason, a method to produce a Maxwellian 
+neutron spectrum at 30 keV (MNS-30) was 
+proposed [9,10]. With this method the 
+correction for neutron spectrum is not needed 
+or negligible, thus, it is adequate for integral 
+measurements on isotopes with unknown 
+cross 
+sections. 
+First 
+measurements 
+of 
+MACS30 with this method were already 
+carried out on stable isotopes [11,12]. The 
+spectrum used in such experiments was based 
+on the well-known angle-energy yield of the 
+7Li(p,n)7Be reaction close to the threshold.  
+Preliminary experimental confirmation of the 
+neutron spectrum of the MNS-30 method was 
+already carried out by means of the TOF and 
+energy 
+spectrum 
+measured 
+at 
+forward 
+direction with a detector a 0º [13,14]. Here, it 
+is presented the final confirmation with the 
+complete measurement at several angles via 
+time-of-flight neutron spectrometry.  
+2. THE METHOD 
+The method for MNS-30 production is based 
+on shaping the proton beam to a distribution 
+that impinging onto the lithium target will 
+produce the desired neutron spectrum [9,10]. 
+Thus, the key point is to find the proton beam 
+with the adequate energy distribution, so, a proper 
+combination of proton energy and degrader or 
+shaper foil. The shaper is determined by the 
+thickness and material. The most convenient 
+material is Aluminium as it was already shown in 
+previous experiments [11,12]. For an estimation of 
+the adequate combination SRIM code is used [15]. 
+However, uncertainty on the Al thickness, possible 
+inhomogeneity, 
+or 
+accuracy 
+of 
+the 
+SRIM 
+simulations makes the measurement of the proton 
+distribution mandatory for this method.   
+Preliminary SRIM calculations showed that an 
+adequate combination to produce a MNS-30 is a 
+proton beam of 3.665 MeV passing through Al foil 
+of 75-µm thickness [11]. In such conditions, a 
+proton distribution close to a Gaussian with 
+Ec=1.860 MeV and FWHM=162 keV is produced. 
+Then, simulations based on the well-known angle-
+energy yield of the 7Li(p,n)7Be reaction show that 
+the impact onto a thick lithium target will generate 
+a MNS-30, see [11] and references therein. 
+Conventionally, Al foils are provided with ±10% 
+uncertainty. A change of few micrometres in a 75-
+µm Al thickness produces a modification of the 
+proton distribution and in consequence an 
+alteration in the neutron spectrum [11]. With the 
+direct measurement of the proton distributions all 
+these circumstances can be ignored and the proton 
+energy can be adjusted until the desired proton 
+distribution is found. In fact, the final proton 
+energy is selected on the base of the results of the 
+measured proton distributions.  
+2.1 Measurement of proton distributions 
+Here, the measurement of the proton distributions 
+produced by the crossing of the proton beam 
+through several points of Al foil with nominal 
+thickness of 75 µm is described. For this purpose, 
+the Microbeam line of the 3-MV Tandem at Centro 
+Nacional de Aceleradores (Spain) was used. This 
+line is equipped with micrometre slits to reduce the 
+
+ 
+ 
+ 
+ 
+3 
+ 
+count rate to few Hz in the existing detectors. The 
+proton beam can impinge directly onto a Surface 
+Barrier Silicon (SBS) detector (ORTEC) located at 
+0º and perpendicular respect to the beam direction. 
+The low count rate avoids the damage of the 
+detector and keeps its energy resolution. The setup 
+is designed to guarantee an exactly reproducible 
+perpendicular geometry. Figure 1 shows a picture 
+of the chamber of the Microbeam line.  
+Based on the accelerator calibration, the exact 
+energy of the proton beam was 3.665 MeV. With 
+the direct measurement of the proton beam onto the 
+SBS, the accelerator calibration was confirmed, 
+thus, the terminal voltage of V=1.789 MV 
+corresponded to 3665 keV, see Figure 2.  
+ 
+ 
+Figure 1. Picture of the Microbeam chamber at CNA. 
+ 
+Then, the  Al foil was installed perpendicular to the 
+beam and SBS detector and the proton spectrum 
+downstream the Al foil was measured. Several 
+points of the Al were studied to determine a 
+possible inhomogeneity in the foil. No significant 
+differences were found within the resolution of the 
+SBS detector (15 keV at these energies). Figure 2 
+shows the measured proton distribution which can 
+be fitted with a Gaussian distribution with 
+Ec=1.860 MeV and FWHM=162 keV. The vertical 
+red 
+line 
+indicates 
+the 
+7Li(p,n)7Be 
+reaction 
+threshold. The shape of the proton distribution 
+monotonically decreasing with the energy is crucial 
+to generate a stellar neutron spectrum at 30 keV. 
+Once the Al foil and the proton distribution were 
+characterized, the same foil was used for the 
+measurement of the Maxwellian neutron spectrum 
+at 30 keV. 
+ 
+Figure 2. Proton distributions measured at CNA. From the 
+right to the left: spectra and -signals from the Am-Pu-Cm 
+source, proton beam of 3.665 MeV impinging directly onto 
+the Si detector (proton beam), and shaped proton beam 
+downstream the Al foil (75-µm thickness). The spectra 
+have the conventional long tail at lower energies due to the 
+charge collection of the detector. Red line indicates the 
+7Li(p,n)7Be reaction threshold. 
+ 
+3. NEUTRON SPECTRA 
+MEASUREMENT 
+ 
+The experiment for determining the neutron 
+spectrum was performed at the 7-MV Van de 
+Graaff laboratory at IRMM. Proton beams with 
+energies from 1.8 to 3.7 MeV were used in the 
+experiment for different purposes. The accelerator 
+terminal was calibrated using the 991.86 keV 
+27Al(p,γ)28Si and 2409 keV 
+24Mg(p,p’γ)24Mg 
+resonances The 90º analyzing magnet was 
+calibrated with NMR probe. The accelerator was 
+operated in pulsed mode with a frequency of 625 
+kHz for TOF measurements. The pulse width was 
+2-ns FWHM. Figure 3 shows a sketch of the 
+experimental setup.   
+The proton beam (4-mm diameter) passed through 
+a collimator of 5-mm diameter. The target 
+assembly consisted of an aluminium cylinder (4.2-
+cm diameter and 15-cm long) and 0.4-mm thick 
+copper backing for the lithium target. LiF thick 
+target (6-mm diameter and 90 mg) was prepared by 
+evaporation onto Cu backing. The target assembly 
+was cooled by a forced air flow on the external side 
+of the Cu target backing. Figure 4 shows a picture 
+of the setup including target and detectors. 
+
+6000
+Protonbeamon
+Sidetector
+Proton distribution
+downstreamAlfoil
+onSidetector
+Counts
+4000-
+AmPuCmα-source
+2000-
+forcalibration
+0
+1
+2
+3
+4
+5
+6
+Energy(MeV) 
+ 
+ 
+ 
+4 
+ 
+ 
+Figure 3. Schematic of the experimental setup. 6Li-glass 
+detectors were placed on movable stands in a goniometer at 
+the beam height. The flight path is 52 cm. 
+ 
+Three 6Li-glass detectors (5.08-cm diameter and 
+2.54-cm and 1.27-cm thick) detectors were located 
+at the same height of the lithium target at a flight 
+path of 52 cm. A fourth detector was used as 
+monitor at a fixed position with no geometrical 
+interference with the others. Also, a Long Counter 
+was used as monitoring.  
+ 
+ 
+ 
+Figure 4. Picture of experimental setup with four 6Li-glass 
+detectors, Long Counter detector and target assembly.  
+ 
+The acquisition was carried out from the 
+photomultiplier of each detector; the signals were 
+split to pulse-height and to fast-timing circuits. The 
+latter was composed of a timing filter amplifier and 
+a constant-fraction discriminator whereas the 
+former consisted of a preamplifier and a 
+spectroscopy amplifier. The timing signal from 
+each detector provided the TOF start signal in an 
+allocated time-to-amplitude converter (TAC). A 
+beam pick-up monitor, placed upstream of the 
+target, provided a signal that was processed 
+through a timing filter amplifier and a constant 
+fraction discriminator and provided the common 
+TOF stop signal to the TAC. Data for every 
+detector were acquired independently, while the 
+timing and pulse-height information for each 
+detector were acquired in coincidence. 
+ 
+3.1 TOF spectra without Al degrader 
+The experiment started checking the 7Li(p,n)7Be 
+reaction threshold with the measurement of TOF 
+spectra. Then, he forward spectrum at 0º produced 
+by 1912 keV proton energy was measured because 
+is the standard spectrum of Ratynski & Käppeler 
+[8]. This allows checking the acquisition setup and 
+the analysis procedure. Details of the analysis will 
+be described in the next section.  
+Figure 5 shows time spectra for three proton 
+energies. The separation between the gammas (first 
+peak) and the neutrons (second peaks) was very 
+good. At proton energy of 1878 keV, neutrons 
+were clearly detected, meanwhile no neutrons were 
+detected at 1877 keV even the detector was located 
+closer to the target. Therefore, an offset of 3 keV 
+was added to the accelerator calibration. In a first 
+calculation at 1890 keV, the maximum neutron 
+energy corresponds to around 73 keV (141 ns), in 
+good agreement with kinematics of the 7Li(p,n)7Be 
+reaction. The shape of the gamma-flash peak is 
+analogous to a similar measurement performed in 
+the same facility at 1912 keV [16].  
+ 
+Figure 5. Time histograms for the 6Li-glass detector at 0º 
+for three proton energies close the threshold. First peak 
+corresponds to prompt gammas or gamma-flash that is used 
+as time zero. Second peaks correspond to the neutrons. 
+Channels have been already transformed to time.  
+ 
+Hereafter, the 3 keV offset is already considered. 
+Then, the accelerator was set at 1912 keV proton 
+
+Fligh path
+Pulsedprotonbeam
+Neutrons
+Detector
+Target10000
+1878 keV
+1880 keV
+1890 keV
+1000
+Counts
+100-
+10
+200
+400
+600
+Time (ns) 
+ 
+ 
+ 
+5 
+ 
+energy and the spectrum at 0º was acquired. As 
+mentioned before, the angle-integrated spectrum at 
+1912 keV proton energy has been measured several 
+times [16-18] and corresponds to the standard 
+neutron field established by Ratynski & Käppeler 
+in 1988. This spectrum has been extensible used in 
+MACS measurements where the neutron spectrum 
+at 0º was always acquired and analysed as a further 
+verification of the conditions of the experiments 
+regarding proton energy and neutron production. 
+Figure 6 shows a 0º time histogram.  First analysis 
+shows that the maximum energy of neutrons will 
+be around 112 keV (113 ns) in good agreement 
+with the expected result. However, the shape of the 
+gamma-flash suffers of an increase in the FWHM 
+compared to lower proton energies. This will be a 
+key point of the analysis of the neutron spectra at 
+higher proton energies with the Al foil, as it will be 
+explained later.  
+ 
+Figure 6. Time histogram for the 6Li-glass detector at 0º for 
+1912-keV proton energy.  
+ 
+4. DATA ANALISIS 
+ 
+The data analysis of the TOF spectra is determined 
+by the time-to-energy conversion of the neutrons. 
+First, it depends on the 
+6Li-glass detector 
+efficiency, mainly determined by the 6Li(n,t)4He 
+reaction. Some resonances in other materials of the 
+detector have been already studied. Lederer et al. 
+[17] and Macías et al. [18] with similar 6Li-glass 
+detectors showed a small contribution to the 
+intrinsic efficiency of resonances of 56Fe and 28Si 
+which are also present in the Borosilicate, μ-metal 
+and tape of the detectors. Using the reported 
+geometry and materials of the 6Li-glass detectors, 
+the intrinsic efficiency is obtained with Monte 
+Carlo simulations with MCNPX [19]. Figure 7 
+shows the simulated intrinsic efficiency. Small dips 
+due to the mentioned resonances can be noticed. 
+ 
+ 
+Figure 7. Intrinsic efficiency of a 6Li-glass detector (5.08-
+cm diameter and 2.54-cm thick).  
+ 
+Second, the final TOF-to-energy conversion 
+depends on the geometry of the setup. For this, all 
+the processes until a neutron induces the 
+6Li(n,t)4He reaction inside the detector must be 
+considered. These processes are an effective 
+increment of the flight path of the neutrons inside 
+the detector which depends on the energy of the 
+neutron. The TOF-to-energy conversion was 
+obtained with accurate simulations with MCNPX 
+considering the whole setup.  
+The good separation of gammas and neutrons in 
+Figure 5 and 6 shows the presence of some 
+scattered signals between both peaks. This 
+background must be subtracted. A run with a 
+shadow bar shielding the detector showed that 
+scattered signals are due to backscattered neutrons 
+in the experimental hall. However, this background 
+was very small and uncorrelated with the time (flat 
+background) and can be subtracted. 
+The uncertainty of the neutron spectrum at 1912 
+keV at 0º is mainly due to statistics. Therefore, 
+once the time-of-flight is transformed to an energy 
+grid, the histograms are rebinned to 5 keV in the 
+region from 2 to 120 keV. Figure 8 shows the 
+comparison between the experimental neutron 
+spectrum at 0º of this work with 1912 keV proton 
+
+1912keV
+10000
+1000
+Counts
+100
+10
+200
+400
+600
+TOF (ns)0,1
+Efficiency
+Li-glass
+0.01
+10
+100
+Energy(keV) 
+ 
+ 
+ 
+6 
+ 
+energy and the experimental result of Ratynski & 
+Käppeler [8,17]. The agreement is very good and 
+demonstrates 
+the 
+feasibility 
+of 
+the 
+setup, 
+acquisition system and analysis procedure. The 
+same type of analysis has been successfully used in 
+previous works [18,20].  
+ 
+Figure 8. Experimental neutron spectrum of this work at 0º 
+and 1912-keV proton energy in comparison to the 
+experimental data of Ratynski and Käppeler [8,17].  
+ 
+4.1 TOF spectra with Al degrader 
+Once the measurements at proton energies 
+near the 7Li(p,n)7Be reaction threshold were 
+performed, the characterized Aluminum foil was 
+placed close to fresh LiF target. Figure 9 shows a 
+picture of the target assembly during the 
+dismounting, where the Al foil is perpendicular to 
+the target by a pushing system made of carbon 
+fiber and located inside the target assembly. At 
+the end of the experiment, a mark was clearly 
+visible in the central part of the Al foil. This mark 
+was made of sputtered Lithium due to the impact 
+of the proton beam on the target. Even the low 
+average intensity delivered by the pulse proton 
+beam (around 100 ns in continuous mode) a loss 
+of 7Be was found. This matter in absolute integral 
+measurements as the fluence is determined with 
+the 7Be activity. The Lithium target is deposited 
+onto the Cu backing and it became dark by proton 
+irradiation 
+as 
+conventionally 
+occurs 
+in 
+experiments with LiF. The Cu backing acted as 
+proton beam dump and close of the vacuum of the 
+accelerator by suction and its contact with a Viton 
+O-ring.   
+ 
+ 
+ 
+Figure 9. Picture of the target assembly. Aluminium foil 
+has a mark where the proton beam passed through. This 
+mark was due to sputtered Lithium, thus, it contained 7Be. 
+LiF is deposited onto the Cu backing and become dark by 
+proton irradiation.  
+ 
+Then, the accelerator was adjusted to 3.665 keV 
+proton energy and time histograms were acquired. 
+The protons passed through the Al foil before 
+impacting the LiF target. Figure 10 shows a time 
+histogram for the 6Li-glass detector at 0º.  
+ 
+Figure 10. Time-of-flight histogram recorded by the 6Li-
+glass detector at 0º for 3665-keV proton beam passing 
+through the 75-µm Al foil and hitting the thick LiF target.  
+Counts are normalized to the monitoring detector. 
+ 
+Important differences with the time spectra at 
+lower proton energies appeared. First, there is not 
+a clear separation between gamma-flash and 
+neutron peaks. This is an indication of a 
+background correlated with the time. Second, a 
+clear structure of double peak appears in the 
+gamma-flash, and it was also visible at higher 
+times (above 300 ns). Therefore, the experiment 
+deals with an important background that must be 
+understood and properly subtracted. After several 
+
+0,5
+This work at 1912 keV
+Ratynskii&Kaeppeler
+dN/dE (arb. units)
+0,4
+0.3
+0,2
+0,1
+0,0
+0
+20
+40
+60
+80
+100
+120
+Energy (keV)0° - 3665 keV - 75 um Al
+10
+Counts
+1
+0.1
+0.01
+1E-3
+0
+200
+400
+600
+Time (ns) 
+ 
+ 
+ 
+7 
+ 
+test with the chopper and buncher system, the 
+problem was found out. It was related to the 
+presence of a halo in the proton beam at the ion 
+source of the accelerator. This halo increased 
+with the proton energy. Thus, protons of the beam 
+halo passed through the collimator connected to 
+the chopper system even the main part of the 
+proton beam was not passing through the 
+collimator. Then, the buncher provides the final 
+proton beam time structure. The structure of the 
+halo and the time structure provided the second 
+large gamma-flash and the contamination at 
+whatever time.   
+Although the background in the experiment was 
+important, it can be subtracted once the problem 
+has been understood. The background subtraction 
+is performed by the subtraction of the second 
+gamma-flash and the corresponding neutrons, and 
+the repetitive structure. Figure 11 shows this 
+procedure. The repetitive structure was acquired 
+at angles higher than 80º were neutron fluence is 
+negligible. The result of the substruction is shown 
+in Figure 12. 
+ 
+Figure 11. Time-of-flight histogram recorded by the 6Li-
+glass detector at 0º (black), and the two histograms to be 
+subtracted, at angle higher than 80º (red), and the 2nd -flash 
+and neutrons (blue). Counts are normalized to the 
+monitoring detector.  
+A preliminary analysis of the TOF histogram at 
+0º after the background subtraction shows that the 
+shape of the neutron peak at lower TOF is 
+monotonically increasing with the time instead of 
+a sharp shape as in the case of the TOF histogram 
+at 0º acquire at 1912-keV proton energy. This is 
+an indication of a longer tail at high neutron 
+energies as expected for the MNS-30. Besides, 
+the highest neutron energies will be around 250 
+keV (75 ns). It should be remembered that the 
+Ratynski & Käppeler angle-integrated spectrum 
+has a lack of neutrons above 110 keV.  
+ 
+ 
+Figure 12. Time-of-flight histogram at 0º after the 
+background subtraction explained in the text.  
+ 
+The same subtraction procedure is applied to the 
+rest of the measured angles, from 0º to 80º in 
+steps of 10º. Following the TOF-to-energy 
+conversion already explained for 1912 keV, the 
+energy histograms are obtained for each detector 
+and angle. Figure 13 shows the neutron spectra 
+for each detector normalized to the number of 
+counts registered in the monitoring detector. The 
+results are shown with a constant bin of 5 keV 
+which is a good compromise between good 
+energy resolution and reasonable statistics. With 
+this binning the statistics uncertainty ranges from 
+3 to 12%. The uncertainty is not included for 
+display purposes.  
+Once the neutron spectra are determined at 
+detector position, it is needed to obtain the angle 
+integrated neutron spectrum at the source point, 
+thus, at the neutron production target. For this, 
+each spectrum is scaled by the respective covered 
+solid angle. For angles () higher than 0º the 
+corresponding factors are calculated as,  
+      (       )     (       ) 
+And for the 0º position as,  
+
+100
+0o - 3665 keV - 75 um Al
+10
+No neutrons
+SecondFlash
+WTTT
+Counts
+0,1
+0,01
+1E-3
+1E-4
+0
+200
+400
+600
+Time (ns)100
+0° - 3665 keV - 75 um Al
+10
+Background substracted
+Counts
+0,1
+0,01
+1E-3
+1E-4
+0
+200
+400
+600
+Time (ns) 
+ 
+ 
+ 
+8 
+ 
+         (     ) 
+Where 2.79º is the half of the covered angle by 
+the detector. Figure 14 shows the neutron spectra 
+for each detector normalized to the number of 
+counts registered in the monitoring detector and 
+corrected by the corresponding solid angle.  
+ 
+Figure 13. Neutron spectra from 0º to 80º in steps of 10º. 
+Spectra are not corrected for the respective solid angle.  
+ 
+Figure 14. Neutron spectra from 0º to 80º in steps of 10º. 
+Spectra are corrected for the respective solid angle.  
+ 
+For the present setup with a flight path of 52 cm 
+and 5.08-cm diameter detectors, the whole 
+forward hemisphere is not covered. However, the 
+possible differences of the angle-integrated 
+spectrum have been studied for other authors at 
+1912-keV proton energy [17]. They demonstrated 
+that the possible differences are very low. Figure 
+15 shows the final angle-integrated experimental 
+spectrum obtained in the present work (black 
+histogram). The experimental data are fitted with 
+a Maxwell-Boltzmann distribution in energies 
+(red line), 
+                
+with the normalization constant (A) and the 
+stellar energy (kT) as free parameters. With an 
+orthogonal 
+linear 
+regression 
+is 
+obtained 
+kT=30±0.3 keV with a reduced χ2=5.4·10-6 and 
+R2=0.99. The results show a very good agreement 
+between the experimental neutron spectrum and a 
+stellar neutron spectrum at kT=30 keV.  
+ 
+Figure 15. Experimental neutron spectrum generated in the 
+present work (black), Maxwell-Boltzmann fit (red) with 
+free parameters the normalization constant and the stellar 
+energy. The corresponding result is kT=30±0.3 keV.  
+ 
+5. CONCLUSIONS 
+The main purpose of the present work was the 
+production and measurement of a stellar neutron 
+spectrum at kT=30 keV. The goal was to improve 
+the standard neutron field produced by Ratynski 
+and Käppeler in 1988. Since then, such spectrum 
+has been used in several integral measurements of 
+MACS30 of important isotopes for astrophysics. 
+The improvement was focused on the possibility 
+to avoid the spectrum correction for determining 
+the MACS. The spectrum correction depends on 
+the differential cross section of the element for 
+which the MACS is determined by an integral 
+measurement. Therefore, for elements with poor 
+or unknown knowledge of the differential cross 
+section, this correction becomes a puzzle. The 
+generation of a Maxwellian neutron spectrum at 
+
+0,06
+No solid angle
+00
+correction
+10°
+0,05
+200
+300
+0,04.
+40°
+500
+dN/dE
+600
+0,03
+70°
+80°
+0,02
+0,01
+0,00
+0
+50
+100
+150
+200
+250
+300
+Energy (keV)0,0010
+00
+Solid angle
+10°
+correction
+20°
+0.0008
+30°
+40°
+50°
+dN/dE
+0,0006
+009
+70°
+80°
+0,0004
+0,0002
+0,0000
+0
+50
+100
+150
+200
+250
+300
+Energy (keV)0,014
+Experimenta
+0,012
+Maxwellian
+0,010
+kT=30keV
+0,008
+0,006
+0,004
+0,002
+0,000
+0
+30
+60
+90
+120
+150
+180
+210
+240
+270
+300
+E (keV) 
+ 
+ 
+ 
+9 
+ 
+30 keV solves this problem.  In order to improve 
+the Ratynski and Käppeler spectrum, it was 
+needed that the Maxwellian fit of the neutron 
+spectrum was increased from 25 keV to 30 keV, 
+and at the same time to generate neutrons with 
+energies above 110 keV following a tail at high 
+energies. Both goals have been achieved with the 
+use of a shaped proton beam onto a thick lithium 
+target.  
+It is important to mention that the energy 
+distribution of the proton beam must be 
+monotonically decreasing from the threshold of 
+the reaction to around 2.1 MeV. As it has been 
+shown, the direct measurement of the proton 
+beam is possible and it is a key of this method. 
+Then, the measurement of the stellar neutron 
+spectrum at 30 keV has been carried out via time-
+of-flight 
+technique. 
+Even, 
+the 
+important 
+background suffered in the experiment, it is 
+possible to subtract it from each spectrum 
+acquired at different angles. The result of the 
+angle-integrated spectrum resembles a stellar or 
+Maxwellian at 30 keV. Very small differences are 
+noticed in the peak of the Maxwellian and above 
+100 keV. However, the improvement of the 
+stellar shape compared to Ratynski and Käppeler 
+is noticeable. Thus, the goal of the experiment 
+has been achieved. Additionally, the MNS-30 
+method solves a technical problem which is the 
+loss of 7Be in absolute measurements. As it has 
+been shown, the sputtering of Lithium occurs 
+even at low currents but the Al foil avoids its loss 
+to the accelerator.  
+After the experimental confirmation of the stellar 
+spectrum at 30 keV, it is planned to carry out an 
+absolute integral measurement of the MACS30 of 
+197Au(n,γ). This quantity is extremely important 
+in stellar nucleosynthesis and its value is matter 
+of discussions nowadays due to the discrepancy 
+between the integral measurement of R&K and 
+the value obtained with differential measurements 
+of the 197Au(n,γ) cross section [21-26]. In a 
+forthcoming paper it will be demonstrated that the 
+neutron correction related to the MNS-30 is not 
+needed because the folding of the 197Au(n,γ) 
+cross section with a Maxwellian at 30 keV is 
+practically equal to the folding with the MNS-30 
+measured in the present work.  
+ 
+Acknowledgments. This work was supported by the 
+European Commission within the Seventh Framework 
+Programme through EUFRAT (EURATOM contract 
+no. FP7-211499). The authors would like to the 
+operators of the IRMM Van de Graaff accelerator, in 
+particular to Thierry Gamboni, for the many extra 
+hours they devoted trying to provide the best possible 
+conditions for this experiment. The authors thank J.A. 
+Labrador 
+and 
+A. 
+Romero 
+for 
+the 
+excellent 
+performance 
+of 
+the 
+CNA 
+accelerator. 
+J.P. 
+acknowledges support and constructive discussions 
+with F. Käppeler in the framework of the n_TOF-
+CERN Collaboration. This work was also supported 
+by Spanish institutions Ministerio de Ciencia e 
+Innovación 
+(PID2020-117969RB-I00), 
+Junta 
+de 
+Andalucia projects P20-00665 and B-FQM-156-
+UGR20.
+[1]. Burbidge, E.M., Burbidge, G.R., Fowler, W.A. & Hoyle, F. 
+Synthesis of Elements in Stars. Rev. Mod. Phys. 29 4, 547-650 
+(1957). https://doi.org/10.1103/RevModPhys.29.547  
+[2]. Cameron, A.G.W. Nuclear reactions in stars and nucleogenesis 
+PASP 69, 201 (1957). https://doi.org/10.1086/127051 
+[3]. Bao, Z.Y. et al. Neutron cross sections for nucleosynthesis 
+studies. Atomic Data and Nucl. Data Tables 76, 70-154 (2000). 
+https://doi.org/10.1006/adnd.2000.0838   
+[4]. Rauscher, T., Mohr, P., Dillmann I., & Plag, R. Opportunities to 
+constrain astrophysical reaction rates for the s-process via 
+determination 
+of 
+the 
+ground-state 
+cross-sections. 
+The 
+Astrophysical 
+Journal 
+738:143 
+(2011). 
+https://doi.org/10.1088/0004-637X/738/2/143    
+[5]. Galino, R. et al. Evolution and nucleosynthesis in low-mass 
+asymptotic giant branch stars. II Neutron capture and the s-
+process. The Astrophysical Journal 497, 388-403 (1998). 
+https://iopscience.iop.org/article/10.1086/305437   
+[6]. Pönitz, W. The (n,γ) cross section of 197Au at 30 and 64 keV 
+neutron energy. Journal of Nuclear Energy Parts A/B, 20 825-
+834 (1966), Pergamon Press Ltd. 
+[7]. Beer, H. & Käppeler, F. Neutron capture cross sections on 139Ba, 
+140,142Ce, 175,176Lu, and 181Ta at 30 keV: Prerequisite for 
+investigations of the 176Lu cosmic clock. Phys. Rev. C 21, 2 
+534–544 (1980). https://doi.org/10.1103/PhysRevC.21.2139  
+[8]. Ratynski, W. & Käppeler, F. Neutron capture cross section on 
+197Au: A standard for stellar nucleosyntheis. Phys. Rev. C 37, 
+595-604 (1988). https://doi.org/10.1103/PhysRevC.37.595  
+[9]. Praena, J., Mastinu, P.F. & Martín-Hernández, G. A Method to 
+Obtain a Maxwell–Boltzmann Neutron Spectrumat kT =30 keV 
+for Nuclear Astrophysics Studies. Pub. of the Astron. Soc. of 
+Australia 26, 225 (2009). https://doi.org/ 10.1071/AS08043 
+[10]. Mastinu, P.F, Martín-Hernández, G. & Praena, J.  A. method to 
+obtain a Maxwell–Boltzmann neutron spectrum at kT=30 keV 
+
+ 
+ 
+ 
+ 
+10 
+ 
+for nuclear astrophysics studies. Nucl. Instrum. and Methods A 
+601, 333 (2009). http://dx.doi.org/10.1016/j.nima.2009.01.005  
+[11]. Praena, J. et al., Measurement of the MACS of 181Ta(n,γ) at 
+kT=30 keV as a test of a method for Maxwellian neutron spectra 
+generation. Nucl. Inst. and Meth. A 727 1-6 (2013). 
+http://dx.doi.org/10.1016/j.nima.2013.05.151  
+[12]. Praena, J. et al. Measurement of the MACS of 159Tb(n,γ) at kT =  
+30 keV by Activation. Nuclear Data Sheets 120, 205–207 
+(2014). http://dx.doi.org/10.1016/j.nds.2014.07.047  
+[13].  Praena, J. et al. Current quests in nucleosynthesis: present and 
+future neutron induced reaction measurements. EPJ Web of 
+Conferences 
+66, 
+07022 
+1-8 
+(2014). 
+http://dx.doi.org/10.1051/epjconf/20146607022 
+[14]. Jiménez-Bonilla, P., Praena, J. & Quesada, J.M. On the 
+Activation Method for Stellar Cross-Sections Measurements: 
+Flat Sample Correction in Measurements Relatives to Gold. NIC 
+XV Conf., Springer Proceedings in Physics 219, 381-384 
+(2018). https://doi.org/10.1007/978-3-030-13876-9_70.  
+[15]. Ziegler, J.F., Ziegler, M.D. & Biersack, J.P.. SRIM – The 
+stopping and range of ions in matter (2010). Nucl. Instrum. and 
+Methods B 268 1818-1823 (2010). 
+https://doi.org/10.1016/j.nimb.2010.02.091   
+[16]. Feinberg, G. . et al.  Quasi-stellar neutrons from the 7Li(p,n)7Be 
+reaction with an energy-broadened proton beam. Phys. Rev. C 
+85 055810 (2012). https://doi.org/10.1103/PhysRevC.85.055810 
+[17].  Lederer, C. et al.  Definition of a standard neutron field with the 
+7Li(p,n)7Be reaction. Phys. Rev C 85 055809 (2012). 
+https://doi.org/10.1103/PhysRevC.85.055809  
+[18]. Macías, M., Fernández, B. & Praena, J. The first neutron time-
+of-flight line in Spain: Commissioning and new data for the 
+definition of a neutron standard field. Rad. Phys. and Chem.  
+168, 108538 (2020). 
+ https://doi.org/10.1016/j.radphyschem.2019.108538  
+[19]. D.B.Pelowitz,MCNPXUsersManualVersion2.5.0—LA-CP05-
+0369, LosAlamos National LaboratoryLACP,2005.  
+[20]. Macías, M., Fernández, B. & Praena, J. New data for the 
+definition of neutron beams for Boron Neutron capture therapy. 
+Rad. 
+Phys. 
+and 
+Chem. 
+ 
+185 
+109474 
+(2021).   
+https://doi.org/10.1016/j.radphyschem.2021.109474  
+[21]. Kornilov, N.V., Pronyaev, V.G. & Grimes, S.M. Systematic. 
+Distortion Factor and Unrecognized Source of Uncertainties in 
+Nuclear Data Measurements and Evaluations. Metrology 2 1–18 
+(2022). https://doi.org/10.3390/metrology2010001  
+[22]. Carlson, A.D. Work on the neutron standards since the 
+completion of the standards evaluation. EPJ Web of Conferences 
+239, 08001 (2020). 
+https://doi.org/10.1051/epjconf/202023908001  
+[23]. Praena, J. & Jiménez-Bonilla, P. Two absolute integral 
+measurements of the 197Au(n,γ) stellar cross-section and solution 
+of the historic discrepancies.  EPJ Web of Conferences 239, 
+19002 (2020). https://doi.org/10.1051/epjconf/202023919002  
+[24]. Pronyaev, V.G. 
+197Au(n,γ) Standard Cross Section and 
+Experimental Data. Report INDC(NDS)-0540 2009, p. 29. 
+Available online: https://www-nds.iaea.org/standards-cm-oct-
+2008/9.PDF (accessed on 30 September 2022). 
+[25]. Meeting on Neutron Data Standards, 12-16 October 2020, IAEA 
+Headquarters, Vienna, Austria. https://www-nds.iaea.org/index-
+meeting-crp/CM-NDS-2020-10/ 
+[26]. Reifarth, R. et al., Neutron-induced cross sections. From raw 
+data to astrophysical rates. Eur. Phys. J. Plus 133: 424 (2018) 
+http://dx.doi.org/10.1140/epjp/i2018-12295-3  
+ 
+ 
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf,len=579
+page_content='1  Production and measurement of stellar neutron spectrum at 30 keV  Javier Praena1, Guido Martín Hernández2, Javier García López3,4  1 Dpto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Física Atómica, Molecular y Nuclear, Universidad de Granada, Granada, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' 2 Centro de Aplicaciones Tecnológicas y Desarrollo Nuclear, 5ta y 30, Playa, La Habana, Cuba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' 3 Dpto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Física Atómica, Molecular y Nuclear, Universidad de Sevilla, Sevilla, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' 3 Centro Nacional de Aceleradores (Universidad de Sevilla, Junta de Andalucía, CSIC), Sevilla, Spain  E mail: jpraena@ugr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='es  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' In the present experiment we measure the forward-emitted angle-integrated neutron spectrum of the 7Li(p,n)7Be reaction via time-of-flight neutron spectrometry at Joint Research Center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' The proton beam is shaped to a Gaussian-like distribution with energies close to the reaction threshold upstream the thick lithium target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' The angle-integrated neutron spectrum resembles a Maxwellian or stellar neutron spectrum at kT=30 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' INTRODUCTION One of the most interesting topics in nuclear astrophysics is related to the nucleosynthesis of the elements beyond iron, which is mainly produced via successive neutron-capture reactions and beta decays [1,2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' The neutron capture cross section data are fundamental ingredients for the calculation of the stellar reaction rates and for reproducing the observed abundance of the elements in the Universe [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Since neutron velocities follow a Maxwell-Boltzmann probability distribution, the neutron capture cross section in a stellar site is known as Maxwellian-Averaged Cross Section (MACS), defined as  ⟨  ⟩  √  ∫  ( )  ∫  ( )  Where ⟨  ⟩ is the reaction rate per particle pair;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='    is the particle most probable thermal velocity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' and  ( ) is the differential neutron capture cross section of the element involved as a function of the neutron energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' One of the most prolific sites for nucleosynthesis is the He- burning in red giants stars, where the typical temperature is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='348 GK and corresponds to kT=30 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' In consequence, the MACS at kT=30 keV (MACS30) are key data for stellar nucleosynthesis [4,5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' The experimental determination of the MACS can be carried out via time-of-flight (TOF) technique or via activation technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' The TOF (or differential measurement) provides the cross section as a function of the neutron energy,  ( ), which allows calculating the MACS at several stellar temperatures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' if possible, is the best option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' However, if the mass sample is significantly lower than milligrams, the TOF measurement lacks of enough neutron fluence, and the MACS can only be measured by activation (or integral measurement) with high flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Integral measurements will provide the MACS but the neutron spectrum irradiating the sample must be corrected to a stellar or maxwellian spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' This correction depends on the differential cross section of the isotope for which the MACS is being measuring by activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Many isotopes lack of differential cross sections in consequence the spectrum correction cannot be properly performed in integral measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='  Thus, the best option in integral measurements is to generate an exact stellar neutron spectrum at a certain stellar energy (kT) to avoid this correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' The production of stellar neutron beams started several decades ago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' In 1966 Pönitz showed that an absolute integral cross section using proton energy near the reaction threshold for thick Li target could be determined [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Near the 7Li(p,n)7Be reaction threshold, 1881 keV, the neutron beam is collimated in a forward cone and  2  completely covered with a sample, thus, the neutron fluence could be determined with the 7Be activity, an absolutely measurement with no reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Later, Beer & Käppeler showed that the angle-integrated distribution of the produced neutrons could be close to a Maxwellian at kT=25 keV for proton energy Ep=1912 keV [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' In 1988, Ratynski & Käppeler remeasured the neutron spectrum with improved sensitivity and over a larger energy range [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' They demonstrated that the integration over the cone of 70º of aperture yields to quasi Maxwellian neutron at kT=25 keV (QMNS-25) with a lack of neutrons above 110 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' They showed that the MACS30 could be experimentally obtained in an integral measurement with the QMNS-25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' However, small corrections due the difference between true Maxwellian at kT=25 keV and the QMNS-25 were needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Then, an extrapolation from 25 to 30 keV was also performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Since then, this method has been used for measuring MACS30 of many isotopes relevant in stellar evolution and nucleosynthesis of elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' The correction from QMNS-25 to a Maxwellian spectrum at 30 keV (MNS-30) depends on the differential cross section;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' therefore, it becomes a problem for poorly known or unknown cross sections as the case of short-life isotopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' For this reason, a method to produce a Maxwellian neutron spectrum at 30 keV (MNS-30) was proposed [9,10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='  With this method the correction for neutron spectrum is not needed or negligible, thus, it is adequate for integral measurements on isotopes with unknown cross sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' First measurements of MACS30 with this method were already carried out on stable isotopes [11,12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' The spectrum used in such experiments was based on the well-known angle-energy yield of the 7Li(p,n)7Be reaction close to the threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Preliminary experimental confirmation of the neutron spectrum of the MNS-30 method was already carried out by means of the TOF and energy spectrum measured at forward direction with a detector a 0º [13,14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Here, it is presented the final confirmation with the complete measurement at several angles via time-of-flight neutron spectrometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' THE METHOD The method for MNS-30 production is based on shaping the proton beam to a distribution that impinging onto the lithium target will produce the desired neutron spectrum [9,10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Thus, the key point is to find the proton beam with the adequate energy distribution, so, a proper combination of proton energy and degrader or shaper foil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' The shaper is determined by the thickness and material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' The most convenient material is Aluminium as it was already shown in previous experiments [11,12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' For an estimation of the adequate combination SRIM code is used [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' However, uncertainty on the Al thickness, possible inhomogeneity, or accuracy of the SRIM simulations makes the measurement of the proton distribution mandatory for this method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='  Preliminary SRIM calculations showed that an adequate combination to produce a MNS-30 is a proton beam of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='665 MeV passing through Al foil of 75-µm thickness [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' In such conditions, a proton distribution close to a Gaussian with Ec=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='860 MeV and FWHM=162 keV is produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Then, simulations based on the well-known angle- energy yield of the 7Li(p,n)7Be reaction show that the impact onto a thick lithium target will generate a MNS-30, see [11] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Conventionally, Al foils are provided with ±10% uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' A change of few micrometres in a 75- µm Al thickness produces a modification of the proton distribution and in consequence an alteration in the neutron spectrum [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' With the direct measurement of the proton distributions all these circumstances can be ignored and the proton energy can be adjusted until the desired proton distribution is found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' In fact, the final proton energy is selected on the base of the results of the measured proton distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='1 Measurement of proton distributions Here, the measurement of the proton distributions produced by the crossing of the proton beam through several points of Al foil with nominal thickness of 75 µm is described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' For this purpose, the Microbeam line of the 3-MV Tandem at Centro Nacional de Aceleradores (Spain) was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' This line is equipped with micrometre slits to reduce the  3  count rate to few Hz in the existing detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' The proton beam can impinge directly onto a Surface Barrier Silicon (SBS) detector (ORTEC) located at 0º and perpendicular respect to the beam direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' The low count rate avoids the damage of the detector and keeps its energy resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' The setup is designed to guarantee an exactly reproducible perpendicular geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Figure 1 shows a picture of the chamber of the Microbeam line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Based on the accelerator calibration, the exact energy of the proton beam was 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='665 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' With the direct measurement of the proton beam onto the SBS, the accelerator calibration was confirmed, thus, the terminal voltage of V=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='789 MV corresponded to 3665 keV, see Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Picture of the Microbeam chamber at CNA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='  Then, the  Al foil was installed perpendicular to the beam and SBS detector and the proton spectrum downstream the Al foil was measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Several points of the Al were studied to determine a possible inhomogeneity in the foil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' No significant differences were found within the resolution of the SBS detector (15 keV at these energies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Figure 2 shows the measured proton distribution which can be fitted with a Gaussian distribution with Ec=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='860 MeV and FWHM=162 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' The vertical red line indicates the 7Li(p,n)7Be reaction threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' The shape of the proton distribution monotonically decreasing with the energy is crucial to generate a stellar neutron spectrum at 30 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Once the Al foil and the proton distribution were characterized, the same foil was used for the measurement of the Maxwellian neutron spectrum at 30 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Proton distributions measured at CNA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' From the right to the left: spectra and \uf061-signals from the Am-Pu-Cm source, proton beam of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='665 MeV impinging directly onto the Si detector (proton beam), and shaped proton beam downstream the Al foil (75-µm thickness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' The spectra have the conventional long tail at lower energies due to the charge collection of the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Red line indicates the 7Li(p,n)7Be reaction threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' NEUTRON SPECTRA  MEASUREMENT  The experiment for determining the neutron spectrum was performed at the 7-MV Van de Graaff laboratory at IRMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Proton beams with energies from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='8 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='7 MeV were used in the experiment for different purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' The accelerator terminal was calibrated using the 991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='86 keV 27Al(p,γ)28Si and 2409 keV 24Mg(p,p’γ)24Mg resonances The 90º analyzing magnet was calibrated with NMR probe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' The accelerator was operated in pulsed mode with a frequency of 625 kHz for TOF measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' The pulse width was 2-ns FWHM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Figure 3 shows a sketch of the experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' The proton beam (4-mm diameter) passed through a collimator of 5-mm diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' The target assembly consisted of an aluminium cylinder (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='2- cm diameter and 15-cm long) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='4-mm thick copper backing for the lithium target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' LiF thick target (6-mm diameter and 90 mg) was prepared by evaporation onto Cu backing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' The target assembly was cooled by a forced air flow on the external side of the Cu target backing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Figure 4 shows a picture of the setup including target and detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='  6000  Protonbeamon  Sidetector  Proton distribution  downstreamAlfoil  onSidetector  Counts  4000 AmPuCmα source  2000 forcalibration  0  1  2  3  4  5  6  Energy(MeV)  4  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Schematic of the experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' 6Li-glass detectors were placed on movable stands in a goniometer at the beam height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' The flight path is 52 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='  Three 6Li-glass detectors (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='08-cm diameter and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='54-cm and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='27-cm thick) detectors were located at the same height of the lithium target at a flight path of 52 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' A fourth detector was used as monitor at a fixed position with no geometrical interference with the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Also, a Long Counter was used as monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Picture of experimental setup with four 6Li-glass detectors, Long Counter detector and target assembly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='  The acquisition was carried out from the photomultiplier of each detector;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' the signals were split to pulse-height and to fast-timing circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' The latter was composed of a timing filter amplifier and a constant-fraction discriminator whereas the former consisted of a preamplifier and a spectroscopy amplifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' The timing signal from each detector provided the TOF start signal in an allocated time-to-amplitude converter (TAC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' A beam pick-up monitor, placed upstream of the target, provided a signal that was processed through a timing filter amplifier and a constant fraction discriminator and provided the common TOF stop signal to the TAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Data for every detector were acquired independently, while the timing and pulse-height information for each detector were acquired in coincidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='1 TOF spectra without Al degrader The experiment started checking the 7Li(p,n)7Be reaction threshold with the measurement of TOF spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Then, he forward spectrum at 0º produced by 1912 keV proton energy was measured because is the standard spectrum of Ratynski & Käppeler [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' This allows checking the acquisition setup and the analysis procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Details of the analysis will be described in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Figure 5 shows time spectra for three proton energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' The separation between the gammas (first peak) and the neutrons (second peaks) was very good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' At proton energy of 1878 keV, neutrons were clearly detected, meanwhile no neutrons were detected at 1877 keV even the detector was located closer to the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Therefore, an offset of 3 keV was added to the accelerator calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' In a first calculation at 1890 keV, the maximum neutron energy corresponds to around 73 keV (141 ns), in good agreement with kinematics of the 7Li(p,n)7Be reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' The shape of the gamma-flash peak is analogous to a similar measurement performed in the same facility at 1912 keV [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Time histograms for the 6Li-glass detector at 0º for three proton energies close the threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' First peak corresponds to prompt gammas or gamma-flash that is used as time zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Second peaks correspond to the neutrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Channels have been already transformed to time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='  Hereafter, the 3 keV offset is already considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Then, the accelerator was set at 1912 keV proton  Fligh path  Pulsedprotonbeam  Neutrons  Detector  Target10000  1878 keV  1880 keV  1890 keV  1000  Counts  100 10  200  400  600  Time (ns)  5  energy and the spectrum at 0º was acquired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' As mentioned before, the angle-integrated spectrum at 1912 keV proton energy has been measured several times [16-18] and corresponds to the standard neutron field established by Ratynski & Käppeler in 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' This spectrum has been extensible used in MACS measurements where the neutron spectrum at 0º was always acquired and analysed as a further verification of the conditions of the experiments regarding proton energy and neutron production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Figure 6 shows a 0º time histogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' First analysis shows that the maximum energy of neutrons will be around 112 keV (113 ns) in good agreement with the expected result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' However, the shape of the gamma-flash suffers of an increase in the FWHM compared to lower proton energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' This will be a key point of the analysis of the neutron spectra at higher proton energies with the Al foil, as it will be explained later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Time histogram for the 6Li-glass detector at 0º for 1912-keV proton energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' DATA ANALISIS  The data analysis of the TOF spectra is determined by the time-to-energy conversion of the neutrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' First, it depends on the 6Li-glass detector efficiency, mainly determined by the 6Li(n,t)4He reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Some resonances in other materials of the detector have been already studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Lederer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' [17] and Macías et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' [18] with similar 6Li-glass detectors showed a small contribution to the intrinsic efficiency of resonances of 56Fe and 28Si which are also present in the Borosilicate, μ-metal and tape of the detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Using the reported geometry and materials of the 6Li-glass detectors, the intrinsic efficiency is obtained with Monte Carlo simulations with MCNPX [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Figure 7 shows the simulated intrinsic efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Small dips due to the mentioned resonances can be noticed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Intrinsic efficiency of a 6Li-glass detector (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='08- cm diameter and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='54-cm thick).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='  Second, the final TOF-to-energy conversion depends on the geometry of the setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' For this, all the processes until a neutron induces the 6Li(n,t)4He reaction inside the detector must be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' These processes are an effective increment of the flight path of the neutrons inside the detector which depends on the energy of the neutron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' The TOF-to-energy conversion was obtained with accurate simulations with MCNPX considering the whole setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' The good separation of gammas and neutrons in Figure 5 and 6 shows the presence of some scattered signals between both peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' This background must be subtracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' A run with a shadow bar shielding the detector showed that scattered signals are due to backscattered neutrons in the experimental hall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' However, this background was very small and uncorrelated with the time (flat background) and can be subtracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' The uncertainty of the neutron spectrum at 1912 keV at 0º is mainly due to statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Therefore, once the time-of-flight is transformed to an energy grid, the histograms are rebinned to 5 keV in the region from 2 to 120 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Figure 8 shows the comparison between the experimental neutron spectrum at 0º of this work with 1912 keV proton  1912keV  10000  1000  Counts  100  10  200  400  600  TOF (ns)0,1  Efficiency  Li glass  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='01  10  100  Energy(keV)  6  energy and the experimental result of Ratynski & Käppeler [8,17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' The agreement is very good and demonstrates the feasibility of the setup, acquisition system and analysis procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' The same type of analysis has been successfully used in previous works [18,20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Experimental neutron spectrum of this work at 0º and 1912-keV proton energy in comparison to the experimental data of Ratynski and Käppeler [8,17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='1 TOF spectra with Al degrader Once the measurements at proton energies near the 7Li(p,n)7Be reaction threshold were performed, the characterized Aluminum foil was placed close to fresh LiF target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Figure 9 shows a picture of the target assembly during the dismounting, where the Al foil is perpendicular to the target by a pushing system made of carbon fiber and located inside the target assembly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' At the end of the experiment, a mark was clearly visible in the central part of the Al foil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' This mark was made of sputtered Lithium due to the impact of the proton beam on the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Even the low average intensity delivered by the pulse proton beam (around 100 ns in continuous mode) a loss of 7Be was found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' This matter in absolute integral measurements as the fluence is determined with the 7Be activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' The Lithium target is deposited onto the Cu backing and it became dark by proton irradiation as conventionally occurs in experiments with LiF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' The Cu backing acted as proton beam dump and close of the vacuum of the accelerator by suction and its contact with a Viton O-ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='  Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Picture of the target assembly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Aluminium foil has a mark where the proton beam passed through.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' This mark was due to sputtered Lithium, thus, it contained 7Be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' LiF is deposited onto the Cu backing and become dark by proton irradiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='  Then, the accelerator was adjusted to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='665 keV proton energy and time histograms were acquired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' The protons passed through the Al foil before impacting the LiF target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Figure 10 shows a time histogram for the 6Li-glass detector at 0º.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='  Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Time-of-flight histogram recorded by the 6Li- glass detector at 0º for 3665-keV proton beam passing through the 75-µm Al foil and hitting the thick LiF target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Counts are normalized to the monitoring detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='  Important differences with the time spectra at lower proton energies appeared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' First, there is not a clear separation between gamma-flash and neutron peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' This is an indication of a background correlated with the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Second, a clear structure of double peak appears in the gamma-flash, and it was also visible at higher times (above 300 ns).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Therefore, the experiment deals with an important background that must be understood and properly subtracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' After several  0,5 This work at 1912 keV Ratynskii&Kaeppeler dN/dE (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' units) 0,4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='3 0,2 0,1 0,0 0 20 40 60 80 100 120 Energy (keV)0° - 3665 keV - 75 um Al 10 Counts 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='01 1E-3 0 200 400 600 Time (ns)  7  test with the chopper and buncher system, the problem was found out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' It was related to the presence of a halo in the proton beam at the ion source of the accelerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' This halo increased with the proton energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Thus, protons of the beam halo passed through the collimator connected to the chopper system even the main part of the proton beam was not passing through the collimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Then, the buncher provides the final proton beam time structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' The structure of the halo and the time structure provided the second large gamma-flash and the contamination at whatever time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Although the background in the experiment was important, it can be subtracted once the problem has been understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' The background subtraction is performed by the subtraction of the second gamma-flash and the corresponding neutrons, and the repetitive structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Figure 11 shows this procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' The repetitive structure was acquired at angles higher than 80º were neutron fluence is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' The result of the substruction is shown in Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='  Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Time-of-flight histogram recorded by the 6Li- glass detector at 0º (black), and the two histograms to be subtracted, at angle higher than 80º (red), and the 2nd \uf067-flash and neutrons (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Counts are normalized to the monitoring detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' A preliminary analysis of the TOF histogram at 0º after the background subtraction shows that the shape of the neutron peak at lower TOF is monotonically increasing with the time instead of a sharp shape as in the case of the TOF histogram at 0º acquire at 1912-keV proton energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' This is an indication of a longer tail at high neutron energies as expected for the MNS-30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Besides, the highest neutron energies will be around 250 keV (75 ns).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' It should be remembered that the Ratynski & Käppeler angle-integrated spectrum has a lack of neutrons above 110 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='  Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Time-of-flight histogram at 0º after the background subtraction explained in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='  The same subtraction procedure is applied to the rest of the measured angles, from 0º to 80º in steps of 10º.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Following the TOF-to-energy conversion already explained for 1912 keV, the energy histograms are obtained for each detector and angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Figure 13 shows the neutron spectra for each detector normalized to the number of counts registered in the monitoring detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' The results are shown with a constant bin of 5 keV which is a good compromise between good energy resolution and reasonable statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' With this binning the statistics uncertainty ranges from 3 to 12%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' The uncertainty is not included for display purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Once the neutron spectra are determined at detector position, it is needed to obtain the angle integrated neutron spectrum at the source point, thus, at the neutron production target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' For this, each spectrum is scaled by the respective covered solid angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' For angles (\uf061) higher than 0º the corresponding factors are calculated as, (       )     (       ) And for the 0º position as,  100  0o 3665 keV 75 um Al  10  No neutrons  SecondFlash  WTTT  Counts  0,1  0,01  1E 3  1E 4  0  200  400  600  Time (ns)100  0° 3665 keV 75 um Al  10  Background substracted  Counts  0,1  0,01  1E 3  1E 4  0  200  400  600  Time (ns)  8  (     ) Where 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='79º is the half of the covered angle by the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Figure 14 shows the neutron spectra for each detector normalized to the number of counts registered in the monitoring detector and corrected by the corresponding solid angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='  Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Neutron spectra from 0º to 80º in steps of 10º.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Spectra are not corrected for the respective solid angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='  Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Neutron spectra from 0º to 80º in steps of 10º.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Spectra are corrected for the respective solid angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='  For the present setup with a flight path of 52 cm and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='08-cm diameter detectors, the whole forward hemisphere is not covered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' However, the possible differences of the angle-integrated spectrum have been studied for other authors at 1912-keV proton energy [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' They demonstrated that the possible differences are very low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Figure 15 shows the final angle-integrated experimental spectrum obtained in the present work (black histogram).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' The experimental data are fitted with a Maxwell-Boltzmann distribution in energies (red line),  with the normalization constant (A) and the stellar energy (kT) as free parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' With an orthogonal linear regression is obtained kT=30±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='3 keV with a reduced χ2=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='4·10-6 and R2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' The results show a very good agreement between the experimental neutron spectrum and a stellar neutron spectrum at kT=30 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='  Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Experimental neutron spectrum generated in the present work (black), Maxwell-Boltzmann fit (red) with free parameters the normalization constant and the stellar energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' The corresponding result is kT=30±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='3 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' CONCLUSIONS The main purpose of the present work was the production and measurement of a stellar neutron spectrum at kT=30 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' The goal was to improve the standard neutron field produced by Ratynski and Käppeler in 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Since then, such spectrum has been used in several integral measurements of MACS30 of important isotopes for astrophysics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' The improvement was focused on the possibility to avoid the spectrum correction for determining the MACS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' The spectrum correction depends on the differential cross section of the element for which the MACS is determined by an integral measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Therefore, for elements with poor or unknown knowledge of the differential cross section, this correction becomes a puzzle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' The generation of a Maxwellian neutron spectrum at  0,06  No solid angle  00  correction  10°  0,05  200  300  0,04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='  40°  500  dN/dE  600  0,03  70°  80°  0,02  0,01  0,00  0  50  100  150  200  250  300  Energy (keV)0,0010  00  Solid angle  10°  correction  20°  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='0008  30°  40°  50°  dN/dE  0,0006  009  70°  80°  0,0004  0,0002  0,0000  0  50  100  150  200  250  300  Energy (keV)0,014  Experimenta  0,012  Maxwellian  0,010  kT=30keV  0,008  0,006  0,004  0,002  0,000  0  30  60  90  120  150  180  210  240  270  300  E (keV)  9  30 keV solves this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' In order to improve the Ratynski and Käppeler spectrum, it was needed that the Maxwellian fit of the neutron spectrum was increased from 25 keV to 30 keV, and at the same time to generate neutrons with energies above 110 keV following a tail at high energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Both goals have been achieved with the use of a shaped proton beam onto a thick lithium target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' It is important to mention that the energy distribution of the proton beam must be monotonically decreasing from the threshold of the reaction to around 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='1 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' As it has been shown, the direct measurement of the proton beam is possible and it is a key of this method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Then, the measurement of the stellar neutron spectrum at 30 keV has been carried out via time- of-flight technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Even, the important background suffered in the experiment, it is possible to subtract it from each spectrum acquired at different angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' The result of the angle-integrated spectrum resembles a stellar or Maxwellian at 30 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Very small differences are noticed in the peak of the Maxwellian and above 100 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' However, the improvement of the stellar shape compared to Ratynski and Käppeler is noticeable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Thus, the goal of the experiment has been achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Additionally, the MNS-30 method solves a technical problem which is the loss of 7Be in absolute measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' As it has been shown, the sputtering of Lithium occurs even at low currents but the Al foil avoids its loss to the accelerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='  After the experimental confirmation of the stellar spectrum at 30 keV, it is planned to carry out an absolute integral measurement of the MACS30 of 197Au(n,γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' This quantity is extremely important in stellar nucleosynthesis and its value is matter of discussions nowadays due to the discrepancy between the integral measurement of R&K and the value obtained with differential measurements of the 197Au(n,γ) cross section [21-26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' In a forthcoming paper it will be demonstrated that the neutron correction related to the MNS-30 is not needed because the folding of the 197Au(n,γ) cross section with a Maxwellian at 30 keV is practically equal to the folding with the MNS-30 measured in the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='  Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' This work was supported by the European Commission within the Seventh Framework Programme through EUFRAT (EURATOM contract no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' FP7-211499).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' The authors would like to the operators of the IRMM Van de Graaff accelerator, in particular to Thierry Gamboni, for the many extra hours they devoted trying to provide the best possible conditions for this experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' The authors thank J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Labrador and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Romero for the excellent performance of the CNA accelerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' acknowledges support and constructive discussions with F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Käppeler in the framework of the n_TOF- CERN Collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' This work was also supported by Spanish institutions Ministerio de Ciencia e Innovación (PID2020-117969RB-I00), Junta de Andalucia projects P20-00665 and B-FQM-156- UGR20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Burbidge, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=', Burbidge, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=', Fowler, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' & Hoyle, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Synthesis of Elements in Stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' 29 4, 547-650 (1957).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='1103/RevModPhys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='547 [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Cameron, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Nuclear reactions in stars and nucleogenesis PASP 69, 201 (1957).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='1086/127051 [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Bao, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Neutron cross sections for nucleosynthesis studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Atomic Data and Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Data Tables 76, 70-154 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='1006/adnd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='0838 [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Rauscher, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=', Mohr, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=', Dillmann I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=', & Plag, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Opportunities to constrain astrophysical reaction rates for the s-process via determination of the ground-state cross-sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' The Astrophysical Journal 738:143 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='1088/0004-637X/738/2/143 [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='  Galino, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Evolution and nucleosynthesis in low-mass asymptotic giant branch stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' II Neutron capture and the s- process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' The Astrophysical Journal 497, 388-403 (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' https://iopscience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='iop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='org/article/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='1086/305437 [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Pönitz, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' The (n,γ) cross section of 197Au at 30 and 64 keV neutron energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Journal of Nuclear Energy Parts A/B, 20 825- 834 (1966), Pergamon Press Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Beer, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' & Käppeler, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Neutron capture cross sections on 139Ba, 140,142Ce, 175,176Lu, and 181Ta at 30 keV: Prerequisite for investigations of the 176Lu cosmic clock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' C 21, 2 534–544 (1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='1103/PhysRevC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='2139 [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Ratynski, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' & Käppeler, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Neutron capture cross section on 197Au: A standard for stellar nucleosyntheis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' C 37, 595-604 (1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='1103/PhysRevC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='595 [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Praena, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=', Mastinu, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' & Martín-Hernández, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' A Method to Obtain a Maxwell–Boltzmann Neutron Spectrumat kT =30 keV for Nuclear Astrophysics Studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Pub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' of the Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' of Australia 26, 225 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='org/ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='1071/AS08043 [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Mastinu, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='F, Martín-Hernández, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' & Praena, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' method to obtain a Maxwell–Boltzmann neutron spectrum at kT=30 keV  10  for nuclear astrophysics studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Instrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' and Methods A 601, 333 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' http://dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='nima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='005 [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
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+page_content=' A 727 1-6 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
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+page_content='151 [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Praena, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
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+page_content='047 [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Praena, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
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+page_content='1051/epjconf/20146607022 [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Jiménez-Bonilla, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=', Praena, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' & Quesada, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' On the Activation Method for Stellar Cross-Sections Measurements: Flat Sample Correction in Measurements Relatives to Gold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' NIC XV Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=', Springer Proceedings in Physics 219, 381-384 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
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+page_content=' [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Ziegler, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=', Ziegler, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' & Biersack, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='. SRIM – The stopping and range of ions in matter (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
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+page_content=' Instrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' and Methods B 268 1818-1823 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
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+page_content='nimb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='091 [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Feinberg, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Quasi-stellar neutrons from the 7Li(p,n)7Be reaction with an energy-broadened proton beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' C 85 055810 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='1103/PhysRevC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='055810 [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Lederer, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='  Definition of a standard neutron field with the 7Li(p,n)7Be reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
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+page_content=' Rev C 85 055809 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='1103/PhysRevC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='055809 [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Macías, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=', Fernández, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' & Praena, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' The first neutron time- of-flight line in Spain: Commissioning and new data for the definition of a neutron standard field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Rad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' and Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' 168, 108538 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
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+page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
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+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='Pelowitz,MCNPXUsersManualVersion2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='0—LA-CP05- 0369, LosAlamos National LaboratoryLACP,2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
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+page_content=' Macías, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
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+page_content=' & Praena, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' New data for the definition of neutron beams for Boron Neutron capture therapy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
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+page_content='  185 109474 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
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+page_content=' Kornilov, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=', Pronyaev, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' & Grimes, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Systematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Distortion Factor and Unrecognized Source of Uncertainties in Nuclear Data Measurements and Evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Metrology 2 1–18 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='3390/metrology2010001 [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Carlson, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Work on the neutron standards since the completion of the standards evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' EPJ Web of Conferences 239, 08001 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='1051/epjconf/202023908001 [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Praena, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' & Jiménez-Bonilla, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Two absolute integral measurements of the 197Au(n,γ) stellar cross-section and solution of the historic discrepancies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' EPJ Web of Conferences 239, 19002 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='1051/epjconf/202023919002 [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Pronyaev, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' 197Au(n,γ) Standard Cross Section and Experimental Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Report INDC(NDS)-0540 2009, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Available online: https://www-nds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='iaea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='org/standards-cm-oct- 2008/9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='PDF (accessed on 30 September 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Meeting on Neutron Data Standards, 12-16 October 2020, IAEA Headquarters, Vienna, Austria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' https://www-nds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='iaea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='org/index- meeting-crp/CM-NDS-2020-10/ [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Reifarth, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=', Neutron-induced cross sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' From raw data to astrophysical rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content=' Plus 133: 424 (2018) http://dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
+page_content='1140/epjp/i2018-12295-3' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FIT4oBgHgl3EQfcitV/content/2301.11266v1.pdf'}
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+SensePOLAR: Word sense aware interpretability for pre-trained
+contextual word embeddings
+Jan Engler∗
+RWTH Aachen
+jan.engler@rwth-aachen.de
+Sandipan Sikdar∗
+L3S Research Center
+sandipan.sikdar@l3s.de
+Marlene Lutz
+University of Mannheim
+Markus Strohmaier
+University of Mannheim, GESIS, CSH Vienna
+{marlene.lutz, markus.strohmaier}@uni-mannheim.de
+Abstract
+Adding interpretability to word embeddings
+represents an area of active research in text rep-
+resentation. Recent work has explored the po-
+tential of embedding words via so-called po-
+lar dimensions (e.g. good vs. bad, correct vs.
+wrong). Examples of such recent approaches
+include SemAxis, POLAR, FrameAxis, and
+BiImp. Although these approaches provide in-
+terpretable dimensions for words, they have
+not been designed to deal with polysemy, i.e.
+they can not easily distinguish between differ-
+ent senses of words. To address this limitation,
+we present SensePOLAR, an extension of the
+original POLAR framework that enables word-
+sense aware interpretability for pre-trained
+contextual word embeddings. The resulting in-
+terpretable word embeddings achieve a level
+of performance that is comparable to original
+contextual word embeddings across a variety
+of natural language processing tasks including
+the GLUE and SQuAD benchmarks. Our work
+removes a fundamental limitation of existing
+approaches by offering users sense aware in-
+terpretations for contextual word embeddings.
+1
+Introduction
+The overwhelming success of deep neural networks
+(DNN) in the last decade has been accompanied by
+increasing concerns about the lack of interpretabil-
+ity (Ribeiro et al., 2016). This problem is ampli-
+fied in the area of Natural Language Processing
+(NLP) where word embeddings are used as input
+to machine learning models instead of more clas-
+sical, understandable features. Traditional (static)
+word embedding models like Word2Vec (Mikolov
+et al., 2013) or Glove (Pennington et al., 2014),
+that create one embedding for each word, are cur-
+rently being replaced by contextual word embed-
+ding models like BERT (Devlin et al., 2019) which
+have achieved competitive performance in NLP
+∗Equal contribution
+benchmarks such as GLUE (Wang et al., 2018) and
+SQuAD (Rajpurkar et al., 2016).
+To improve interpretability, recent approaches
+such as SemAxis (An et al., 2018), POLAR
+(Mathew et al., 2020), FrameAxis (Kwak et al.,
+2021), and BiImp (¸Senel et al., 2022) have explored
+the potential of embedding words via polar dimen-
+sions (e.g. good vs. bad, correct vs. wrong). While
+these approaches have been useful for interpreting
+word vectors, they have not been designed to deal
+with polysemy, i.e. multiple senses of words.
+Objective: Addressing polysemy, in this paper we
+aim to enable word-sense aware interpretability for
+pre-trained contextual word embeddings.
+Approach: We base our approach on the origi-
+nal POLAR framework (Mathew et al., 2020) and
+the idea of semantic differentials (Osgood et al.,
+1957), which are psychometric scales between two
+antonym words, e.g. “right” ↔ “wrong”. Sense-
+POLAR extends POLAR (Mathew et al., 2020)
+to contextual word embeddings, and defines polar
+sense instead of polar word scales. This enables
+SensePOLAR to offer polar dimensions that distin-
+guish between the correctness sense of “right” and
+the direction sense of “right”, for example.
+Results: SensePOLAR enables word sense aware
+interpretability of contextual embeddings by select-
+ing polar sense dimensions that align reasonably
+well with human judgements, as demonstrated in
+survey experiments. SensePOLAR exhibits com-
+petitive performance on various NLP tasks where
+it is used as input features for a separate model
+(feature-based approach) as well as directly inte-
+grated in the model itself (fine-tuning approach).
+Contributions: SensePOLAR introduces the no-
+tion of sense aware interpretations. To the best of
+our knowledge, SensePOLAR represents the first
+(semi-) supervised method that enables word sense
+aware interpretability for contextual word embed-
+dings. SensePOLAR is publicly available1.
+1https://github.com/JanEnglerRWTH/
+arXiv:2301.04704v1  [cs.CL]  11 Jan 2023
+
+Figure 1: SensePOLAR overview.
+Pre-trained contextual word embeddings are transformed into an inter-
+pretable space where the word’s semantics are rated on scales individually encoded by opposite senses such as
+“good”↔“bad”. The scores across the dimensions are representative of the strength of relationship (between word
+and dimension) which allows us to rank the dimensions and thereby identify the most discriminative dimensions
+for a word. In this example, the word “wave” is used in two senses: hand waving and ocean wave. SensePO-
+LAR not only generates dimensions that are representative of individual contextual meanings, the alignment to
+the respective sense spaces also aligns well with human judgement. SensePOLAR generates neutral scores for
+dimensions not related to the word in the given context (e.g., “idle”↔“work”, “social”↔“unsocial”). We follow
+the WordNet convention to represent a particular sense of a word. For example, “Tide.v.01” represents the word
+“tide” in the sense of surge (rise or move forward).
+2
+SensePOLAR
+The key idea of SensePOLAR is to transform pre-
+trained word embeddings into an interpretable,
+sense aware space. In this space, each dimension
+represents a scale on which words are rated, in-
+spired by the semantic differential technique (Os-
+good et al., 1957). In a departure from the existing
+approaches, we define opposite senses for the poles
+of these scales (e.g. “left direction” ↔ “right di-
+rection”), as opposed to opposite words (e.g. “left”
+↔ “right”), as used in Mathew et al. (2020).
+Given a contextual word embedding model M,
+the interpretable embeddings are obtained through
+the following steps. 1) We use M to obtain the
+(non-interpretable) contextual embedding space. 2)
+We obtain polar senses with contextual information
+from an oracle. 3) We proceed with generating
+representative sense embeddings from which we
+4) construct the interpretable polar sense space. 5)
+The original embedding is transformed into the po-
+lar sense space, which enables interpretation with
+regard to opposite sense pairs. We illustrate each
+step in figure 1 and elaborate them next.
+1. Obtaining contextual embeddings: To obtain
+the embedding of a particular word, we forward the
+word with its context, i.e. an example sentence, to
+the embedding model M. The embedding of the
+corresponding word can then be retrieved from the
+output of M. Because most models deploy sub-
+word tokenization algorithms, such as WordPiece
+SensePOLAR
+(Wu et al., 2016), embeddings of only subword
+tokens, rather than entire words, are generated by
+the contextual embedding models. This provides
+for obtaining representations for out-of-vocabulary
+words but, at the same time, makes embeddings of
+even common words not directly available. Follow-
+ing existing literature (McCormick and Ryan, 2019;
+Bommasani et al., 2020), we compute the embed-
+ding of a word by averaging over the embeddings
+of the constituent tokens.
+2. Selecting opposite polar senses: Each dimen-
+sion in the interpretable space corresponds to a
+scale spanned by opposite polar senses, which we
+define as a polar sense dimension. We assume
+that the poles and corresponding contexts are pro-
+vided by an oracle. In this paper, we use Word-
+Net (Miller, 1995) as an oracle, since the database
+already provides senses, contexts and antonyms
+for many words. Each sense of a word is repre-
+sented by a unique identifier, e.g. “Right.r.0” (a
+convention followed in WordNet) encodes “right”
+in the sense of direction. From over 6000 sense-
+antonym pairs that are available in WordNet, we
+use only a subset (1763) that are annotated with
+example sentences for both words. After various
+post-processing steps (cf. Appendix), these exam-
+ple sentences are used as context in step 3.
+3. Generating polar sense embeddings: We pro-
+pose to generate polar sense embeddings for each
+sense that is chosen by the oracle. Let w denote the
+word of interest and s the word-sense. Furthermore,
+
+: Input: Pretrained contextual embeddings
+SensePOLAR
+Output: Sense aware interpretable embeddings
+Wave(x)
+(1)
+ Polar sense space
+Transformation
+ai = S-i - Si
+Pc = (αT)-1xc
+Pc2
+ail
+Tide.v.01 - Ebb.v.01
+-0.2
+-0.1 0.0
+0.1
+0.2
+-0.25-0.1250.00.1250.25
+: C1: She is waving her hand to the
+crowd.
+ the rocks.
+(4)
+(5) 
+ Dramatic
+Undramatic Tide.v.01.....
+Ebb.v.01
+-a.01
+.a.01
+Xc1
+Xc2
+Polar sense embeddings
+(3)
+: Divided.a.01
+.United.a.01 Fresh.a.06
+[0.12
+Salty.a.02:
+0.25
+m
+-0.8
+0.18
+Pre-trained
++Tide.v.01
+Tide.01
+Offshore
+Onshore :
+0.32
+0.2
+m
+m
+Down.r.02
+Up.r.04
+ embeddings
+.r.01
+-r.01
+j=1
+:
+:
+oracle
+Level.v.02
+Raise.v.09
+Relax.v.01
+(2)
+Tense.v.02 :
+(s-1,S1)
+ Tide.v.01, Ebb.v.01
+(s-2, S2)
+.
+ ldle.v.02, Work.v.01
+Idle.v.02
+0
+0
+Unsocial
+Work.v.01
+Socia.a.01
+.a.01let Cs = {c1, ..., cm} be m context examples for
+the sense s, which we assume are provided by the
+oracle. In each context c ∈ Cs, the word w is used
+in the sense s, e.g. “A strange sound came from the
+right side.” for the word “right” in the sense of di-
+rection (i.e., we intend to embed “Right.r.04”). We
+create polar sense embeddings in two steps. First,
+we input m context examples for a sense s to the
+embedding model M and retrieve an embedding
+ws
+c ∈ Rd of the word w for each context c ∈ Cs. If
+the word w consists of several subword tokens, the
+individual subword embeddings are averaged. We
+also allow for senses consisting of multiple words,
+e.g. “keep track” ↔ “lose track”, where we again
+average the embeddings of the individual tokens.
+Second, we compute the average of the contextual
+word embeddings per sense and define it as the
+sense embedding s ∈ Rd:
+s = 1
+m
+m
+�
+j=1
+ws
+cj
+(1)
+This is a rather straightforward way to represent
+individual senses of words in a (semi-) supervised
+manner. The method is dependent on the quality
+and the number of the example sentences provided
+by the oracle. We observe that more context ex-
+amples lead to a better and stable representation,
+but we usually achieve a satisfactory representation
+with already one suitable example sentence. This is
+motivated by the observations in Reif et al. (2019)
+which provide strong evidence that BERT positions
+the embeddings of senses in individual clusters
+in space and that these clusters are usually suffi-
+ciently spatially separated from each other. A polar
+sense dimension is represented by a pair of opposite
+senses (s−i, si) (e.g., “Right.a.02”, “Wrong.a.01”).
+4. Constructing a polar sense space:
+Given
+n
+polar
+sense
+dimensions
+S = ((s−1, s1), ..., (s−n, sn))
+with
+their
+con-
+texts C
+=
+((Cs−1, Cs1), (Cs−n, Csn)),
+we
+compute the polar sense embedding si for each
+sense si and corresponding context Csi, following
+equation 1.
+We now utilize the representations of individual
+senses to construct the interpretable polar sense
+space. Each polar sense dimension (si, s−i) ∈ S
+defines an interpretable scale, which is encoded by
+the direction vector ai, defined as follows:
+ai = s−i − si
+(2)
+The direction vectors for all polar sense dimen-
+sions are then stacked to obtain the change of basis
+matrix a ∈ Rn×d for the interpretable polar sense
+space.
+5.
+Transformation to interpretable embed-
+dings: Finally, an embedding of a word x in a
+context c, xc can be transformed into the polar
+sense space in the following way. Given a repre-
+sents the change of basis matrix, we can compute
+the polar sense embedding pc following the rules
+of linear algebra:
+aT pc = xc
+(3)
+pc = (aT )−1 xc
+(4)
+The inverse of aT is computed by the Moore-
+Penrose generalized inverse (Ben-Israel and Gre-
+ville, 2003). The resulting contextual word embed-
+ding pc in the polar sense space is of dimension
+n × 1. The absolute value across axis ai corre-
+sponds to the word’s rating on the scale between
+the polar senses (s−i, si) and the sign represents
+the direction of alignment to a particular pole. A
+higher absolute value represents a stronger relation-
+ship to the corresponding polar sense dimension.
+This allows us to obtain the most expressive polar
+sense dimensions for a given word and context.
+Normalization: As a post-processing step, we av-
+erage the word embeddings of all words (from a
+corpus) to get the average-word embedding in our
+interpretable space and subtract this average word
+embedding from each embedding when analyzing
+interpretability. This also allows us to deal with
+the anisotropic nature of contextual word embed-
+dings (Ethayarajh, 2019) whereby the embeddings
+are not randomly distributed but rather lay on a
+high-dimensional cone in space.
+3
+Evaluation
+Note that while SensePOLAR allows for deploy-
+ment across any contextual word embedding model,
+in this work, we consider BERT (Devlin et al.,
+2019) as our model for illustration.
+We con-
+sider a BERT-base model which utilizes 12 trans-
+former (Vaswani et al., 2017) encoder layers and
+generates embeddings of size 768. The pre-trained
+BERT-base model was downloaded from Hugging-
+face2. In addition, we use WordNet as our oracle
+2https://huggingface.co/docs/
+transformers/model_doc/bert,
+we
+used
+“bert-
+base-uncased”
+
+ML Model
+SVM
+FFN
+Task
+Base
+SensePOLAR
+Base
+SensePOLAR
+Sport
+0.941
+0.935↓ 0.6%
+0.961
+0.956↓ 0.5%
+Religion
+0.891
+0.848↑ 4.8%
+0.880
+0.894↑ 1.6%
+Computer
+0.770
+0.750↓ 2.6%
+0.763
+0.727↓ 4.7%
+Table 1:
+Performance of the original BERT (Base)
+embeddings and SensePOLAR embeddings on feature-
+based tasks with a support vector machine (SVM) and
+a feed-forward neural network (FFN) classifier. Sense-
+POLAR achieves performance comparable to the origi-
+nal BERT embeddings across all three tasks.
+with 1763 polar sense pairs.
+3.1
+Performance on downstream tasks
+The goal of SensePOLAR is to add interpretabil-
+ity to word embeddings without major losses in
+performance. Hence, we evaluate SensePOLAR
+on a wide range of NLP downstream tasks. We
+investigate whether replacing the original BERT
+embeddings with SensePOLAR embeddings has
+any effect on performance.
+3.1.1
+Feature-based tasks
+We analyze the effectiveness of SensePOLAR em-
+beddings in a “classical” NLP pipeline, where word
+embeddings are generated beforehand and are used
+as input-features to a separate machine learning
+model. We consider a binary text classification task
+utilizing the 20 Newsgroups dataset (Lang, 1995).
+The dataset consists of ∼ 20K news articles cov-
+ering 20 types of news. Our experiment follows
+the structure of Panigrahi et al. (2019), where we
+only consider the topics sports, computer and re-
+ligion. For each topic, an article must be clas-
+sified into one of two categories (“baseball” or
+“hockey” for sports, “IBM” or “Apple” for com-
+puter, “christianity” or “atheism” for religion). In
+table 1, we present the results in terms of accuracy
+across the three tasks. We use a support vector
+machine (SVM) and a 2-layer feed-forward neural
+network (FFN) as classifier models, which use the
+BERT and SensePOLAR embeddings as features.
+Across all three tasks, SensePOLAR achieves a
+level of performance that is comparable to the orig-
+inal embeddings.
+3.1.2
+Fine-tuning tasks
+Integrating SensePOLAR into fine-tuned mod-
+els: The models achieving state-of-the-art perfor-
+mances on different NLP tasks usually deploy a
+task specific network layer (usually a feed-forward
+network) on top of the embedding layers. The
+SQuAD 1.1
+SQuAD 2.0
+Metric
+Base
+SensePOLAR
+Base
+SensePOLAR
+EM
+86.92
+86.85↓ 0.07%
+80.88
+81.06↑ 0.22%
+F1
+93.15
+93.12↓ 0.03%
+83.87
+83.89↑ 0.02%
+Table 2: Results of fine-tuned BERT embeddings and
+with SensePOLAR transformed embeddings on the
+SQuAD benchmark. The results are competitive and
+even improve marginally after applying SensePOLAR.
+embedding layers and the task specific layers are
+then fine-tuned on the task specific dataset. Con-
+sequently, SensePOLAR embeddings need to be
+computed considering the fine-tuned version of the
+embeddings rather than the original pre-trained ver-
+sion. In this particular setting, we propose to uti-
+lize the embedding layer of the fine-tuned model
+(instead of the original pre-trained version) to con-
+struct the polar sense space. Given an input text,
+each token (including the [CLS] token) can then
+be transformed to a corresponding SensePOLAR
+embedding. Because of the dimensionality mis-
+match between the original embedding and the
+transformed SensePOLAR embedding, we replace
+the first layer of the task specific feed-forward net-
+work and re-fine-tune it on the task specific dataset.
+Note that the weights of the underlying embedding
+model are frozen during this re-fine-tuning proce-
+dure. This is computationally inexpensive as only
+the task specific layers need to be trained, which
+are often just 1 or 2-layered feed-forward network.
+Question answering: This task deals with locat-
+ing an answer to a question in a given paragraph
+and is often referred to as a reading comprehen-
+sion task. We consider the SQuAD benchmark,
+including both SQuAD1.1 (Rajpurkar et al., 2016)
+and SQuAD2.0 (Rajpurkar et al., 2018) versions.
+The BERT-based QA model consists of the em-
+bedding module followed by a span-classification
+head, which is a 1-layer feed-forward network. The
+model takes both the question text and the passage
+text as input. The [CLS] token (a special token
+generated by BERT for classification tasks) ob-
+tained from the embedding module is then passed
+onto the span-classification head, which predicts
+the start and the end position of the span in the text
+passage that contains the answer.
+The polar sense space is computed using the
+BERT embedding module already fine-tuned on
+the task. The [CLS] token is then transformed
+into the interpretable space before being passed
+on to the span-classification head. This classifica-
+tion head, however, needs to be replaced (to match
+
+the dimension of the transformed embedding) and
+re-trained. In table 2, we report the exact match
+(EM) and F1 scores with the original BERT (base)
+and the SensePOLAR model. SensePOLAR again
+achieves comparable performance, even marginally
+outperforming the base model for SQuAD2.0.
+Natural language understanding:
+We utilize
+the General Language Understanding Evaluation
+(GLUE) benchmark, which is designed for compar-
+ing models on the task of natural language under-
+standing (NLU). It consists of nine tasks that cover
+a diverse range of text genres, dataset sizes, and de-
+grees of difficulty (Wang et al., 2018). We point the
+reader to the original paper by Wang et al. (2018)
+for a general overview of the tasks. To evaluate
+SensePOLAR, we follow a similar procedure to the
+previous question answering task. The polar sense
+space is computed using the underlying BERT em-
+bedding module, already fine-tuned on the task.
+This is followed by transforming the [CLS] to-
+ken into the interpretable polar sense space. The
+feed-forward layers on top are then replaced and re-
+trained. In table 3, we report the results on GLUE
+Task
+Train size
+Metric
+Base
+SensePOLAR
+CoLa
+8.5k
+Matthew’s corr.
+56.62
+55.05 ↓ 2.77%
+SST-2
+67k
+Accuracy
+91.51
+91.40 ↓ 0.12%
+MRPC
+3.7k
+Accuracy
+84.31
+82.84 ↓ 1.74%
+F1
+89.00
+87.41 ↓ 1.79%
+STS-B
+7k
+Person corr.
+89.03
+84.17 ↓ 5.46%
+QQP
+364k
+Accuracy
+90.59
+90.15 ↓ 0.49%
+F1
+87.29
+86.82 ↓ 0.54%
+MNLI
+393k
+Accuracy
+84.49
+84.04 ↓ 0.53%
+QNLI
+105k
+Accuracy
+91.54
+91.58 ↑ 0.04%
+RTE
+2.5k
+Accuracy
+63.18
+59.93 ↓ 5.14%
+WNLI
+634
+Accuracy
+56.34
+56.34 ↑↓0%
+Table 3: Comparison of the fine-tuned BERT model
+and the re-fine-tuned BERT model with SensePO-
+LAR embeddings. Mostly, comparable performance is
+achieved. Slightly worse performance is achieved for
+tasks with smaller training datasets.
+tasks with both the original BERT (Base) and the
+SensePOLAR embeddings. SensePOLAR achieves
+competitive performances across all the tasks.
+The results indicate that SensePOLAR is able
+to achieve interpretability without compromising
+performance on downstream tasks.
+3.2
+Interpretability
+We turn our attention to evaluating the interpretabil-
+ity of SensePOLAR.
+Qualitative analysis: We transform the embed-
+dings of the words into a polar sense space and an-
+alyze the position/rating (determined by the signed
+value on that dimension) of different words on se-
+lected dimensions. More specifically, we consider
+a context in which the word is used and pass it
+through the BERT module. The embedding corre-
+sponding to the target word (note that BERT gener-
+ates embeddings corresponding to each word in the
+context) is then transformed into the polar sense
+space through the base change operation. Ana-
+lyzing the ratings of words in a selected dimen-
+sion, allows us to demonstrate the advantages of
+interpreting word embeddings in terms of polar
+sense dimensions. We first consider the dimension
+“Black.a.02”↔“White.a.02” (in the sense of ethnic-
+ity) and transform the embeddings of celebrities
+and nationalities on this dimension. The obser-
+vations mostly match the ethnicities of the indi-
+viduals (see figure 2(a)). We also consider words
+such as milk, coal etc. which are not related to
+“Black.a.02”↔“White.a.02” in the sense of ethnic-
+ity and observe their corresponding scores in this
+dimension to be neutral. However, their representa-
+tion on the dimension “Black.a.01”↔“White.a.01”
+(in the sense of color), captures their semantic well.
+This demonstrates the benefits of using polar senses
+as dimensions instead of words, which would have
+failed to differentiate between the two senses.
+We also consider other dimensions and present
+the connotative meanings of words across these di-
+mensions in figure 2(b) which leads to interesting
+observations. For example, “politician”, “meet-
+ing” are more aligned towards “Hate.a.01” (in the
+sense of disgust). Similarly, “murder” and “devil”
+are aligned towards “Wrong.a.01” (in the sense of
+morality).
+In addition to picking out interesting dimensions
+by hand, we also propose to evaluate interpretabil-
+ity by investigating the most descriptive dimensions
+of a given word. The dimensions for a word are
+ranked based on the absolute value across all di-
+mensions. Ideally, the top dimensions should be
+the most descriptive and fitting for the word. For
+illustration, we provide example words and the cor-
+responding top-5 dimensions in figure 3. The top
+dimensions mostly have a high semantic similarity
+with the word, and they also reasonably align with
+human judgement.
+Survey experiment: For evaluating interpretabil-
+ity on a larger scale, we follow the approach
+by Mathew et al. (2020) and conduct a human judg-
+ment survey. We utilize the crowdsourcing plat-
+
+(a) Benefits of sense dimensions
+(b) Connotative meanings across dimensions
+Figure 2: Illustration of polar sense dimensions. (a) SensePOLAR allows for interpretability along multiple senses.
+“black”↔“white” in the sense of ethnicity (top) can be differentiated from “black”↔“white” in the sense of color
+(bottom). Words like “snow”or “coal” - which are not semantically related to ethnicity - score neutral on the
+upper scale while being clearly distinguishable on the lower scale. (b) The connotative meanings of words can
+also be investigated through SensePOLAR. For example, “politician” is associated with “hate” while “mother” is
+associated with “love”.
+(a) Run
+(b) Music
+(c) Right
+(d) Happy
+Figure 3: Illustration of SensePOLAR embeddings. We show the top 5 dimensions as selected by SensePOLAR
+for exemplary words. The pre-trained embeddings are obtained using BERT. The top dimensions and the word’s
+rating/alignment to the pole reasonably align with human judgement (cf. table 4).
+form Clickworker3 where we randomly select 15
+common English nouns, verbs and adjectives (with
+short context) and compute their interpretable em-
+bedding with SensePOLAR. Then, for each word,
+we extract the top-5 polar sense dimensions (mea-
+sured in absolute value) and additionally five ran-
+dom dimensions from the lower 50%. These 10
+dimensions are then presented to the participants
+in a random order. Participants are asked to select
+five dimensions that are most representative of a
+given word and to rate each dimension based on
+their alignment to one of the poles on a likert scale
+between 1 and 7 (with 4 as neutral). Each word
+is assigned 3 annotators. For a given word, each
+dimension is assigned a score depending on how
+many annotators found it relevant. We then select
+the top 5 dimensions based on this score and we
+consider them as the ground-truth dimension to
+which we compare the ones selected by SensePO-
+3https://www.clickworker.com/
+LAR.
+In table 4, we present the conditional probability
+of the top k dimensions selected by SensePOLAR
+to be also chosen by the human annotators. In the
+same table, we also report the random chance of
+getting selected. For the top-1 dimension, agree-
+ment is roughly 87% and for the top-2 dimension
+it is still around 65%, indicating strong alignment
+with human judgment. We also found that the par-
+ticipant’s ratings on these dimensions were the (ab-
+solute) highest, showing that the word is strongly
+connected to one of the polar senses.
+Differentiating between senses: We also evalu-
+ate the interpretability of SensePOLAR in terms
+of its ability to differentiate between two senses of
+a given word. As an illustrative example, we con-
+sider the word “right” in the sense of both direction
+and correctness (refer to figure 4). The selected
+polar sense dimensions are indeed representative
+of the correct sense. Note that the original POLAR
+
+Kobe Bryant
+Angela Merkel
+German
+Black.a.02
+White.a.02
+-0.08
+-0.04
+0.0
+0.04
+0.08
+Barack Obama
+MichaelJackson
+Swedish
+Space
+Paper
+Snow
+Black.a.01
+White.a.01
+-0.03
+-0.015
+0.0
+0.015
+0.03
+Coal
+MilkPolitician
+Meeting
+King
+Girlfriend
+Mother
+Hate.a.01
+Love.n.01
+-0.03
+-0.02
+-0.01
+0.0
+0.01
+0.02
+0.03
+Work
+Baby
+Bragging
+Devil Murder
+Kicked
+Praying
+Donating
+Wrong.a.02
+Right.a.03
+-0.03
+-0.02
+-0.01
+0.0
+0.01
+0.02
+0.03
+Killed Lying
+Steal Jesus
+LawfulI like to run fast.
+-0.2
+-0.1
+0.1
+0.2
+Running.a.04
+Standing.a.04
+Idle.v.01
+Run.v.13
+Passing.a.02
+Running.a.03
+Delay.v.01
+Rush.v.03
+Stoppable.a.01
+Unstoppable.a.01Music is life
+-0.15
+-0.05
+0.05
+0.15
+Musical.a.02
+Unmusical.a.01
+Atomism.n.02
+Holism.n.01
+Harmonic.a.01
+Non-harmonic.a.01
+Experience.n.01
+Inexperience.n.01
+Absence.n.01
+Presence.n.01The answer is right
+-0.15
+-0.05
+0.05
+0.15
+Right.a.05
+Wrong.a.05
+Due.a.01
+Undue.a.01
+False.a.01
+True.a.01
+Certain.a.03
+Uncertain.a.02
+Dead.a.02
+Live.a.02am happy to see you
+-0.15
+-0.05
+0.05
+0.15
+Unwelcome.a.01
+Welcome.a.01
+Glad.a.01
+Sad.a.01
+Insincerely.r.01
+Sincerely.r.01
+Evil.n.03
+Go0d.n.02
+Troubled.a.01
+Untroubled.a.01Top-k
+1
+2
+3
+4
+5
+SensePOLAR
+0.876
+0.558
+0.312
+0.187
+0.093
+Random
+0.5
+0.22
+0.083
+0.023
+0.004
+Table 4: Alignment with human judgement. The condi-
+tional probability of the top-k dimensions selected by
+SensePOLAR to be also chosen by the human anno-
+tators, together with the random chance of guessing.
+Significantly higher probabilities than random chance
+are achieved, indicating that the chosen dimensions
+are meaningful and match human judgment reasonably
+well.
+Figure 4: Top-5 dimensions of the word “left” for two
+different contexts in the sense of going away (left) and
+direction (right). The top dimensions are indeed dif-
+ferent for the different word-senses and are reasonably
+descriptive of the correct sense.
+framework would not be able to differentiate be-
+tween the senses, given it generates exactly one
+embedding for a given word.
+We follow up with another human judgement ex-
+periment where we present the top-10 polar sense
+dimensions of words with multiple meanings, to-
+gether with the word’s score on these dimensions,
+to the annotators. The task is to identify in which
+sense the target word is being used in. We limit this
+experiment to only two common senses for each
+word and present the WordNet definitions as the
+answer possibilities. Thus, by random guessing,
+an accuracy of 50% would be achieved. For our
+hand-picked examples, the correct sense was iden-
+tified in around 95% of the examples. The average
+inter-participant agreement on the result is around
+78%.
+The results in this section indicate that Sense-
+POLAR is indeed able to add interpretability to
+contextual word embeddings and that it aligns rea-
+sonably well with human judgement.
+4
+Applications
+We discuss two potential use cases: (i) analyzing
+bias in contextual embeddings and (ii) explaining
+model decisions on downstream tasks.
+4.1
+Bias Analysis
+Contextual embedding models have been shown
+to capture bias from training data (Nadeem et al.,
+2021; Ahn and Oh, 2021; Bartl et al., 2020; Moza-
+fari et al., 2020). We demonstrate that SensePO-
+LAR can be used for analyzing biases by exam-
+ining the polar sense dimensions which differ the
+most. For our evaluation, we utilize examples from
+the Stereoset dataset from Nadeem et al. (2021).
+Next sentence prediction: BERT can be used to
+predict the probability of a second sentence fol-
+lowing a first one.
+Consider the sentence pair
+“Hispanic people are very diverse.
+All people
+like that are criminals.”.
+Replacing the word
+“Hispanic” by “American” leads to significant de-
+crease in BERT’s confidence score.
+We pick
+the SensePOLAR embeddings of the words “His-
+panic” and “American” from these sentences and
+compute the polar sense dimensions in which
+they differ the most. Noticeably, the most dis-
+criminative dimensions mostly encode senses re-
+lated to being or not being a legal citizen such
+as “Documented.a.01”↔“Undocumented.a.01” or
+“Irregular.a.01”↔“Regular.a.01”. We observe that
+the word “Hispanic” is rated on the side of irregu-
+lar, undocumented, unrighteous and unconcerned,
+as opposed to “American”.
+Masked LM: We conduct a similar experiment on
+the task of masked language modeling. Consider
+the sentence “Because he is black, he works as a
+[MASK].”. Replacing “black” with “white” leads
+to vastly different predictions for the [MASK] to-
+ken. We apply SensePOLAR to analyze the polar
+sense dimensions of the [MASK] token in both con-
+texts. We find that the most discriminative dimen-
+sion is “Employed.a.01”↔“Unemployed.a.01”, in-
+dicating that BERT predicts a word more related to
+unemployed when the word “black” is used.
+4.2
+Explaining classifier results
+SensePOLAR can further be deployed to ex-
+plain decisions of classifier models that make
+use of contextual word embeddings.
+To il-
+lustrate this we consider binary sentiment pre-
+diction (positive or negative) on the SST-2
+dataset (Socher et al., 2013). We sample and av-
+erage the SensePOLAR transformed [CLS] to-
+kens from the positive and negative class sepa-
+rately and examine the most discriminative dimen-
+sions. We find the most discriminative dimensions
+to be “sharp”↔“dull”, “unpleasant”↔“pleasant”,
+“endemic”↔“cosmopolitan”, “soft”↔“loud” and
+“tasteless”↔“tasteful”. BERT is more likely to clas-
+sify a review as negative when it is seen as more
+sharp, unpleasant, endemic, and tasteless.
+
+He left the room.
+He looked right and left.
+-0.15-0.075
+0.0750.15
+-0.25-0.125
+0.125
+0.25
+Exit.v.02
+Enter.v.01
+Left..02 -...
+Right.r.01
+Inner.a.02
+Outer.a.01
+Here.r.02
+There.r.01
+Pop in.v.01
+Pop out.v.01
+Downwind.r.01
+Upwind.r.01
+Shut.a.01
+Open.a.01
+Early.r.02
+Late.r.01
+Disappear.v.01
+Appear.v.01
+Foolish.a.01
+Wise.a.015
+Discussion
+Next, we discuss issues pertinent to SensePOLAR.
+Generalizability: SensePOLAR is applicable to
+any pre-trained contextual embedding model. It
+can also be deployed on top of any of the con-
+stituent transformer layers. This allows for not only
+comparing different contextual word embedding
+models in terms of interpretability or bias analysis
+but also performing similar analysis across trans-
+former layers of the same embedding model.
+Extension to other languages:
+SensePOLAR
+should also be extendable to other languages. The
+only requirement would be to be able to obtain
+suitable sense antonym pairs as well as example
+contexts via an oracle.
+Interpretable decision-making. In section 4.2,
+we demonstrated how SensePOLAR could be used
+to explain decisions of text classifiers. However,
+the design of SensePOLAR allows for deployment
+across any other downstream task as well. This
+is in contrast with existing interpretability meth-
+ods which are often developed with a particular
+downstream task in mind.
+Quantitative
+comparison
+with
+other
+inter-
+pretability methods: An ideal evaluation set up
+would have been to quantitatively compare Sense-
+POLAR to other interpretability methods. How-
+ever, as pointed out in the existing literature (Sun-
+dararajan et al., 2017; Sikdar et al., 2021), when
+two models provide different interpretations, it is
+difficult to judge if one is better than the other. In-
+volving humans makes it even harder, as one now
+needs to tease out a person’s own subjective bi-
+ases. Hence, our crowdsourcing experiments were
+only designed to understand the efficacy of Sense-
+POLAR. Nevertheless, we provide a qualitative
+comparison with the existing methods in section 6.
+SensePOLAR variants: Other variants of Sense-
+POLAR can be devised as well. For example, lin-
+ear transformation instead of base change could
+be used for obtaining SensePOLAR embeddings.
+However, we observed that linear transformation
+does not preserve the original structure of the em-
+bedding space, where the different senses of words
+are already sufficiently separated. One can also ex-
+periment with different normalization techniques,
+such as scaling or standardization. In this paper, we
+concentrated on an exhaustive evaluation setup to
+include more downstream tasks and crowdsourcing
+experiments rather than exploring other variants.
+We consider all the above variants promising av-
+enues for future work.
+6
+Related work
+In this section, we briefly summarize previous re-
+search on enabling interpretability for both static
+and contextual word embeddings.
+Unsupervised methods: The key idea for this
+class of methods is to create sparse embeddings,
+which is achieved through a post-processing step
+on top of the embeddings (Murphy et al., 2012;
+Faruqui et al., 2015; Luo et al., 2015). Addition-
+ally, the idea of creating sparse embeddings can
+also be integrated into the word embedding train-
+ing itself, as demonstrated in Sun et al. (2016);
+Chen et al. (2017). The meaning of the dimen-
+sions are assigned by the model itself (hence un-
+supervised) and are often intelligible to humans.
+Notably, Word2Sense (Panigrahi et al., 2019) pro-
+poses to create sparse non-negative vectors through
+Latent Dirichlet Allocation (LDA). Each dimen-
+sion is assigned a meaning, which is retrieved from
+a training corpus. The methods discussed above
+are specific to static word embeddings. Berend
+(2020) extends some of these ideas to contextual
+word embeddings.
+(Semi-)supervised methods: This class of meth-
+ods aims at adding interpretability to word embed-
+dings by first defining an interpretable space and
+then transforming the pre-trained embeddings to
+this space. In this space, each dimension spans be-
+tween two pole words. While SemAxis (An et al.,
+2018) proposes to use antonym pairs retrieved from
+ConceptNet (Speer et al., 2017), the POLAR frame-
+work (Mathew et al., 2020) utilizes the semantic
+differential technique pioneered by Osgood (Os-
+good et al., 1957). Similarly, BiImp (¸Senel et al.,
+2022) proposes to use opposite semantic concepts
+as poles. Not only are the dimensions interpretable,
+these methods are computationally less expensive.
+Embedding geometry: Part of the existing re-
+search has focussed on analyzing the position of
+words in the embedding space. Ethayarajh (2019)
+provides evidence that the BERT embeddings are
+not uniformly distributed in the space, but rather
+lay on a high dimensional cone. Reif et al. (2019)
+demonstrate that BERT is able to separate fine-
+grained senses of words by placing them in differ-
+ent locations in space. Similar observations are
+made by Schmidt and Hofmann (2020) as well.
+Probing: The goal in probing tasks is to determine
+whether some syntactic or semantic knowledge is
+
+encoded in the produced word embeddings (or at-
+tention heads). The embeddings (or attentions)
+are fed into a simple linear classifier to predict
+unseen linguistic properties. The performance of
+the classifier is indicative of the extent to which
+these linguistic properties are encoded in the em-
+beddings. For BERT, these probing experiments
+have demonstrated that the layers on the top are
+more contextual (Ethayarajh, 2019) and the layers
+at the center contain a large amount of syntactic
+information (Hewitt and Manning, 2019; Goldberg,
+2019; Jawahar et al., 2019; Chi et al., 2020). The
+semantic information is generally spread across the
+entire network (Tenney et al., 2019; Zhao et al.,
+2020; Lin et al., 2019).
+Visual explanations: Finally, recent work has also
+considered visualizing attention in transformer lay-
+ers to explain contextual language models (Hoover
+et al., 2020; Vig, 2019). Similarly, visualizing
+word embeddings can also aid in explaining what a
+model learns as demonstrated in Liu et al. (2017);
+Heimerl and Gleicher (2018); Boggust et al. (2022)
+(static) and Sevastjanova et al. (2021); Berger
+(2020) (contextual).
+Comparison with SensePOLAR: To the best of
+our knowledge, SensePOLAR is the first (semi-)
+supervised method for enabling interpretability for
+contextual word embeddings. We extend the idea
+of rating the meaning of words on a scale - defined
+between two polar words - to two polar senses.
+SensePOLAR can also be integrated into task spe-
+cific fine-tuned models as well. In comparison
+to unsupervised methods, our method enables us
+to understand the individual dimensions and ac-
+tively choose and adjust polar sense dimensions
+for the task at hand. While probing and visualiza-
+tion methods can reveal whether specific linguistic
+information is encoded in the embeddings, analyz-
+ing the embedding geometry can help in uncover-
+ing the model characteristics. However, none of
+these methods can directly augment interpretabil-
+ity to the embeddings. Since with SensePOLAR
+interpretability is directly incorporated into the em-
+beddings, it is applicable to any downstream task.
+This is in contrast to most of the existing methods,
+which are often specific to embedding methods or
+downstream tasks.
+7
+Conclusion
+We introduced SensePOLAR which enables word
+sense aware interpretability for contextual word
+embeddings. The key idea is to project word em-
+beddings onto an interpretable space which is con-
+structed from polar sense pairs obtained from an
+oracle. SensePOLAR extends the original POLAR
+framework developed for static word embeddings
+to contextual word embeddings. We demonstrated
+that the obtained interpretable embeddings align
+well with human judgement. Moreover, SensePO-
+LAR could be integrated into fine-tuned models
+and can be deployed to specific applications like
+bias analysis and explaining prediction results of
+classifier models.
+8
+Limitations
+Underlying embedding models: SensePOLAR
+uses embeddings of polar senses to build an inter-
+pretable subspace. Thereby, we assume that the un-
+derlying embedding model captures the semantics
+of words from which we construct the sense em-
+beddings. As a result, SensePOLAR is dependent
+on the quality of the underlying contextual word
+embedding model. Compared to the original PO-
+LAR framework proposed in (Mathew et al., 2020),
+the present approach also depends on the ability of
+the model to capture individual word-senses with
+sufficient accuracy.
+Presence of bias: Naturally, our model inherits the
+biases of the underlying embedding model. The
+word “physics”, for example, has a high rating
+towards “male” on the polar sense scale of “male”
+↔ “female”. However, SensePOLAR could be
+used to make these biases visible and potentially
+help to remove them. One can also tap into state-of-
+the-art bias mitigation methods (e.g. Ahn and Oh
+(2021); Bartl et al. (2020); Mozafari et al. (2020))
+to address this issue.
+Dependence on oracles: The construction of the
+polar sense space depends largely on the choice of
+polar opposite senses and the quality of the con-
+text examples. Using the example of WordNet, we
+have shown how a general model can be created.
+However, we observed that rare senses and low-
+quality example sentences can lead to poor results.
+Moreover, it is not clear how the optimal number
+of polar dimensions can be determined. Empiri-
+cally, we observed that adding more pairs does not
+necessarily lead to improvement in performance.
+For a particular downstream task, it may also be
+appropriate to discard polar sense pairs that are not
+relevant to the task (e.g. if they never occur in the
+corpus).
+
+Counter-intuitive rating of words. We find that
+in some cases the rating of words on the polar sense
+scales does not coincide with human judgement.
+The word “doctor”, for example, is highly skewed
+towards “guilty” on a scale from “innocent” ↔
+“guilty”, which does not match the typical percep-
+tion of doctors. We believe this is because word
+embeddings by design are shaped by their context.
+There are probably more articles and stories about
+“guilty doctors” than “innocent doctors”, because
+these stories would be less interesting.
+Acknowledgements
+Sandipan Sikdar was supported in part by RWTH
+Aachen Startup Grant No. StUpPD384-20.
+References
+Jaimeen Ahn and Alice Oh. 2021. Mitigating language-
+dependent ethnic bias in BERT. In Proceedings of
+the 2021 Conference on Empirical Methods in Nat-
+ural Language Processing, pages 533–549, Online
+and Punta Cana, Dominican Republic.
+Jisun An, Haewoon Kwak, and Yong-Yeol Ahn. 2018.
+SemAxis: A lightweight framework to characterize
+domain-specific word semantics beyond sentiment.
+In Proceedings of the 56th Annual Meeting of the As-
+sociation for Computational Linguistics (Volume 1:
+Long Papers), pages 2450–2461, Melbourne, Aus-
+tralia.
+Marion Bartl, Malvina Nissim, and Albert Gatt. 2020.
+Unmasking contextual stereotypes: Measuring and
+mitigating BERT’s gender bias. In Proceedings of
+the Second Workshop on Gender Bias in Natural
+Language Processing, pages 1–16.
+Adi Ben-Israel and Thomas NE Greville. 2003. Gen-
+eralized inverses:
+Theory and Applications, vol-
+ume 15. Springer Science & Business Media.
+Gábor Berend. 2020.
+Sparsity makes sense: Word
+sense disambiguation using sparse contextualized
+word representations. In Proceedings of the 2020
+Conference on Empirical Methods in Natural Lan-
+guage Processing, pages 8498–8508.
+Matthew Berger. 2020. Visually analyzing contextual-
+ized embeddings. In 2020 IEEE Visualization Con-
+ference (VIS), pages 276–280. IEEE.
+Angie Boggust, Brandon Carter, and Arvind Satya-
+narayan. 2022.
+Embedding comparator: Visualiz-
+ing differences in global structure and local neigh-
+borhoods via small multiples. In 27th International
+Conference on Intelligent User Interfaces, pages
+746–766.
+Rishi Bommasani, Kelly Davis, and Claire Cardie.
+2020.
+Interpreting Pretrained Contextualized
+Representations via Reductions to Static Embed-
+dings.
+In Proceedings of the 58th Annual Meet-
+ing of the Association for Computational Linguistics
+, pages 4758–4781.
+Yunchuan Chen, Ge Li, and Zhi Jin. 2017. Learning
+sparse overcomplete word vectors without interme-
+diate dense representations. In International Confer-
+ence on Knowledge Science, Engineering and Man-
+agement, pages 3–15. Springer.
+Ethan A Chi, John Hewitt, and Christopher D Manning.
+2020.
+Finding universal grammatical relations in
+multilingual bert. arXiv preprint arXiv:2005.04511.
+Jacob Devlin, Ming-Wei Chang, Kenton Lee, and
+Kristina Toutanova. 2019.
+BERT: Pre-training of
+deep bidirectional transformers for language under-
+standing. In Proceedings of the 2019 Conference of
+the North American Chapter of the Association for
+Computational Linguistics: Human Language Tech-
+nologies, Volume 1 (Long and Short Papers), pages
+4171–4186.
+Kawin Ethayarajh. 2019. How contextual are contextu-
+alized word representations? Comparing the geom-
+etry of BERT, ELMo, and GPT-2 embeddings. In
+Proceedings of the 2019 Conference on Empirical
+Methods in Natural Language Processing and the
+9th International Joint Conference on Natural Lan-
+guage Processing, pages 55–65, Hong Kong, China.
+Manaal Faruqui, Yulia Tsvetkov, Dani Yogatama, Chris
+Dyer, and Noah A. Smith. 2015. Sparse overcom-
+plete word vector representations. In Proceedings
+of the 53rd Annual Meeting of the Association for
+Computational Linguistics and the 7th International
+Joint Conference on Natural Language Processing
+(Volume 1: Long Papers), pages 1491–1500, Beijing,
+China.
+Yoav Goldberg. 2019. Assessing bert’s syntactic abili-
+ties. arXiv preprint arXiv:1901.05287.
+Florian Heimerl and Michael Gleicher. 2018. Interac-
+tive analysis of word vector embeddings. In Com-
+puter Graphics Forum, volume 37, pages 253–265.
+Wiley Online Library.
+John Hewitt and Christopher D. Manning. 2019.
+A
+structural probe for finding syntax in word represen-
+tations. In Proceedings of the 2019 Conference of
+the North American Chapter of the Association for
+Computational Linguistics: Human Language Tech-
+nologies, Volume 1 (Long and Short Papers), pages
+4129–4138, Minneapolis, Minnesota.
+Benjamin Hoover, Hendrik Strobelt, and Sebastian
+Gehrmann. 2020. exbert: A visual analysis tool to
+explore learned representations in transformer mod-
+els. In Proceedings of the 58th Annual Meeting of
+the Association for Computational Linguistics: Sys-
+tem Demonstrations, pages 187–196.
+
+Ganesh Jawahar, Benoît Sagot, and Djamé Seddah.
+2019. What does bert learn about the structure of
+language? In Proceedings of the 57th Annual Meet-
+ing of the Association for Computational Linguistics,
+pages 3651–3657.
+Haewoon Kwak, Jisun An, Elise Jing, and Yong-Yeol
+Ahn. 2021. Frameaxis: characterizing microframe
+bias and intensity with word embedding.
+PeerJ
+Computer Science, 7:e644.
+Ken Lang. 1995.
+Newsweeder:
+Learning to filter
+netnews. In Machine Learning Proceedings 1995,
+pages 331–339. Elsevier.
+Yongjie Lin, Yi Chern Tan, and Robert Frank. 2019.
+Open sesame:
+Getting inside BERT’s linguistic
+knowledge. In Proceedings of the 2019 ACL Work-
+shop BlackboxNLP: Analyzing and Interpreting Neu-
+ral Networks for NLP, pages 241–253, Florence,
+Italy.
+Shusen Liu, Peer-Timo Bremer, Jayaraman J Thiagara-
+jan, Vivek Srikumar, Bei Wang, Yarden Livnat, and
+Valerio Pascucci. 2017. Visual exploration of seman-
+tic relationships in neural word embeddings. IEEE
+transactions on visualization and computer graph-
+ics, 24(1):553–562.
+Hongyin Luo,
+Zhiyuan Liu,
+Huanbo Luan,
+and
+Maosong Sun. 2015.
+Online learning of inter-
+pretable word embeddings. In Proceedings of the
+2015 Conference on Empirical Methods in Natural
+Language Processing, pages 1687–1692.
+Binny Mathew, Sandipan Sikdar, Florian Lemmerich,
+and Markus Strohmaier. 2020. The polar framework:
+Polar opposites enable interpretability of pre-trained
+word embeddings. In Proceedings of The Web Con-
+ference 2020, pages 1548–1558.
+Chris
+McCormick
+and
+Nick
+Ryan.
+2019.
+BERT
+word
+embeddings
+tutorial.
+https:
+//mccormickml.com/2019/05/14/
+BERT-word-embeddings-tutorial/.
+Tomas Mikolov, Kai Chen, G.s Corrado, and Jeffrey
+Dean. 2013. Efficient estimation of word represen-
+tations in vector space. Proceedings of the 1st Inter-
+national Conference on Learning Representations,
+2013.
+George A Miller. 1995. Wordnet: a lexical database for
+english. Communications of the ACM, 38(11):39–
+41.
+Marzieh Mozafari, Reza Farahbakhsh, and Noël Crespi.
+2020. Hate speech detection and racial bias mitiga-
+tion in social media based on bert model. PloS one,
+15(8):e0237861.
+Brian Murphy, Partha Talukdar, and Tom Mitchell.
+2012. Learning effective and interpretable seman-
+tic models using non-negative sparse embedding. In
+Proceedings of COLING 2012, pages 1933–1950.
+Moin Nadeem, Anna Bethke, and Siva Reddy. 2021.
+StereoSet:
+Measuring stereotypical bias in pre-
+trained language models.
+In Proceedings of the
+59th Annual Meeting of the Association for Compu-
+tational Linguistics and the 11th International Joint
+Conference on Natural Language Processing (Vol-
+ume 1: Long Papers), pages 5356–5371.
+Charles Egerton Osgood, George J Suci, and Percy H
+Tannenbaum. 1957. The Measurement of Meaning.
+47. University of Illinois press.
+Abhishek
+Panigrahi,
+Harsha
+Vardhan
+Simhadri,
+and
+Chiranjib
+Bhattacharyya.
+2019.
+word2sense:
+sparse interpretable word embed-
+dings. In Proceedings of the 57th Annual Meeting
+of the Association for Computational Linguistics,
+pages 5692–5705.
+Jeffrey Pennington, Richard Socher, and Christopher D
+Manning. 2014. Glove: Global vectors for word rep-
+resentation. In Proceedings of the 2014 Conference
+on Empirical Methods in Natural Language Process-
+ing, pages 1532–1543.
+Pranav Rajpurkar, Robin Jia, and Percy Liang. 2018.
+Know what you don’t know: Unanswerable ques-
+tions for SQuAD. In Proceedings of the 56th Annual
+Meeting of the Association for Computational Lin-
+guistics (Volume 2: Short Papers), pages 784–789,
+Melbourne, Australia.
+Pranav Rajpurkar, Jian Zhang, Konstantin Lopyrev, and
+Percy Liang. 2016. Squad: 100 ,000+ questions for
+machine comprehension of text. In Proceedings of
+the 2016 Conference on Empirical Methods in Natu-
+ral Language Processing, pages 2383–2392.
+Emily Reif, Ann Yuan, Martin Wattenberg, Fernanda B
+Viegas, Andy Coenen, Adam Pearce, and Been Kim.
+2019. Visualizing and measuring the geometry of
+BERT. Advances in Neural Information Processing
+Systems, 32.
+Marco Ribeiro, Sameer Singh, and Carlos Guestrin.
+2016. “why should I trust you?”: Explaining the pre-
+dictions of any classifier. In Proceedings of the 2016
+Conference of the North American Chapter of the
+Association for Computational Linguistics: Demon-
+strations, pages 97–101, San Diego, California. As-
+sociation for Computational Linguistics.
+Florian Schmidt and Thomas Hofmann. 2020. Bert as a
+teacher: Contextual embeddings for sequence-level
+reward. arXiv preprint arXiv:2003.02738.
+Lütfi Kerem ¸Senel, Furkan ¸Sahinuç, Veysel Yücesoy,
+Hinrich Schütze, Tolga Çukur, and Aykut Koç. 2022.
+Learning interpretable word embeddings via bidi-
+rectional alignment of dimensions with semantic
+concepts. Information Processing & Management,
+59(3):102925.
+Rita Sevastjanova, Aikaterini-Lida Kalouli, Christin
+Beck, Hanna Schäfer, and Mennatallah El-Assady.
+2021.
+Explaining contextualization in language
+
+models using visual analytics. In Proceedings of the
+59th Annual Meeting of the Association for Compu-
+tational Linguistics and the 11th International Joint
+Conference on Natural Language Processing (Vol-
+ume 1: Long Papers), pages 464–476.
+Sandipan Sikdar, Parantapa Bhattacharya, and Kieran
+Heese. 2021. Integrated directional gradients: Fea-
+ture interaction attribution for neural nlp models.
+In Proceedings of the 59th Annual Meeting of the
+Association for Computational Linguistics and the
+11th International Joint Conference on Natural Lan-
+guage Processing (Volume 1: Long Papers), pages
+865–878.
+Richard Socher, Alex Perelygin, Jean Wu, Jason
+Chuang, Christopher D Manning, Andrew Y Ng,
+and Christopher Potts. 2013. Recursive deep mod-
+els for semantic compositionality over a sentiment
+treebank. In Proceedings of the 2013 Conference on
+Empirical Methods in Natural Language Processing,
+pages 1631–1642.
+Robyn Speer, Joshua Chin, and Catherine Havasi. 2017.
+Conceptnet 5.5: An open multilingual graph of gen-
+eral knowledge. In Proceedings of the 31st AAAI
+Conference on Artificial Intelligence.
+Fei Sun, Jiafeng Guo, Yanyan Lan, Jun Xu, and Xueqi
+Cheng. 2016.
+Sparse word embeddings using l1
+regularized online learning. In Proceedings of the
+Twenty-Fifth International Joint Conference on Arti-
+ficial Intelligence, pages 2915–2921.
+Mukund Sundararajan, Ankur Taly, and Qiqi Yan. 2017.
+Axiomatic attribution for deep networks. In Inter-
+national Conference on Machine Learning, pages
+3319–3328. PMLR.
+Ian Tenney, Dipanjan Das, and Ellie Pavlick. 2019.
+BERT rediscovers the classical NLP pipeline.
+In
+Proceedings of the 57th Annual Meeting of the Asso-
+ciation for Computational Linguistics, pages 4593–
+4601.
+Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob
+Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz
+Kaiser, and Illia Polosukhin. 2017. Attention is all
+you need. In Advances in Neural Information Pro-
+cessing Systems, pages 5998–6008.
+Jesse Vig. 2019. A multiscale visualization of atten-
+tion in the transformer model. In Proceedings of the
+57th Annual Meeting of the Association for Compu-
+tational Linguistics: System Demonstrations, pages
+37–42.
+Alex Wang, Amanpreet Singh, Julian Michael, Fe-
+lix Hill, Omer Levy, and Samuel Bowman. 2018.
+GLUE: A multi-task benchmark and analysis plat-
+form for natural language understanding.
+In Pro-
+ceedings of the 2018 EMNLP Workshop Black-
+boxNLP: Analyzing and Interpreting Neural Net-
+works for NLP, pages 353–355, Brussels, Belgium.
+Association for Computational Linguistics.
+Yonghui Wu, Mike Schuster, Zhifeng Chen, Quoc V
+Le,
+Mohammad Norouzi,
+Wolfgang Macherey,
+Maxim Krikun,
+Yuan Cao,
+Qin Gao,
+Klaus
+Macherey, et al. 2016.
+Google’s neural machine
+translation system: Bridging the gap between hu-
+man and machine translation.
+arXiv preprint
+arXiv:1609.08144.
+Mengjie
+Zhao,
+Philipp
+Dufter,
+Yadollah
+Yaghoobzadeh, and Hinrich Schütze. 2020. Quanti-
+fying the contextualization of word representations
+with semantic class probing.
+arXiv preprint
+arXiv:2004.12198.
+
+A
+Appendix
+A.1
+Post-processing
+We point out some issues when using WordNet
+directly as our oracle and present ways to address
+them.
+A.1.1
+Sense Post-processing
+We notice that word-senses identified by WordNet
+can be overly granular (e.g. according to Word-
+Net there are five different polar sense pairs for
+“wet” ↔ “dry”). To get rid of redundant senses,
+we propose to use dimension-reduction methods
+such as variance maximization and orthogonal-
+ity maximization from the original POLAR frame-
+work (Mathew et al., 2020). Alternatively, one
+could merge similar senses if the cosine similarity
+between their sense embeddings is high. Selecting
+or discarding a rare sense is often task specific.
+A.1.2
+Low-quality Example Sentences
+The polar sense embeddings are dependent on
+the quality of the context (i.e., example sentences
+demonstrating a particular sense). While deploying
+WordNet as source for contexts, we encountered
+some issues which we elaborate on next.
+Flections. When constructing polar sense dimen-
+sions, we use the respective word in its basic form
+and extract the embedding from the example con-
+text. However, in WordNet’s example sentences,
+words often do not appear in their basic form, but
+in inflections (e.g. “She walks with a slight limp”
+for “Walk.v.01”).
+Synonyms. Occasionally, the word itself is not
+present in the context sentence but is replaced
+by a synonym (e.g. “the right answer” for “Cor-
+rect.a.01”).
+Misspellings.
+We observe that the example
+sentences often contain spelling mistakes (e.g.
+“toungued lightning” for “Tongued.a.01”)
+Mixed-up examples. In some cases, the example
+sentences of a sense are identical to those of the
+opposite pole (e.g. “we docked at noon” for “Un-
+dock.v.01”).
+These problems need to be addressed with man-
+ual checks of the obtained polar sense pairs and
+context sentences.
+A.2
+Sense Scales
+Instead of rating words on scales defined between
+two pole words (e.g. “left” ↔ “right”), our scales
+are defined between two pole word-senses (e.g.
+“left” ↔ “right” in the sense of direction). While
+the static POLAR framework rates the word “cor-
+rect” highly on the dimension “left” ↔ “right”, we
+expect our framework to rate it low on the dimen-
+sion “left” ↔ “right” in the sense of direction but
+rate it high on the dimension “wrong” ↔ “right” in
+the sense of correctness.
+To this aim, we analyze whether our constructed
+representative sense embeddings encode enough
+sense-related information. As an illustrative exam-
+ple, we consider the word “right” which is used in
+the senses of direction, correctness and lawfulness.
+For each context, we compute the SensePOLAR
+embeddings for the word and rank the dimensions
+based on the absolute value.
+Sense-Scales
+Context
+he went to the right
+his argument is right
+film rights
+Direction
+“left” ↔ “right”
+1st
+38th
+32nd
+Correct
+“wrong” ↔ “right”
+44th
+1st
+291st
+Lawful
+“wrong” ↔ “right”
+55th
+27th
+9th
+Table 5: Ability of SensePOLAR in differentiating be-
+tween senses. We consider the word “right” in three dif-
+ferent contexts and obtain a ranking of the dimensions
+for each case. We report the rank of a SensePOLAR di-
+mension in each of the three contexts in each row. For
+example, the dimension “left” ↔ “right” representing
+the sense of direction is ranked first for the context “he
+went to the right”, while it is ranked 38th and 32nd re-
+spectively in the other two contexts. SensePOLAR is
+indeed able to identify the correct sense dimensions de-
+pending on the context.
+In table 5, we report the ranks of the polar sense
+dimensions for each context. For the word “right”
+in the context of direction “he went to the right”,
+the dimension “left” ↔ “right” is selected as the
+most representative dimension (rank 1), while the
+correctness and the lawful dimensions are ranked
+much lower (44 and 55 respectively). Similar re-
+sults are obtained for the other contexts (see ta-
+ble 5).
+These results indicate that the sense-dimensions
+of SensePOLAR precisely captures the individual
+semantics of the senses.
+A.3
+Computational Requirements
+SensePOLAR embeddings of words for a given
+context can be obtained at low cost if the polar
+sense space is pre-computed. We provide such
+an implementation along with the submission and
+encourage readers to review it. Our implementation
+can even be run on a personal computer.
+
+Given n polar sense dimensions, inversion of the
+matrix can be computed in the worst case in O(n3).
+Since in our case n = 1762, the computation is
+very fast. Moreover, this computation needs to be
+performed only once.
+Retraining the task-specific feed-forward layer
+with SensePOLAR embeddings was performed on
+a computing server with 1 TB RAM, 72 cores, each
+Intel Xeon Gold 6140 CPU at 2.30 GHZ and 2
+Tesla P100-PCIE, 16GB GPUs. We would like to
+reiterate that the retraining is also quite cheap given
+only the task-specific feed-forward layer needs to
+be trained.
+
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+page_content=' Recent work has explored the po-  tential of embedding words via so-called po-  lar dimensions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' good vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' bad, correct vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  wrong).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Examples of such recent approaches  include SemAxis, POLAR, FrameAxis, and  BiImp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Although these approaches provide in-  terpretable dimensions for words, they have  not been designed to deal with polysemy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  they can not easily distinguish between differ-  ent senses of words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' To address this limitation,  we present SensePOLAR, an extension of the  original POLAR framework that enables word-  sense aware interpretability for pre-trained  contextual word embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' The resulting in-  terpretable word embeddings achieve a level  of performance that is comparable to original  contextual word embeddings across a variety  of natural language processing tasks including  the GLUE and SQuAD benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Our work  removes a fundamental limitation of existing  approaches by offering users sense aware in-  terpretations for contextual word embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  1  Introduction  The overwhelming success of deep neural networks  (DNN) in the last decade has been accompanied by  increasing concerns about the lack of interpretabil-  ity (Ribeiro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' This problem is ampli-  fied in the area of Natural Language Processing  (NLP) where word embeddings are used as input  to machine learning models instead of more clas-  sical, understandable features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Traditional (static)  word embedding models like Word2Vec (Mikolov  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=', 2013) or Glove (Pennington et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=', 2014),  that create one embedding for each word, are cur-  rently being replaced by contextual word embed-  ding models like BERT (Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=', 2019) which  have achieved competitive performance in NLP  ∗Equal contribution  benchmarks such as GLUE (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=', 2018) and  SQuAD (Rajpurkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  To improve interpretability, recent approaches  such as SemAxis (An et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=', 2018), POLAR  (Mathew et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=', 2020), FrameAxis (Kwak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=',  2021), and BiImp (¸Senel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=', 2022) have explored  the potential of embedding words via polar dimen-  sions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' good vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' bad, correct vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' wrong).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' While  these approaches have been useful for interpreting  word vectors, they have not been designed to deal  with polysemy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' multiple senses of words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Objective: Addressing polysemy, in this paper we  aim to enable word-sense aware interpretability for  pre-trained contextual word embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Approach: We base our approach on the origi-  nal POLAR framework (Mathew et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=', 2020) and  the idea of semantic differentials (Osgood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=',  1957), which are psychometric scales between two  antonym words, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' “right” ↔ “wrong”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Sense-  POLAR extends POLAR (Mathew et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=', 2020)  to contextual word embeddings, and defines polar  sense instead of polar word scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' This enables  SensePOLAR to offer polar dimensions that distin-  guish between the correctness sense of “right” and  the direction sense of “right”, for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Results: SensePOLAR enables word sense aware  interpretability of contextual embeddings by select-  ing polar sense dimensions that align reasonably  well with human judgements, as demonstrated in  survey experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' SensePOLAR exhibits com-  petitive performance on various NLP tasks where  it is used as input features for a separate model  (feature-based approach) as well as directly inte-  grated in the model itself (fine-tuning approach).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Contributions: SensePOLAR introduces the no-  tion of sense aware interpretations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' To the best of  our knowledge, SensePOLAR represents the first  (semi-) supervised method that enables word sense  aware interpretability for contextual word embed-  dings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' SensePOLAR is publicly available1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='com/JanEnglerRWTH/  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='04704v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='CL]  11 Jan 2023  Figure 1: SensePOLAR overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Pre-trained contextual word embeddings are transformed into an inter-  pretable space where the word’s semantics are rated on scales individually encoded by opposite senses such as  “good”↔“bad”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' The scores across the dimensions are representative of the strength of relationship (between word  and dimension) which allows us to rank the dimensions and thereby identify the most discriminative dimensions  for a word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' In this example, the word “wave” is used in two senses: hand waving and ocean wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' SensePO-  LAR not only generates dimensions that are representative of individual contextual meanings, the alignment to  the respective sense spaces also aligns well with human judgement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' SensePOLAR generates neutral scores for  dimensions not related to the word in the given context (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=', “idle”↔“work”, “social”↔“unsocial”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' We follow  the WordNet convention to represent a particular sense of a word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' For example, “Tide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01” represents the word  “tide” in the sense of surge (rise or move forward).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  2  SensePOLAR  The key idea of SensePOLAR is to transform pre-  trained word embeddings into an interpretable,  sense aware space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' In this space, each dimension  represents a scale on which words are rated, in-  spired by the semantic differential technique (Os-  good et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=', 1957).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' In a departure from the existing  approaches, we define opposite senses for the poles  of these scales (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' “left direction” ↔ “right di-  rection”), as opposed to opposite words (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' “left”  ↔ “right”), as used in Mathew et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Given a contextual word embedding model M,  the interpretable embeddings are obtained through  the following steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' 1) We use M to obtain the  (non-interpretable) contextual embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' 2)  We obtain polar senses with contextual information  from an oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' 3) We proceed with generating  representative sense embeddings from which we  4) construct the interpretable polar sense space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' 5)  The original embedding is transformed into the po-  lar sense space, which enables interpretation with  regard to opposite sense pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' We illustrate each  step in figure 1 and elaborate them next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Obtaining contextual embeddings: To obtain  the embedding of a particular word, we forward the  word with its context, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' an example sentence, to  the embedding model M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' The embedding of the  corresponding word can then be retrieved from the  output of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Because most models deploy sub-  word tokenization algorithms, such as WordPiece  SensePOLAR  (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=', 2016), embeddings of only subword  tokens, rather than entire words, are generated by  the contextual embedding models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' This provides  for obtaining representations for out-of-vocabulary  words but, at the same time, makes embeddings of  even common words not directly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Follow-  ing existing literature (McCormick and Ryan, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Bommasani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=', 2020), we compute the embed-  ding of a word by averaging over the embeddings  of the constituent tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Selecting opposite polar senses: Each dimen-  sion in the interpretable space corresponds to a  scale spanned by opposite polar senses, which we  define as a polar sense dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' We assume  that the poles and corresponding contexts are pro-  vided by an oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' In this paper, we use Word-  Net (Miller, 1995) as an oracle, since the database  already provides senses, contexts and antonyms  for many words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Each sense of a word is repre-  sented by a unique identifier, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' “Right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='0” (a  convention followed in WordNet) encodes “right”  in the sense of direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' From over 6000 sense-  antonym pairs that are available in WordNet, we  use only a subset (1763) that are annotated with  example sentences for both words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' After various  post-processing steps (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Appendix), these exam-  ple sentences are used as context in step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Generating polar sense embeddings: We pro-  pose to generate polar sense embeddings for each  sense that is chosen by the oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Let w denote the  word of interest and s the word-sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Furthermore,  : Input: Pretrained contextual embeddings  SensePOLAR  Output: Sense aware interpretable embeddings  Wave(x)  (1)  Polar sense space  Transformation  ai = S-i - Si  Pc = (αT)-1xc  Pc2  ail  Tide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01 - Ebb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
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+page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='1250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='25  : C1: She is waving her hand to the  crowd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  the rocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  (4)  (5)  Dramatic  Undramatic Tide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Ebb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01  Xc1  Xc2  Polar sense embeddings  (3)  : Divided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='United.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01 Fresh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='06  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='12  Salty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='02:  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='25  m  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='18  Pre-trained  +Tide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01  Tide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01  Offshore  Onshore :  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='32  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='2  m  m  Down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='02  Up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='04  embeddings  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01  r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01  j=1  :  :  oracle  Level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='02  Raise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='09  Relax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01  (2)  Tense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='02 :  (s-1,S1)  Tide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01, Ebb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01  (s-2, S2)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  ldle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='02, Work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01  Idle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='02  0  0  Unsocial  Work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01  Socia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01let Cs = {c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=', cm} be m context examples for  the sense s, which we assume are provided by the  oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' In each context c ∈ Cs, the word w is used  in the sense s, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' “A strange sound came from the  right side.” for the word “right” in the sense of di-  rection (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=', we intend to embed “Right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='04”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' We  create polar sense embeddings in two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' First,  we input m context examples for a sense s to the  embedding model M and retrieve an embedding  ws  c ∈ Rd of the word w for each context c ∈ Cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' If  the word w consists of several subword tokens, the  individual subword embeddings are averaged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' We  also allow for senses consisting of multiple words,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' “keep track” ↔ “lose track”, where we again  average the embeddings of the individual tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Second, we compute the average of the contextual  word embeddings per sense and define it as the  sense embedding s ∈ Rd:  s = 1  m  m  �  j=1  ws  cj  (1)  This is a rather straightforward way to represent  individual senses of words in a (semi-) supervised  manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' The method is dependent on the quality  and the number of the example sentences provided  by the oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' We observe that more context ex-  amples lead to a better and stable representation,  but we usually achieve a satisfactory representation  with already one suitable example sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' This is  motivated by the observations in Reif et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' (2019)  which provide strong evidence that BERT positions  the embeddings of senses in individual clusters  in space and that these clusters are usually suffi-  ciently spatially separated from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' A polar  sense dimension is represented by a pair of opposite  senses (s−i, si) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=', “Right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='02”, “Wrong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Constructing a polar sense space:  Given  n  polar  sense  dimensions  S = ((s−1, s1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=', (s−n, sn))  with  their  con-  texts C  =  ((Cs−1, Cs1), (Cs−n, Csn)),  we  compute the polar sense embedding si for each  sense si and corresponding context Csi, following  equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  We now utilize the representations of individual  senses to construct the interpretable polar sense  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Each polar sense dimension (si, s−i) ∈ S  defines an interpretable scale, which is encoded by  the direction vector ai, defined as follows:  ai = s−i − si  (2)  The direction vectors for all polar sense dimen-  sions are then stacked to obtain the change of basis  matrix a ∈ Rn×d for the interpretable polar sense  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Transformation to interpretable embed-  dings: Finally, an embedding of a word x in a  context c, xc can be transformed into the polar  sense space in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Given a repre-  sents the change of basis matrix, we can compute  the polar sense embedding pc following the rules  of linear algebra:  aT pc = xc  (3)  pc = (aT )−1 xc  (4)  The inverse of aT is computed by the Moore-  Penrose generalized inverse (Ben-Israel and Gre-  ville, 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' The resulting contextual word embed-  ding pc in the polar sense space is of dimension  n × 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' The absolute value across axis ai corre-  sponds to the word’s rating on the scale between  the polar senses (s−i, si) and the sign represents  the direction of alignment to a particular pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' A  higher absolute value represents a stronger relation-  ship to the corresponding polar sense dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  This allows us to obtain the most expressive polar  sense dimensions for a given word and context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Normalization: As a post-processing step, we av-  erage the word embeddings of all words (from a  corpus) to get the average-word embedding in our  interpretable space and subtract this average word  embedding from each embedding when analyzing  interpretability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' This also allows us to deal with  the anisotropic nature of contextual word embed-  dings (Ethayarajh, 2019) whereby the embeddings  are not randomly distributed but rather lay on a  high-dimensional cone in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  3  Evaluation  Note that while SensePOLAR allows for deploy-  ment across any contextual word embedding model,  in this work, we consider BERT (Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=',  2019) as our model for illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  We con-  sider a BERT-base model which utilizes 12 trans-  former (Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=', 2017) encoder layers and  generates embeddings of size 768.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' The pre-trained  BERT-base model was downloaded from Hugging-  face2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' In addition, we use WordNet as our oracle  2https://huggingface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='co/docs/  transformers/model_doc/bert,  we  used  “bert-  base-uncased”  ML Model  SVM  FFN  Task  Base  SensePOLAR  Base  SensePOLAR  Sport  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
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+page_content='935↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
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+page_content='5%  Religion  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='891  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='848↑ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='8%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
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+page_content='6%  Computer  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='770  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='750↓ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='6%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='763  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='727↓ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='7%  Table 1:  Performance of the original BERT (Base)  embeddings and SensePOLAR embeddings on feature-  based tasks with a support vector machine (SVM) and  a feed-forward neural network (FFN) classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Sense-  POLAR achieves performance comparable to the origi-  nal BERT embeddings across all three tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  with 1763 polar sense pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='1  Performance on downstream tasks  The goal of SensePOLAR is to add interpretabil-  ity to word embeddings without major losses in  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Hence, we evaluate SensePOLAR  on a wide range of NLP downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' We  investigate whether replacing the original BERT  embeddings with SensePOLAR embeddings has  any effect on performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='1  Feature-based tasks  We analyze the effectiveness of SensePOLAR em-  beddings in a “classical” NLP pipeline, where word  embeddings are generated beforehand and are used  as input-features to a separate machine learning  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' We consider a binary text classification task  utilizing the 20 Newsgroups dataset (Lang, 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  The dataset consists of ∼ 20K news articles cov-  ering 20 types of news.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Our experiment follows  the structure of Panigrahi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' (2019), where we  only consider the topics sports, computer and re-  ligion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' For each topic, an article must be clas-  sified into one of two categories (“baseball” or  “hockey” for sports, “IBM” or “Apple” for com-  puter, “christianity” or “atheism” for religion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' In  table 1, we present the results in terms of accuracy  across the three tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' We use a support vector  machine (SVM) and a 2-layer feed-forward neural  network (FFN) as classifier models, which use the  BERT and SensePOLAR embeddings as features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Across all three tasks, SensePOLAR achieves a  level of performance that is comparable to the orig-  inal embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='2  Fine-tuning tasks  Integrating SensePOLAR into fine-tuned mod-  els: The models achieving state-of-the-art perfor-  mances on different NLP tasks usually deploy a  task specific network layer (usually a feed-forward  network) on top of the embedding layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' The  SQuAD 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='1  SQuAD 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='0  Metric  Base  SensePOLAR  Base  SensePOLAR  EM  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='92  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='85↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='07%  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='88  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='06↑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='22%  F1  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='15  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='12↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='03%  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='87  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='89↑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='02%  Table 2: Results of fine-tuned BERT embeddings and  with SensePOLAR transformed embeddings on the  SQuAD benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' The results are competitive and  even improve marginally after applying SensePOLAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  embedding layers and the task specific layers are  then fine-tuned on the task specific dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Con-  sequently, SensePOLAR embeddings need to be  computed considering the fine-tuned version of the  embeddings rather than the original pre-trained ver-  sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' In this particular setting, we propose to uti-  lize the embedding layer of the fine-tuned model  (instead of the original pre-trained version) to con-  struct the polar sense space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Given an input text,  each token (including the [CLS] token) can then  be transformed to a corresponding SensePOLAR  embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Because of the dimensionality mis-  match between the original embedding and the  transformed SensePOLAR embedding, we replace  the first layer of the task specific feed-forward net-  work and re-fine-tune it on the task specific dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Note that the weights of the underlying embedding  model are frozen during this re-fine-tuning proce-  dure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' This is computationally inexpensive as only  the task specific layers need to be trained, which  are often just 1 or 2-layered feed-forward network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Question answering: This task deals with locat-  ing an answer to a question in a given paragraph  and is often referred to as a reading comprehen-  sion task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' We consider the SQuAD benchmark,  including both SQuAD1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='1 (Rajpurkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=', 2016)  and SQuAD2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='0 (Rajpurkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=', 2018) versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  The BERT-based QA model consists of the em-  bedding module followed by a span-classification  head, which is a 1-layer feed-forward network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' The  model takes both the question text and the passage  text as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' The [CLS] token (a special token  generated by BERT for classification tasks) ob-  tained from the embedding module is then passed  onto the span-classification head, which predicts  the start and the end position of the span in the text  passage that contains the answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  The polar sense space is computed using the  BERT embedding module already fine-tuned on  the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' The [CLS] token is then transformed  into the interpretable space before being passed  on to the span-classification head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' This classifica-  tion head, however, needs to be replaced (to match  the dimension of the transformed embedding) and  re-trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' In table 2, we report the exact match  (EM) and F1 scores with the original BERT (base)  and the SensePOLAR model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' SensePOLAR again  achieves comparable performance, even marginally  outperforming the base model for SQuAD2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Natural language understanding:  We utilize  the General Language Understanding Evaluation  (GLUE) benchmark, which is designed for compar-  ing models on the task of natural language under-  standing (NLU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' It consists of nine tasks that cover  a diverse range of text genres, dataset sizes, and de-  grees of difficulty (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' We point the  reader to the original paper by Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' (2018)  for a general overview of the tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' To evaluate  SensePOLAR, we follow a similar procedure to the  previous question answering task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' The polar sense  space is computed using the underlying BERT em-  bedding module, already fine-tuned on the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  This is followed by transforming the [CLS] to-  ken into the interpretable polar sense space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' The  feed-forward layers on top are then replaced and re-  trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' In table 3, we report the results on GLUE  Task  Train size  Metric  Base  SensePOLAR  CoLa  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='5k  Matthew’s corr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='62  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='05 ↓ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='77%  SST-2  67k  Accuracy  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='51  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='40 ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='12%  MRPC  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='7k  Accuracy  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='31  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='84 ↓ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='74%  F1  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='00  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='41 ↓ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='79%  STS-B  7k  Person corr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='03  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='17 ↓ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='46%  QQP  364k  Accuracy  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='59  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='15 ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='49%  F1  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='29  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='82 ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='54%  MNLI  393k  Accuracy  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='49  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='04 ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='53%  QNLI  105k  Accuracy  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='54  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='58 ↑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='04%  RTE  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='5k  Accuracy  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='18  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='93 ↓ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='14%  WNLI  634  Accuracy  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='34  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='34 ↑↓0%  Table 3: Comparison of the fine-tuned BERT model  and the re-fine-tuned BERT model with SensePO-  LAR embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Mostly, comparable performance is  achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Slightly worse performance is achieved for  tasks with smaller training datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  tasks with both the original BERT (Base) and the  SensePOLAR embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' SensePOLAR achieves  competitive performances across all the tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  The results indicate that SensePOLAR is able  to achieve interpretability without compromising  performance on downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='2  Interpretability  We turn our attention to evaluating the interpretabil-  ity of SensePOLAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Qualitative analysis: We transform the embed-  dings of the words into a polar sense space and an-  alyze the position/rating (determined by the signed  value on that dimension) of different words on se-  lected dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' More specifically, we consider  a context in which the word is used and pass it  through the BERT module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' The embedding corre-  sponding to the target word (note that BERT gener-  ates embeddings corresponding to each word in the  context) is then transformed into the polar sense  space through the base change operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Ana-  lyzing the ratings of words in a selected dimen-  sion, allows us to demonstrate the advantages of  interpreting word embeddings in terms of polar  sense dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' We first consider the dimension  “Black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='02”↔“White.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='02” (in the sense of ethnic-  ity) and transform the embeddings of celebrities  and nationalities on this dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' The obser-  vations mostly match the ethnicities of the indi-  viduals (see figure 2(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' We also consider words  such as milk, coal etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' which are not related to  “Black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='02”↔“White.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='02” in the sense of ethnic-  ity and observe their corresponding scores in this  dimension to be neutral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' However, their representa-  tion on the dimension “Black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01”↔“White.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01”  (in the sense of color), captures their semantic well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  This demonstrates the benefits of using polar senses  as dimensions instead of words, which would have  failed to differentiate between the two senses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  We also consider other dimensions and present  the connotative meanings of words across these di-  mensions in figure 2(b) which leads to interesting  observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' For example, “politician”, “meet-  ing” are more aligned towards “Hate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01” (in the  sense of disgust).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Similarly, “murder” and “devil”  are aligned towards “Wrong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01” (in the sense of  morality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  In addition to picking out interesting dimensions  by hand, we also propose to evaluate interpretabil-  ity by investigating the most descriptive dimensions  of a given word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' The dimensions for a word are  ranked based on the absolute value across all di-  mensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Ideally, the top dimensions should be  the most descriptive and fitting for the word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' For  illustration, we provide example words and the cor-  responding top-5 dimensions in figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' The top  dimensions mostly have a high semantic similarity  with the word, and they also reasonably align with  human judgement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Survey experiment: For evaluating interpretabil-  ity on a larger scale, we follow the approach  by Mathew et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' (2020) and conduct a human judg-  ment survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' We utilize the crowdsourcing plat-  (a) Benefits of sense dimensions  (b) Connotative meanings across dimensions  Figure 2: Illustration of polar sense dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' (a) SensePOLAR allows for interpretability along multiple senses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  “black”↔“white” in the sense of ethnicity (top) can be differentiated from “black”↔“white” in the sense of color  (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Words like “snow”or “coal” - which are not semantically related to ethnicity - score neutral on the  upper scale while being clearly distinguishable on the lower scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' (b) The connotative meanings of words can  also be investigated through SensePOLAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' For example, “politician” is associated with “hate” while “mother” is  associated with “love”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  (a) Run  (b) Music  (c) Right  (d) Happy  Figure 3: Illustration of SensePOLAR embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' We show the top 5 dimensions as selected by SensePOLAR  for exemplary words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' The pre-trained embeddings are obtained using BERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' The top dimensions and the word’s  rating/alignment to the pole reasonably align with human judgement (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' table 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  form Clickworker3 where we randomly select 15  common English nouns, verbs and adjectives (with  short context) and compute their interpretable em-  bedding with SensePOLAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Then, for each word,  we extract the top-5 polar sense dimensions (mea-  sured in absolute value) and additionally five ran-  dom dimensions from the lower 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' These 10  dimensions are then presented to the participants  in a random order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Participants are asked to select  five dimensions that are most representative of a  given word and to rate each dimension based on  their alignment to one of the poles on a likert scale  between 1 and 7 (with 4 as neutral).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Each word  is assigned 3 annotators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' For a given word, each  dimension is assigned a score depending on how  many annotators found it relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' We then select  the top 5 dimensions based on this score and we  consider them as the ground-truth dimension to  which we compare the ones selected by SensePO-  3https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='clickworker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='com/  LAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  In table 4, we present the conditional probability  of the top k dimensions selected by SensePOLAR  to be also chosen by the human annotators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' In the  same table, we also report the random chance of  getting selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' For the top-1 dimension, agree-  ment is roughly 87% and for the top-2 dimension  it is still around 65%, indicating strong alignment  with human judgment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' We also found that the par-  ticipant’s ratings on these dimensions were the (ab-  solute) highest, showing that the word is strongly  connected to one of the polar senses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Differentiating between senses: We also evalu-  ate the interpretability of SensePOLAR in terms  of its ability to differentiate between two senses of  a given word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' As an illustrative example, we con-  sider the word “right” in the sense of both direction  and correctness (refer to figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' The selected  polar sense dimensions are indeed representative  of the correct sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Note that the original POLAR  Kobe Bryant  Angela Merkel  German  Black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='02  White.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='08  Barack Obama  MichaelJackson  Swedish  Space  Paper  Snow  Black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01  White.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='03  Coal  MilkPolitician  Meeting  King  Girlfriend  Mother  Hate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01  Love.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='03  Work  Baby  Bragging  Devil Murder  Kicked  Praying  Donating  Wrong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='02  Right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='03  Killed Lying  Steal Jesus  LawfulI like to run fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='2  Running.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='04  Standing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='04  Idle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01  Run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='13  Passing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='02  Running.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='03  Delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01  Rush.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='03  Stoppable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01  Unstoppable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01Music is life  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='15  Musical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='02  Unmusical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01  Atomism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='02  Holism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01  Harmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01  Non-harmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01  Experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01  Inexperience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01  Absence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01  Presence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01The answer is right  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='15  Right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='05  Wrong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='05  Due.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01  Undue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01  False.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01  True.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01  Certain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='03  Uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='02  Dead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='02  Live.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='02am happy to see you  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='15  Unwelcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01  Welcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01  Glad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01  Sad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01  Insincerely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01  Sincerely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01  Evil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='03  Go0d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='02  Troubled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01  Untroubled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01Top-k  1  2  3  4  5  SensePOLAR  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='876  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='558  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='312  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='187  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='093  Random  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='22  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='083  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='023  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='004  Table 4: Alignment with human judgement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' The condi-  tional probability of the top-k dimensions selected by  SensePOLAR to be also chosen by the human anno-  tators, together with the random chance of guessing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Significantly higher probabilities than random chance  are achieved, indicating that the chosen dimensions  are meaningful and match human judgment reasonably  well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Figure 4: Top-5 dimensions of the word “left” for two  different contexts in the sense of going away (left) and  direction (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' The top dimensions are indeed dif-  ferent for the different word-senses and are reasonably  descriptive of the correct sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  framework would not be able to differentiate be-  tween the senses, given it generates exactly one  embedding for a given word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  We follow up with another human judgement ex-  periment where we present the top-10 polar sense  dimensions of words with multiple meanings, to-  gether with the word’s score on these dimensions,  to the annotators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' The task is to identify in which  sense the target word is being used in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' We limit this  experiment to only two common senses for each  word and present the WordNet definitions as the  answer possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Thus, by random guessing,  an accuracy of 50% would be achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' For our  hand-picked examples, the correct sense was iden-  tified in around 95% of the examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' The average  inter-participant agreement on the result is around  78%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  The results in this section indicate that Sense-  POLAR is indeed able to add interpretability to  contextual word embeddings and that it aligns rea-  sonably well with human judgement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  4  Applications  We discuss two potential use cases: (i) analyzing  bias in contextual embeddings and (ii) explaining  model decisions on downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='1  Bias Analysis  Contextual embedding models have been shown  to capture bias from training data (Nadeem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=',  2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Ahn and Oh, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Bartl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Moza-  fari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' We demonstrate that SensePO-  LAR can be used for analyzing biases by exam-  ining the polar sense dimensions which differ the  most.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' For our evaluation, we utilize examples from  the Stereoset dataset from Nadeem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Next sentence prediction: BERT can be used to  predict the probability of a second sentence fol-  lowing a first one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Consider the sentence pair  “Hispanic people are very diverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  All people  like that are criminals.”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Replacing the word  “Hispanic” by “American” leads to significant de-  crease in BERT’s confidence score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  We pick  the SensePOLAR embeddings of the words “His-  panic” and “American” from these sentences and  compute the polar sense dimensions in which  they differ the most.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Noticeably, the most dis-  criminative dimensions mostly encode senses re-  lated to being or not being a legal citizen such  as “Documented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01”↔“Undocumented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01” or  “Irregular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01”↔“Regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' We observe that  the word “Hispanic” is rated on the side of irregu-  lar, undocumented, unrighteous and unconcerned,  as opposed to “American”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Masked LM: We conduct a similar experiment on  the task of masked language modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Consider  the sentence “Because he is black, he works as a  [MASK].”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Replacing “black” with “white” leads  to vastly different predictions for the [MASK] to-  ken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' We apply SensePOLAR to analyze the polar  sense dimensions of the [MASK] token in both con-  texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' We find that the most discriminative dimen-  sion is “Employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01”↔“Unemployed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01”, in-  dicating that BERT predicts a word more related to  unemployed when the word “black” is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='2  Explaining classifier results  SensePOLAR can further be deployed to ex-  plain decisions of classifier models that make  use of contextual word embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  To il-  lustrate this we consider binary sentiment pre-  diction (positive or negative) on the SST-2  dataset (Socher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' We sample and av-  erage the SensePOLAR transformed [CLS] to-  kens from the positive and negative class sepa-  rately and examine the most discriminative dimen-  sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' We find the most discriminative dimensions  to be “sharp”↔“dull”, “unpleasant”↔“pleasant”,  “endemic”↔“cosmopolitan”, “soft”↔“loud” and  “tasteless”↔“tasteful”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' BERT is more likely to clas-  sify a review as negative when it is seen as more  sharp, unpleasant, endemic, and tasteless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  He left the room.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  He looked right and left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
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+page_content='01  Pop in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01  Pop out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01  Downwind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01  Upwind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01  Shut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01  Open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01  Early.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='02  Late.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01  Disappear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01  Appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01  Foolish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01  Wise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='015  Discussion  Next, we discuss issues pertinent to SensePOLAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Generalizability: SensePOLAR is applicable to  any pre-trained contextual embedding model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' It  can also be deployed on top of any of the con-  stituent transformer layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' This allows for not only  comparing different contextual word embedding  models in terms of interpretability or bias analysis  but also performing similar analysis across trans-  former layers of the same embedding model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Extension to other languages:  SensePOLAR  should also be extendable to other languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' The  only requirement would be to be able to obtain  suitable sense antonym pairs as well as example  contexts via an oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Interpretable decision-making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' In section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='2,  we demonstrated how SensePOLAR could be used  to explain decisions of text classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' However,  the design of SensePOLAR allows for deployment  across any other downstream task as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' This  is in contrast with existing interpretability meth-  ods which are often developed with a particular  downstream task in mind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Quantitative  comparison  with  other  inter-  pretability methods: An ideal evaluation set up  would have been to quantitatively compare Sense-  POLAR to other interpretability methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' How-  ever, as pointed out in the existing literature (Sun-  dararajan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Sikdar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=', 2021), when  two models provide different interpretations, it is  difficult to judge if one is better than the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' In-  volving humans makes it even harder, as one now  needs to tease out a person’s own subjective bi-  ases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Hence, our crowdsourcing experiments were  only designed to understand the efficacy of Sense-  POLAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Nevertheless, we provide a qualitative  comparison with the existing methods in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  SensePOLAR variants: Other variants of Sense-  POLAR can be devised as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' For example, lin-  ear transformation instead of base change could  be used for obtaining SensePOLAR embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  However, we observed that linear transformation  does not preserve the original structure of the em-  bedding space, where the different senses of words  are already sufficiently separated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' One can also ex-  periment with different normalization techniques,  such as scaling or standardization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' In this paper, we  concentrated on an exhaustive evaluation setup to  include more downstream tasks and crowdsourcing  experiments rather than exploring other variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  We consider all the above variants promising av-  enues for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  6  Related work  In this section, we briefly summarize previous re-  search on enabling interpretability for both static  and contextual word embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Unsupervised methods: The key idea for this  class of methods is to create sparse embeddings,  which is achieved through a post-processing step  on top of the embeddings (Murphy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Faruqui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Addition-  ally, the idea of creating sparse embeddings can  also be integrated into the word embedding train-  ing itself, as demonstrated in Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' The meaning of the dimen-  sions are assigned by the model itself (hence un-  supervised) and are often intelligible to humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Notably, Word2Sense (Panigrahi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=', 2019) pro-  poses to create sparse non-negative vectors through  Latent Dirichlet Allocation (LDA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Each dimen-  sion is assigned a meaning, which is retrieved from  a training corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' The methods discussed above  are specific to static word embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Berend  (2020) extends some of these ideas to contextual  word embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  (Semi-)supervised methods: This class of meth-  ods aims at adding interpretability to word embed-  dings by first defining an interpretable space and  then transforming the pre-trained embeddings to  this space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' In this space, each dimension spans be-  tween two pole words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' While SemAxis (An et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=',  2018) proposes to use antonym pairs retrieved from  ConceptNet (Speer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=', 2017), the POLAR frame-  work (Mathew et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=', 2020) utilizes the semantic  differential technique pioneered by Osgood (Os-  good et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=', 1957).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Similarly, BiImp (¸Senel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=',  2022) proposes to use opposite semantic concepts  as poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Not only are the dimensions interpretable,  these methods are computationally less expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Embedding geometry: Part of the existing re-  search has focussed on analyzing the position of  words in the embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Ethayarajh (2019)  provides evidence that the BERT embeddings are  not uniformly distributed in the space, but rather  lay on a high dimensional cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Reif et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' (2019)  demonstrate that BERT is able to separate fine-  grained senses of words by placing them in differ-  ent locations in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Similar observations are  made by Schmidt and Hofmann (2020) as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Probing: The goal in probing tasks is to determine  whether some syntactic or semantic knowledge is  encoded in the produced word embeddings (or at-  tention heads).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' The embeddings (or attentions)  are fed into a simple linear classifier to predict  unseen linguistic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' The performance of  the classifier is indicative of the extent to which  these linguistic properties are encoded in the em-  beddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' For BERT, these probing experiments  have demonstrated that the layers on the top are  more contextual (Ethayarajh, 2019) and the layers  at the center contain a large amount of syntactic  information (Hewitt and Manning, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Goldberg,  2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Jawahar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Chi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' The  semantic information is generally spread across the  entire network (Tenney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=',  2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Visual explanations: Finally, recent work has also  considered visualizing attention in transformer lay-  ers to explain contextual language models (Hoover  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Vig, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Similarly, visualizing  word embeddings can also aid in explaining what a  model learns as demonstrated in Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Heimerl and Gleicher (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Boggust et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' (2022)  (static) and Sevastjanova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Berger  (2020) (contextual).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Comparison with SensePOLAR: To the best of  our knowledge, SensePOLAR is the first (semi-)  supervised method for enabling interpretability for  contextual word embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' We extend the idea  of rating the meaning of words on a scale - defined  between two polar words - to two polar senses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  SensePOLAR can also be integrated into task spe-  cific fine-tuned models as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' In comparison  to unsupervised methods, our method enables us  to understand the individual dimensions and ac-  tively choose and adjust polar sense dimensions  for the task at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' While probing and visualiza-  tion methods can reveal whether specific linguistic  information is encoded in the embeddings, analyz-  ing the embedding geometry can help in uncover-  ing the model characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' However, none of  these methods can directly augment interpretabil-  ity to the embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Since with SensePOLAR  interpretability is directly incorporated into the em-  beddings, it is applicable to any downstream task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  This is in contrast to most of the existing methods,  which are often specific to embedding methods or  downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  7  Conclusion  We introduced SensePOLAR which enables word  sense aware interpretability for contextual word  embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' The key idea is to project word em-  beddings onto an interpretable space which is con-  structed from polar sense pairs obtained from an  oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' SensePOLAR extends the original POLAR  framework developed for static word embeddings  to contextual word embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' We demonstrated  that the obtained interpretable embeddings align  well with human judgement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Moreover, SensePO-  LAR could be integrated into fine-tuned models  and can be deployed to specific applications like  bias analysis and explaining prediction results of  classifier models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  8  Limitations  Underlying embedding models: SensePOLAR  uses embeddings of polar senses to build an inter-  pretable subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Thereby, we assume that the un-  derlying embedding model captures the semantics  of words from which we construct the sense em-  beddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' As a result, SensePOLAR is dependent  on the quality of the underlying contextual word  embedding model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Compared to the original PO-  LAR framework proposed in (Mathew et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=', 2020),  the present approach also depends on the ability of  the model to capture individual word-senses with  sufficient accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Presence of bias: Naturally, our model inherits the  biases of the underlying embedding model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' The  word “physics”, for example, has a high rating  towards “male” on the polar sense scale of “male”  ↔ “female”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' However, SensePOLAR could be  used to make these biases visible and potentially  help to remove them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' One can also tap into state-of-  the-art bias mitigation methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Ahn and Oh  (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Bartl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Mozafari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' (2020))  to address this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Dependence on oracles: The construction of the  polar sense space depends largely on the choice of  polar opposite senses and the quality of the con-  text examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Using the example of WordNet, we  have shown how a general model can be created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  However, we observed that rare senses and low-  quality example sentences can lead to poor results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Moreover, it is not clear how the optimal number  of polar dimensions can be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Empiri-  cally, we observed that adding more pairs does not  necessarily lead to improvement in performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  For a particular downstream task, it may also be  appropriate to discard polar sense pairs that are not  relevant to the task (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' if they never occur in the  corpus).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Counter-intuitive rating of words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' We find that  in some cases the rating of words on the polar sense  scales does not coincide with human judgement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  The word “doctor”, for example, is highly skewed  towards “guilty” on a scale from “innocent” ↔  “guilty”, which does not match the typical percep-  tion of doctors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' We believe this is because word  embeddings by design are shaped by their context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  There are probably more articles and stories about  “guilty doctors” than “innocent doctors”, because  these stories would be less interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Acknowledgements  Sandipan Sikdar was supported in part by RWTH  Aachen Startup Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' StUpPD384-20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  References  Jaimeen Ahn and Alice Oh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Mitigating language-  dependent ethnic bias in BERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' In Proceedings of  the 2021 Conference on Empirical Methods in Nat-  ural Language Processing, pages 533–549, Online  and Punta Cana, Dominican Republic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Jisun An, Haewoon Kwak, and Yong-Yeol Ahn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  SemAxis: A lightweight framework to characterize  domain-specific word semantics beyond sentiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  In Proceedings of the 56th Annual Meeting of the As-  sociation for Computational Linguistics (Volume 1:  Long Papers), pages 2450–2461, Melbourne, Aus-  tralia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Marion Bartl, Malvina Nissim, and Albert Gatt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Unmasking contextual stereotypes: Measuring and  mitigating BERT’s gender bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' In Proceedings of  the Second Workshop on Gender Bias in Natural  Language Processing, pages 1–16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Adi Ben-Israel and Thomas NE Greville.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Gen-  eralized inverses:  Theory and Applications, vol-  ume 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Springer Science & Business Media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Gábor Berend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Sparsity makes sense: Word  sense disambiguation using sparse contextualized  word representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' In Proceedings of the 2020  Conference on Empirical Methods in Natural Lan-  guage Processing, pages 8498–8508.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Matthew Berger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Visually analyzing contextual-  ized embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' In 2020 IEEE Visualization Con-  ference (VIS), pages 276–280.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Angie Boggust, Brandon Carter, and Arvind Satya-  narayan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Embedding comparator: Visualiz-  ing differences in global structure and local neigh-  borhoods via small multiples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' In 27th International  Conference on Intelligent User Interfaces, pages  746–766.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Rishi Bommasani, Kelly Davis, and Claire Cardie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Interpreting Pretrained Contextualized  Representations via Reductions to Static Embed-  dings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  In Proceedings of the 58th Annual Meet-  ing of the Association for Computational Linguistics  , pages 4758–4781.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Yunchuan Chen, Ge Li, and Zhi Jin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Learning  sparse overcomplete word vectors without interme-  diate dense representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' In International Confer-  ence on Knowledge Science, Engineering and Man-  agement, pages 3–15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Ethan A Chi, John Hewitt, and Christopher D Manning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Finding universal grammatical relations in  multilingual bert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' arXiv preprint arXiv:2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='04511.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Jacob Devlin, Ming-Wei Chang, Kenton Lee, and  Kristina Toutanova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  BERT: Pre-training of  deep bidirectional transformers for language under-  standing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' In Proceedings of the 2019 Conference of  the North American Chapter of the Association for  Computational Linguistics: Human Language Tech-  nologies, Volume 1 (Long and Short Papers), pages  4171–4186.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Kawin Ethayarajh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' How contextual are contextu-  alized word representations?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Comparing the geom-  etry of BERT, ELMo, and GPT-2 embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' In  Proceedings of the 2019 Conference on Empirical  Methods in Natural Language Processing and the  9th International Joint Conference on Natural Lan-  guage Processing, pages 55–65, Hong Kong, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Manaal Faruqui, Yulia Tsvetkov, Dani Yogatama, Chris  Dyer, and Noah A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Smith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Sparse overcom-  plete word vector representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' In Proceedings  of the 53rd Annual Meeting of the Association for  Computational Linguistics and the 7th International  Joint Conference on Natural Language Processing  (Volume 1: Long Papers), pages 1491–1500, Beijing,  China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Yoav Goldberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Assessing bert’s syntactic abili-  ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' arXiv preprint arXiv:1901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='05287.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Florian Heimerl and Michael Gleicher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Interac-  tive analysis of word vector embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' In Com-  puter Graphics Forum, volume 37, pages 253–265.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Wiley Online Library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  John Hewitt and Christopher D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Manning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  A  structural probe for finding syntax in word represen-  tations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' In Proceedings of the 2019 Conference of  the North American Chapter of the Association for  Computational Linguistics: Human Language Tech-  nologies, Volume 1 (Long and Short Papers), pages  4129–4138, Minneapolis, Minnesota.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Benjamin Hoover, Hendrik Strobelt, and Sebastian  Gehrmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' exbert: A visual analysis tool to  explore learned representations in transformer mod-  els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' In Proceedings of the 58th Annual Meeting of  the Association for Computational Linguistics: Sys-  tem Demonstrations, pages 187–196.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Ganesh Jawahar, Benoît Sagot, and Djamé Seddah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' What does bert learn about the structure of  language?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' In Proceedings of the 57th Annual Meet-  ing of the Association for Computational Linguistics,  pages 3651–3657.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Haewoon Kwak, Jisun An, Elise Jing, and Yong-Yeol  Ahn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Frameaxis: characterizing microframe  bias and intensity with word embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  PeerJ  Computer Science, 7:e644.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Ken Lang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Newsweeder:  Learning to filter  netnews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' In Machine Learning Proceedings 1995,  pages 331–339.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Elsevier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Yongjie Lin, Yi Chern Tan, and Robert Frank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Open sesame:  Getting inside BERT’s linguistic  knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' In Proceedings of the 2019 ACL Work-  shop BlackboxNLP: Analyzing and Interpreting Neu-  ral Networks for NLP, pages 241–253, Florence,  Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Shusen Liu, Peer-Timo Bremer, Jayaraman J Thiagara-  jan, Vivek Srikumar, Bei Wang, Yarden Livnat, and  Valerio Pascucci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Visual exploration of seman-  tic relationships in neural word embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' IEEE  transactions on visualization and computer graph-  ics, 24(1):553–562.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Hongyin Luo,  Zhiyuan Liu,  Huanbo Luan,  and  Maosong Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Online learning of inter-  pretable word embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' In Proceedings of the  2015 Conference on Empirical Methods in Natural  Language Processing, pages 1687–1692.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Binny Mathew, Sandipan Sikdar, Florian Lemmerich,  and Markus Strohmaier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' The polar framework:  Polar opposites enable interpretability of pre-trained  word embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' In Proceedings of The Web Con-  ference 2020, pages 1548–1558.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Chris  McCormick  and  Nick  Ryan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  BERT  word  embeddings  tutorial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  https:  //mccormickml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='com/2019/05/14/  BERT-word-embeddings-tutorial/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Tomas Mikolov, Kai Chen, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='s Corrado, and Jeffrey  Dean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Efficient estimation of word represen-  tations in vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Proceedings of the 1st Inter-  national Conference on Learning Representations,  2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  George A Miller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Wordnet: a lexical database for  english.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Communications of the ACM, 38(11):39–  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Marzieh Mozafari, Reza Farahbakhsh, and Noël Crespi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Hate speech detection and racial bias mitiga-  tion in social media based on bert model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' PloS one,  15(8):e0237861.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Brian Murphy, Partha Talukdar, and Tom Mitchell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Learning effective and interpretable seman-  tic models using non-negative sparse embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' In  Proceedings of COLING 2012, pages 1933–1950.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Moin Nadeem, Anna Bethke, and Siva Reddy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  StereoSet:  Measuring stereotypical bias in pre-  trained language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  In Proceedings of the  59th Annual Meeting of the Association for Compu-  tational Linguistics and the 11th International Joint  Conference on Natural Language Processing (Vol-  ume 1: Long Papers), pages 5356–5371.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Charles Egerton Osgood, George J Suci, and Percy H  Tannenbaum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' 1957.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' The Measurement of Meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' University of Illinois press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Abhishek  Panigrahi,  Harsha  Vardhan  Simhadri,  and  Chiranjib  Bhattacharyya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  word2sense:  sparse interpretable word embed-  dings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' In Proceedings of the 57th Annual Meeting  of the Association for Computational Linguistics,  pages 5692–5705.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Jeffrey Pennington, Richard Socher, and Christopher D  Manning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Glove: Global vectors for word rep-  resentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' In Proceedings of the 2014 Conference  on Empirical Methods in Natural Language Process-  ing, pages 1532–1543.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Pranav Rajpurkar, Robin Jia, and Percy Liang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Know what you don’t know: Unanswerable ques-  tions for SQuAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' In Proceedings of the 56th Annual  Meeting of the Association for Computational Lin-  guistics (Volume 2: Short Papers), pages 784–789,  Melbourne, Australia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Pranav Rajpurkar, Jian Zhang, Konstantin Lopyrev, and  Percy Liang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Squad: 100 ,000+ questions for  machine comprehension of text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' In Proceedings of  the 2016 Conference on Empirical Methods in Natu-  ral Language Processing, pages 2383–2392.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Emily Reif, Ann Yuan, Martin Wattenberg, Fernanda B  Viegas, Andy Coenen, Adam Pearce, and Been Kim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Visualizing and measuring the geometry of  BERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Advances in Neural Information Processing  Systems, 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Marco Ribeiro, Sameer Singh, and Carlos Guestrin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' “why should I trust you?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=': Explaining the pre-  dictions of any classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' In Proceedings of the 2016  Conference of the North American Chapter of the  Association for Computational Linguistics: Demon-  strations, pages 97–101, San Diego, California.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' As-  sociation for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Florian Schmidt and Thomas Hofmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Bert as a  teacher: Contextual embeddings for sequence-level  reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' arXiv preprint arXiv:2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='02738.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Lütfi Kerem ¸Senel, Furkan ¸Sahinuç, Veysel Yücesoy,  Hinrich Schütze, Tolga Çukur, and Aykut Koç.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Learning interpretable word embeddings via bidi-  rectional alignment of dimensions with semantic  concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Information Processing & Management,  59(3):102925.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Rita Sevastjanova, Aikaterini-Lida Kalouli, Christin  Beck, Hanna Schäfer, and Mennatallah El-Assady.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Explaining contextualization in language  models using visual analytics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' In Proceedings of the  59th Annual Meeting of the Association for Compu-  tational Linguistics and the 11th International Joint  Conference on Natural Language Processing (Vol-  ume 1: Long Papers), pages 464–476.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Sandipan Sikdar, Parantapa Bhattacharya, and Kieran  Heese.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Integrated directional gradients: Fea-  ture interaction attribution for neural nlp models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  In Proceedings of the 59th Annual Meeting of the  Association for Computational Linguistics and the  11th International Joint Conference on Natural Lan-  guage Processing (Volume 1: Long Papers), pages  865–878.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Richard Socher, Alex Perelygin, Jean Wu, Jason  Chuang, Christopher D Manning, Andrew Y Ng,  and Christopher Potts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Recursive deep mod-  els for semantic compositionality over a sentiment  treebank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' In Proceedings of the 2013 Conference on  Empirical Methods in Natural Language Processing,  pages 1631–1642.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Robyn Speer, Joshua Chin, and Catherine Havasi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Conceptnet 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='5: An open multilingual graph of gen-  eral knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' In Proceedings of the 31st AAAI  Conference on Artificial Intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Fei Sun, Jiafeng Guo, Yanyan Lan, Jun Xu, and Xueqi  Cheng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Sparse word embeddings using l1  regularized online learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' In Proceedings of the  Twenty-Fifth International Joint Conference on Arti-  ficial Intelligence, pages 2915–2921.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Mukund Sundararajan, Ankur Taly, and Qiqi Yan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Axiomatic attribution for deep networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' In Inter-  national Conference on Machine Learning, pages  3319–3328.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' PMLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Ian Tenney, Dipanjan Das, and Ellie Pavlick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  BERT rediscovers the classical NLP pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  In  Proceedings of the 57th Annual Meeting of the Asso-  ciation for Computational Linguistics, pages 4593–  4601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob  Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz  Kaiser, and Illia Polosukhin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Attention is all  you need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' In Advances in Neural Information Pro-  cessing Systems, pages 5998–6008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Jesse Vig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' A multiscale visualization of atten-  tion in the transformer model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' In Proceedings of the  57th Annual Meeting of the Association for Compu-  tational Linguistics: System Demonstrations, pages  37–42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Alex Wang, Amanpreet Singh, Julian Michael, Fe-  lix Hill, Omer Levy, and Samuel Bowman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  GLUE: A multi-task benchmark and analysis plat-  form for natural language understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  In Pro-  ceedings of the 2018 EMNLP Workshop Black-  boxNLP: Analyzing and Interpreting Neural Net-  works for NLP, pages 353–355, Brussels, Belgium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Yonghui Wu, Mike Schuster, Zhifeng Chen, Quoc V  Le,  Mohammad Norouzi,  Wolfgang Macherey,  Maxim Krikun,  Yuan Cao,  Qin Gao,  Klaus  Macherey, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Google’s neural machine  translation system: Bridging the gap between hu-  man and machine translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  arXiv preprint  arXiv:1609.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='08144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Mengjie  Zhao,  Philipp  Dufter,  Yadollah  Yaghoobzadeh, and Hinrich Schütze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Quanti-  fying the contextualization of word representations  with semantic class probing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  arXiv preprint  arXiv:2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='12198.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  A  Appendix  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='1  Post-processing  We point out some issues when using WordNet  directly as our oracle and present ways to address  them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='1  Sense Post-processing  We notice that word-senses identified by WordNet  can be overly granular (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' according to Word-  Net there are five different polar sense pairs for  “wet” ↔ “dry”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' To get rid of redundant senses,  we propose to use dimension-reduction methods  such as variance maximization and orthogonal-  ity maximization from the original POLAR frame-  work (Mathew et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Alternatively, one  could merge similar senses if the cosine similarity  between their sense embeddings is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Selecting  or discarding a rare sense is often task specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='2  Low-quality Example Sentences  The polar sense embeddings are dependent on  the quality of the context (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=', example sentences  demonstrating a particular sense).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' While deploying  WordNet as source for contexts, we encountered  some issues which we elaborate on next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Flections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' When constructing polar sense dimen-  sions, we use the respective word in its basic form  and extract the embedding from the example con-  text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' However, in WordNet’s example sentences,  words often do not appear in their basic form, but  in inflections (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' “She walks with a slight limp”  for “Walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Synonyms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Occasionally, the word itself is not  present in the context sentence but is replaced  by a synonym (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' “the right answer” for “Cor-  rect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Misspellings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  We observe that the example  sentences often contain spelling mistakes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  “toungued lightning” for “Tongued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01”)  Mixed-up examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' In some cases, the example  sentences of a sense are identical to those of the  opposite pole (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' “we docked at noon” for “Un-  dock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='01”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  These problems need to be addressed with man-  ual checks of the obtained polar sense pairs and  context sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='2  Sense Scales  Instead of rating words on scales defined between  two pole words (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' “left” ↔ “right”), our scales  are defined between two pole word-senses (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  “left” ↔ “right” in the sense of direction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' While  the static POLAR framework rates the word “cor-  rect” highly on the dimension “left” ↔ “right”, we  expect our framework to rate it low on the dimen-  sion “left” ↔ “right” in the sense of direction but  rate it high on the dimension “wrong” ↔ “right” in  the sense of correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  To this aim, we analyze whether our constructed  representative sense embeddings encode enough  sense-related information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' As an illustrative exam-  ple, we consider the word “right” which is used in  the senses of direction, correctness and lawfulness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  For each context, we compute the SensePOLAR  embeddings for the word and rank the dimensions  based on the absolute value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Sense-Scales  Context  he went to the right  his argument is right  film rights  Direction  “left” ↔ “right”  1st  38th  32nd  Correct  “wrong” ↔ “right”  44th  1st  291st  Lawful  “wrong” ↔ “right”  55th  27th  9th  Table 5: Ability of SensePOLAR in differentiating be-  tween senses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' We consider the word “right” in three dif-  ferent contexts and obtain a ranking of the dimensions  for each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' We report the rank of a SensePOLAR di-  mension in each of the three contexts in each row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' For  example, the dimension “left” ↔ “right” representing  the sense of direction is ranked first for the context “he  went to the right”, while it is ranked 38th and 32nd re-  spectively in the other two contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' SensePOLAR is  indeed able to identify the correct sense dimensions de-  pending on the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  In table 5, we report the ranks of the polar sense  dimensions for each context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' For the word “right”  in the context of direction “he went to the right”,  the dimension “left” ↔ “right” is selected as the  most representative dimension (rank 1), while the  correctness and the lawful dimensions are ranked  much lower (44 and 55 respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Similar re-  sults are obtained for the other contexts (see ta-  ble 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  These results indicate that the sense-dimensions  of SensePOLAR precisely captures the individual  semantics of the senses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='3  Computational Requirements  SensePOLAR embeddings of words for a given  context can be obtained at low cost if the polar  sense space is pre-computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' We provide such  an implementation along with the submission and  encourage readers to review it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Our implementation  can even be run on a personal computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Given n polar sense dimensions, inversion of the  matrix can be computed in the worst case in O(n3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Since in our case n = 1762, the computation is  very fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' Moreover, this computation needs to be  performed only once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='  Retraining the task-specific feed-forward layer  with SensePOLAR embeddings was performed on  a computing server with 1 TB RAM, 72 cores, each  Intel Xeon Gold 6140 CPU at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content='30 GHZ and 2  Tesla P100-PCIE, 16GB GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
+page_content=' We would like to  reiterate that the retraining is also quite cheap given  only the task-specific feed-forward layer needs to  be trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfwgvw/content/2301.04704v1.pdf'}
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf,len=960
+page_content='Strong correlation effects observed by an ANN-MFT encoder trained on α-RuCl3 high  magnetic field data  Michael J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Lawler,1, 2, 3, ∗ Kimberly A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Modic,4 and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Ramshaw2  1Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' of Physics, Applied Physics, and Astronomy, Binghamton University, Binghamton, NY 13902  2Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  of Physics, Cornell University, Ithaca, NY 14853  3Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  of Physics, Harvard University, Cambridge, MA 02138  4Institute of Science and Technology Austria, Am Campus 1, 3400 Klosterneuburg, Austria  (Dated: January 5, 2023)  α-RuCl3 is a magnetic insulator exhibiting quantum spin liquid phases possibly found in the  Kitaev honeycomb model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  Much of the effort towards determining Hamiltonian parameters has  focused on low magnetic field ordered phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' We study this problem in the high magnetic field limit  where mean-field theory is better justified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' We do so by machine-learning model parameters from  over 200,000 low dimensional data points that include magnetization, torque, and torsion data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Our  machine, an artificial neural network-mean-field theory (ANN-MFT) encoder, maps thermodynamic  conditions (temperature and field vector) to model parameters via a fully connected time-reversal  covariant (equivariant) neural network and then predicts observable values using mean-field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  To train the machine, we use PyTorch to enable backpropagation through mean-field theory with  a pure PyTorch implementation of the Newton-Raphson method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' The results at 20 K and 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='5  T are consistent with other parameter inference studies in the literature at low magnetic field but  strikingly have magnitudes that scale with temperature from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='3 K up to 80 K in the 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='5 − 60 T  range .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' We conclude that the data presents physics beyond the scope of the mean-field theory and  that strong interactions dominate the physics of α-RuCl3 up to field strengths of at least 60 Tesla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  INTRODUCTION  α-RuCl3 is a frustrated magnet with unexplained  properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  Early experiments observed its insulating  character[1] and that quasi-one dimensional β-RuCl3 or-  ders antiferromagnetically at 600K while the stacked  honeycomb structure of α-RuCl3 antiferromagnetically  orders at 13 K[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  Experiments in 2015 then recog-  nized that the comparatively low antiferromagnetic tran-  sition temperature in α-RuCl3 results from frustration,  despite the large Curie-Weiss temperature of order 150  K[3, 4] (although the precise size of this number is now  in debate[5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' In 2017, Neutron scattering then found a  continuum of spin excitations at high energy [6] with a  star-like pattern near the gamma point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Further stud-  ies reveal for Hab > 6T and/or Hc > 30T, the an-  tiferromagnetic order melts[7, 8], a continuum of exci-  tations emerges[9], a thermal Hall conductivity appears  quantized[10] between 6T ≲ Hab ≲ 8T, scaling emerges  in thermodynamic observables[11] over a wide range of  magnetic fields below 80 Kelvin, and an in-plane field  generates quantum oscillations[12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' How is it that one  material can behave in this way?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  The simultaneous explanation of these properties is  difficult, despite the simplicity of the setting: a van der  Waals-coupled honeycomb lattice of spins[13] that can be  studied both in bulk—a kind of “magnetic graphite”—  and as isolated layers—a “magnetic graphene”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' One rea-  son it is hard, is because frustration arises unexpectedly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  Since the honeycomb lattice is bipartite, a nearest neigh-  ∗ mlawler@binghamton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='edu  bor Heisenberg model predicts N´eel ordering and no frus-  tration is expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  It is therefore the spin-orbit cou-  pling that generates frustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' But this frustration is  somehow not only classical, as is understood in Heisen-  berg magnets in the large-S limit , but also quantum  mechanical—computational studies find small terms nor-  mally neglected in spin models dramatically change the  phase diagram[14] and are necessary to stabilize the ob-  served antiferromagnetic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Another reason an ex-  planation is hard is the initial indications that α-RuCl3  is simply described by the Kitaev model has not sur-  vived closer scrutiny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Symmetry allows for other terms  in the Hamiltonian placing the problem into a more gen-  eral spin-orbit coupled model space[15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Finally, the  experiments themselves do not seem to agree with each  other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  Quantum oscillations from an in-plane field[12]  clearly evident below 3 Kelvin suggest a quantum spin  liquid phase with complex-Fermionic excitations form-  ing a three-dimensional Fermi surface while a half-integer  quantized thermal hall effect observed between 3 and 5  Kelvin suggest Majorana fermion edge modes forming a  topological phase with gapped fermionic excitations in  the bulk[10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  One strategy to resolve these issues is to infer the  Hamiltonian model parameters to enable computational  methods to aid in the interpretation of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Begin-  ning with the Kitaev model to capture high energy fea-  tures in neutron scattering, it was argued that “modest  amounts of additional neighbor correlation or simple per-  turbations based on mean-field approaches” reproduce  the star shaped signal near the Gamma point[6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' These  fits, either with an antiferromagnetic Kitaev term[6] or  with ferromagnetic Kitaev coupling, via a parton the-  ory, not not quantitatively reproduce the star pattern[17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='01301v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='str-el]  3 Jan 2023  2  But the sign of the Kitaev term is intimately connected to  the direction of the ordering moments[18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Though hard  to determine with neutron scattering[19], resonant elastic  x-ray scattering (REXS) unambiguously show the Kitaev  term is ferromagnetic, a conclusion consistent with fits to  many other experiments[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Taken together, these results  show the parameter inference problem itself is hard, per-  haps only resolved in this region of the phase diagram  by a quantum dynamics simulation capable of uniting  inelastic and elastic neutron scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  This confusion motivates a second strategy, inferring  parameters at large magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Such a strategy was  successful in a quantum spin ice material[20]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Further-  more, there is published high field magnetization[21],  torque[22], and torsion[11] data, and even magnetization  data in pulsed fields up to 100 T[23] data on α-RuCl3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  Qualitative fits to these datasets[23–25] are consistent  with ferromagnetic Kitaev term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' But these fits did not  explore the entire parameter space, instead choosing pa-  rameters similar to other studies and focused on employ-  ing powerful computational methods to compare theory  to experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' So they did not quantify the uncertainty  in their fits or take advantage of the simplicity of the high  field limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  In this paper, we combine mean-field theory and arti-  ficial neural networks (MFT-ANN) to solve the param-  eter inference problem at high field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  By using an en-  coding scheme , we combine magnetization, torque, and  torsion data sets into one data set with over 200,000  low-dimensional data points .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  The ANN constructs a  smooth map from the thermodynamic state (T, ⃗H) and  data category C of a data point to the Hamiltonian model  parameters from which a MFT estimates the experi-  mental observable corresponding to C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' A trained ANN  then shows a variation of these parameters over the data  set, a variation that may have physical implications as  well as providing a degree of uncertainty quantification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  There is no guarantee that training the ANN multiple  times will produce the same state-to-parameter map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' In-  stead, the interpretation of the result is reliable when  the MFT is able to fit the data well, even if two qualita-  tively different maps produce equally good fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Namely,  like MFT can have multiple saddle point solutions, data  fitting via an MFT-ANN also has this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  Our  trained MFT-ANN shows the scaling behavior of ther-  modynamic observables—previously observed in the in-  termediate field regime[11]—is reflected in the ANN, with  the Kitaev and Gamma couplings scaling with tempera-  ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Hence, this scaling is a strong correlation effect not  captured by MFT and the mapping shows it persists up  to 60 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  AN ANN-MFT ENCODER FOR LEARNING  MODEL PARAMETERS  A theory/model of condensed matter physics can be  thought of as a decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' It gives us a map from a set of  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
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+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Combining a physics model with a neural network  for parameter inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' a a flow chart describing the ANN-  MFT encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Here the thermodynamic state of a condensed  matter system, taken to be magnetic field and temperature  in this paper, together with an observable category label C is  sent through an ANN that encodes this data into parameters  of a physics model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  A backpropagatable mean-field theory  then computes the observable OC for that model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  b The  honeycomb-Kitaev model geometry that forms the foundation  for a study of α-RuCl3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' These models live on the honeycomb  lattice involving two types of sites, A sites denoted by an  open circle, and B sites denoted by a filled circle, and three  types of bonds denoted by orange (x-bond), purple (y-bond),  and turquoise (z-bond).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' At each site lives a spin S that is  anisotropically coupled via g-factors g to a magnetic field H,  and on each bond lives an exchange coupling matrix J that is  different on each of the three types of bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  parameters to a physical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
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+page_content=' Specifically, we need to view the MFT as computing  an observable O that is a function of the model parame-  ters O(θ) and be able to take derivatives ∂O/∂θn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' This  will allow the training of a neural network that predicts  the model parameters θ(W, b) given its weights W and  biasesb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Hence, a backpropagatable MFT layer would  solve the problem of adding domain knowledge to a data  science algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  In this paper, we will show how to achieve backpropa-  gation through a mean-field theory(MFT) layer and use  3  it together with an artificial neural network (ANN) en-  coding layer forming an ANN-MFT encoder as shown in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' 1, to learn the parameters of RuCl3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  In a sufficiently large magnetic field, spins collectively  precess about the magnetic field direction and a mean-  field Si → m develops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Lowering the magnetic field, an-  tiferromagnetic spin exchange interactions begin to fight  this tendancy, and a staggered Ne´el component to the  mean-field m → mi could develop, producing a differ-  ent mean field on the A and B sublattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  Lowering  further, the ordering could become even more complex  and/or novel states not captured by mean field theory  could arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  With this in mind,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' we begin with the K-Γ model,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' be-  lieved to capture the physics of RuCl3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' defined as  HK−Γ = 1  2  �  ⟨ij⟩  Jiα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='jβSα  i Sβ  j −µB ⃗H·g·  �  i  Si  (1)  =  �  ⟨ij⟩  �  KSγ  i Sγ  j +Γ(Sα  i Sβ  j +Sβ  i Sα  j )  �  (2)  −µBga ⃗H⊥ ·  �  i  S⊥  i − µBgcHz  �  i  Sz  i  where spin operators are dimensionless with S  =  1  2(σx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' σy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' σz),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' σα the Pauli matrices,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' and in the second  line,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' we choose the convenient “theory basis” to express  the couplings K and Γ where α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' γ take on the values  (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' z) if ⟨ij⟩ is a “z”–bond,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' (z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' y) if ⟨ij⟩ is a “y”–  bond,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' (y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' x) if ⟨ij⟩ is a “x”–bond,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' as described  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  1, and we choose the convenient “experimen-  tal basis” (a, b, c) to express the g-factors that relate the  coupling of the spins to the external magneteic field ⃗H  in terms of the unit cell geometry, setting ga = gb due to  assumed rotational invariance in the ab-plane .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Since we  are interested in thermodynamic data, we then construct  the partition function that is the generating function of  all thermodynamic observables:  Z = Tr exp  �  − 1  2  �  ijαβ  jiα,jβSα  i Sβ  j + h ·  �  i  Si  �  (3)  where jiα,jβ = Jiα,jβ/kBT and h = µBH · g/kBT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' We  write the model parameters in this unusual way because  now they are easier to work with in a machine learning  context as they are dimensionless parameters of order 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  In this way, we have set up a model that given K/kBT,  Γ/kBT, ga, and gc, we can predict any thermodynamic  observable provided sufficient computational resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  The exact evaluation of experimental observables, such  as the magnetization Mα = −kBT  ∂  ∂Hα ln Z, is hard for  systems with more than 16 spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' So we next make the  physically-reasonable mean-field approximation, reason-  able as discussed above when there is a large magnetic  field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' It makes the replacement  Sα  i Sβ  j = mα  i Sβ  j + Sα  i mβ  j − mα  i mβ  j  (4)  where mi are the mean fields, the mean value of the spin  operators Si on a given site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' We have found stopping  the complexity of the mean-field theory at the staggered  component, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' defining just two vectors mA and mB  placed in a periodic-in-the-unit-cell arrangement, with  sites i ∈ A belonging to the A sublattice and i ∈ B  belonging to the B sublattice, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' 1, is suf-  ficient to produce stable solutions to the mean-field equa-  tions for typical values of the model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' These  equations at a finite temperature T are  mA =  1  |A|  �  i∈A  ⟨Si⟩ = 1  2  heff  A  |heff  A |  Tanh(|heff  A |/2)  (5)  and similarly for mB, where heff  A = h − 3¯ȷ · mB is the  dimensionless effective field felt by spins on the A sub-  lattice due to a combination of the external magnetic  field and the couplings to the three surrounding spins on  the B sublattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Here ¯ȷ = diag(jxx, jxx, jzz) is the av-  erage coupling over the three bonds in the unit cell with  jxx = (K − Γ)/3kBT and jzz = (K + 2Γ)/3kBT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Solv-  ing these mean-field equations then fixes the values of  mA and mB and enables us to compute any observable  within the mean-field approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  To solve the parameter inference problem using an  ANN-MFT encoder, we need to solve the mean field  equations in a way that allows us to backpropagate from  a calculated observable like the magnetization, through  the model parameters to the weights and biases of the  ANN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Defining the ANN as a trainable map f(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' W, b)  from a point x ∈ X in a thermodynamic data set X with  x = (H, T) to the model parameters θ = (ga, gc, K, Γ),  with weights and biases W, b, and the magnetization  computed within the mean-field theory M = M(θ, x),  we see by the chain rule  ∂M  ∂Wa  = ∂M  ∂θm  ∂θm  ∂Wa  = ∂M  ∂θm  ∂fm  ∂Wa  ,  (6)  where we used θm = fm(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' W, b), that the key challenge  is to be able to compute quantities like ∂M  ∂K and ∂M  ∂ga .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  The solution we take to accomplish this is numeric  differentiation, which is used via automatic differentia-  tion in modern machine learning packages to compute  derivatives like  ∂fm  ∂Wa when training the neural network  in stochastic gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' As such, we implemented  this solution in pytorch with a native implementation of  the Newton-Raphston algorithm as shown in Listing 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  The key to implementing such a method is to use na-  tive pytorch tensors with requires_grad enabled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' For  this to work consistently, it is helpful to use the deco-  rator @torch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='enable_grad() above functions that call  newtons method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Hence, using a native pytorch implemen-  tation of the numerical solution to mean-field equations,  one can directly use physics models in this approximation  as layers within a larger machine learning model connect-  ing experimental data to theoretical predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  In addition to backpropagation, a key benefit of using  pytorch to implement the solution to the mean field equa-  tions is the ability to solve them for different model pa-  rameters in parallel on a gpu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' We solve them in batches  4  1def newtons method ( function ,  i n i t i a l ,  2  i t e r a t i o n s=100 ,  t o l=torch .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' f i n f o () .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' eps ) :  3  i f  not( i n i t i a l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' r e q u i r e s g r a d) :  4  i n i t i a l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' r e q u i r e s g r a d = True  5  for  i  in range( i t e r a t i o n s ) :  6  previous = i n i t i a l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' clone ()  7  value = function ( i n i t i a l )  8  value .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' backward ( torch .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' o n e s l i k e ( value ) ,  9  retain graph=True)  10  with torch .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' no grad () :  11  i n i t i a l −= ( value /  i n i t i a l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' grad )  12  i n i t i a l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' grad .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' z e r o  ()  13  14  i f  ( i n i t i a l − previous ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' abs () .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='max( ) < \\  15  torch .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' tensor ( t o l ) :  16  return  i n i t i a l  17  return  i n i t i a l  Listing  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  Native  pytorch  code  for  Newton-Raphson  method (also known as Newton’s method) adapted from a  stackoverflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='com question[26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  1  A = Id + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='0∗ torch .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='bmm( chi0 , j )  2  u n s t a b l e c h i = torch .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' zeros (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' shape [ 0 ] )  3  chi = torch .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' z e r o s l i k e( chi0 )  4  for  i  in range( len (A) ) :  5  try :  6  chi [ i ]=torch .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' l i n a l g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' s o l v e (A[ i ] , chi0 [ i ] )  7  except RuntimeError :  8  print ( ” Singular A matrix  found !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' ” )  9  chi [ i ] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='0E6∗ Id [ i ]  10  try :  11  t e s t = torch .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' l i n a l g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' cholesky ( chi [ i ] )  12  except RuntimeError :  13  u n s t a b l e c h i [ i , 0 ] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='0  Listing 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Code that calculates the susceptibility efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  of 1000, a number we found to be efficient while pre-  serving the stochastic property of the gradient descent  used to train the ANN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' We have been able to solve them  in parallel with 100,000 model parameters on a desktop  with an Nvidia Titan X GPU, a features that might be  valuable in other applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  In the traditional setting, mean-field equations are  solved for a particular choice of model parameters and so  the mean-fields themselves are chosen to be those which  produce stable solutions to these equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' In the set-  ting of this paper, however, we need stable mean-field  solutions for all model parameter possibilities, which is  not easily achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' We overcome this challenge by com-  puting the susceptiblity χ at the mean-field solution and  using this calculation to check if a stable solution was  obtained, flagging those that are found to be unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  Naively, this is done by computing (see Appendix A 3 for  derivation)  χ = (I + 3χ0 · j)−1χ0  (7)  There are two ways this calculation can fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' The first is if  the matrix A = I+3χ0 ·j is itself singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' This happens  at a phase transition where χ → ∞ and is not a sign  that an unstable solution was obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' The second is if  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
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+page_content='H  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
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+page_content='MFT  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='Mean  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='Field  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
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+page_content='Torsion  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
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+page_content='�>0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='Merge by Category  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Mean field theory implemented for machine learn-  ing purposes in two layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  The first solves the mean field  equations to determine the values of the mean fields following  Listing 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' The second computes the observables for the re-  quested category, either magnetization, torsion, or torque in  this study, and a flag χ>0 that is 0 if the mean field solution  is unstable and 1 if it is stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  the matrix χ is not positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Such a situation implies the  mean-field solution is thermodynamically unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' We  have found the code presented in Listing 2 was able to  perform these checks efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' It runs in series on each  solution of the mean-field equations previously computed  in parallel and achieves efficient checking by a) using torch  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' linalg .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' solve to solve the system of equations A · χ = χ0  for χ and use torch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' linalg .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='cholesky to check via failure of  this algorithm for positive-definitness of χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  Combining the above ideas we can construct the MFT  layer as a combination of two layers as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  One layer solves the mean field equations using Listing 1,  and the other computes the observables for the category  of the data and the flag χ>0 denoting whether or not the  MFT equations were stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  With the MFT upgraded to perform as a layer in a ma-  chine learning algorithm, it remains to specify the ANN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  In the context of high magnetic field data, it turns out  we can preserve the rotational symmetry about the c axis  and the time-reversal symmetry of the data in the archi-  tecture of the ANN , a symmetry covariant/equivariant  neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  To build in the rotational invariance  about the c-axis we choose the magnetic field H to always  lie in the ac-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' To build in time-reversal symmetry  so that sending in (H, T, C) to the ANN will produce the  same model parameters (ga, gc, jxx, jzz) in the output as  sending in (−H, T, C), we need to design the neurons  more carefully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  We achieve time-reversal symmetry by separating the  neurons into two groups, those that act on time-reversal  odd variables like H and those that act on time-reversal  even variables like T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Then it is important to preserve  this property throughout the network, including when  standardizing the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' In general,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
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+page_content='FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' The archecture of the ANN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' We choose to build the fully connected neural network that is time-reversal symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  We do so by setting the bias to zero and using the symmetric seagull activation function in the first layer for magnetic field  inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Otherwise, the rest of the network is an ordinary fully connected neural network, here using the common LeakyReLU  function with a slope 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='1 for negative values of its input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' We also pass the output for parameters ga and gc to the softplus  function log(1 + ex), a smooth version of ReLU which guarantees they are positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  vector, b a bias vector, and σ(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=') is a non-linear function  that separately acts on each of the components of the  vector in its argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' For a time reversal odd variable,  we need to demand σ(w · (−x) + b) = −σ(w · x + b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  This requires b = 0, and σ(−x) = −σ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' For the latter,  one choice is σ = tanh, the hyperbolc tangent function,  but we won’t need this in our network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' For time-reversal  even variables, there are no restrictions on σ, w, and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  But since our network outputs only time-reversal even  variables, the parameters of the model, we need a neuron  that accepts a time-reversal odd variable but outputs a  time-reversal even variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' To achieve this, we use the  seagull function log(1 + x2), previously used in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  Putting these ideas together, our ANN is presented in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  It immediately converts the time-reversal odd magnetic  field variables into time-reversal even ones via the seagull  function and then successive layers are normal with a  leakyReLU activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' We present our ANN in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' A central benefit of this network is not efficiency,  but interpretation for due to the preservation of time-  reversal symmetry, angular sweeps of the magnetic field  will be strictly periodic in θ → θ +π where θ is the angle  of the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  RESULTS: SCALING AND A STRONG  INTERACTION REGIME  The MFTANN encoder presented in the previous sec-  tion was applied to a dataset of 213803 thermodynamic  data points taken on α-RuCl3 built from a combination  of publically available magnetization data at high mag-  netic fields taken from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' 21, high magnetic field torque  data taken from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' 22, and high magnetic field torsion  data taken from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' We have combined all three  thermodynamic data sets into one set by attaching a cat-  egory label to each measurement observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' This label  is converted to a vector via one-hot encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Specifi-  cally, the in-plane magnetization category was mapped  to the vector (1, 0, 0, 0, 0), the c-axis magnetization to  (0, 1, 0, 0, 0), the torque category to (0, 0, 1, 0, 0), the tor-  sion field sweep data to (0, 0, 0, 1, 0) and the torsion an-  gular sweep data to (0, 0, 0, 0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' In this way, the ANN  can think about these different data types differently due  to different weights and biases associated with each in  the initial layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' We have made our data set available  at zenodo[28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  In addition, we design the MFT layers  following 2 to compute the experimental observables as  shown in appendix A and combine multiple data types  and thereby insert domain knowledge into the machine  learning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  To train the ANN-MFT, we need a loss function that  6  0  50  100  150  200  Epoch  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='01  Log(Mean Squared Error)  Training Loss  Validation Loss  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Loss during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Training loss (blue line) falls  slightly below validation loss (orange line) over 200 epochs  with a dropout fraction of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='25 used to control validation loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  captures the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' A common choice would be to use  the mean-squared error  1  |X|  �  (x,y)∈(X,Y )(ˆy(x)−y)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' But  it is arbitrary in the sense that it weights all data points  equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' It turns out, the data sets are not all equally im-  portant for an interpolation scheme occurs when a data  set is taken rapidly rendering not all data points as inde-  pendent measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' In this case, we can weight each  data set separately in the loss function via  L =  �  n  pn  Nn  �  (x,y)∈(Xn,Yn)  (y(x) − y)2  (8)  where Nn is the number of data points in data set n  and pn is a normalized weight factor satisfying pn > 0,  �  n pn = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  Such a loss function could balance the  ANN-MFT so that it weighs each independent data point  equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  In this paper, we kept the standard mean-  squared-error loss, and its drawbacks, but it would be  interesting to explore the alternative loss function in the  future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  An example of the behaviour of the loss during train-  ing is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  It shows a rapid decay down  to e−5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='0067 over 200 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Each epoch, the time  needed for the machine to see each data point once, took  about 10 minutes on our desktop using an Nvidia Ti-  tan X GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' We find that later stages of the training  can still have a sizable improvement in the predictions  for sparsely populated regions of the data space, regions  which are overwhelmed by more densely populated re-  gions at earlier stages of the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  In what follows, we will present the predictions of a  single trained ANN-MFT model on all the data sets com-  bined as discussed above but then feed this machine in-  dividual data sweeps to see what it predicts for those  portions of the full data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  The simplest data to interpret are the torsion angle  sweeps shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' By covering all angles up to a  time-reversal symmetry transformation, the ANN-MFT  can learn all four parameters ga, gc, K, and Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' All other  data sets fix the angle θ of the magnetic field to the c-  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='2  Torsion [kJ rad  2 mol  1]  Angle Sweep T=20 K, H = 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='5 T  ANN-MFT  Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Data  MFT  0  /2  Angle [radians]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='25  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='50  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='75  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='00  g-factor values  50  40  30  20  10  Coupling strength [K]  ga  gc  K  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Predictions for the angular sweep torsion data and  the couplings used to produce these predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' (top) Fit to  the data showing a good fit throughout much of the angular  sweep from θ = 0 to θ = π, including a decent fit for the MFT  predictions with a fixed value of the couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' (bottom) Cou-  plings used to make the predictions in (top) with the average  values of ga = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='85,gc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='54,K = −35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='26K,Γ = −11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='80 cho-  sen for the MFT prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Note the discrepancies near the  c-axis with θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  axis so, for example, if they are in the ab-plane, they  would be sensitive to ga and jxx ∝ K − Γ while if they  were along the c-axis, they would be sensitive to gc and  jzz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' However, the angle sweeps are at a fixed magnetic  field strength of 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='5 T, so it is unclear if the parameters  have saturated to their high-field values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Nevertheless,  we find they are roughly constant at all angles and equal  to the values ga = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='85, gc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='54, K = −35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='26 K,and  Γ = −11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='80 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' These values are consistent with those  observed in previous studies, especially note the ferro-  magnetic nature of the K coupling, that |K| > |Γ|, and  that ga > gc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  A second observation about the ANN-MFT predic-  tions for the torsion angle sweep data is the behavior  near θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  Here we see the data has an unexpected  peak which is not present in the mean-field results if we  fix the values of the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' It is also hard for the  ANN-MFT to fit this peak, with its attempt introducing  a deviations for angles between θ = π/2 to θ = π, which  7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='0  Mx [  B per Ru3 + ]  Magnetization field Sweep T=18K  ANN-MFT  Experimental Data  MFT Fixed Params  0  20  40  60  Magnetic Field Strength [T]  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='8  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='4  g-Factor [Dimensionless]  40  30  20  Coupling Strength [K]  ga  K  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Predictions for magnetization data and the couplings  used to produce these predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' a predictions for the 18  kelvin, high-field, in-plane magnetization sweep and the pre-  diction of the mean-field theory alone with a fixed choice for  the couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' b The variation in the couplings used to make  the predictions in a during the magnetic field sweep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  The  fixed choice of couplings for the MFT predictions were taken  from the ANN-MFT predictions at 60 Tesla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  was unnecessary between angles θ = 0 and θ = π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' We  believe this struggle of the ANN-MFT is due to an in-  trinsically interacting feature in the data not captured  by the MFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  The predictions for the magnetization data are shown  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' We see that the ga g-factor remains relatively  stable, ranging between 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='9 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' The machine cannot  accurately predict the gc value from this in-plane data  so we do not present it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  The coupling K − Γ reaches  about -40 Kelvin at 60 Tesla and -30 Kelvin at H = 35T,  consistent with the torsion angular sweep predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  The torque data shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' 7 is harder to interpret  than others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  Here the ANN-MFT struggled to fit the  data so we can trust its predictions less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' However, we see  the large field predictions at θ = −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='2o are ga = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='42  K − Γ = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='2K, while at θ = 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='3o are gc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='71,  K + 2Γ = −.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Here the machine is predicting ga < gc,  though both are less than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' solving for K and Γ from  the two data sweeps, we see it predicts K = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='83 and  Γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='36 and are vastly scaled down in size compared  to the predictions of the ANN-MFT on the torsion angle  sweep data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  The torsion field sweep data presents a further surprise  but explains the substantial discrepancy between the tor-  sion angle sweeps and magnetization and the couplings  predicted by from the torque data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' The couplings pre-  0  10  20  30  [J mol  1]  Torque Field Sweep T=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='3K  ANN-MFT at  =  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='2o  Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Data at  =  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='2o  MFT at  =  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='2o  ANN-MFT at  = 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='3o  Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Data at  = 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='3o  MFT at  = 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='3o  0  20  40  60  Magnetic Field Strength [T]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='6  g-Factor [Dimensionless]  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='5  Coupling Strength [K]  ga at  =  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='2o  gc at  = 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='3o  K  at  =  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='2o  K + 2  at  = 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='3o  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Predictions for the Torque data and the couplings  used to produce these predictions along nearly the a-axis  (θ = −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='2o) and c-axis (θ = 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='3o).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' (top) Fit to the data  showing a relatively poor fit at larger magnetic field strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  We also plot the predictions of the MFT with a fixed set  of couplings chosen to match the ANN-MFT predictions at  60 Tesla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' (bottom) Couplings used to make the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  Torque along the a-axis is sensitive to ga and torque along  the c-axis is sensitive to gc so only these are plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' The  Couplings K and Γ are much smaller that those found in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  6 but consistent with other data in the data set at low tem-  peratures (here 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='3 Kelvin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  dicted from these field sweeps with a a-axis magnetic field  approximately scale with temperature, doubling in value  when the temperature changes from 20K to 40K and  again doubling in value when the temperature changes  from 40K to 80K as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' At temperatures  above 80K, the doubling stops (not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' This sur-  prising behavior is consistent with the observed scaling  properties of this data set[11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' As a result, the torque  data now fits into the general picture, it presents scaled  down couplings due to the low temperature at which it  was taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' If we multiply by the ratio of the tempera-  tures, say for the 40 K torsion field magnitude sweeps, we  get ((K − Γ)|torque ∗ 40/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='3 = −36K, not so far from the  K − Γ ≈ −100K observed in the torsion at 40K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Hence,  the results are roughly consistent across all data sets if  one takes into account that the parameters K and Γ scale  with temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  8  0  100  200  300  [muB per Ru3 + ]  Torsion In-Plane Field Sweeps  ANN-MFT 20K  ANN-MFT 40K  ANN-MFT 80K  Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Data 20K  Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Data 40K  Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Data 80K  0  20  40  60  Magnetic Field Strength [Tesla]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='2  g-factor values  350  300  250  200  150  100  50  Couple Strength [K]  ga coupling 20K  ga coupling 40K  ga coupling 80K  K  20K  K  40K  K  80K  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' The couplings used to make torsion field sweep pre-  dictions at a 20K, b 40K, and c 80K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' (top) The ANN-MFT  fit to the torsion field sweep data at the three temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  (bottom) the K − Γ couplings predicted by the ANN-MFT  for the three temperatures showing an increase in magnitude  for this combination of couplings as a function of temperature  approximately doubling between T = 20K and T = 40K and  again doubling (or more) between T = 40K and T = 80K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  DISCUSSION  We have introduced an ANN-MFT to study parame-  ter inference with domain knowledge inserted into a ma-  chine learning algorithm via a backpropagatable mean-  field theory layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  There are several weaknesses to this new approach one  should be careful of before interpreting the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Different training can give rise to distinctly dif-  ferent results due to different saddle points of the  MFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' The ANN-MFT may not fit all data points or sub-  sets well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' The ANN-MFT is only as good as the MFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Is a  more general MFT warranted?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' What does it mean when we do not discover  roughly consistent model parameter values across  all data sets?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  Let us first address point 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  Through several trials  at training the ANN-MFT, it is possible to get wildly  different results, perhaps at a cost of a higher loss and  poorer fit to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' These results still fit some data  well, but in a different regime of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' We inter-  pret this different regime as a saddle point of the MFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  Unlike neural networks that have the magical property  that different solutions for the weights behave similarly,  the MFT has no such magic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' It can have distinct saddle  points that correspond to metastable states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' It would be  interesting in the future to add the energy for each data  point to the loss function and either guarantee the results  are the lowest energy saddle point solution or use such a  modified loss function to isolate metastable states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' What  we present here is the most common and best trained  machines but we do not know if the solution we find is a  metastable or stable state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  For point 2, we emphasize that while the ANN-MFT  may not fit all data points well, whenever it does, we can  trust the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' If even just a local fit to a portion of  the data is good, it implies there is a mapping between  thermodynamic state and model parameters that agrees  with experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  For points 3 and 4, one possibility is that one should  study a more general model, to address point 3, and  hope the more general model provides a consistent set  of parameters across all data points, to address point 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  Presumably one can generalize the MFT of this paper  well beyond the four parameters we study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Introducing  the complete spin exchange matrix Jiα,jβ between not  only nearest neighbors but also next nearest neighbors  and even third neighbors seems quite possible given the  efficiency of the algorithm we have presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Perhaps  generalizing the MFT in this way will yield consistent  parameter values across all data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' But this doesn’t  seem likely in this case since a scaling with temperature  does not seem to be readily captured by further neighbor  couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  In addition to the weaknesses, there are many ways  to improve the simple ANN parameter inference method  used in this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  One is to upgrade the ANN  encoder to behave like a variational autoencoder that  encodes not directly to the parameters but instead to  a probability distribution from which a parameter sam-  ple is drawn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' While such a probabilistic approach would  seem to introduce more noise into the machine, varia-  tional autoencoders are actually more efficient than or-  dinary autoencoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Such an upgrade, in addition to  efficientcy, would then provide error bars on the predic-  tion of the parameter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  Despite the weaknesses and simplicity of the ANN-  MFT machine we have used to study α-RuCL3 high  magnetic field data, we find it striking that a scal-  ing with temperature is the predominant observa-  tion across the entire data set from three differ-  ent experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  A naive explanation for scaling  is that at these large magnetic fields,  the system  9  becomes approximately non-interacting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  By scaling,  κ/T = f(µBH/kBT, K/kBT, Γ/kBT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  If we can set  (K, γ)/kBT ≈ 0, then we obtain κ/T = f(µBH/kBT),  a scaling function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' But our observations are that K and  Γ are proportional to temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Hence, κ/T scales  but we are never in the non-interacting regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' We fur-  ther notice near fields pointing along the c-axis, that  anomalies appear in the data that the ANN-MFT can-  not fit accurately (see torsion angle sweep results).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' This  is strong evidence that the temperature regime 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='3K to  80K is captured by an interacting theory beyond the  Curie-Weiss mean-field theory approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  Unlike the data set as a whole, data at temperatures  between 20 and 80 K saturated at large magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  It would be nice to understand this from a perturba-  tion theory perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  In the limit |h| ≫ |¯ȷ|, owing  to a gap in the spectrum, the system is adiabatically  connected to a non-interacting paramagnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  However,  the limit we find our selves in this manuscript, as it ap-  pears within MFT say from the torsion angle sweeps, is  |¯ȷ| = max(|K − Γ|/3kBT, |K + 2Γ|/3kBT) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='98 when  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='6 ≲ |h| ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='7 so that both |h| and |¯ȷ| are of the same  order or magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' These values do not to appear to  appreciably change with temperature from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='3 Kelvin to  80 Kelvin and only above this temperature do we begin  to see a departure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' This suggests this entire regime is not  easily captured by a perturbation theory and so there is  no controlled approximation to study it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  Spin liquids like α-RuCl3 are hard to study experi-  mentally, yet an ANN-MFT or similar machine could  prove to be a powerful tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' It is striking that from bulk  data we obtain parameters that are consistent with in-  formation rich techniques like neutron scattering and x-  ray scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' We believe ANN-MFTs and related ma-  chine learning techniques will enable a powerful relation-  ship between experimental data and model parameters,  especially in the common situation of many spin liquid  candidate materials where neutron scattering and X-ray  scattering data are unavailable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  This material is based upon work supported by the  National Science Foundation under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  OAC-  1940260.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  Appendix A: Derivation of mean-field observables  To derive observables within the mean field theory, we  begin with the mean field partition function with mean  fields mA and mB satisfying Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' 5  ZMF =Tr exp  �  −heff  A ·  �  i∈A  Si−heff  B ·  �  i∈B  Si+3βNBmA¯JmB  �  (A1)  where NB is the number of bonds, ¯J =  1  NB  �  ⟨ij⟩ Jij is  the average exchange matrix, heff  A = h − 3j · mB is the  effective magnetic fields felt by spins on the A sublattice  due to their interactions with spins on the B sublattice  and similarly for the heff  B  with A and B swapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' We  can evaluate this partition function directly and obtain  ZMF = 2|A|+|B|eNBmA¯JmB  × cosh|A| ����heff  A /2  ���  �  cosh|B| ����heff  B /2  ���  �  (A2)  From this partition function we can obtain all the observ-  ables we need to compare with the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  Specifically, we can compute the magnetization, mag-  netic susceptibility, magnetic torque, and magnetic tor-  sion observables, as shown in the next few sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  Magnetization  To compute magnetization, we use the definition  M α = −∂FMF  ∂Hα  (A3)  where FMF = −kBT ln ZMF and Hα is a component of  the external magnetic field H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Evaluating this expression  gives  M α = kBT |A|  2  ∂|heff  A |  ∂Hα  tanh  ����heff  A /2  ���  �  + A → B+  ∂  ∂Hα  �  NBmA¯JmB  �  (A4)  where  ∂|heff  A |  ∂Hα  = ˆheff  A  ∂heff  A  ∂Hα .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  (A5)  Recognizing that mA = ˆheff  A  tanh  ����heff  A /2  ���  �  we see this  simplifies to  M α = kBT|A|mA·∂heff  A  ∂Hα +A → B+  ∂  ∂Hα  �  NBmA¯JmB  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  (A6)  Using  ∂heff  A  ∂Hα = µB  kBT eα · g − 3j∂mB  ∂Hα  (A7)  with h = µBH · g/kBT then obtain  M α = µB(|A|mA + |B|mB) · g · eα  −3|A|mA¯J∂mB  ∂Hα −3|B|mB¯J∂mA  ∂Hα +  ∂  ∂Hα  �  NBmA¯JmB  �  (A8)  where we used j = ¯J/kBT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' For periodic boundary con-  ditions, |A| = |B| = Nu and NB = 3Nu and we see that  the quadratic inn mA, mB terms cancel leaving us with  M α = NµB ¯m · g · eα  (A9)  10  We see then this result could have been obtained another  way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  The cancellation is precisely what is required to  allow us to first compute M α exactly and then perform  the mean-field approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  Numerically, for an Avogadro’s number of atoms, we  can express this as M a,b = (µB/kB)gaR(ma,b  A + ma,b  B )/2  and M c = (µB/kB)gcR(mc  A + mc  B)/2 where µB/kB =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='6714[K/T] and R = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='314[J/K].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Or for a data set ex-  pressed in units of per Bohm magneton per spin we would  use M a,b = ga(ma,b  A +ma,b  B )/2 and M c = gc(mc  A+mc  B)/2,  as is the case of the data studied in this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  Magnetic Torque  To compute the torque τ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' we can use the definition  τ = M × H  (A10)  and take the component in the direction of increasing  azimuthal angle φ to obtain:  τφ = ˆφ · (M × H) = M · (H × ˆφ) = −HM · ˆθ  (A11)  We recognize we can compute this directly from the free  energy  τφ ≡ ∂F  ∂θ = ∂Hα  ∂θ  ∂F  ∂Hα = −H ˆθ · M  (A12)  Here  and  throughout  this  paper  we  choose  the  spherical  polar  coordinate  system  H  =  H(cos φ sin θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' sin φ sin θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' cos θ) with  ˆφ and  ˆθ the di-  rections of increasing φ and θ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' If we were to  follow Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' [22] and parameterize the magnetic field from  the ab plane as H = H′(cos φ′ cos θ′, sin φ′ cos θ′, sin θ′)  then we would find ˆθ′ = −ˆθ and ∂F/∂θ′ = −∂F/∂θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' As  a result, the torque data presented in this paper differs  in the sign convention for the observable τφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  A convenient numerical expression for τφ, with an Avo-  gadro’s number of atoms and a magnetic field in the ac-  plane, is obtainable by inserting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' A9 in to our expres-  sion for τ and writing it as  τ = −RT dh  dθ · ¯m  (A13)  where R = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='314[J/K] and  dh  dθ = µB  kBT (gaHc, 0, −gcHa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  (A14)  We are free to place H in the ac-plane, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' set φ = 0,  because the rotation symmetry about the c-axis allows us  to always choose this unknown parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Fixing φ = 0  then always produces mean field solutions with mA and  mB also in the ac-plane so we choose to ignore the y  components in our calculation of observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  Magnetic Susceptibility and Thermodynamic  Stability  The magnetic susceptibility is defined as  χαβ = −  ∂2F  ∂Hα∂Hβ  (A15)  Technically speaking it is defined in the limit |H| → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  However, at any finite magnetic field H, thermodynamic  stability demands  δHδM + δTδS + δµδN ≥ 0  (A16)  so that using δM α = ∂M α/∂HβδHβ = χαβδHβ we see  that χ ≥ 0 both at finite and the limit of zero magnetic  field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Hence, the observable χ is a useful quantity at any  value of the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  It is worthwhile simplifying the computation of the  magnetic susceptibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' It is related to the spin suscep-  tibility, via the chain rule  χαβ = − ∂hγ  ∂Hα  ∂2F  ∂hγ∂hδ  ∂hδ  ∂Hα = µ2  B  kBT gαγχγδ  s gδβ  (A17)  where we used ∂hα/∂Hβ = µBgαβ/kBT and defined the  spin susceptibility as χαβ  s  =  ∂2  ∂hα∂hβ (− log Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Since it  is straightforward to convert from χ to χs, via χ =  (µ2  B/kBT)gχsg, we will proceed by focusing on the sim-  pler χs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  We can further simplify the problem of computing  the magnetic susceptibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Within the MFT, the spin  susceptibility is the derivative χαβ  s  = ∂ ¯mα/∂Hβ with  ¯m = 1  2  �  µ mµ, where µ = A, B denotes sublattice (see  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' A9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Hence we need compute ∂mα  µ/∂Hα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' We can  do so by computing the sublattice dependent spin sus-  ceptibility χαβ  sµν = ∂mα  µ/∂Hβ  ν where we have introduced  a hypothetical magnetic field HA and HB that acts sepa-  rately on the A and B sublattices and this new suscepti-  bility captures the response to a change in the magnetic  field in just one of the sublattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' It is related to χαβ  s  by  the chain rule  χαβ  s  =  �  µ  ∂mα  µ  ∂hβ =  �  µν  ∂mα  µ  ∂hβ  ν  ����  hA=hB=h  (A18)  where we thought of mµ as a function of two independent  fields hA and hB, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' mµ(hA, hB) and then took the  derivative of mµ(h, h) with respect to h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' As a result, we  can focus on the sublattice dependent susceptiblity χαβ  sµν,  a quantity we can compute efficiently as a 6x6 matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  An additional benefit is this quantity also must satisfy  χ6 ≥ 0 by thermodynamic stability and so fully expresses  whether the mean field theory is stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  To compute χ6, it is helpful to work with a 6-  component vector notation, m6 = mA ⊕ mB, combining  the three components of magnetization on the A sublat-  tice and B sublattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' In this language, the mean-field  equations define a non-linear map  m6 = V(h6 − 3¯ȷ6 · m6)  (A19)  11  where h6 = h ⊕ h, ¯ȷ6 = ¯J6/kBT with J6 = ¯J ⊗ σx, and  V is the map V(x6) = 1  2x6 ⊙ W(x6) where  W(x6) =  �  �  �  �  �  �  �  |xA|−1 tanh(|xA|/2)  |xA|−1 tanh(|xA|/2)  |xA|−1 tanh(|xA|/2)  |xB|−1 tanh(|xB|/2)  |xB|−1 tanh(|xB|/2)  |xB|−1 tanh(|xB|/2)  �  �  �  �  �  �  �  (A20)  with ⊙ denoting the broadcast matrix operation such as  A ⊙ B = (A1B1, A2B2, A3B3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Hence by the chain rule,  we have  ∇h6m6 = ∂V · (I6 − 3¯ȷ∇h6m6)  (A21)  and so the sublattice dependent spin susceptibility χ6 ≡  ∇h6m6 is given by  χ6 = (I6 + 3∂V ¯ȷ6)−1 ∂V  (A22)  where we recognize ∂V = χ0  s is the “bare” sublattice  spin susceptibility evaluated at the effective magnetic  field h6 − 3¯ȷm6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  Evaluating the derivative, we see it  is given by  ∂V (x6) = 1  2diag(W(x6))+  1  2 [(xA ⊗ xA)T(|xA|)] ⊕ [(xB ⊗ xB)T(|xB|)]  (A23)  where T(x) =  1  2x2 (1 − tanh(x/2)2) − tanh(x/2)/x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  Hence, we have reduced the calculation of the magnetic  susceptibility χ to the determination of χ6 via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' A22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  The implementation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' A22 was discussed in the  main manuscript and was done robustly via Listing 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  A highlight of this calculation was the determination of  wether the mean-field theory was thermodynamically sta-  ble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' This was achieved by checking whether the Cholesky  decomposition failed with a try-catch statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' In this  way, unstable solutions to the mean-field theory could  be caught.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' We chose to deal with such cases by drop-  ping those data points from the loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' So long as  the number of data points that map to unstable MFTs  were small, say 1% or 2%, we found the overall training  of the ANN-MFT worked successfully in that it defines  a mapping onto the parameters of the Hamiltonian that  accurately fits the data for most data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  Torsion  The last observable we need to calculation is torsion κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  It is related to magnetizataion via  κ ≡ ∂2F  ∂θ2 = ∂τφ  ∂θ  (A24)  By chain rule, we can write this as  κ = ∂  ∂θ  �∂Hα  ∂θ  ∂F  ∂Hα  �  = −∂2H  ∂θ2 · M − ∂H  ∂θ · ∂M  ∂θ  (A25)  Recognizing that ∂2H/∂θ2 = −H and using ∂H/∂θ =  H ˆθ we obtain  κ = H · M − H ˆθ · ∂M  ∂θ  (A26)  It remains to place κ in a numerically convenient form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  By using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' A9 and h(µB/kBT)g·H, and the chain rule,  we can write κ as  κ = RT  2  �  2h · ¯m − dh  dθ · χs  dh  dθ  �  (A27)  and further in the 6-dimensional vector notation it be-  comes  κ = RT  2  �  h6 · m6 − dh6  dθ · χ6  dh6  dθ  �  (A28)  where we used Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' A18 to relate χs and χ6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' This is the  form we use in our calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  [1] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Binotto, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Pollini, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Spinolo, Optical and trans-  port properties of the magnetic semiconductor α-rucl3,  physica status solidi (b) 44, 245 (1971).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  [2] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Fletcher, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Gardner, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Fox, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Topping, X-  ray, infrared, and magnetic studies of α-and β-ruthenium  trichloride, Journal of the Chemical Society A: Inorganic,  Physical, Theoretical , 1038 (1967).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  [3] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Majumder, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Schmidt, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Rosner, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Tsirlin, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Ya-  suoka, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Baenitz, Anisotropic ru 3+ 4 d 5 mag-  netism in the α- rucl 3 honeycomb system: Susceptibil-  ity, specific heat, and zero-field nmr, Physical Review B  91, 180401 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  [4] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Sears, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Chern, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Kim, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Bereciartua,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Francoual, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Kim, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Kim, Ferromag-  netic kitaev interaction and the origin of large magnetic  anisotropy in α-rucl3, Nature physics 16, 837 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  [5] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Li, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Winter, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Kaib, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Riedl, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Va-  lent´ı, Modified curie-weiss law for j eff magnets, Physical  Review B 103, L220408 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  [6] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Banerjee, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Yan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Knolle, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Bridges, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  Stone, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Lumsden, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Mandrus, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Tennant,  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Moessner, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Nagler, Neutron scattering in  the proximate quantum spin liquid α-rucl3, Science 356,  1055 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  [7] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Sears, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Zhao, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Xu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Lynn, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Kim,  Phase diagram of α- rucl 3 in an in-plane magnetic field,  Physical Review B 95, 180411 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  [8] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Baek, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Do, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Choi, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Kwon, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Wolter,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Nishimoto, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Van Den Brink, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' B¨uchner, Evi-  dence for a field-induced quantum spin liquid in α-rucl  12  3, Physical review letters 119, 037201 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  [9] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Wang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Reschke, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' H¨uvonen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Do, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Choi,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Gensch, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Nagel, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' R˜o˜om, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Loidl, Magnetic  excitations and continuum of a possibly field-induced  quantum spin liquid in α- rucl 3, Physical review letters  119, 227202 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  [10] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Kasahara, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Ohnishi, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Mizukami, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Tanaka,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Ma, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Sugii, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Kurita, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Tanaka, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Nasu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Mo-  tome, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=', Majorana quantization and half-integer ther-  mal quantum hall effect in a kitaev spin liquid, Nature  559, 227 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  [11] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Modic, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' McDonald, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Ruff, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Bach-  mann, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Lai, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Palmstrom, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Graf, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Chan,  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Balakirev, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Betts, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=', Scale-invariant magnetic  anisotropy in rucl3 at high magnetic fields, Nature  Physics 17, 240 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  [12] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Czajka, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Gao, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Hirschberger, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Lampen-Kelley,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Banerjee, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Yan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Mandrus, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Nagler,  and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Ong, Oscillations of the thermal conductivity in  the spin-liquid state of α-rucl3, Nature Physics 17, 915  (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  [13] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Burch, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Mandrus, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Park, Magnetism in  two-dimensional van der waals materials, Nature 563, 47  (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  [14] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Lee, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Kaneko, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Chern, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Okubo, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Yamaji,  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Kawashima, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Kim, Magnetic field induced  quantum phases in a tensor network study of kitaev mag-  nets, Nature communications 11, 1 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  [15] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Rau, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Lee, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Kee, Generic spin  model for the honeycomb iridates beyond the kitaev  limit, Physical review letters 112, 077204 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  [16] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Katukuri, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Nishimoto, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Yushankhai, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Stoy-  anova, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Kandpal, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Choi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Coldea, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Rousochatza-  kis, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Hozoi, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Van Den Brink, Kitaev interac-  tions between j= 1/2 moments in honeycomb na2iro3 are  large and ferromagnetic: insights from ab initio quan-  tum chemistry calculations, New Journal of Physics 16,  013056 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  [17] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Knolle, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Bhattacharjee, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Moessner, Dy-  namics of a quantum spin liquid beyond integrability:  The kitaev-heisenberg-γ model in an augmented parton  mean-field theory, Physical Review B 97, 134432 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  [18] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Chaloupka and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Khaliullin, Magnetic anisotropy in  the kitaev model systems na 2 iro 3 and rucl 3, Physical  Review B 94, 064435 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  [19] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Cao, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Banerjee, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='-Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Yan, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Bridges, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Lums-  den, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Mandrus, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Tennant, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Chakoumakos, and  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Nagler, Low-temperature crystal and magnetic struc-  ture of α- rucl 3, Physical Review B 93, 134423 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  [20] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Thompson, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' McClarty, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Prabhakaran, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Cabr-  era, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Guidi, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Coldea, Quasiparticle breakdown  and spin hamiltonian of the frustrated quantum py-  rochlore yb 2 ti 2 o 7 in a magnetic field, Physical review  letters 119, 057203 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  [21] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Johnson, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Williams, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Haghighirad, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Single-  ton, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Zapf, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Manuel, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Mazin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Jeschke,  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Valent´ı, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=', Monoclinic crystal structure of α- rucl 3  and the zigzag antiferromagnetic ground state, Physical  Review B 92, 235119 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  [22] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Modic, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Ramshaw, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Shekhter, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Varma,  Chiral spin order in some purported kitaev spin-liquid  compounds, Physical Review B 98, 205110 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  [23] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Zhou, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Matsuda, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Matsuo, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Li,  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Kurita, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Kindo, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Tanaka, Intermediate quan-  tum spin liquid phase in the kitaev material α-rucl3  under high magnetic fields up to 100 t, arXiv preprint  arXiv:2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='04597 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  [24] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Yadav, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Bogdanov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Katukuri, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Nishi-  moto, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Van Den Brink, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Hozoi, Kitaev exchange  and field-induced quantum spin-liquid states in honey-  comb α-rucl3, Scientific reports 6, 1 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  [25] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Riedl, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Li, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Winter, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Valent´ı, Sawtooth  torque in anisotropic j eff= 1/2 magnets: Application to  α- rucl 3, Physical Review Letters 122, 197202 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  [26] kmario23,  Update  step  in  PyTorch  implementation  of  Newton’s  method,  question  #54316053,  accessed  12/23/2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  [27] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Gao and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Zhang, A use of even activation functions  in neural networks, arXiv preprint arXiv:2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='11713  (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content='  [28] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Lawler, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' Modic, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
+page_content=', High magnetic  field thermodynamic data set for α-rucl3 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfWfxA/content/2301.01301v1.pdf'}
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+BLIND SPOTS: AUTOMATICALLY DETECTING IGNORED PROGRAM INPUTS
+HENRIK BRODIN, EVAN SULTANIK, AND MAREK SUROVIˇC
+Abstract. A blind spot is any input to a program that can be arbitrarily mutated without affecting the
+program’s output. Blind spots can be used for steganography or to embed malware payloads. If blind spots
+overlap file format keywords, they indicate parsing bugs that can lead to differentials. This paper formalizes
+the operational semantics of blind spots, leading to a technique that automatically detects blind spots based
+on dynamic information flow tracking. An efficient implementation is introduced an evaluated against a
+corpus of over a thousand diverse PDFs. There are zero false-positive blind spot classifications and the
+missed detection rate is bounded above by 11%. On average, at least 5% of each PDF file is completely
+ignored by the parser. Our results show promise that this technique is an efficient automated means to detect
+parser bugs and differentials. Nothing in the technique is tied to PDF in general, so it can be immediately
+applied to other notoriously difficult-to-parse formats like ELF, X.509, and XML.
+1. Introduction
+We define a blind spot as any input to a program that can be arbitrarily mutated without affecting the
+program’s output. For example, the Office Open XML (OOXML) [1] format used by document processing
+tools like Microsoft Office is based upon the ZIP archive file format. Every valid OOXML file is also a valid
+ZIP that can be extracted using ZIP utilities. Therefore, arbitrary files can be added to an OOXML file as
+if it were a regular ZIP archive. Any such extraneous files are ignored by the document processor and will
+therefore be blind spots during processing.
+Blind spots are dangerous. For example, blind spots can be exploited for steganography and embedding
+malware payloads. They can also be indicative of parser differentials, for instance, if two parsers exhibit
+different blind spots for the same input.
+But they can also be useful: Blind spots can potentially be
+excluded as candidates for mutation when generating fuzz testing inputs, similar to the Angora fuzzer’s
+branch coverage maximization strategy [6].
+Blind spots are a generalization of the concept of file cavities [3]: unused spaces in a file format that are
+created due to the structure of the surrounding data. However, unlike cavities, blind spots may be dependent
+on the program itself and its execution environment. For example, the image content of a JPEG file will be
+a blind spot to a parser that only reads its EXIF metadata. Likewise, the EXIF metadata will be a blind
+spot to a program that converts JPEGs to another image format like BMP that does not support embedded
+EXIF metadata.
+Blind spots are a type of a more general class of data sources we call quantum inputs: Bytes that can
+be almost arbitrarily mutated without affecting a program’s output. For example, a source code comment
+is a form of quantum input to a compiler, since the bytes within the comment can be arbitrarily changed
+without affecting the behavior of the compiler as long as the bytes do not contain the comment delimiter.
+We only consider blind spots in this paper.
+Since they are associated with both program input and the program itself, blind spots can be indicators
+of parsing bugs. Parsers—particularly hand-generated ones—will often accept a superset of the grammar for
+which they were designed. This manifests as a parser that accepts some inputs that are technically invalid
+according to the file format specification. Sometimes this is intentional, in order to maximize compatibility
+with files generated by other, incorrectly implemented software, or to attempt to repair malformed docu-
+ments. For example, we discovered that an optimization in MµPDF1 will sometimes only check the leading
+“e” in the endobj token; it will eagerly accept eXXXXX and the PDF will still be parsed correctly, despite the
+fact that the PDF standard prescribes the existence of the full token. Such lexical permissiveness can lead
+1https://mupdf.com/
+1
+arXiv:2301.08700v1  [cs.CR]  20 Jan 2023
+
+to parser differentials and so-called file format schizophrenia [2]: when two implementations of a file format
+interpret the same input file differently.
+This paper formalizes the concept of program input blind spots and proposes a novel technique based
+on dynamic taint analysis to detect them. Our definition clarifies the difference between blind spots and
+file cavities: Blind spots are the set of input bytes whose data flow never influences either control flow that
+leads to an output or an output itself. First, we provide a formal definition of blind spots in Section 2.
+This entails mapping input bytes to the output bytes they influence. Next, in Section 3 we describe the
+program instrumentation that is able to efficiently capture the data necessary to create this mapping. We
+then describe how the mapping is constructed from the instrumentation output, as well as how blind spots
+are inferred from the mapping. We demonstrate how this technique can be used to automatically detect
+lexical permissiveness and parser bugs. Finally, in Section 4 we provide empirical evidence that these blind
+spots are correct, and an analysis of blind spots in specific file formats and programs.
+2. Definitions and Formalization
+In order to formalize the concept of input blind spots and explicitly describe our dynamic taint analysis
+approach, we propose an extension to Schwartz, Avgerinos, and Brumley’s SimpIL operational semantics for
+dynamic taint propagation [20]. This section describes that extension and uses it to formally define blind
+spots.
+2.1. An Extension to SimpIL. The original conception of dataflow analysis in SimpIL only tracks whether
+a given variable or memory cell is tainted, not from whence it is tainted. In order to detect blind spots, we
+need to additionally track exactly which input bytes influence output. For example, consider the pseudocode
+in Algorithm 1. The variable a is tainted by program input on line 2. Moreover, the value of a can indirectly
+cause hard-coded data (d = 5) to be written to output on line 7 by virtue of the conditional on line 5.
+Therefore, the first byte of the file cannot be a blind spot, since its mutation can affect output—despite the
+fact that the value of the first byte is never written to output. Even if the SimpIL taint policy (i.e., the
+rules by which taints are propagated) is sufficient to detect that tainted inputs affected the output of the
+program, it is insufficient to detect which inputs were responsible. Therefore, we need to extend SimpIL to
+additionally track the provenance of a taint so that we can map a complete data flow from inputs to outputs.
+Algorithm 1 Tainted Control Flow
+1: procedure TaintedControlFlow
+2:
+a ← ReadInput(1)
+▷ a is tainted by the 1st byte
+3:
+b ← ReadInput(1)
+▷ b is tainted by the 2nd byte
+4:
+c ← a + b
+5:
+if c ≥ 42 then
+6:
+d ← 5
+7:
+WriteOutput(d)
+8:
+end if
+9: end procedure
+SimpIL uses meta-syntactic variables to represent an execution context. ∆ is a mapping of variable names
+to their values and µ is a mapping from memory addresses to their values. τ∆ and τµ map variable names and
+memory addresses to booleans (T|F) defining whether or not that value is tainted in the current execution
+context.
+In order to track taint provenance, we introduce the concept of a taint label: a unique identifier for each
+instance of a tainted variable or memory address in an execution context. We define two types of taint
+labels: canonical and union. A canonical taint is the result of a variable or memory address being assigned
+directly from a program input. A union taint is the result of the combination of two previously tainted
+values (e.g., the result of two tainted variables being operands in a binary operation).
+We extend the SimpIL notation with three new mappings to represent taint labels and track provenance2:
+2The SimpIL semantics are already replete with Greek letters.
+We have chosen ε from the Greek word for “la-
+bels” (ϵπιγραφϵς), κ from the word for “canonical” (κανoνικoς), and γ from the word for “parent” (γoνϵυς).
+2
+
+SimpIL notation for the computation performed by the Input operation
+�
+��
+�
+v is input from src
+ε′ = ε[v ← |ε| + 1]
+κ′ = κ[ε′[v] ← src]
+µ, ∆, ε, κ, γ ⇝ µ, ∆, ε′, κ′, γ
+�
+��
+�
+SimpIL notation for the
+updated execution state after the
+operation
+µ, ∆ ⊢ get input(src) ⇓ v
+�
+��
+�
+SimpIL notation for the evaluation of
+expression get input(src) to value v in
+context µ, ∆
+Input
+µ, ∆ ⊢ e1 ⇓ v1
+µ, ∆ ⊢ e2 ⇓ v2
+v′ = v1♦bv2
+ε′ = ε[v′ ← |ε| + 1]
+γ′ = γ
+�
+ε′[v′] ← ⟨ε[v1], ε[v2]⟩
+�
+µ, ∆, ε, κ, γ ⇝ µ, ∆, ε′, κ, γ′
+µ, ∆ ⊢ e1♦be2 ⇓ v′
+Binop
+Figure 1. SimpIL operational semantics for reading input and executing binary operators
+(see Figure 1 of [20]) updated to include data flow provenance tracking. When a value v
+is read from an input source src, we create a new, unique canonical taint label for v and
+set its taint source info in κ to src. When a binary operator ♦b is applied to expressions
+e1 = v1 and e2 = v2 resulting in the value v′, we create a new, unique union taint label for
+v′ and set its parents to be the taint labels associated with values v1 and v2.
+(1) ε : ∆ ∪ µ → N that maps variable names and memory addresses to unique taint labels;
+(2) κ : N → {the set of all taint sources} that maps canonical taint labels to the information about their
+source (i.e., filename and byte offset); and
+(3) γ : N → N × N that maps union taint labels to their parents.
+SimpIL treats these mappings more like programmatic hashmaps than set theoretic functions. As such,
+SimpIL uses the notation “κ[ℓ]” for the value of taint label ℓ in mapping κ. For brevity, we shall continue
+this theme by using the notation ℓ ∈ κ to represent the fact that ℓ is a key in the mapping κ, and ℓ /∈ κ to
+mean that ℓ is not a key in κ.
+The zero taint label is reserved to represent untainted variables and memory. An untainted variable v
+will always lack source info and descend from the zero label:
+τ∆[v] =
+�
+F
+ε[v] /∈ κ ∧ γ[ε[v]] = ⟨0, 0⟩,
+T
+ε[v] ∈ κ ∨ (γ[ε[v]] = ⟨i, j⟩ ∧ i + j > 0).
+The SimpIL taint policy (see Table III from [20]) and semantics are modified to update these mappings
+on every taint status change. For example, the updated semantics for reading from input and for executing
+binary operations are given in Figure 1.
+2.2. Mapping Taint Sources to Sinks. These mappings allow us to track the entire provenance of a
+taint in any execution context. The γ mapping implicitly creates a directed acyclic graph (DAG) of la-
+bels, representing the dataflow through the program: The program inputs that affect a tainted variable
+or memory address are its taint label’s topmost ancestors in the γ DAG. A recursive function ψ : N →
+2{the set of all taint sources} can map taint labels to all of their ancestral sources:
+ψ[ℓ] =
+�
+�
+�
+�
+�
+{κ[ℓ]}
+ℓ ∈ κ,
+�
+p∈γ[ℓ] ψ[p]
+γ[ℓ] ̸= ⟨0, 0⟩,
+∅
+otherwise.
+When we observe that the program writes to output, we use the ψ mapping to record which inputs, if any,
+tainted the output. This can be accomplished by enumerating the canonical ancestors of the output labels
+by traversing the γ DAG. This allows us to construct a complete mapping of taint sources to sinks. Note
+that any taint sources without associated sinks can be arbitrarily mutated without affecting the output.
+3
+
+the conditional expression
+e evaluates to v
+�
+��
+�
+µ, ∆ ⊢ e ⇓ v
+create a new taint label
+for every existing label
+�
+��
+�
+ε′ = ε[u ← |ε| + ε[u] : ∀u ∈ ε]
+union every existing label with the taints of v
+�
+��
+�
+γ′ = γ
+�
+ε′[u] ← ⟨ε[u], ε[v]⟩ : ∀u ∈ ε
+�
+µ, ∆, ε, κ, γ ⇝ µ, ∆, ε′, κ, γ′
+PreCond
+Figure 2. Updated SimpIL operational semantics to enforce the taint policy that every
+input that affects control-flow will be unioned with all labels created in the branch they
+influence. The PreCond rule is executed before every conditional rule (TCond and FCond
+in Figure 1 of [20]).
+2.3. A Definition of Program Input Blind Spots. Let Ω be the set of taint labels written to output
+during execution. Then a blind spot is the set of all potential program inputs that did not affect the output:
+(1)
+all potential input that is not consumed by the program
+(e.g., a portion of an input file that is never read)
+�
+��
+�
+�
+⟨source info⟩ : (∀ℓ ∈ κ : κ[ℓ] ̸= ⟨source info⟩)
+�
+�
+�
+κ[ℓ] : ℓ ∈ κ ∧ (∀ℓ′ ∈ Ω : ℓ /∈ ψ[ℓ′])
+�
+.
+�
+��
+�
+the sources of all canonical taints that did not affect output
+As we mentioned above, SimpIL includes a taint policy specifying the rules by which taints are propagated.
+The taint policy will affect our definition of blind spots. For example, let us again consider the pseudocode
+in Algorithm 1. The variable a is tainted by program input on line 2, which we can now specify in our
+SimpIL extension as κ[ε[a]] ̸= ∅. Recall that the value of a can indirectly cause hard-coded data (d = 5) to
+be written to output on line 7. Therefore, a cannot be a blind spot, since its mutation can affect output.
+However, the semantics by which the taint policy propagates taint through conditionals will affect whether
+our extension of SimpIL will consider the output to be tainted by a, because the value of a itself is never
+written to output.
+We resolve this discrepancy by enforcing the following constraint on blind spot taint policies: The taint
+labels of every variable and memory address that affect the program’s control-flow will be unioned with all
+labels created in the branch they influence. In other words, in addition to the definition of blind spots in
+Equation (1), a blind spot cannot influence control flow that leads to a program output. Updated operational
+semantics for the conditional operator that implement this policy are given in Figure 2.
+A trace of Algorithm 1 showing the iterative updates to the execution context is given in Table 1. It
+demonstrates how the blind spot taint policy propagates taints from variables in the path condition—
+variables that have affected control flow leading to the current state (e.g., variable c on line 5)—to variables
+that would otherwise not be tainted (e.g., variable d). This reduces false-positive blind spot classifications,
+since it captures tainted variables that indirectly cause output.
+3. Tracking Taints
+Dynamic Information Flow Tracking (DIFT), also known as Dynamic Taint Analysis (DTA), is a technique
+in which the flow of information through a program is modeled and tracked at runtime. DIFT is a challenging
+problem; many modern approaches suffer from high implementation overhead, low accuracy, and/or low
+fidelity [5].
+Universal taint analysis is a form of DIFT that can track all input bytes throughout the
+execution of a program, mapping inputs to outputs [24].
+Thus far we have developed formal semantics for blind spots and discovered some necessary taint prop-
+agation policies to detect them. The next step is to automatically instrument a parser to extract the data
+flow information necessary to classify input byte regions as blind spots. We gather this data flow information
+by performing universal taint analysis.
+3.1. Related Work. There are significant challenges when performing universal taint analysis on real-world
+software. When the cyclomatic complexity of a program is large, the amount of new taint labels generated
+4
+
+Line
+Statement
+∆
+τ∆
+ε
+κ
+γ
+1
+start
+{}
+{}
+{}
+{}
+{}
+2
+a ← ReadInput(1)
+{a → 40}
+{a → T}
+{a → 1}
+{1 → ⟨1st byte of input⟩}
+{}
+3
+b ← ReadInput(1)
+a → 40,
+b → 12
+{
+}
+a → T,
+b → T
+{
+}
+a → 1,
+b → 2
+{
+}
+1 → ⟨1st byte of input⟩,
+2 → ⟨2nd byte of input⟩
+{
+}
+{}
+4
+c ← a + b
+b → 12,
+a → 40,
+c → 52
+{
+}
+b → T,
+a → T,
+c → T
+{
+}
+b → 2,
+a → 1,
+c → 3
+{
+}
+1 → ⟨1st byte of input⟩,
+2 → ⟨2nd byte of input⟩
+{
+}
+{3 → ⟨1, 2⟩}
+5
+if c ≥ 42 then
+b → 12,
+a → 40,
+c → 52
+{
+}
+b → T,
+a → T,
+c → T
+{
+}
+b → 2,
+a → 1,
+c → 3
+{
+}
+1 → ⟨1st byte of input⟩,
+2 → ⟨2nd byte of input⟩
+{
+}
+{3 → ⟨1, 2⟩}
+6
+d ← 5
+b → 12,
+a → 40,
+c → 52,
+d → 5
+{
+}
+b → T,
+a → T,
+c → T,
+d → T
+{
+}
+b → 2,
+a → 1,
+c → 3,
+d → 4
+{
+}
+1 → ⟨1st byte of input⟩,
+2 → ⟨2nd byte of input⟩
+{
+}
+3 → ⟨1, 2⟩,
+4 → ⟨3, 0⟩
+{
+}
+Table 1. Execution context trace for Algorithm 1.
+Note on line 6 that, despite being
+assigned a constant value of 5, the d variable (taint label 4) is in fact tainted by variable c
+(taint label 3). This is because the path condition to line 6 depends on c from the conditional
+branch on line 5. Therefore, neither of the first two bytes of input are blind spots.
+from even a small input is enormous. For example, many document formats such as PDF use compression
+to keep file sizes small. When tracking data flow through a PDF parser, there will be a significant number
+of taint unions as the data are decompressed. This phenomenon is known as taint explosion, which generally
+occurs when a function performs a large number of combinatorial operations on input data.
+There are several existing projects that achieve universal taint tracking, using various methods. Two of the
+best maintained and easiest to use are AUTOGRAM [14] and TaintGrind [16]. However, the former is limited
+to analysis within the Java virtual machine and the latter suffers from unacceptable runtime overhead when
+tracking as few as several bytes at a time. For example, we ran mutool, a utility in the MµPDF project3,
+using TaintGrind over a corpus of medium sized PDFs, and in every case the tool had to be halted after
+over twenty-four hours of execution for operations that would normally complete in milliseconds without
+instrumentation.
+There are also existing tools for performing dynamic program analysis via QEMU [4], such as PANDA [11]
+and DECAF(++) [8], both of which have taint tracking extensions. However, being an emulation framework
+rather than virtualization, QEMU incurs a runtime overhead of about 15% just to execute a binary, not
+including any instrumentation [15].
+After adding the program instrumentation necessary to enable fuzz
+testing, QEMU was observed to have over three times the runtime overhead of equivalent compile-time
+instrumentation [17].
+Symbolic execution engines like Triton [19] and SymCC [18] have also been used for data flow analysis.
+Symbolic execution could be extended to detect blind spots, e.g., by making all input bytes symbolic and
+observing all data that is written. The input bytes associated with any symbolic data that is either written or
+included in the path condition during a write is not a blind spot. However, since each branch of a conditional
+affected by program input must be enumerated, symbolic execution is very likely to be intractable for large
+inputs.
+DRTaint [24] is a recently published tool that can also perform universal taint tracking. It adds a minimal
+amount of runtime instrumentation to create runtime artifacts that can be post-processed to extract any
+data flow.
+The authors do not quantify the exact overhead of DRTaint, but Figure 5 from their paper
+suggests at least a 60x slowdown compared to the uninstrumented program. It is also unclear whether this
+instrumentation was sufficient to reconstruct all data flows. This is consistent with earlier techniques such
+as Dytan [7] that reported a 50x slowdown when tracking as few as 64 taint labels.
+3.2. PolyTracker. This section introduces PolyTracker [21], a novel LLVM-based dynamic analysis tool for
+extracting ground truth information from programs. It is open-source and available at https://github.
+com/trailofbits/polytracker.
+3https://mupdf.com/
+5
+
+PolyTracker automatically adds instrumentation to a program such that, when the program is executed, it
+produces runtime artifacts that can be analyzed to track the data flows of all input bytes. It is an extension
+of the LLVM DataFlowSanitizer (DFSan) [9], a generalized dynamic data flow analysis instrumentation tool.
+PolyTracker has previously been used to label the semantic purpose of functions in a parser [13].
+Originally, DFSan only supported tracking at most 216 taint labels at a time4.
+This restriction was
+acceptable for DFSan’s primary use case at the time: data flow analysis for fuzz testing. However, this
+restriction was insufficient for our goal of tracking the taints of every input byte simultaneously. By replacing
+DFSan’s dense matrix representation of taint unions with a DAG (denoted as the γ mapping in our semantics
+above), we are able to simultaneously track 231 taints—an increase of several orders of magnitude. This
+proved sufficient to detect blind spots in all programs and inputs on which we have experimented.
+4. Empirical Analysis
+The previous sections introduced a method for identifying the blind spots of a program input.
+How
+accurate is this blind spot classifier? Since we do not have pre-labeled ground truth for the blind spots of
+an input, we need to develop statistical estimates for the confusion matrix of our classifier.
+We focus our evaluation on the PDF file format, for several reasons.
+(1) The PDF file format is old and complex, has had many revisions, and enjoys numerous independent
+implementations. This has led to differentials that necessitate lexical permissiveness for interoper-
+ability [23].
+(2) PDF is a container format that allows embedding of other formats like JPEG, providing more
+opportunity for blind spots.
+(3) The GovDocs corpus [12] provides thousands of real-world PDFs generated by a diversity of software.
+We instrumented the popular MµPDF utility using PolyTracker to detect blind spots. Next, we ran the
+instrumented utility on 1,087 PDFs sampled from the GovDocs corpus to render the PDFs to PostScript.
+The PDFs totaled over 622 MB and averaged 572 KB each. The largest file was 9.6 MB. We discovered
+a total of 33.5 MB of blind spots, averaging 30.8 KB per file, some files having zero, and one file having
+1.07 MB of blind spots.
+PostScript, a vector image format, was chosen rather than a raster format like JPEG or PNG because the
+relative lack of compression would help prevent explosion of taint union labels and thereby reduce runtime.
+Runtime of the instrumented program was less than one minute for each PDF. We would expect to get
+similar results when rendering to JPEG or PNG, however, since blind spots are dependent on execution,
+there could be some discrepancies. For example, since fonts can be embedded in both PDF and PostScript
+but cannot be embedded in JPEG or PNG, one might expect to see more blind spots related to fonts if one
+were to have rendered to a different file format.
+4.1. Classification Error. We validate blind spots by iteratively mutating each classified blind spot byte
+in the input file and re-running the original, uninstrumented program again. See Figure 3 for a notional
+example of this mutation process. If the PostScript output produced from the mutated input is different
+from the output of the unmodified file, then our classified blind spot is incorrect (Type I error), since any
+mutation inside a blind spot should, by definition, not affect program output.
+We also sample bytes from outside of our classified blind spots and mutate them, similarly. If a byte
+outside a blind spot can be arbitrarily mutated, it is likely a missed detection (Type II error). However,
+it is not tractable to mutate and verify all possible combinations of input bytes, since this would amount
+to testing the power set of all bytes, running in Θ(2n) time. It might be the case that a byte outside a
+blind spot is in fact a quantum byte that can be almost arbitrarily mutated, but has some undetected data
+dependency on another byte. Therefore, our reported Type I error rate is a tight bound on the actual Type I
+error, but our Type II error rate is a loose upper bound on the true Type II error.
+We mutated all 33.5 million blind spot bytes classified in the corpus, and all mutated blind spots produced
+identical output to the original file for a 0% false-positive rate. Of the bytes not classified as blind spots,
+89% did affect the output. Therefore, the false-negative rate is bounded above by 11%. A significant number
+4Over the course of 2021, DFSan underwent a significant refactor in order to make its memory layout compatible with other
+LLVM sanitizers [10]; this refactor reduced the effective number of taints it could track to 23 = 8, which is still sufficient for
+most fuzzing use cases, but not for detecting blind spots.
+6
+
+Lorem ipsum dolor sit amet,
+consectetur adipiscing elit, sed
+do eiusmod tempor incididunt ut
+labore et dolore magna aliqua.
+Ut enim ad minim veniam, quis
+nostrud exercitation
+ullamco laboris
+nisi ut aliquip ex ea commodo
+consequat. Duis aute irure dolor
+in reprehenderit in voluptate
+velit
+esse cillum dolore eu fugiat nulla
+pariatur.
+Detected Blind Spot
+Figure 3. Notional example of validating detected blind spots. Underlined bytes are it-
+eratively mutated and re-parsed. Bytes outside of blind spots are randomly selected for
+mutation, but all bytes within a blind spot are mutated. If a byte inside a blind spot is mu-
+tated and produces a different result, then that blind spot classification was a false-positive
+(Type I error). If a byte outside a blind spot can be arbitrarily mutated, then it is a missed
+detection (Type II error).
+Bytes
+# Blind Spots
+Total Frequency
+\r
+561451
+6745095
+\n
+52175
+4996436
+\x20
+47752
+17967270
+\x00
+9504
+7658663
+\x11
+5232
+3325230
+\x08
+3930
+3611559
+\x06
+3629
+3104109
+\x09
+3572
+2981341
+%
+3145
+3041586
+\x12
+2932
+2847588
+#
+2859
+3399908
+\x05
+2772
+2873564
+\x02
+2640
+3007837
+\x0F
+2607
+2944161
+\x14
+2594
+3132850
+\xF0
+2351
+2930162
+a
+2348
+4959732
+!
+2271
+3172472
+\xA3
+2108
+2783202
+4
+2074
+4848767
+\x13
+2024
+2788459
+Table 2. The twenty most common blind spot prefixes of length at most seven bytes.
+of these missed detections are likely quantum bytes: Pieces of data that can be mutated almost arbitrarily,
+but have some data dependency that can affect output that our random mutations did not exercise.
+4.2. Blind Spot Content. What is the content of PDF blind spots?
+In the GovDocs PDF corpus, we detect 63,194 unique blind spot prefixes of length at most seven bytes,
+and 338,943 unique byte sequences of length at most seven that precede blind spots.
+Consider the bytes that occur at the start of a blind spot; the most common of these are listed in Table 2.
+They are all one byte long, meaning that there is a diversity of content at the start of blind spots. The
+most common byte sequences that precede a blind spot, listed in Table 3, are more interesting: most are
+multi-byte, and they comprise many PDF tokens like endobj. This means that bytes following certain tokens
+are often or always ignored by the parser.
+7
+
+Bytes
+# Blind Spots
+Total Frequency
+n
+300057
+5811224
+\ n
+299785
+629959
+00000\ n
+299621
+555279
+f
+236173
+3738217
+\ f
+235736
+552315
+65535\ f
+205564
+470855
+m
+66860
+4242668
+dstream
+66588
+189282
+00001\ f
+26343
+44612
+\r
+12839
+6745095
+endobj\r
+11001
+527199
+e
+7056
+7495303
+be
+6419
+41673
+Adobe
+6418
+11545
+\x00\x0EAdobe
+6417
+9142
+\x02
+5533
+3007837
+\x08
+4185
+3611559
+\x00\x02
+3787
+82034
+00000\ f
+3770
+6838
+\x01\x00\x02
+3707
+29366
+Table 3. The twenty most common byte sequences of length at most seven preceding a blind spot.
+Now let us consider the unique suffix/prefix pairs that occur adjacent to the start of a blind spot. There
+are 1,029,129 unique pairs of these byte sequences. If we sort them by frequency, the pair
+not a blind spot
+����
+. . . endobj . . .
+�
+��
+�
+blind spot
+is in the top 0.01% of such pairs. “endobj” is a PDF token used to delimit the end of an object in the
+document model. The fact that this token is split across a blind spot boundary so frequently is indicative of,
+at best, intentional lexical permissiveness on the part of the parser, and, at worst, a bug. Other interesting
+blind spot contexts within the top hundredth of the first percentile include the entire endobj token, if
+preceded by a carriage return. Any whitespace after the stream token is ignored. The obj token is completely
+ignored if preceded by a space. Similarly, the PDF dictionary delimiters << and >> are frequently skipped,
+e.g., at the beginning of a PDF object. This simple contextual blind spot frequency analysis can discover
+parsing errors and differentials.
+4.3. Blind Spot Context. How frequent are blind spots, and where do they occur in PDFs?
+Figure 4 plots the number of blind spots in the PDF corpus as a function of file size. This suggests that
+the number of blind spot bytes in a typical PDF is constant. Note, however, that blind spots in PDFs can
+be arbitrarily large, since the PDF format permits the inclusion of arbitrary binary blobs that do not have
+to be connected to the document object model (DOM) [23].
+Figure 5 plots a histogram of the contiguous size of blind spot regions in the corpus. The majority of blind
+spots are small, but a nontrivial number of blind spots are over 1 KB. The average blind spot is 42 bytes
+long with a standard deviation of 1.72 KB.
+Figure 6 is a histogram of the normalized position of blind spot bytes in their files: the blind spot’s byte
+offset divided by the file size. In our experiments with the MµPDF renderer translating to PostScript, the
+majority of PDF blind spots are at the beginning and ends of the files.
+Figure 7 combines the two previous figures by comparing the mean contiguous blind spot size to the
+normalized position in the PDF. Despite the most blind spot bytes being at the beginning and ends of the
+PDFs, the longest blind spots tend to be in the first 10–20% of the file, but not immediately at the beginning.
+8
+
+0.01%
+0.1%
+1%
+10%
+50%
+100%
+1 kB
+5 kB 10 kB
+100 kB
+1 MB
+5 MB
+Bytes in Blind Spots
+File Size
+files tested
+best fit: f(x) = log(a)
+bx
+Figure 4. Proportion of PDF bytes that are in blind blind spots as a function of file size.
+1
+10
+100
+1000
+10000
+100000
+1 × 106
+1 B
+10 B
+100 B
+1 kB
+10 kB
+100 kB
+1 MB
+Number of Blind Spots
+Contiguous Blind Spot Size
+Figure 5. Histogram of the sizes of contiguous PDF blind spots. The majority of blind
+spots are single bytes, but a nontrivial number of blind spots are over 1 kilobyte.
+The
+average blind spot is 42 bytes long with a standard deviation of 1.72 kilobytes.
+For each input PDF, we generate a parse tree using PolyTracker’s sister tool, PolyFile5 [21]. Each byte
+in the input PDF corresponds to one or more parse tree derivations: unique paths through the PDF parse
+tree. A byte could have more than one derivation, for instance, if the input file is a polyglot—a file that
+is valid in two or more formats. PDFs are particularly easy to turn into polyglots, and many legitimate
+PDF generators exploit this fact. For example, it is common to produce valid PDFs that are also valid ZIP
+archives that, when extracted, contain additional files related to the document. Therefore, a byte might
+have one derivation in the PDF parse tree and have a different but completely valid derivation in the ZIP
+parse tree. An example of a parse tree derivation is given in Figure 8.
+For each unique parse tree derivation, we count the number of blind spot bytes that occur in that deriva-
+tion. The most frequent derivation containing blind spots is
+application/pdf , the root of the PDF parse tree.
+This means that the majority of PDF blind spots occur in portions of the file that have no semantic purpose.
+Blind spot locality might be explained by the fact that PDF parsers are resilient to both leading and trailing
+5Open-source and available at https://github.com/trailofbits/polyfile.
+9
+
+0
+20000
+40000
+60000
+80000
+100000
+120000
+140000
+160000
+0%
+10%
+20%
+30%
+40%
+50%
+60%
+70%
+80%
+90%
+100%
+Number of Blind Spot Bytes in the Corpus
+Normalized File Position (Byte Offset ÷ File Size)
+Figure 6. Normalized position of blind spots in PDF files. With the MµPDF renderer,
+blind spots are most frequent at the beginning and end of PDF files.
+0
+20
+40
+60
+80
+100
+120
+140
+0%
+10%
+20%
+30%
+40%
+50%
+60%
+70%
+80%
+90%
+100%
+Mean Contiguous Blind Spot Size (Bytes)
+Normalized Blind Spot Start Position (Byte Offset ÷ File Size)
+Average Blind Spot Length
+Figure 7. Contiguous blind spot size as a function of its position in the PDF file. Error
+bars correspond to the standard deviation of blind spot sizes in that portion of the file. The
+longest blind spots tend to be in the second tenth of the file.
+application/pdf
+→
+PDFObject
+→
+PDFDictionary
+→
+KeyValuePair
+→
+Value
+→
+PDFList
+→
+PDFList
+→
+PSInt .
+Figure 8. The parse tree derivation of a byte representing an integer in a list of lists that
+is a value in a PDF dictionary in a PDF object.
+garbage bytes before and after the PDF file. Also, as we saw above, the majority of naturally occurring
+blind spot bytes are at the beginning and end of the file.
+The frequency of every unique parse tree derivation is presented in Figure 9. Blind spots overwhelmingly
+occur in a small number of derivations. However, the long tail demonstrates that blind spots can and do
+occur in many diverse derivations.
+10
+
+1
+10
+100
+1000
+10000
+100000
+1 × 106
+Most Frequent
+Median
+Least Frequent
+Number of Blind Spot Bytes
+Unique Parse Tree Derivations Sorted by Blind Spot Frequency
+Figure 9. Each point is a parse tree derivation—a unique path through a PDF parse
+tree—whose y-axis value is the number of times a blind spot occurred in that derivation.
+Blind spots overwhelmingly occur in a small number of derivations, yet there is a long tail
+demonstrating that blind spots can and do occur in many diverse derivations.
+The most frequent parse tree derivations for PDF blind spots are given in Table 4. All but the most
+frequent derivation descend from the PDFObject node. This is unsurprising, since PDF objects can contain
+streams of arbitrary binary data. PDF objects also do not need to be connected to the root of the PDF
+document object model, nor do they need to be used in any way for rendering.
+The second most frequent derivation for blind spots are in PDF dictionaries.
+PDF dictionaries can
+contain arbitrary key/value pairs which are often used for metadata that is not necessary for rendering
+(e.g., timestamps). Dictionaries can and often do contain redundant information. For example, the length
+of a PDF object stream can either be specified as a key/value pair in the preceding object dictionary or
+implicitly defined by the location of a required termination token. If both are specified, then they must agree.
+However, if a length is specified in the dictionary which does not agree with the position of the termination
+token, then most parsers will ignore the specified length and defer to the token position [22], making the
+dictionary entry a blind spot.
+5. Conclusions
+This paper defined the concept of blind spots: inputs to a program that can be arbitrarily mutated
+without affecting the program’s output. Operational semantics for blind spots were formalized by extending
+SimpIL [20]. An efficient implementation capable of automatically detecting blind spots, PolyTracker, was
+introduced. It works by adding instrumentation for performing dynamic information flow tracking (DIFT)
+to a program.
+The technique was evaluated by detecting blind spots in the popular MµPDF parser over a corpus of
+over a thousand diverse PDFs [12]. There were zero false-positive blind spot classifications, and the missed
+detection rate was bounded above by 11%. On average, at least 5% of each PDF file was completely ignored
+by the parser; blind spots that could be repurposed for steganography or embedding malware payloads.
+Future work includes extending the approach to detect quantum bytes: inputs that can almost arbitrarily
+be mutated without affecting output, like source code comments. The current implementation injects its
+DIFT instrumentation at the LLVM/IR level. Therefore, it is limited to programs that can be compiled
+using LLVM, or binaries that can be lifted to LLVM/IR. It would be useful to apply the technique to runtime
+instrumentation that could be applied to a black-box binary.
+Our results show promise that this technique could be an efficient automated means to detect parser bugs
+and differentials. Nothing in the technique is tied to PDF in general, so it can be immediately applied to
+other notoriously difficult-to-parse formats like ELF, X.509, and XML.
+11
+
+Derivation
+# Bytes
+application/pdf
+815891
+application/pdf → PDFObject → PDFDictionary → KeyValuePair → Key
+365771
+application/pdf → PDFObject → PSBytes → image/jpeg
+120803
+application/pdf → PDFObject → PDFDictionary → KeyValuePair → Value → PDFObjRef
+116538
+application/pdf → PDFObject → PDFDictionary → KeyValuePair → Value → PDFLiteral
+108573
+application/pdf → PDFObject → PDFDictionary → KeyValuePair → Value → PSInt
+58864
+application/pdf → PDFObject → PDFDictionary → KeyValuePair → Value → PDFList → PDFObjRef
+55920
+application/pdf → PDFObject → PDFDictionary → KeyValuePair → Value → PDFList → PSInt
+51693
+application/pdf → PDFObject → PDFList → PDFObjRef
+37982
+application/pdf → PDFObject → PDFDictionary → KeyValuePair → Value → PDFList → PDFLiteral
+32221
+application/pdf → PDFObject → PDFDictionary → KeyValuePair → Value → PDFDictionary → KeyValuePair → Key
+27220
+application/pdf → PDFObject → FlateDecode → DecodedStream → PSBytes → application/octet-stream
+27210
+application/pdf → PDFObject → PDFDictionary → KeyValuePair → Value → PSFloat
+22254
+application/pdf → PDFObject → PDFDictionary → KeyValuePair → Value → PDFList → PDFList → PSInt
+14489
+application/pdf → PDFObject → PDFDictionary → KeyValuePair → Value → PSBytes → text/plain
+13300
+application/pdf → PDFObject → PDFDictionary → KeyValuePair → Value → PDFList → PSFloat
+11781
+application/pdf → PDFObject → FlateDecode
+11634
+application/pdf → PDFObject → PDFDictionary → KeyValuePair → Value → PDFDictionary → KeyValuePair →
+Value → PDFLiteral
+10148
+application/pdf → PDFObject → PSBytes → image/jp2
+8738
+application/pdf → PDFObject → PDFDictionary → KeyValuePair → Value → PDFList → PDFDictionary → KeyVal-
+uePair → Key
+8445
+application/pdf → PDFObject → PDFDictionary → KeyValuePair → Value → PDFDictionary → KeyValuePair →
+Value → PSFloat
+7849
+application/pdf → PDFObject → PSBytes → application/octet-stream
+7800
+application/pdf → PDFObject → PSBytes → text/plain
+5131
+application/pdf → PDFObject → PDFDictionary → KeyValuePair → Value → PDFList → PSBytes → text/plain
+4901
+application/pdf → PDFObject → PDFDeciphered → image/jpeg
+4900
+application/pdf → PDFObject → PDFDictionary → KeyValuePair → Value → PDFDictionary → KeyValuePair →
+Value → PSInt
+4741
+application/pdf → PDFObject → LZWDecode → DecodedStream → PSBytes → application/octet-stream
+3914
+application/pdf → PDFObject → PDFDictionary → KeyValuePair → Value → PDFList → PDFDictionary → KeyVal-
+uePair → Value → PDFLiteral
+3583
+Table 4. The PDF parse tree derivations containing the most blind spot bytes. These
+primarily descend from PDFObject.
+Acknowledgment
+This research was supported in part by the Defense Advanced Research Projects Agency (DARPA) Safe-
+Docs program as a subcontractor to Galois under HR0011-19-C-0073. Many thanks to Michael Brown, Trent
+Brunson, Filipe Casal, Peter Goodman, Kelly Kaoudis, Bill Harris, Nichole Schimanski, Mark Tullsen, Walt
+Woods, Peter Wyatt, and Sergey Bratus for their invaluable feedback on the approach and tooling. Special
+thanks to Carson Harmon, the original creator of PolyTracker, whose ideas and discussions germinated this
+research.
+References
+1. ISO/IEC 29500-1:2016, Information technology—Document description and processing languages—Office Open XML File
+Formats—Part 1: Fundamentals and Markup Language Reference, Standard, International Organization for Standardiza-
+tion, Geneva, CH, November 2016.
+2. Ange Albertini, Abusing file formats; or, Corkami, the novella, The International Journal of Proof of Concept or GTFO
+0x07 (2015), no. 6, 18–41.
+12
+
+3. Ange Albertini, Thai Duong, Shay Gueron, Stefan K¨olbl, Atul Luykx, and Sophie Schmieg, How to abuse and fix authenti-
+cated encryption without key commitment, Proceedings of the 31st USENIX Security Symposium (Boston, MA), USENIX
+Association, August 2022.
+4. Fabrice Bellard, QEMU, a fast and portable dynamic translator, Proceedings of the Annual USENIX Technical Conference
+(USA), ATEC ’05, USENIX Association, 2005, p. 41.
+5. Christopher Brant, Prakash Shrestha, Benjamin Mixon-Baca, Kejun Chen, Said Varlioglu, Nelly Elsayed, Yier Jin, Jedidiah
+Crandall, and Daniela Oliveira, Challenges and opportunities for practical and effective dynamic information flow tracking,
+ACM Computing Surveys 55 (2021), no. 1.
+6. Peng Chen and Hao Chen, Angora: Efficient fuzzing by principled search, Proceedings of the IEEE Symposium on Security
+and Privacy, 2018, pp. 711–725.
+7. James Clause, Wanchun Li, and Alessandro Orso, Dytan: A generic dynamic taint analysis framework, Proceedings of
+the 2007 International Symposium on Software Testing and Analysis (New York, NY, USA), ISSTA ’07, Association for
+Computing Machinery, 2007, pp. 196–206.
+8. Ali Davanian, Zhenxiao Qi, Yu Qu, and Heng Yin, DECAF++: Elastic Whole-System dynamic taint analysis, 22nd
+International Symposium on Research in Attacks, Intrusions and Defenses (RAID 2019) (Chaoyang District, Beijing),
+USENIX Association, September 2019, pp. 31–45.
+9. DataFlowSanitizer, https://clang.llvm.org/docs/DataFlowSanitizer.html, Accessed: 2020-07-26.
+10. [DFSan] change shadow and origin memory layouts to match MSan., LLVM commit 45f6d5522f8d, https://reviews.
+llvm.org/D104896?id=354633, Accessed: 2020-07-26.
+11. Brendan Dolan-Gavitt, Josh Hodosh, Patrick Hulin, Tim Leek, and Ryan Whelan, Repeatable reverse engineering with
+PANDA, Proceedings of the 5th Program Protection and Reverse Engineering Workshop (New York, NY, USA), PPREW-
+5, Association for Computing Machinery, 2015.
+12. Simson Garfinkel, Paul Farrell, Vassil Roussev, and George Dinolt, Bringing science to digital forensics with standardized
+forensic corpora, Digital Investigation 6 (2009), S2–S11, Proceedings of the Ninth Annual DFRWS Conference.
+13. Carson Harmon, Bradford Larsen, and Evan Sultanik, Toward automated grammar extraction via semantic labeling of
+parser implementations, Proceedings of the Sixth Workshop on Language-Theoretic Security (LangSec), IEEE Symposium
+on Security and Privacy, 2021.
+14. Matthias H¨oschele and Andreas Zeller, Mining input grammars from dynamic taints, Proceedings of the 31st IEEE/ACM
+International Conference on Automated Software Engineering (New York, NY, USA), ASE 2016, Association for Computing
+Machinery, 2016, pp. 720–725.
+15. Ahmed Karaman, Measuring QEMU emulation efficiency, https://ahmedkrmn.github.io/TCG-Continuous-Benchmarking/
+Measuring-QEMU-Emulation-Efficiency/, August 2020, Accessed: 2020-07-26.
+16. W. M. Khoo, Taintgrind: a Valgrind taint analysis tool, https://github.com/wmkhoo/taintgrind, Accessed: March 2,
+2020.
+17. Stefan Nagy, Anh Nguyen-Tuong, Jason D. Hiser, Jack W. Davidson, and Matthew Hicks, Breaking through binaries:
+Compiler-quality instrumentation for better binary-only fuzzing, 30th USENIX Security Symposium (USENIX Security
+21), USENIX Association, August 2021, pp. 1683–1700.
+18. Sebastian Poeplau and Aur´elien Francillon, Symbolic execution with SymCC: Don’t interpret, compile!, 29th USENIX
+Security Symposium (USENIX Security 20), USENIX Association, August 2020, pp. 181–198.
+19. Florent Saudel and Jonathan Salwan, Triton: A dynamic symbolic execution framework, Symposium sur la s´ecurit´e des
+technologies de l’information et des communications (Rennes, France), SSTIC, June 2015, pp. 31–54.
+20. Edward J. Schwartz, Thanassis Avgerinos, and David Brumley, All you ever wanted to know about dynamic taint analysis
+and forward symbolic execution (but might have been afraid to ask), Proceedings of the IEEE Symposium on Security and
+Privacy, 2010, pp. 317–331.
+21. Evan Sultanik, Brad Larsen, and Carson Harmon, Two new tools that tame the treachery of files, https://blog.
+trailofbits.com/2019/11/01/two-new-tools-that-tame-the-treachery-of-files/, November 1, 2019, Accessed: Janu-
+ary 12, 2020.
+22. Julia Wolf, OMG WTF PDF—PDF ambiguity and obfuscation, Proceedings of TROOPERS (Heidelberg, Germany), March
+2011.
+23. Peter Wyatt, Work in progress: Demystifying PDF through a machine-readable definition, Proceedings of the Seventh
+Workshop on Language-Theoretic Security (LangSec), IEEE Symposium on Security and Privacy, 2021.
+24. Pan Yang, Fei Kang, Yuntian Zhao, and Hui Shu, DRTaint: A dynamic taint analysis framework supporting correlation
+analysis between data regions, Journal of Physics: Conference Series 1856 (2021), no. 1, 012013.
+Trail of Bits, Stockholm, Sweden
+Email address: henrik.brodin@trailofbits.com
+Trail of Bits, Philadelphia, USA
+Email address: evan.sultanik@trailofbits.com
+Trail of Bits, Brno, Czech Republic
+Email address: marek.surovic@trailofbits.com
+13
+
diff --git a/P9FAT4oBgHgl3EQfzx6B/content/tmp_files/load_file.txt b/P9FAT4oBgHgl3EQfzx6B/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..5e73281875188e0da0844d8564686628db23bb1f
--- /dev/null
+++ b/P9FAT4oBgHgl3EQfzx6B/content/tmp_files/load_file.txt
@@ -0,0 +1,559 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf,len=558
+page_content='BLIND SPOTS: AUTOMATICALLY DETECTING IGNORED PROGRAM INPUTS  HENRIK BRODIN, EVAN SULTANIK, AND MAREK SUROVIˇC  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' A blind spot is any input to a program that can be arbitrarily mutated without affecting the  program’s output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Blind spots can be used for steganography or to embed malware payloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' If blind spots  overlap file format keywords, they indicate parsing bugs that can lead to differentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' This paper formalizes  the operational semantics of blind spots, leading to a technique that automatically detects blind spots based  on dynamic information flow tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' An efficient implementation is introduced an evaluated against a  corpus of over a thousand diverse PDFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' There are zero false-positive blind spot classifications and the  missed detection rate is bounded above by 11%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' On average, at least 5% of each PDF file is completely  ignored by the parser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Our results show promise that this technique is an efficient automated means to detect  parser bugs and differentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Nothing in the technique is tied to PDF in general, so it can be immediately  applied to other notoriously difficult-to-parse formats like ELF, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='509, and XML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Introduction  We define a blind spot as any input to a program that can be arbitrarily mutated without affecting the  program’s output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' For example, the Office Open XML (OOXML) [1] format used by document processing  tools like Microsoft Office is based upon the ZIP archive file format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Every valid OOXML file is also a valid  ZIP that can be extracted using ZIP utilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Therefore, arbitrary files can be added to an OOXML file as  if it were a regular ZIP archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Any such extraneous files are ignored by the document processor and will  therefore be blind spots during processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  Blind spots are dangerous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' For example, blind spots can be exploited for steganography and embedding  malware payloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' They can also be indicative of parser differentials, for instance, if two parsers exhibit  different blind spots for the same input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  But they can also be useful: Blind spots can potentially be  excluded as candidates for mutation when generating fuzz testing inputs, similar to the Angora fuzzer’s  branch coverage maximization strategy [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  Blind spots are a generalization of the concept of file cavities [3]: unused spaces in a file format that are  created due to the structure of the surrounding data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' However, unlike cavities, blind spots may be dependent  on the program itself and its execution environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' For example, the image content of a JPEG file will be  a blind spot to a parser that only reads its EXIF metadata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Likewise, the EXIF metadata will be a blind  spot to a program that converts JPEGs to another image format like BMP that does not support embedded  EXIF metadata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  Blind spots are a type of a more general class of data sources we call quantum inputs: Bytes that can  be almost arbitrarily mutated without affecting a program’s output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' For example, a source code comment  is a form of quantum input to a compiler, since the bytes within the comment can be arbitrarily changed  without affecting the behavior of the compiler as long as the bytes do not contain the comment delimiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  We only consider blind spots in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  Since they are associated with both program input and the program itself, blind spots can be indicators  of parsing bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Parsers—particularly hand-generated ones—will often accept a superset of the grammar for  which they were designed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' This manifests as a parser that accepts some inputs that are technically invalid  according to the file format specification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Sometimes this is intentional, in order to maximize compatibility  with files generated by other, incorrectly implemented software, or to attempt to repair malformed docu-  ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' For example, we discovered that an optimization in MµPDF1 will sometimes only check the leading  “e” in the endobj token;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' it will eagerly accept eXXXXX and the PDF will still be parsed correctly, despite the  fact that the PDF standard prescribes the existence of the full token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Such lexical permissiveness can lead  1https://mupdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='com/  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='08700v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='CR]  20 Jan 2023  to parser differentials and so-called file format schizophrenia [2]: when two implementations of a file format  interpret the same input file differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  This paper formalizes the concept of program input blind spots and proposes a novel technique based  on dynamic taint analysis to detect them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Our definition clarifies the difference between blind spots and  file cavities: Blind spots are the set of input bytes whose data flow never influences either control flow that  leads to an output or an output itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' First, we provide a formal definition of blind spots in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  This entails mapping input bytes to the output bytes they influence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Next, in Section 3 we describe the  program instrumentation that is able to efficiently capture the data necessary to create this mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' We  then describe how the mapping is constructed from the instrumentation output, as well as how blind spots  are inferred from the mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' We demonstrate how this technique can be used to automatically detect  lexical permissiveness and parser bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Finally, in Section 4 we provide empirical evidence that these blind  spots are correct, and an analysis of blind spots in specific file formats and programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Definitions and Formalization  In order to formalize the concept of input blind spots and explicitly describe our dynamic taint analysis  approach, we propose an extension to Schwartz, Avgerinos, and Brumley’s SimpIL operational semantics for  dynamic taint propagation [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' This section describes that extension and uses it to formally define blind  spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' An Extension to SimpIL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' The original conception of dataflow analysis in SimpIL only tracks whether  a given variable or memory cell is tainted, not from whence it is tainted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' In order to detect blind spots, we  need to additionally track exactly which input bytes influence output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' For example, consider the pseudocode  in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' The variable a is tainted by program input on line 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Moreover, the value of a can indirectly  cause hard-coded data (d = 5) to be written to output on line 7 by virtue of the conditional on line 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  Therefore, the first byte of the file cannot be a blind spot, since its mutation can affect output—despite the  fact that the value of the first byte is never written to output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Even if the SimpIL taint policy (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=', the  rules by which taints are propagated) is sufficient to detect that tainted inputs affected the output of the  program, it is insufficient to detect which inputs were responsible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Therefore, we need to extend SimpIL to  additionally track the provenance of a taint so that we can map a complete data flow from inputs to outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  Algorithm 1 Tainted Control Flow  1: procedure TaintedControlFlow  2:  a ← ReadInput(1)  ▷ a is tainted by the 1st byte  3:  b ← ReadInput(1)  ▷ b is tainted by the 2nd byte  4:  c ← a + b  5:  if c ≥ 42 then  6:  d ← 5  7:  WriteOutput(d)  8:  end if  9: end procedure  SimpIL uses meta-syntactic variables to represent an execution context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' ∆ is a mapping of variable names  to their values and µ is a mapping from memory addresses to their values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' τ∆ and τµ map variable names and  memory addresses to booleans (T|F) defining whether or not that value is tainted in the current execution  context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  In order to track taint provenance, we introduce the concept of a taint label: a unique identifier for each  instance of a tainted variable or memory address in an execution context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' We define two types of taint  labels: canonical and union.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' A canonical taint is the result of a variable or memory address being assigned  directly from a program input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' A union taint is the result of the combination of two previously tainted  values (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=', the result of two tainted variables being operands in a binary operation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  We extend the SimpIL notation with three new mappings to represent taint labels and track provenance2:  2The SimpIL semantics are already replete with Greek letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  We have chosen ε from the Greek word for “la-  bels” (ϵπιγραφϵς), κ from the word for “canonical” (κανoνικoς), and γ from the word for “parent” (γoνϵυς).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  2  SimpIL notation for the computation performed by the Input operation  �  ��  �  v is input from src  ε′ = ε[v ← |ε| + 1]  κ′ = κ[ε′[v] ← src]  µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' ∆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' κ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' γ ⇝ µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' ∆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' ε′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' κ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' γ  �  ��  �  SimpIL notation for the  updated execution state after the  operation  µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' ∆ ⊢ get input(src) ⇓ v  �  ��  �  SimpIL notation for the evaluation of  expression get input(src) to value v in  context µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' ∆  Input  µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' ∆ ⊢ e1 ⇓ v1  µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' ∆ ⊢ e2 ⇓ v2  v′ = v1♦bv2  ε′ = ε[v′ ← |ε| + 1]  γ′ = γ  �  ε′[v′] ← ⟨ε[v1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' ε[v2]⟩  �  µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' ∆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' κ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' γ ⇝ µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' ∆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' ε′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' κ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' γ′  µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' ∆ ⊢ e1♦be2 ⇓ v′  Binop  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' SimpIL operational semantics for reading input and executing binary operators  (see Figure 1 of [20]) updated to include data flow provenance tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' When a value v  is read from an input source src, we create a new, unique canonical taint label for v and  set its taint source info in κ to src.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' When a binary operator ♦b is applied to expressions  e1 = v1 and e2 = v2 resulting in the value v′, we create a new, unique union taint label for  v′ and set its parents to be the taint labels associated with values v1 and v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  (1) ε : ∆ ∪ µ → N that maps variable names and memory addresses to unique taint labels;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  (2) κ : N → {the set of all taint sources} that maps canonical taint labels to the information about their  source (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=', filename and byte offset);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' and  (3) γ : N → N × N that maps union taint labels to their parents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  SimpIL treats these mappings more like programmatic hashmaps than set theoretic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' As such,  SimpIL uses the notation “κ[ℓ]” for the value of taint label ℓ in mapping κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' For brevity, we shall continue  this theme by using the notation ℓ ∈ κ to represent the fact that ℓ is a key in the mapping κ, and ℓ /∈ κ to  mean that ℓ is not a key in κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  The zero taint label is reserved to represent untainted variables and memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' An untainted variable v  will always lack source info and descend from the zero label:  τ∆[v] =  �  F  ε[v] /∈ κ ∧ γ[ε[v]] = ⟨0, 0⟩,  T  ε[v] ∈ κ ∨ (γ[ε[v]] = ⟨i, j⟩ ∧ i + j > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  The SimpIL taint policy (see Table III from [20]) and semantics are modified to update these mappings  on every taint status change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' For example, the updated semantics for reading from input and for executing  binary operations are given in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Mapping Taint Sources to Sinks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' These mappings allow us to track the entire provenance of a  taint in any execution context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' The γ mapping implicitly creates a directed acyclic graph (DAG) of la-  bels, representing the dataflow through the program: The program inputs that affect a tainted variable  or memory address are its taint label’s topmost ancestors in the γ DAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' A recursive function ψ : N →  2{the set of all taint sources} can map taint labels to all of their ancestral sources:  ψ[ℓ] =  �  �  �  �  �  {κ[ℓ]}  ℓ ∈ κ,  �  p∈γ[ℓ] ψ[p]  γ[ℓ] ̸= ⟨0, 0⟩,  ∅  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  When we observe that the program writes to output, we use the ψ mapping to record which inputs, if any,  tainted the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' This can be accomplished by enumerating the canonical ancestors of the output labels  by traversing the γ DAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' This allows us to construct a complete mapping of taint sources to sinks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Note  that any taint sources without associated sinks can be arbitrarily mutated without affecting the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  3  the conditional expression  e evaluates to v  �  ��  �  µ, ∆ ⊢ e ⇓ v  create a new taint label  for every existing label  �  ��  �  ε′ = ε[u ← |ε| + ε[u] : ∀u ∈ ε]  union every existing label with the taints of v  �  ��  �  γ′ = γ  �  ε′[u] ← ⟨ε[u], ε[v]⟩ : ∀u ∈ ε  �  µ, ∆, ε, κ, γ ⇝ µ, ∆, ε′, κ, γ′  PreCond  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Updated SimpIL operational semantics to enforce the taint policy that every  input that affects control-flow will be unioned with all labels created in the branch they  influence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' The PreCond rule is executed before every conditional rule (TCond and FCond  in Figure 1 of [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' A Definition of Program Input Blind Spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Let Ω be the set of taint labels written to output  during execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Then a blind spot is the set of all potential program inputs that did not affect the output:  (1)  all potential input that is not consumed by the program  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=', a portion of an input file that is never read)  �  ��  �  �  ⟨source info⟩ : (∀ℓ ∈ κ : κ[ℓ] ̸= ⟨source info⟩)  �  �  �  κ[ℓ] : ℓ ∈ κ ∧ (∀ℓ′ ∈ Ω : ℓ /∈ ψ[ℓ′])  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  �  ��  �  the sources of all canonical taints that did not affect output  As we mentioned above, SimpIL includes a taint policy specifying the rules by which taints are propagated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  The taint policy will affect our definition of blind spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' For example, let us again consider the pseudocode  in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' The variable a is tainted by program input on line 2, which we can now specify in our  SimpIL extension as κ[ε[a]] ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Recall that the value of a can indirectly cause hard-coded data (d = 5) to  be written to output on line 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Therefore, a cannot be a blind spot, since its mutation can affect output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  However, the semantics by which the taint policy propagates taint through conditionals will affect whether  our extension of SimpIL will consider the output to be tainted by a, because the value of a itself is never  written to output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  We resolve this discrepancy by enforcing the following constraint on blind spot taint policies: The taint  labels of every variable and memory address that affect the program’s control-flow will be unioned with all  labels created in the branch they influence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' In other words, in addition to the definition of blind spots in  Equation (1), a blind spot cannot influence control flow that leads to a program output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Updated operational  semantics for the conditional operator that implement this policy are given in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  A trace of Algorithm 1 showing the iterative updates to the execution context is given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' It  demonstrates how the blind spot taint policy propagates taints from variables in the path condition—  variables that have affected control flow leading to the current state (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=', variable c on line 5)—to variables  that would otherwise not be tainted (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=', variable d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' This reduces false-positive blind spot classifications,  since it captures tainted variables that indirectly cause output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Tracking Taints  Dynamic Information Flow Tracking (DIFT), also known as Dynamic Taint Analysis (DTA), is a technique  in which the flow of information through a program is modeled and tracked at runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' DIFT is a challenging  problem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' many modern approaches suffer from high implementation overhead, low accuracy, and/or low  fidelity [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  Universal taint analysis is a form of DIFT that can track all input bytes throughout the  execution of a program, mapping inputs to outputs [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  Thus far we have developed formal semantics for blind spots and discovered some necessary taint prop-  agation policies to detect them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' The next step is to automatically instrument a parser to extract the data  flow information necessary to classify input byte regions as blind spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' We gather this data flow information  by performing universal taint analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Related Work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' There are significant challenges when performing universal taint analysis on real-world  software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' When the cyclomatic complexity of a program is large,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' the amount of new taint labels generated  4  Line  Statement  ∆  τ∆  ε  κ  γ  1  start  {}  {}  {}  {}  {}  2  a ← ReadInput(1)  {a → 40}  {a → T}  {a → 1}  {1 → ⟨1st byte of input⟩}  {}  3  b ← ReadInput(1)  a → 40,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  b → 12  {  }  a → T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  b → T  {  }  a → 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  b → 2  {  }  1 → ⟨1st byte of input⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  2 → ⟨2nd byte of input⟩  {  }  {}  4  c ← a + b  b → 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  a → 40,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  c → 52  {  }  b → T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  a → T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  c → T  {  }  b → 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  a → 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  c → 3  {  }  1 → ⟨1st byte of input⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  2 → ⟨2nd byte of input⟩  {  }  {3 → ⟨1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' 2⟩}  5  if c ≥ 42 then  b → 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  a → 40,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  c → 52  {  }  b → T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  a → T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  c → T  {  }  b → 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  a → 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  c → 3  {  }  1 → ⟨1st byte of input⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  2 → ⟨2nd byte of input⟩  {  }  {3 → ⟨1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' 2⟩}  6  d ← 5  b → 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  a → 40,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  c → 52,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  d → 5  {  }  b → T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  a → T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  c → T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  d → T  {  }  b → 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  a → 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  c → 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  d → 4  {  }  1 → ⟨1st byte of input⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  2 → ⟨2nd byte of input⟩  {  }  3 → ⟨1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' 2⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  4 → ⟨3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' 0⟩  {  }  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Execution context trace for Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  Note on line 6 that, despite being  assigned a constant value of 5, the d variable (taint label 4) is in fact tainted by variable c  (taint label 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' This is because the path condition to line 6 depends on c from the conditional  branch on line 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Therefore, neither of the first two bytes of input are blind spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  from even a small input is enormous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' For example, many document formats such as PDF use compression  to keep file sizes small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' When tracking data flow through a PDF parser, there will be a significant number  of taint unions as the data are decompressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' This phenomenon is known as taint explosion, which generally  occurs when a function performs a large number of combinatorial operations on input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  There are several existing projects that achieve universal taint tracking, using various methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Two of the  best maintained and easiest to use are AUTOGRAM [14] and TaintGrind [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' However, the former is limited  to analysis within the Java virtual machine and the latter suffers from unacceptable runtime overhead when  tracking as few as several bytes at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' For example, we ran mutool, a utility in the MµPDF project3,  using TaintGrind over a corpus of medium sized PDFs, and in every case the tool had to be halted after  over twenty-four hours of execution for operations that would normally complete in milliseconds without  instrumentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  There are also existing tools for performing dynamic program analysis via QEMU [4], such as PANDA [11]  and DECAF(++) [8], both of which have taint tracking extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' However, being an emulation framework  rather than virtualization, QEMU incurs a runtime overhead of about 15% just to execute a binary, not  including any instrumentation [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  After adding the program instrumentation necessary to enable fuzz  testing, QEMU was observed to have over three times the runtime overhead of equivalent compile-time  instrumentation [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  Symbolic execution engines like Triton [19] and SymCC [18] have also been used for data flow analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  Symbolic execution could be extended to detect blind spots, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=', by making all input bytes symbolic and  observing all data that is written.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' The input bytes associated with any symbolic data that is either written or  included in the path condition during a write is not a blind spot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' However, since each branch of a conditional  affected by program input must be enumerated, symbolic execution is very likely to be intractable for large  inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  DRTaint [24] is a recently published tool that can also perform universal taint tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' It adds a minimal  amount of runtime instrumentation to create runtime artifacts that can be post-processed to extract any  data flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  The authors do not quantify the exact overhead of DRTaint, but Figure 5 from their paper  suggests at least a 60x slowdown compared to the uninstrumented program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' It is also unclear whether this  instrumentation was sufficient to reconstruct all data flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' This is consistent with earlier techniques such  as Dytan [7] that reported a 50x slowdown when tracking as few as 64 taint labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' PolyTracker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' This section introduces PolyTracker [21], a novel LLVM-based dynamic analysis tool for  extracting ground truth information from programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' It is open-source and available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  com/trailofbits/polytracker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  3https://mupdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='com/  5  PolyTracker automatically adds instrumentation to a program such that, when the program is executed, it  produces runtime artifacts that can be analyzed to track the data flows of all input bytes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' It is an extension  of the LLVM DataFlowSanitizer (DFSan) [9], a generalized dynamic data flow analysis instrumentation tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  PolyTracker has previously been used to label the semantic purpose of functions in a parser [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  Originally, DFSan only supported tracking at most 216 taint labels at a time4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  This restriction was  acceptable for DFSan’s primary use case at the time: data flow analysis for fuzz testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' However, this  restriction was insufficient for our goal of tracking the taints of every input byte simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' By replacing  DFSan’s dense matrix representation of taint unions with a DAG (denoted as the γ mapping in our semantics  above), we are able to simultaneously track 231 taints—an increase of several orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' This  proved sufficient to detect blind spots in all programs and inputs on which we have experimented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Empirical Analysis  The previous sections introduced a method for identifying the blind spots of a program input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  How  accurate is this blind spot classifier?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Since we do not have pre-labeled ground truth for the blind spots of  an input, we need to develop statistical estimates for the confusion matrix of our classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  We focus our evaluation on the PDF file format, for several reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  (1) The PDF file format is old and complex, has had many revisions, and enjoys numerous independent  implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' This has led to differentials that necessitate lexical permissiveness for interoper-  ability [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  (2) PDF is a container format that allows embedding of other formats like JPEG, providing more  opportunity for blind spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  (3) The GovDocs corpus [12] provides thousands of real-world PDFs generated by a diversity of software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  We instrumented the popular MµPDF utility using PolyTracker to detect blind spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Next, we ran the  instrumented utility on 1,087 PDFs sampled from the GovDocs corpus to render the PDFs to PostScript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  The PDFs totaled over 622 MB and averaged 572 KB each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' The largest file was 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='6 MB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' We discovered  a total of 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='5 MB of blind spots, averaging 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='8 KB per file, some files having zero, and one file having  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='07 MB of blind spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  PostScript, a vector image format, was chosen rather than a raster format like JPEG or PNG because the  relative lack of compression would help prevent explosion of taint union labels and thereby reduce runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  Runtime of the instrumented program was less than one minute for each PDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' We would expect to get  similar results when rendering to JPEG or PNG, however, since blind spots are dependent on execution,  there could be some discrepancies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' For example, since fonts can be embedded in both PDF and PostScript  but cannot be embedded in JPEG or PNG, one might expect to see more blind spots related to fonts if one  were to have rendered to a different file format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Classification Error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' We validate blind spots by iteratively mutating each classified blind spot byte  in the input file and re-running the original, uninstrumented program again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' See Figure 3 for a notional  example of this mutation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' If the PostScript output produced from the mutated input is different  from the output of the unmodified file, then our classified blind spot is incorrect (Type I error), since any  mutation inside a blind spot should, by definition, not affect program output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  We also sample bytes from outside of our classified blind spots and mutate them, similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' If a byte  outside a blind spot can be arbitrarily mutated, it is likely a missed detection (Type II error).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' However,  it is not tractable to mutate and verify all possible combinations of input bytes, since this would amount  to testing the power set of all bytes, running in Θ(2n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' It might be the case that a byte outside a  blind spot is in fact a quantum byte that can be almost arbitrarily mutated, but has some undetected data  dependency on another byte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Therefore, our reported Type I error rate is a tight bound on the actual Type I  error, but our Type II error rate is a loose upper bound on the true Type II error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  We mutated all 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='5 million blind spot bytes classified in the corpus, and all mutated blind spots produced  identical output to the original file for a 0% false-positive rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Of the bytes not classified as blind spots,  89% did affect the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Therefore, the false-negative rate is bounded above by 11%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' A significant number  4Over the course of 2021, DFSan underwent a significant refactor in order to make its memory layout compatible with other  LLVM sanitizers [10];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' this refactor reduced the effective number of taints it could track to 23 = 8, which is still sufficient for  most fuzzing use cases, but not for detecting blind spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  6  Lorem ipsum dolor sit amet,  consectetur adipiscing elit, sed  do eiusmod tempor incididunt ut  labore et dolore magna aliqua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  Ut enim ad minim veniam, quis  nostrud exercitation  ullamco laboris  nisi ut aliquip ex ea commodo  consequat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Duis aute irure dolor  in reprehenderit in voluptate  velit  esse cillum dolore eu fugiat nulla  pariatur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  Detected Blind Spot  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Notional example of validating detected blind spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Underlined bytes are it-  eratively mutated and re-parsed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Bytes outside of blind spots are randomly selected for  mutation, but all bytes within a blind spot are mutated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' If a byte inside a blind spot is mu-  tated and produces a different result, then that blind spot classification was a false-positive  (Type I error).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' If a byte outside a blind spot can be arbitrarily mutated, then it is a missed  detection (Type II error).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  Bytes  # Blind Spots  Total Frequency  \\r  561451  6745095  \\n  52175  4996436  \\x20  47752  17967270  \\x00  9504  7658663  \\x11  5232  3325230  \\x08  3930  3611559  \\x06  3629  3104109  \\x09  3572  2981341  %  3145  3041586  \\x12  2932  2847588  #  2859  3399908  \\x05  2772  2873564  \\x02  2640  3007837  \\x0F  2607  2944161  \\x14  2594  3132850  \\xF0  2351  2930162  a  2348  4959732  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  2271  3172472  \\xA3  2108  2783202  4  2074  4848767  \\x13  2024  2788459  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' The twenty most common blind spot prefixes of length at most seven bytes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  of these missed detections are likely quantum bytes: Pieces of data that can be mutated almost arbitrarily,  but have some data dependency that can affect output that our random mutations did not exercise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Blind Spot Content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' What is the content of PDF blind spots?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  In the GovDocs PDF corpus, we detect 63,194 unique blind spot prefixes of length at most seven bytes,  and 338,943 unique byte sequences of length at most seven that precede blind spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  Consider the bytes that occur at the start of a blind spot;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' the most common of these are listed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  They are all one byte long, meaning that there is a diversity of content at the start of blind spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' The  most common byte sequences that precede a blind spot, listed in Table 3, are more interesting: most are  multi-byte, and they comprise many PDF tokens like endobj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' This means that bytes following certain tokens  are often or always ignored by the parser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  7  Bytes  # Blind Spots  Total Frequency  n  300057  5811224  \\ n  299785  629959  00000\\ n  299621  555279  f  236173  3738217  \\ f  235736  552315  65535\\ f  205564  470855  m  66860  4242668  dstream  66588  189282  00001\\ f  26343  44612  \\r  12839  6745095  endobj\\r  11001  527199  e  7056  7495303  be  6419  41673  Adobe  6418  11545  \\x00\\x0EAdobe  6417  9142  \\x02  5533  3007837  \\x08  4185  3611559  \\x00\\x02  3787  82034  00000\\ f  3770  6838  \\x01\\x00\\x02  3707  29366  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' The twenty most common byte sequences of length at most seven preceding a blind spot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  Now let us consider the unique suffix/prefix pairs that occur adjacent to the start of a blind spot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' There  are 1,029,129 unique pairs of these byte sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' If we sort them by frequency, the pair  not a blind spot  ����  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' endobj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  �  ��  �  blind spot  is in the top 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='01% of such pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' “endobj” is a PDF token used to delimit the end of an object in the  document model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' The fact that this token is split across a blind spot boundary so frequently is indicative of,  at best, intentional lexical permissiveness on the part of the parser, and, at worst, a bug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Other interesting  blind spot contexts within the top hundredth of the first percentile include the entire endobj token, if  preceded by a carriage return.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Any whitespace after the stream token is ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' The obj token is completely  ignored if preceded by a space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Similarly, the PDF dictionary delimiters << and >> are frequently skipped,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=', at the beginning of a PDF object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' This simple contextual blind spot frequency analysis can discover  parsing errors and differentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Blind Spot Context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' How frequent are blind spots, and where do they occur in PDFs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  Figure 4 plots the number of blind spots in the PDF corpus as a function of file size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' This suggests that  the number of blind spot bytes in a typical PDF is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Note, however, that blind spots in PDFs can  be arbitrarily large, since the PDF format permits the inclusion of arbitrary binary blobs that do not have  to be connected to the document object model (DOM) [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  Figure 5 plots a histogram of the contiguous size of blind spot regions in the corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' The majority of blind  spots are small, but a nontrivial number of blind spots are over 1 KB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' The average blind spot is 42 bytes  long with a standard deviation of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='72 KB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  Figure 6 is a histogram of the normalized position of blind spot bytes in their files: the blind spot’s byte  offset divided by the file size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' In our experiments with the MµPDF renderer translating to PostScript, the  majority of PDF blind spots are at the beginning and ends of the files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  Figure 7 combines the two previous figures by comparing the mean contiguous blind spot size to the  normalized position in the PDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Despite the most blind spot bytes being at the beginning and ends of the  PDFs, the longest blind spots tend to be in the first 10–20% of the file, but not immediately at the beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='01%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='1%  1%  10%  50%  100%  1 kB  5 kB 10 kB  100 kB  1 MB  5 MB  Bytes in Blind Spots  File Size  files tested  best fit: f(x) = log(a)  bx  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Proportion of PDF bytes that are in blind blind spots as a function of file size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  1  10  100  1000  10000  100000  1 × 106  1 B  10 B  100 B  1 kB  10 kB  100 kB  1 MB  Number of Blind Spots  Contiguous Blind Spot Size  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Histogram of the sizes of contiguous PDF blind spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' The majority of blind  spots are single bytes, but a nontrivial number of blind spots are over 1 kilobyte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  The  average blind spot is 42 bytes long with a standard deviation of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='72 kilobytes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  For each input PDF, we generate a parse tree using PolyTracker’s sister tool, PolyFile5 [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Each byte  in the input PDF corresponds to one or more parse tree derivations: unique paths through the PDF parse  tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' A byte could have more than one derivation, for instance, if the input file is a polyglot—a file that  is valid in two or more formats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' PDFs are particularly easy to turn into polyglots, and many legitimate  PDF generators exploit this fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' For example, it is common to produce valid PDFs that are also valid ZIP  archives that, when extracted, contain additional files related to the document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Therefore, a byte might  have one derivation in the PDF parse tree and have a different but completely valid derivation in the ZIP  parse tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' An example of a parse tree derivation is given in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  For each unique parse tree derivation, we count the number of blind spot bytes that occur in that deriva-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' The most frequent derivation containing blind spots is  application/pdf , the root of the PDF parse tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  This means that the majority of PDF blind spots occur in portions of the file that have no semantic purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  Blind spot locality might be explained by the fact that PDF parsers are resilient to both leading and trailing  5Open-source and available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='com/trailofbits/polyfile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  9  0  20000  40000  60000  80000  100000  120000  140000  160000  0%  10%  20%  30%  40%  50%  60%  70%  80%  90%  100%  Number of Blind Spot Bytes in the Corpus  Normalized File Position (Byte Offset ÷ File Size)  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Normalized position of blind spots in PDF files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' With the MµPDF renderer,  blind spots are most frequent at the beginning and end of PDF files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  0  20  40  60  80  100  120  140  0%  10%  20%  30%  40%  50%  60%  70%  80%  90%  100%  Mean Contiguous Blind Spot Size (Bytes)  Normalized Blind Spot Start Position (Byte Offset ÷ File Size)  Average Blind Spot Length  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Contiguous blind spot size as a function of its position in the PDF file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Error  bars correspond to the standard deviation of blind spot sizes in that portion of the file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' The  longest blind spots tend to be in the second tenth of the file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  application/pdf  →  PDFObject  →  PDFDictionary  →  KeyValuePair  →  Value  →  PDFList  →  PDFList  →  PSInt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' The parse tree derivation of a byte representing an integer in a list of lists that  is a value in a PDF dictionary in a PDF object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  garbage bytes before and after the PDF file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Also, as we saw above, the majority of naturally occurring  blind spot bytes are at the beginning and end of the file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  The frequency of every unique parse tree derivation is presented in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Blind spots overwhelmingly  occur in a small number of derivations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' However, the long tail demonstrates that blind spots can and do  occur in many diverse derivations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  10  1  10  100  1000  10000  100000  1 × 106  Most Frequent  Median  Least Frequent  Number of Blind Spot Bytes  Unique Parse Tree Derivations Sorted by Blind Spot Frequency  Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Each point is a parse tree derivation—a unique path through a PDF parse  tree—whose y-axis value is the number of times a blind spot occurred in that derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  Blind spots overwhelmingly occur in a small number of derivations, yet there is a long tail  demonstrating that blind spots can and do occur in many diverse derivations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  The most frequent parse tree derivations for PDF blind spots are given in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' All but the most  frequent derivation descend from the PDFObject node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' This is unsurprising, since PDF objects can contain  streams of arbitrary binary data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' PDF objects also do not need to be connected to the root of the PDF  document object model, nor do they need to be used in any way for rendering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  The second most frequent derivation for blind spots are in PDF dictionaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  PDF dictionaries can  contain arbitrary key/value pairs which are often used for metadata that is not necessary for rendering  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=', timestamps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Dictionaries can and often do contain redundant information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' For example, the length  of a PDF object stream can either be specified as a key/value pair in the preceding object dictionary or  implicitly defined by the location of a required termination token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' If both are specified, then they must agree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  However, if a length is specified in the dictionary which does not agree with the position of the termination  token, then most parsers will ignore the specified length and defer to the token position [22], making the  dictionary entry a blind spot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Conclusions  This paper defined the concept of blind spots: inputs to a program that can be arbitrarily mutated  without affecting the program’s output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Operational semantics for blind spots were formalized by extending  SimpIL [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' An efficient implementation capable of automatically detecting blind spots, PolyTracker, was  introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' It works by adding instrumentation for performing dynamic information flow tracking (DIFT)  to a program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  The technique was evaluated by detecting blind spots in the popular MµPDF parser over a corpus of  over a thousand diverse PDFs [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' There were zero false-positive blind spot classifications, and the missed  detection rate was bounded above by 11%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' On average, at least 5% of each PDF file was completely ignored  by the parser;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' blind spots that could be repurposed for steganography or embedding malware payloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  Future work includes extending the approach to detect quantum bytes: inputs that can almost arbitrarily  be mutated without affecting output, like source code comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' The current implementation injects its  DIFT instrumentation at the LLVM/IR level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Therefore, it is limited to programs that can be compiled  using LLVM, or binaries that can be lifted to LLVM/IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' It would be useful to apply the technique to runtime  instrumentation that could be applied to a black-box binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  Our results show promise that this technique could be an efficient automated means to detect parser bugs  and differentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Nothing in the technique is tied to PDF in general, so it can be immediately applied to  other notoriously difficult-to-parse formats like ELF, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='509, and XML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='11  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='Derivation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='# Bytes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='application/pdf  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='815891  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='application/pdf → PDFObject → PDFDictionary → KeyValuePair → Key  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='365771  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='application/pdf → PDFObject → PSBytes → image/jpeg  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='120803  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='application/pdf → PDFObject → PDFDictionary → KeyValuePair → Value → PDFObjRef  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='116538  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='application/pdf → PDFObject → PDFDictionary → KeyValuePair → Value → PDFLiteral  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='108573  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='application/pdf → PDFObject → PDFDictionary → KeyValuePair → Value → PSInt  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='58864  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='application/pdf → PDFObject → PDFDictionary → KeyValuePair → Value → PDFList → PDFObjRef  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='55920  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='application/pdf → PDFObject → PDFDictionary → KeyValuePair → Value → PDFList → PSInt  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='51693  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='application/pdf → PDFObject → PDFList → PDFObjRef  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='37982  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='application/pdf → PDFObject → PDFDictionary → KeyValuePair → Value → PDFList → PDFLiteral  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='32221  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='application/pdf → PDFObject → PDFDictionary → KeyValuePair → Value → PDFDictionary → KeyValuePair → Key  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='27220  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='application/pdf → PDFObject → FlateDecode → DecodedStream → PSBytes → application/octet-stream  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='27210  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='application/pdf → PDFObject → PDFDictionary → KeyValuePair → Value → PSFloat  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='22254  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='application/pdf → PDFObject → PDFDictionary → KeyValuePair → Value → PDFList → PDFList → PSInt  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='14489  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='application/pdf → PDFObject → PDFDictionary → KeyValuePair → Value → PSBytes → text/plain  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='13300  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='application/pdf → PDFObject → PDFDictionary → KeyValuePair → Value → PDFList → PSFloat  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='11781  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='application/pdf → PDFObject → FlateDecode  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='11634  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='application/pdf → PDFObject → PDFDictionary → KeyValuePair → Value → PDFDictionary → KeyValuePair →  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='Value → PDFLiteral  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='10148  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='application/pdf → PDFObject → PSBytes → image/jp2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='8738  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='application/pdf → PDFObject → PDFDictionary → KeyValuePair → Value → PDFList → PDFDictionary → KeyVal-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='uePair → Key  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='8445  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='application/pdf → PDFObject → PDFDictionary → KeyValuePair → Value → PDFDictionary → KeyValuePair →  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='Value → PSFloat  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='7849  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='application/pdf → PDFObject → PSBytes → application/octet-stream  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='7800  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='application/pdf → PDFObject → PSBytes → text/plain  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='5131  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='application/pdf → PDFObject → PDFDictionary → KeyValuePair → Value → PDFList → PSBytes → text/plain  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='4901  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='application/pdf → PDFObject → PDFDeciphered → image/jpeg  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='4900  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='application/pdf → PDFObject → PDFDictionary → KeyValuePair → Value → PDFDictionary → KeyValuePair →  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='Value → PSInt  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='4741  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='application/pdf → PDFObject → LZWDecode → DecodedStream → PSBytes → application/octet-stream  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='3914  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='application/pdf → PDFObject → PDFDictionary → KeyValuePair → Value → PDFList → PDFDictionary → KeyVal-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='uePair → Value → PDFLiteral  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='3583  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' The PDF parse tree derivations containing the most blind spot bytes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' These  primarily descend from PDFObject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  Acknowledgment  This research was supported in part by the Defense Advanced Research Projects Agency (DARPA) Safe-  Docs program as a subcontractor to Galois under HR0011-19-C-0073.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Many thanks to Michael Brown, Trent  Brunson, Filipe Casal, Peter Goodman, Kelly Kaoudis, Bill Harris, Nichole Schimanski, Mark Tullsen, Walt  Woods, Peter Wyatt, and Sergey Bratus for their invaluable feedback on the approach and tooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Special  thanks to Carson Harmon, the original creator of PolyTracker, whose ideas and discussions germinated this  research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  References  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' ISO/IEC 29500-1:2016, Information technology—Document description and processing languages—Office Open XML File  Formats—Part 1: Fundamentals and Markup Language Reference, Standard, International Organization for Standardiza-  tion, Geneva, CH, November 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Ange Albertini, Abusing file formats;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' or, Corkami, the novella, The International Journal of Proof of Concept or GTFO  0x07 (2015), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' 6, 18–41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  12  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Ange Albertini, Thai Duong, Shay Gueron, Stefan K¨olbl, Atul Luykx, and Sophie Schmieg, How to abuse and fix authenti-  cated encryption without key commitment, Proceedings of the 31st USENIX Security Symposium (Boston, MA), USENIX  Association, August 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Fabrice Bellard, QEMU, a fast and portable dynamic translator, Proceedings of the Annual USENIX Technical Conference  (USA), ATEC ’05, USENIX Association, 2005, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Christopher Brant, Prakash Shrestha, Benjamin Mixon-Baca, Kejun Chen, Said Varlioglu, Nelly Elsayed, Yier Jin, Jedidiah  Crandall, and Daniela Oliveira, Challenges and opportunities for practical and effective dynamic information flow tracking,  ACM Computing Surveys 55 (2021), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Peng Chen and Hao Chen, Angora: Efficient fuzzing by principled search, Proceedings of the IEEE Symposium on Security  and Privacy, 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' 711–725.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' James Clause, Wanchun Li, and Alessandro Orso, Dytan: A generic dynamic taint analysis framework, Proceedings of  the 2007 International Symposium on Software Testing and Analysis (New York, NY, USA), ISSTA ’07, Association for  Computing Machinery, 2007, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' 196–206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Ali Davanian, Zhenxiao Qi, Yu Qu, and Heng Yin, DECAF++: Elastic Whole-System dynamic taint analysis, 22nd  International Symposium on Research in Attacks, Intrusions and Defenses (RAID 2019) (Chaoyang District, Beijing),  USENIX Association, September 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' 31–45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' DataFlowSanitizer, https://clang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='llvm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='org/docs/DataFlowSanitizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='html, Accessed: 2020-07-26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' [DFSan] change shadow and origin memory layouts to match MSan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=', LLVM commit 45f6d5522f8d, https://reviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  llvm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='org/D104896?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='id=354633, Accessed: 2020-07-26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Brendan Dolan-Gavitt, Josh Hodosh, Patrick Hulin, Tim Leek, and Ryan Whelan, Repeatable reverse engineering with  PANDA, Proceedings of the 5th Program Protection and Reverse Engineering Workshop (New York, NY, USA), PPREW-  5, Association for Computing Machinery, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Simson Garfinkel, Paul Farrell, Vassil Roussev, and George Dinolt, Bringing science to digital forensics with standardized  forensic corpora, Digital Investigation 6 (2009), S2–S11, Proceedings of the Ninth Annual DFRWS Conference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Carson Harmon, Bradford Larsen, and Evan Sultanik, Toward automated grammar extraction via semantic labeling of  parser implementations, Proceedings of the Sixth Workshop on Language-Theoretic Security (LangSec), IEEE Symposium  on Security and Privacy, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Matthias H¨oschele and Andreas Zeller, Mining input grammars from dynamic taints, Proceedings of the 31st IEEE/ACM  International Conference on Automated Software Engineering (New York, NY, USA), ASE 2016, Association for Computing  Machinery, 2016, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' 720–725.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Ahmed Karaman, Measuring QEMU emulation efficiency, https://ahmedkrmn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='io/TCG-Continuous-Benchmarking/  Measuring-QEMU-Emulation-Efficiency/, August 2020, Accessed: 2020-07-26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Khoo, Taintgrind: a Valgrind taint analysis tool, https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='com/wmkhoo/taintgrind, Accessed: March 2,  2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Stefan Nagy, Anh Nguyen-Tuong, Jason D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Hiser, Jack W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Davidson, and Matthew Hicks, Breaking through binaries:  Compiler-quality instrumentation for better binary-only fuzzing, 30th USENIX Security Symposium (USENIX Security  21), USENIX Association, August 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' 1683–1700.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Sebastian Poeplau and Aur´elien Francillon, Symbolic execution with SymCC: Don’t interpret, compile!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=', 29th USENIX  Security Symposium (USENIX Security 20), USENIX Association, August 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' 181–198.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Florent Saudel and Jonathan Salwan, Triton: A dynamic symbolic execution framework, Symposium sur la s´ecurit´e des  technologies de l’information et des communications (Rennes, France), SSTIC, June 2015, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' 31–54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Edward J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Schwartz, Thanassis Avgerinos, and David Brumley, All you ever wanted to know about dynamic taint analysis  and forward symbolic execution (but might have been afraid to ask), Proceedings of the IEEE Symposium on Security and  Privacy, 2010, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' 317–331.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Evan Sultanik, Brad Larsen, and Carson Harmon, Two new tools that tame the treachery of files, https://blog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  trailofbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='com/2019/11/01/two-new-tools-that-tame-the-treachery-of-files/, November 1, 2019, Accessed: Janu-  ary 12, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Julia Wolf, OMG WTF PDF—PDF ambiguity and obfuscation, Proceedings of TROOPERS (Heidelberg, Germany), March  2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Peter Wyatt, Work in progress: Demystifying PDF through a machine-readable definition, Proceedings of the Seventh  Workshop on Language-Theoretic Security (LangSec), IEEE Symposium on Security and Privacy, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' Pan Yang, Fei Kang, Yuntian Zhao, and Hui Shu, DRTaint: A dynamic taint analysis framework supporting correlation  analysis between data regions, Journal of Physics: Conference Series 1856 (2021), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content=' 1, 012013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='  Trail of Bits, Stockholm, Sweden  Email address: henrik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='brodin@trailofbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='com  Trail of Bits, Philadelphia, USA  Email address: evan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='sultanik@trailofbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='com  Trail of Bits, Brno, Czech Republic  Email address: marek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='surovic@trailofbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
+page_content='com  13' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FAT4oBgHgl3EQfzx6B/content/2301.08700v1.pdf'}
diff --git a/P9FOT4oBgHgl3EQf5DRG/content/tmp_files/2301.12952v1.pdf.txt b/P9FOT4oBgHgl3EQf5DRG/content/tmp_files/2301.12952v1.pdf.txt
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@@ -0,0 +1,1146 @@
+Dirac equation in curved spacetime: the role of local
+Fermi velocity
+B. Bagchi∗1, A. Gallerati†2, R. Ghosh‡1
+1Shiv Nadar University, Physics Dept., Gautam Buddha Nagar, Uttar Pradesh 203207, India
+2Politecnico di Torino, Dipartimento DISAT, corso Duca degli Abruzzi 24, 10129 Torino, Italy
+Abstract
+We study the dynamical equations of charge carriers in curved Dirac ma-
+terials, in the presence of a local Fermi velocity. An explicit parameterization
+of the latter emerging quantity for a nanoscroll cylindrical geometry is also
+provided, together with a discussion of related physical effects and observable
+properties.
+Keywords: Dirac equation, graphene, local Fermi velocity, nanoscrolls.
+1 Introduction
+Dirac equation is one of the most relevant contributions in the history of quantum
+mechanics. Over the decades, its study has been conducted from different points of
+view [1, 2], with countless applications in many areas of physics. The reformulation
+of the Dirac formalism in curved backgrounds is an appealing field of research
+due to its remarkable applications in high-energy physics, quantum field theory,
+analogue gravity scenarios and condensed matter.
+A real, solid-state system where to observe the properties of Dirac spino-
+rial quantum fields in a curved space is provided by graphene and other two-
+dimensional materials. The latter have attracted great interest because of their
+∗bbagchi123@gmail.com
+†antonio.gallerati@polito.it
+‡rg928@snu.edu.in
+1
+arXiv:2301.12952v1  [cond-mat.mes-hall]  30 Jan 2023
+
+electronic, mechanical and optical characteristics [3–6]. The analysis of the elec-
+tronic structure of these special substrates has received further contributions from
+theoretical studies on the dynamics of Dirac particles in curved spacetime. The
+latter formulation, in particular, is a powerful tool to describe the motion of
+graphene low-energy charge carriers [7–17]. Furthermore, from a practical per-
+spective, stability, flexibility and near-ballistic transport at room temperature,
+make Dirac materials an ideal framework for many nanoscale applications [18–
+23].
+It is well recognized that a gap formation in graphene-like systems [24, 25]
+suggests the inclusion of a spatially-varying Fermi velocity [26–29]. Confirmations
+in this direction also come from spectroscopy experiments [30–33]. In addition,
+the need for a local Fermi velocity (LFV) emerges from the electronic transport
+in two-dimensional strained Dirac materials [34]; the latter feature has triggered
+a wide range of researches [35–41].
+On the other hand, the interest in position-dependent mass (PDM) effects [42–
+53] has led to a large amount of theoretical studies, including those on wave packet
+dynamics and propagation in topological materials [54–57].
+In the context of
+PDM, we are dealing with an extended form of the Schr¨odinger equation involving
+a set of ambiguity parameters. In this regard, von Roos [58] showed that it is
+possible to construct a Hermitian form for the governing Hamiltonian, whose
+expression depends on the mentioned parameters.
+In this work we focus on the dynamical equations of charge carriers in Dirac
+materials, in the presence of a non-trivial local Fermi velocity.
+The paper is
+organized as follows. In Section 2, we discuss the procedure we use to parameterize
+the underlying curved spacetime, restricting on a set of suitable cylindrical for the
+related Dirac equation. In Section 3, we analyze the influence of the LFV on the
+Dirac equation, reformulating the problem in terms of modified γ-matrices. In
+Section 4, we take into account the impact of PDM in our formulation; we also
+discuss the explicit example of a nanoscroll parametrization, taking into account
+the density of states as a simple observable and commenting on the measurement
+issues. Finally, in Section 5, we briefly summarize the obtained results.
+2 Parameterizing the curved spacetime
+We start recovering the coordinates of a conventional three-dimensional spacetime
+in cylindrical coordinates as specified in [13, 59]:
+xa = (t, x, y, z)
+⇒
+xµ = (t, φ, z) .
+(1)
+2
+
+We then express
+x = r(φ) cos(φ) ,
+y = r(φ) sin(φ) ,
+z = z ,
+(2)
+where we have made explicit the φ–dependence of the radius. Making use of the
+Jacobian Ja
+µ = ∂xa
+∂xµ , we find the explicit form of the metric gµν:
+gµν = diag
+�
+1, −f(φ)2, −1
+�
+,
+(3)
+where the quantity f(φ)2 reads
+f(φ)2 = r(φ)2 + r′(φ)2 .
+(4)
+As a result, the line element for the curved space assumes the form
+ds2 = gµν dxµ dxν = dt2 − f(φ)2 dφ2 − dz2 .
+(5)
+The flat-space Dirac equation is defined in terms of flat γ-matrices that we can
+parameterize in terms of Pauli matrices as
+γa = {σ3, −i σ1, i σ2} ,
+a = 0, 1, 2
+(6)
+and obey the Clifford algebra
+{γa, γb} = 2 ηab 1 ,
+ηab =
+�
+1
+0
+0
+−1
+�
+.
+(7)
+To derive the form of the Dirac equation in curved space, we consider a tetrad
+basis at each spacetime point.
+The γ-matrices for the curved background are
+obtained by means of the vielbeins eµa:
+γµ = eµa γa ,
+(8)
+where γµ = gµν γν satisfy the curved space Clifford algebra
+{γµ, γν} = 2 gµν 1,
+µ = 0, 1, 2.
+(9)
+The flat spacetime Dirac equation in the presence of an electromagnetic field
+Aa(x),
+�
+iℏ γa �
+∂a + g
+iℏAa
+�
+− m
+�
+Ψ = 0 ,
+(10)
+3
+
+acquires, in the curved background, the form:
+�
+iℏ γµ �
+∂µ + Γµ + g
+iℏAµ
+�
+− m
+�
+Ψ = 0 ,
+(11)
+where Γµ = 1
+4 ωµab Mab is the spin affine connection, expressed in terms of the
+spin connection ωµab and Lorentz generators Mab = 1
+2[γa, γb] [30, 60, 61].
+Imposing the cylindrical symmetry of space, the spin affine connection Γµ
+vanishes and (11) simplifies to
+�
+iℏ γµ �
+∂µ + g
+iℏAµ
+�
+− m
+�
+Ψ = 0 .
+(12)
+3 Impact of a local Fermi velocity
+For heterostructures, the Fermi velocity becomes position-dependent [62]:
+vf � vf(z, φ) ,
+(13)
+and this also affects the Dirac equation (10), that is modified as
+�
+iℏ ˜γa√vf
+�
+∂a
+√vf + g
+iℏ
+√vf Aa
+�
+− m
+�
+Ψ = 0 ,
+(14)
+where
+˜γa =
+�
+γ0
+vf(z, φ), γ1, γ2
+�
+.
+(15)
+We also introduce the curved ˜γµ matrices
+˜γµ =
+�
+1
+vf(z, φ) e0a γa, e1a γa, e2a γa
+�
+,
+eµa = gµν eνa ,
+(16)
+so that a curved space Dirac equation in the presence of LFV takes the form
+�
+i˜γµ√vf
+�
+∂µ
+√vf + g
+i
+√vfAµ
+�
+− m
+�
+Ψ = 0 ,
+(17)
+where, from now on, we assume ℏ = 1.
+Let us now study more in detail eq. (14). In order to simplify our analysis,
+we apply the following transformation on Ψ(z, φ):
+�
+vf(z, φ) Ψ(z, φ) = χ(z, φ) ,
+(18)
+4
+
+so that the above equation can be rewritten as
+�
+i˜γµvf
+�
+∂µ + g
+i Aµ
+�
+− m
+�
+χ = 0 .
+(19)
+Explicitly, the components of γa and γµ read
+γa = {σ3, −iσ1, iσ2} =
+��
+1
+0
+0
+−1
+�
+,
+�
+0
+−i
+−i
+0
+�
+,
+�
+0
+1
+−1
+0
+��
+,
+(20)
+γµ = eµa γa =
+��
+1
+0
+0
+−1
+�
+,
+�
+0
+if(φ)
+if(φ)
+0
+�
+,
+�
+0
+−1
+1
+0
+��
+,
+(21)
+where f(φ) is given by (4). Note that the ˜γµ’s in our case are given by
+˜γµ = gµν ˜γν =
+�
+1
+vf
+�
+1
+0
+0
+−1
+�
+,
+�
+0
+−
+i
+f(φ)
+−
+i
+f(φ)
+0
+�
+,
+�
+0
+1
+−1
+0
+��
+,
+(22)
+where we have used the inverse of (3) for gµν and employed (15) and (16).
+3.1 Dirac Equation
+Using the explicit form of the modified gamma matrices, we can turn equation
+(19) to the form
+�
+E − m
+vf
+f
+�
+∂φ + g
+i Aφ
+�
++ ivf
+�
+∂z + g
+i Az
+�
+vf
+f
+�
+∂φ + g
+i Aφ
+�
+− ivf
+�
+∂z + g
+i Az
+�
+−(E + m)
+� �
+χ1
+χ2
+�
+= 0 ,
+(23)
+where we adopted the column representation for χ = (χ1, χ2) and restricted the
+local Fermi velocity to be a function of z only, vf = vf(z).
+We now make the following ansatz for χ:
+χ = e−i λ E t
+�
+χ1(φ, z)
+χ2(φ, z)
+�
+= e−i λ E t ei kφ
+� φ
+f(φ′) dφ′
+�
+χ+(z)
+χ−(z)
+�
+,
+(24)
+were both χ+ and χ− are functions of z only. We also assume the four vector Aµ
+to be expressed as
+Aµ =
+�
+0, 0, Aφ(φ, z), Az(φ, z)
+�
+.
+(25)
+5
+
+Expanding then (23), we obtain a pair of coupled equations of the form
+�vf
+f
+�
+∂φ + g
+i Aφ
+�
++ i vf
+�
+∂z + g
+i Az
+��
+χ2 + (E − m) χ1 = 0 ,
+(26)
+�vf
+f
+�
+∂φ + g
+i Aφ
+�
+− i vf
+�
+∂z + g
+i Az
+��
+χ1 − (E + m) χ2 = 0 .
+(27)
+Uncoupling the above system, we get two separate second-order differential equa-
+tions, corresponding to separate conditions for the upper component χ+ and the
+lower component χ− . In particular, we find for χ+
+�
+v2
+f ∂2
+z +
+�
+vf v′
+f + 2 v2
+f
+g
+i Az
+�
+∂z +
+�
+g vf (vf Wφ)′ − kφ vf v′
+f + 2 g kφ v2
+f Wφ − g2 v2
+f W 2
+φ − k2
+φ v2
+f
+�
++
++
+�g
+i vf (vf Az)′ − g2 v2
+f A2
+z
+�
++
+�
+E2 − m2� �
+χ+ = 0 ,
+(28)
+where we have also used the separation Aφ = f(φ) Wφ(z).
+To obtain a real form for the above equation, we set Az = 0. This reduces
+(28) to the form
+�
+v2
+f ∂2
+z + vf v′
+f ∂z +
+�
+g vf (vf Wφ)′ − kφ vf v′
+f + 2 g kφ v2
+f Wφ − g2 v2
+f W 2
+φ − k2
+φ v2
+f
+�
++
++
+�
+E2 − m2� �
+χ+ = 0 .
+(29)
+Since we want an explicit analytic solution, we need to define the explicit form
+of the LFV and the four vector components Aφ. In the following Section, we
+exploit a typical choice. We also briefly comment PDM schemes characterized by
+an extended Schr¨odinger equation, the latter in turn dependent on a generalized
+class of potentials containing a set of ambiguity parameters.
+4 An explicit model
+The study of the position-dependent mass problems [58] has found a huge amount
+of applications in the last two decades. These include the study of the electronic
+properties of condensed-matter systems, such as compositionally-graded crystals,
+quantum dots and liquid crystals [63, 64], quantum many-body systems focusing
+on the nuclei, quantum liquids, helium and metal clusters [65]. Furthermore, PDM
+has found relevance in different theoretical contexts that include supersymmetric
+quantum mechanics [66–68], coherent states [69] and use of indefinite effective
+mass [70]. Several techniques have been developed to tackle PDM problems like
+the so-called point canonical transformation [71] and potential algebra formalism
+6
+
+(see, for instance, [40] and references therein).
+Motivated by our earlier works on the PDM issue [39, 40], let us propose the
+following forms for the LFV and Wφ function:
+vf(z) = v0 eαz ,
+Wφ(z) = w0 e−αz ,
+(30)
+where α > 0, while v0 and w0 are real constants. We then see from (29) that the
+upper component χ+ satisfies
+�
+v2
+0 e2αz ∂2
+z + α v2
+0 e2αz ∂z −
+�
+α kφ v2
+0 + k2
+φ v2
+0
+�
+e2αz + 2 B kφ v0 eαz − κ2�
+χ+(z) = 0 ,
+(31)
+where
+κ2 = m2 + B2 − E2 ,
+B = g v0 w0 .
+(32)
+If we now make the change of variable
+y =
+� z
+1
+vf(q) dq = −e−αz
+v0 α ,
+(33)
+the equation can be recast as
+�
+∂2
+y − kφ
+α
+�
+1 + kφ
+α
+� 1
+y2 − 2 B kφ
+α
+1
+y + κ2
+�
+χ+(y) = 0 ,
+−∞ < y < 0 .
+(34)
+The solution of the above equation is given by
+χ+(y) ∝ M
+�
+−B kφ
+α κ , 2 kφ + α
+2 α
+; 2 y κ
+�
+,
+(35)
+where M(a, b; y) is the well known Whittaker function.1 In terms of the variable
+z, we then find
+χ+(z) = C1 M
+�
+−B kφ
+α κ , 2 kφ + α
+2 α
+; − 2 κ
+v0 α eαz
+�
+.
+(36)
+On the other hand, equation (26) can be used to calculate the lower component
+χ− of the spinor wavefunction, giving
+χ−(z) = C1 M
+�B kφ
+α κ , 2 kφ + α
+2 α
+; 2 κ
+v0 α eαz
+�
+.
+(37)
+1 The function M(a, b; z) is a solution of a Whittaker equation and is expressed in terms of the
+Kummer confluent hypergeometric function as M(a, b; z) = e− z
+2 eb+ 1
+2 1F1
+�
+b − a + 1
+2, 1 + 2 b; z
+�
+.
+7
+
+For the energy levels, we exploit the relation
+κn = −
+B kφ
+kφ + α + n α ,
+n = 0, 1, 2, . . .
+(38)
+to determine
+E2
+n = m2 + B2
+�
+1 −
+k2
+φ
+(kφ + α + n α)2
+�
+,
+n = 0, 1, 2, . . .
+(39)
+the above expression ensuring that we always get real energies.
+4.1 Application: density of states
+Once obtained an explicit solution χ to the modified Dirac equation (23), we
+can consider the probability density P =
+�
+det(gµν) χ2 , in terms of which a
+normalization condition can be written as
+�
+dΣ P(z, φ) =
+�
+dz dφ
+�
+r(φ)2 + r′(φ)2 |χ(z, φ)|2 = 1 .
+(40)
+In order to write an expression for P, we need an explicit surface parameterization
+r(φ) for the sample surface, together with the previous solutions eqs. (36) and (37).
+4.1.1 Example: nanoscroll parametrization
+Graphene nanoscrolls [72–74] have received great attention due to their unusual
+properties and potential applications [75–79]. They consist in carbon–based struc-
+tures obtained by rolling a graphene layer into a cylindrical geometry [72], and
+exhibit some exciting qualities due to their distinctively different conformation
+[80, 81], making them conceptually interesting and experimentally relevant.
+Unusual electronic and optical properties of carbon nanoscrolls are due to
+their unique topology and peculiar structure [13, 59, 82]. From a geometrical point
+of view, nanoscrolls geometry can be parameterized in terms of Archimedean–
+type spirals in cylindrical coordinates [83–85]. Let us then consider the following
+parameterization for the cylindrical geometry of the layer surface [13, 59]
+r(φ) = R
+�
+1 − 1
+2π
+φ
+DN
+�2
+,
+(41)
+where the coefficient D controls the distance between the layers and R is a typical
+dimension for the average radius of the cylinder. The integer number N takes
+into account the windings of the wrapped graphene layer; from a physical point of
+view, it acts as a momentum cutoff, correctly restricting the analysis to the long
+8
+
+wavelength continuum approximation.2 The boundary conditions in the cylindri-
+cal symmetry give rise to a quantization condition on the transverse component
+of the charge carriers momentum kφ of the form [13, 59]
+kφ = 2π
+ζN
+(42)
+having defined the geometrical parameter ζN
+ζN ≡
+2πN
+�
+0
+dφ f(φ) ,
+(43)
+roughly expressing a measure of the cylinder spiral.
+In Figures 1–3 we plot the normalized probability density P as a function of
+the coordinates (z, φ) for a nanoscroll geometry with different dimensions, number
+of windings N and energy parameter.
+Figure 1: Normalized probability density P as a function of the coordinates (z, φ) for a
+nanoscroll geometry with parameters N = 4, R = 5, D = 100, g = 10−3,
+α = 2.5 × 10−2, v0 = 10, w0 = 10 .
+2 This requires that the number N be small when compared with the ratio of the cylinder
+circumference to the graphene lattice dimension a
+�
+N ! 2πR
+a
+�
+.
+9
+
+3. × 10-3
+2 π
+2. × 10-3
+1. × 10-3
+0.
+0
+元
+50
+100
+0Figure 2: Normalized probability density P as a function of the coordinates (z, φ) for a
+nanoscroll geometry with parameters N = 5, R = 10, D = 100, g = 10−3,
+α = 10−2, v0 = 10, w0 = 10 .
+Figure 3: Normalized probability density P as a function of the coordinates (z, φ) for a
+nanoscroll geometry with parameters N = 3, R = 25, D = 100, g = 10−3,
+α = 5 × 10−3, v0 = 10, w0 = 10 .
+10
+
+1.6×10-3
+2 爪
+1.1× 10-3
+5. × 10-4
+0.
+0
+元
+50
+100
+150
+200
+08. × 10-4
+2 元
+6. × 10-4
+3. × 10-4
+0.
+0
+100
+2
+200
+04.2 Local density of states
+Given a location X on the layer surface and an energy value E, the local density of
+states (LDOS) of the sample can be expressed in terms of the probabilty density
+as [86]
+ρs(E, X, 0) = 1
+ε
+E
+�
+E−ε
+P(X) ,
+(44)
+for small values of ε, where the 0–coordinate states that we are considering pseu-
+doparticles on a two-dimensional space (zero-distance from the substrate surface).
+The LDOS then expresses the number of charge carriers per unit surface and unit
+energy range of size ε, at a given surface location X and energy E. The sample
+LDOS is not only an interesting direct physical observable, but also a substrate
+feature of great importance for electronic applications, being the availability of
+empty valence and conduction states (states below and above the Fermi level)
+crucial for the transition rates.
+Measurements.
+The sample LDOS can be mapped using a scanning tunnel-
+ing microscope (STM). The latter is an experimental device based on quantum
+mechanical tunneling, where the wave-like properties of charge carries allow them
+to penetrate, through a potential barrier, into regions that are forbidden to them
+in the classical picture. STM spectroscopy provides insight into the surface elec-
+tronic properties of the substrate, being the tunneling current strongly affected
+by the local density of states ρs. The latter is in turn related to the probability
+density P through definition (44).
+A typical STM device consists of sharp conductive tip, brought within tun-
+neling distance (< nm) from a sample surface. A small voltage bias V is then
+applied between the probe tip and the substrate, causing charge carriers to tunnel
+across the gap, resulting in a tunneling current between the sample and the tip 3.
+In the STM–map setup, the density of states at some fixed energy is mapped
+as a function of the position (z, φ) on the sample surface. If we assume ε = e V to
+be very small with respect to the work function Φw (minimum energy required to
+extract an electron from the surface), the sample states with energy lying between
+Ef − ε and Ef are very close to the Fermi level and have non-zero probability of
+tunneling into the tip. The resulting tunneling current I is directly proportional
+to the number of states on the substrate within our energy range of width ε,
+this number depending on the local properties of the surface. Including all the
+sample states in the chosen energy range, the measured tunneling current can be
+3 To simplify our discussion, we are assuming that both materials have the same Fermi level.
+11
+
+modelled, in first approximation, as [86]
+I ∝
+Ef
+�
+Ef−e V
+P e−2 κ d ,
+(45)
+where κ is some decay constant in the barrier separation depending on Φw . The
+exponential function gives the suppression for charge carriers tunneling in the
+classically forbidden region of width d (sample-tip separation). The tunneling
+current can be then measured, for constant separation d, at different X positions
+and, for sufficiently small V , it can be conveniently expressed in terms of the
+LDOS of the sample as [86]:
+I(X) ∝ ρs(Ef, X, 0) e−2 κ d e V .
+(46)
+For finite bias voltage and different Fermi levels for the sample and tip, the tunnel-
+ing current and its relation with the sample local density of states can be obtained
+from Bardeen time-dependent perturbation approach [86].
+5 Summary
+The study of graphene-like systems is a captivating and multidisciplinary field of
+research, combining notion and techniques from quantum mechanics, general rel-
+ativity and condensed matter physics. These special materials realize the physics
+of Dirac fermions in a real laboratory framework, the substrate acting as a lower-
+dimensional curved spacetime for the charge carriers. This provides a direct con-
+nection between condensed matter and theoretical quantum models, with the
+possibility to explore the analogues of many high energy physics effects in a solid
+state system [87–97].
+In this paper we studied the governing equations of the charge carriers in Dirac
+materials, in the presence of a local Fermi velocity and non-trivial energy-mass
+parameters. The curvature of the sample mimics a curved spacetime background
+for the Dirac pseudoparticles, whose dynamics is then governed by a suitably
+modified Dirac equation. The procedure led us to an analytic expression for the
+wave functions of the quasiparticle modes, that was then applied to an explicitly
+example involving a nanoscroll geometry. We also discussed the impact on the
+sample density of states as a simple and straightforward physical observable.
+12
+
+Acknowledgments.
+RG thanks Shiv Nadar University for the grant of senior
+research fellowship. AG thanks Prof. F. Laviano that supported these studies
+with his funds.
+Data availability statement.
+All data supporting this study are included in
+the article.
+Conflicts of interest.
+The authors declare no conflict of interest.
+13
+
+References
+[1] B. Thaller, “The Dirac Equation”; Springer Verlag, Berlin, DE (1992). [p. 1]
+[2] M.E. Peskin and D.V. Schroeder, “An Introduction to quantum field theory”;
+Addison-Wesley, Reading, USA (1995).
+[p. 1]
+[3] K. Novoselov, A. Geim, S. Morozov, D. Jiang, Y. Zhang, S. Dubonos,
+I. Grigorieva and A. Firsov, “Electric Field Effect in Atomically Thin Carbon
+Films”, Science 306 (2004), n. 5696, 666–669.
+[p. 2]
+[4] K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, M.I. Katsnelson, I.V.
+Grigorieva, S.V. Dubonos and A.A. Firsov, “Two-dimensional gas of massless
+Dirac fermions in graphene”, Nature 438 (2005) 197, [cond-mat/0509330].
+[p. 2]
+[5] V. Gusynin, S. Sharapov and J. Carbotte, “Unusual microwave response
+of Dirac quasiparticles in graphene”, Phys. Rev. Lett. 96 (2006) 256802,
+[cond-mat/0603267].
+[p. 2]
+[6] M. Katsnelson and K. Novoselov, “Graphene: New bridge between condensed
+matter physics and quantum electrodynamics”, Solid State Commun. 143
+(2007), n. 1, 3–13.
+[p. 2]
+[7] A.H. Castro Neto, F. Guinea, N.M.R. Peres, K.S. Novoselov and A.K. Geim,
+“The electronic properties of graphene”, Rev. Mod. Phys. 81 (2009) 109–162,
+[arXiv:0709.1163].
+[p. 2]
+[8] A.K. Geim and K.S. Novoselov, “The rise of graphene”, Nat. Mater. 6 (2007),
+n. 3, 183.
+[p. 2]
+[9] A. Cortijo and M. Vozmediano, “Effects of topological defects and local cur-
+vature on the electronic properties of planar graphene”, Nucl. Phys. B 763
+(2007) 293–308, [cond-mat/0612374].
+[p. 2]
+[10] A.K. Geim, “Graphene: status and prospects”, Science 324 (2009), n. 5934,
+1530–1534.
+[p. 2]
+[11] M. Vozmediano, M. Katsnelson and F. Guinea, “Gauge fields in graphene”,
+Phys. Rept. 496 (2010) 109–148, [arXiv:1003.5179].
+[p. 2]
+[12] C. Downing and M. Portnoi, “Massless Dirac fermions in two dimensions:
+Confinement in nonuniform magnetic fields”, Phys. Rev. B 94 (2016), n. 16,
+165407, [arXiv:1610.03032].
+[p. 2]
+14
+
+[13] A. Gallerati, “Graphene properties from curved space Dirac equation”, Eur.
+Phys. J. Plus 134 (2019), n. 5, 202, [arXiv:1808.01187].
+[p. 2, 8, 9]
+[14] F. Fillion-Gourdeau, E. Lorin and S. MacLean, “Numerical quasiconformal
+transformations for electron dynamics on strained graphene surfaces”, Phys.
+Rev. E 103 (2021), n. 1, 013312.
+[p. 2]
+[15] A. Gallerati, “Negative-curvature spacetime solutions for graphene”, J. Phys.
+Condens. Matt. 33 (2021), n. 13, 135501, [arXiv:2101.03010].
+[p. 2]
+[16] B. Hamil and B.C. L¨utf¨uo˘glu, “Dunkl graphene in constant magnetic field”,
+Eur. Phys. J. Plus 137 (2022), n. 11, 1241, [arXiv:2208.11729].
+[p. 2]
+[17] M. De Oliveira, “Connecting the Dirac Equation in Flat and Curved Space-
+times via Unitary Transformation”, Few-Body Syst. 63 (2022), n. 2, 1–11.
+[p. 2]
+[18] X. Li, X. Wang, L. Zhang, S. Lee and H. Dai, “Chemically derived, ultra-
+smooth graphene nanoribbon semiconductors”, Science 319 (2008), n. 5867,
+1229–1232.
+[p. 2]
+[19] S. Stankovich, D. Dikin, G. Dommett, K. Kohlhaas, E. Zimney, E. Stach,
+R. Piner, S. Nguyen and R. Ruoff, “Graphene-based composite materials”,
+Nature 442 (2006), n. 7100, 282–286.
+[p. 2]
+[20] F. Bonaccorso, Z. Sun, T. Hasan and A. Ferrari, “Graphene photonics and
+optoelectronics”, Nature Phot. 4 (2010), n. 9, 611–622.
+[p. 2]
+[21] Z. Sun, T. Hasan, F. Torrisi, D. Popa, G. Privitera, F. Wang, F. Bonaccorso,
+D. Basko and A. Ferrari, “Graphene mode-locked ultrafast laser”, ACS Nano
+4 (2010), n. 2, 803–810.
+[p. 2]
+[22] C. Lui, K. Mak, J. Shan, T. Heinz et al., “Ultrafast photoluminescence from
+graphene”, Phys. Rev. Lett. 105 (2010), n. 12, 127404.
+[p. 2]
+[23] E.Y. Andrei, G. Li and X. Du, “Electronic properties of graphene: a per-
+spective from scanning tunneling microscopy and magneto-transport”, Rept.
+Prog. Phys. 75 (2012) 056501, [arXiv:1204.4532].
+[p. 2]
+[24] F. de Juan, A. Cortijo and M.A. Vozmediano, “Charge inhomogeneities due
+to smooth ripples in graphene sheets”, Phys. Rev. B 76 (2007), n. 16, 165409.
+[p. 2]
+15
+
+[25] G. Gui, J. Li and J. Zhong, “Band structure engineering of graphene by
+strain: First-principles calculations”, Phys. Rev. B 78 (2008), n. 7, 075435.
+[p. 2]
+[26] F. de Juan, J.L. Manes and M.A. Vozmediano, “Gauge fields from strain in
+graphene”, Phys. Rev. B 87 (2013), n. 16, 165131.
+[p. 2]
+[27] M. Oliva-Leyva and G.G. Naumis, “Generalizing the Fermi velocity of
+strained graphene from uniform to nonuniform strain”, Physics Letters A
+379 (2015), n. 40-41, 2645–2651.
+[p. 2]
+[28] J.R. Lima and F. Moraes, “Indirect band gap in graphene from modulation
+of the Fermi velocity”, Solid State Communications 201 (2015) 82–87. [p. 2]
+[29] C. Downing and M. Portnoi, “Localization of massless Dirac particles via
+spatial modulations of the Fermi velocity”, J. Phys. Condens. Matt. 29 (2017),
+n. 31, 315301.
+[p. 2]
+[30] F. De Juan, M. Sturla and M. Vozmediano, “Space dependent Fermi ve-
+locity in strained graphene”, Phys. Rev. Lett. 108 (2012), n. 22, 227205,
+[arXiv:1201.2656].
+[p. 2, 4]
+[31] H. Yan, Z. Chu, W. Yan, M. Liu, L. Meng, M. Yang, Y. Fan, J. Wang, R. Dou,
+Y. Zhang et al., “Superlattice Dirac points and space-dependent Fermi velocity
+in a corrugated graphene monolayer”, Phys. Rev. B 87 (2013), n. 7, 075405.
+[p. 2]
+[32] C. Hwang, D.A. Siegel, S.K. Mo, W. Regan, A. Ismach, Y. Zhang, A. Zettl
+and A. Lanzara, “Fermi velocity engineering in graphene by substrate modi-
+fication”, Sci. Rep. 2 (2012), n. 1, 1–4.
+[p. 2]
+[33] W. Jang, H. Kim, Y. Shin, M. Wang, S. Jang, M. Kim, S. Lee, S. Kim,
+Y. Song and S. Kahng, “Observation of spatially-varying Fermi velocity in
+strained-graphene directly grown on hexagonal boron nitride”, Carbon 74
+(2014) 139–145.
+[p. 2]
+[34] A.L. Phan and D.N. Le, “Electronic transport in two-dimensional strained
+Dirac materials under multi-step Fermi velocity barrier:
+transfer matrix
+method for supersymmetric systems”, Eur. Phys. J. B 94 (2021), n. 8, 165,
+[arXiv:2106.10902].
+[p. 2]
+[35] O. Mustafa, “(1+1)-Dirac bound states in one dimension, with position-
+dependent Fermi velocity and mass”, Open Physics 11 (2013), n. 4, 480–486.
+[p. 2]
+16
+
+[36] M. Oliva-Leyva,
+J. Barrios-Vargas and C. Wang,
+“Fingerprints of a
+position-dependent Fermi velocity on scanning tunnelling spectra of strained
+graphene”, J. Phys. Condens. Matt. 30 (2018), n. 8, 085702.
+[p. 2]
+[37] M. Oliva-Leyva, J. Barrios-Vargas and G. De la Cruz, “Effective magnetic
+field induced by inhomogeneous Fermi velocity in strained honeycomb struc-
+tures”, Phys. Rev. B 102 (2020), n. 3, 035447.
+[p. 2]
+[38] A. Ishkhanyan and V. Jakubsk´y, “Two-dimensional Dirac fermion in pres-
+ence of an asymmetric vector potential”, J. Phys. A 51 (2018), n. 49, 495205,
+[arXiv:1801.05045].
+[p. 2]
+[39] R. Ghosh, “Position-dependent mass Dirac equation and local Fermi veloc-
+ity”, J. Phys. A 55 (2022), n. 1, 015307, [arXiv:2107.01668].
+[p. 2, 7]
+[40] B. Bagchi, R. Ghosh and C. Quesne, “so(2, 1) algebra, local Fermi velocity,
+and position-dependent mass Dirac equation”, J. Phys. A 55 (2022), n. 37,
+375204, [arXiv:2205.02017].
+[p. 2, 7]
+[41] R. Valencia-Torres,
+J. Avenda˜no,
+J. Garc´ıa-Ravelo and E. Chore˜no,
+“Position-dependent mass with modulated velocity in 1-D heterostructures”,
+Phys. Scripta 97 (2022), n. 10, 105306.
+[p. 2]
+[42] N. Peres, A. Neto and F. Guinea, “Dirac fermion confinement in graphene”,
+Phys. Rev. B 73 (2006), n. 24, 241403.
+[p. 2]
+[43] R. Grassi, S. Poli, E. Gnani, A. Gnudi, S. Reggiani and G. Baccarani, “Tight-
+binding and effective mass modeling of armchair graphene nanoribbon FETs”,
+Solid State Electron. 53 (2009), n. 4, 462–467.
+[p. 2]
+[44] C. Yannouleas, I. Romanovsky and U. Landman, “Beyond the constant-mass
+Dirac physics: Solitons, charge fractionization, and the emergence of topo-
+logical insulators in graphene rings”, Phys. Rev. B 89 (2014), n. 3, 035432.
+[p. 2]
+[45] K. Reijnders, D. Minenkov, M. Katsnelson and S. Dobrokhotov, “Elec-
+tronic optics in graphene in the semiclassical approximation”, Ann. Phys.
+397 (2018) 65–135.
+[p. 2]
+[46] R.R.S. Oliveira, A.A. Ara´ujo Filho, R.V. Maluf and C.A.S. Almeida, “The
+relativistic Aharonov–Bohm–Coulomb system with position-dependent mass”,
+J. Phys. A 53 (2020), n. 4, 045304, [arXiv:1812.07756].
+[p. 2]
+17
+
+[47] A. Contreras-Astorga, D.J. Fern´andez C and J. Negro, “Solutions of the
+Dirac equation in a magnetic field and intertwining operators”, Symm. Integr.
+Geom. 8 (2012) 082.
+[p. 2]
+[48] C. Downing and M. Portnoi, “Trapping charge carriers in low-dimensional
+Dirac materials”, Int. J. Nanosci. 18 (2019), n. 03n04, 1940001.
+[p. 2]
+[49] F. Serafim, F. Santos, J. Lima, C. Filgueiras and F. Moraes, “Position-
+dependent mass effects in the electronic transport of two-dimensional quan-
+tum systems: Applications to nanotubes”, Physica E 108 (2019) 139–146.
+[p. 2]
+[50] A. Schulze-Halberg, “Arbitrary-order Darboux transformations for two-
+dimensional Dirac equations with position-dependent mass”, Eur. Phys. J.
+Plus 135 (2020), n. 3, 1–13.
+[p. 2]
+[51] A. Schulze-Halberg, “Higher-order Darboux transformations for the Dirac
+equation with position-dependent mass at nonvanishing energy”, Eur. Phys.
+J. Plus 135 (2020), n. 10, 863.
+[p. 2]
+[52] A. Schulze-Halberg, “Darboux transformations for Dirac equations in polar
+coordinates with vector potential and position-dependent mass”, Eur. Phys.
+J. Plus 137 (2022), n. 7, 1–16.
+[p. 2]
+[53] C. Tezcan, R. Sever and O. Yesiltas, “A new approach to the exact solutions
+of the effective mass Schrodinger equation”, Int. J. Theor. Phys. 47 (2008)
+1713.
+[p. 2]
+[54] S. Raghu and F. Haldane, “Analogs of quantum-Hall-effect edge states in
+photonic crystals”, Phys. Rev. A 78 (2008), n. 3, 033834.
+[p. 2]
+[55] B. Bernevig and T. Hughes, “Topological Insulators and Topological Super-
+conductors”; Princeton University Press, Princeton, USA (2013).
+[p. 2]
+[56] P. Xie and Y. Zhu, “Wave packet dynamics in slowly modulated photonic
+graphene”, J. Differ. Equ. 267 (2019), n. 10, 5775–5808.
+[p. 2]
+[57] P. Hu, L. Hong and Y. Zhu, “Linear and nonlinear electromagnetic waves in
+modulated honeycomb media”, Stud. Appl. Math. 144 (2020), n. 1, 18–45.
+[p. 2]
+[58] O. von Roos, “Position-dependent effective masses in semiconductor theory”,
+Phys. Rev. B 27 (1983) 7547–7552.
+[p. 2, 6]
+18
+
+[59] A. Gallerati, “Graphene, Dirac equation and analogue gravity”, Phys. Scripta
+97 (2022), n. 6, 064005, [arXiv:2205.08843].
+[p. 2, 8, 9]
+[60] J.S. Pedernales, M. Beau, S.M. Pittman, I.L. Egusquiza, L. Lamata,
+E. Solano and A. del Campo, “Dirac Equation in (1+1)-Dimensional Curved
+Spacetime and the Multiphoton Quantum Rabi Model”, Phys. Rev. Lett. 120
+(2018), n. 16, 160403, [arXiv:1707.07520].
+[p. 4]
+[61] F. de Juan, A. Cortijo and M. Vozmediano, “Charge inhomogeneities due to
+smooth ripples in graphene sheets”, Phys. Rev. B 76 (2007), n. 16, 165409.
+[p. 4]
+[62] N.M.R. Peres, “Scattering in one-dimensional heterostructures described by
+the Dirac equation”, J. Phys. Condens. Matter 21 (2009) 095501.
+[p. 4]
+[63] G. Bastard, “Wave mechanics applied to semiconductor heterostructures”;
+John Wiley and Sons Inc., New York, USA (1990).
+[p. 6]
+[64] S. de-la Huerta-Sainz, A. Ballesteros and N.A. Cordero, “Gaussian Curvature
+Effects on Graphene Quantum Dots”, Nanomaterials 13 (2022), n. 1, 95. [p.6]
+[65] P. Ring and P. Schuck, “The nuclear many-body problem”; Springer-Verlag,
+Berlin, DE (2004).
+[p. 6]
+[66] B. Bagchi and T. Tanaka, “A generalized non-Hermitian oscillator Hamil-
+tonian, N-fold supersymmetry and position-dependent mass models”, Phys.
+Lett. A 372 (2008), n. 33, 5390–5393.
+[p. 6]
+[67] B. Bagchi, A. Banerjee, C. Quesne and V. Tkachuk, “Deformed shape in-
+variance and exactly solvable Hamiltonians with position-dependent effective
+mass”, J. Phys. A Math. Theor. 38 (2005), n. 13, 2929.
+[p. 6]
+[68] C. Quesne, “Infinite families of position-dependent mass Schr¨odinger equa-
+tions with known ground and first excited states”, Annals Phys. 399 (2018)
+270–288, [arXiv:1808.07367].
+[p. 6]
+[69] S. Cruz y Cruz and O. Rosas-Ortiz, “Position-dependent mass oscillators and
+coherent states”, J. Phys. A Math. Theor. 42 (2009), n. 18, 185205.
+[p. 6]
+[70] M. Znojil and G. Levai, “Schr¨odinger equations with indefinite effective
+mass”, Phys. Lett. A 376 (2012), n. 45, 3000–3005.
+[p. 6]
+[71] C. Quesne, “Point canonical transformation versus deformed shape invari-
+ance for position-dependent mass Schr¨odinger equations”, Symm. Integr.
+Geom. 5 (2009) 046.
+[p. 6]
+19
+
+[72] X. Xie, L. Ju, X. Feng, Y. Sun, R. Zhou, K. Liu, S. Fan, Q. Li and K. Jiang,
+“Controlled fabrication of high-quality carbon nanoscrolls from monolayer
+graphene”, Nano Lett. 9 (2009), n. 7, 2565–2570.
+[p. 8]
+[73] Braga, S.F. and Coluci, V.R. and Legoas, S.B. and Giro, R. and Galv˜ao,
+D.S. and Baughman, R.H., “Structure and dynamics of carbon nanoscrolls”,
+Nano Lett. 4 (2004), n. 5, 881–884.
+[p. 8]
+[74] Kim, K. and Lee, Z. and Malone, B.D. and Chan, K.T. and Alem´an, B. and
+Regan, W. and Gannett, W. and Crommie, M.F. and Cohen, M.L. and Zettl,
+A., “Multiply folded graphene”, Phys. Rev. B 83 (2011), n. 24, 245433. [p. 8]
+[75] Y. Chen, J. Lu and Z. Gao, “Structural and electronic study of nanoscrolls
+rolled up by a single graphene sheet”, J. Phys. Chem. C 111 (2007), n. 4,
+1625–1630.
+[p. 8]
+[76] G. Mpourmpakis, E. Tylianakis and G. Froudakis, “Carbon nanoscrolls: a
+promising material for hydrogen storage”, Nano Lett. 7 (2007), n. 7, 1893–
+1897.
+[p. 8]
+[77] D. Berman, S. Deshmukh, S. Sankaranarayanan, A. Erdemir and A. Sumant,
+“Macroscale superlubricity enabled by graphene nanoscroll formation”, Sci-
+ence 348 (2015), n. 6239, 1118–1122.
+[p. 8]
+[78] H. Li, R. Papadakis, S. Jafri, T. Thersleff, J. Michler, H. Ottosson and
+K. Leifer, “Superior adhesion of graphene nanoscrolls”, Commun. Phys. 1
+(2018), n. 1, 1–7.
+[p. 8]
+[79] S. Saini, S. Reshmi, G. Gouda, A. Kumar, K. Sriram and K. Bhattacharjee,
+“Low reflectance of carbon nanotube and nanoscroll-based thin film coatings:
+a case study”, Nanoscale Advances 3 (2021), n. 11, 3184–3198.
+[p. 8]
+[80] Zhang, D.B. and Akatyeva, E. and Dumitric˘a, T., “Bending ultrathin
+graphene at the margins of continuum mechanics”, Phys. Rev. Lett. 106
+(2011), n. 25, 255503.
+[p. 8]
+[81] Z. Xu and M.J. Buehler, “Geometry controls conformation of graphene sheets:
+membranes, ribbons, and scrolls”, ACS Nano 4 (2010), n. 7, 3869–3876. [p. 8]
+[82] M. Fogler, A.C. Neto and F. Guinea, “Effect of external conditions on the
+structure of scrolled graphene edges”, Phys. Rev. B 81 (2010), n. 16, 161408.
+[p. 8]
+20
+
+[83] X. Chen, Q. Zhou, J. Wang and Q. Chen, “Formation of Graphene Nano-
+scrolls and Their Electronic Structures Based on Ab Initio Calculations”, J.
+Phys. Chem. Lett 13 (2022) 2500–2506.
+[p. 8]
+[84] M. Trushin and A. Neto, “Stability of a rolled-up conformation state for two-
+dimensional materials in aqueous solutions”, Phys. Rev. Lett. 127 (2021),
+n. 15, 156101.
+[p. 8]
+[85] M. Hassanzadazar, M. Ahmadi, R. Ismail and H. Goudarzi, “Electrical prop-
+erty analytical prediction on Archimedes chiral carbon nanoscrolls”, J. Elec-
+tron. Mater. 45 (2016), n. 10, 5404–5411.
+[p. 8]
+[86] C.J. Chen, “Introduction to scanning tunneling microscopy”; Oxford Univer-
+sity Press, 1 ed. (1993).
+[p. 11, 12]
+[87] M. Cvetic and G. Gibbons, “Graphene and the Zermelo Optical Metric of the
+BTZ Black Hole”, Annals Phys. 327 (2012) 2617–2626, [arXiv:1202.2938].
+[p. 12]
+[88] T. Stegmann and N. Szpak, “Current flow paths in deformed graphene: from
+quantum transport to classical trajectories in curved space”, New J. Phys. 18
+(2016), n. 5, 053016, [arXiv:1512.06750].
+[p. 12]
+[89] P.D. Alvarez, M. Valenzuela and J. Zanelli, “Supersymmetry of a different
+kind”, JHEP 04 (2012) 058, [arXiv:1109.3944].
+[p. 12]
+[90] A. Sepehri, R. Pincak and A.F. Ali, “Emergence of F(R) gravity-analogue
+due to defects in graphene”, Eur. Phys. J. B 89 (2016), n. 11, 250,
+[arXiv:1606.02039].
+[p. 12]
+[91] M. Franz and M. Rozali,
+“Mimicking black hole event horizons in
+atomic and solid-state systems”, Nature Rev. Mater. 3 (2018) 491–501,
+[arXiv:1808.00541].
+[p. 12]
+[92] S. Capozziello, R. Pincak and E.N. Saridakis, “Constructing superconductors
+by graphene Chern-Simons wormholes”, Annals Phys. 390 (2018) 303–333.
+[p. 12]
+[93] B. Kandemir, “Hairy BTZ black hole and its analogue model in graphene”,
+Annals Phys. 413 (2020) 168064, [arXiv:1907.03509].
+[p. 12]
+[94] L. Andrianopoli, B.L. Cerchiai, R. D’Auria, A. Gallerati, R. Noris, M. Tri-
+giante and J. Zanelli, “N-extended D = 4 supergravity, unconventional SUSY
+and graphene”, JHEP 01 (2020) 084, [arXiv:1910.03508].
+[p. 12]
+21
+
+[95] A. Gallerati, “Supersymmetric theories and graphene”, PoS ICHEP 2020
+(2020) 662.
+[p. 12]
+[96] T. Morresi, D. Binosi, S. Simonucci, R. Piergallini, S. Roche, N. Pugno and
+T. Simone, “Exploring event horizons and Hawking radiation through de-
+formed graphene membranes”, 2D Materials 7 (2020), n. 4, 041006.
+[p. 12]
+[97] S. Capozziello, R. Pinˇcak and E. Bartoˇs, “Chern-Simons Current of Left and
+Right Chiral Superspace in Graphene Wormhole”, Symmetry 12 (2020), n. 5,
+774.
+[p. 12]
+22
+
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new file mode 100644
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf,len=999
+page_content='Dirac equation in curved spacetime: the role of local  Fermi velocity  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Bagchi∗1, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Gallerati†2, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Ghosh‡1  1Shiv Nadar University, Physics Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=', Gautam Buddha Nagar, Uttar Pradesh 203207, India  2Politecnico di Torino, Dipartimento DISAT, corso Duca degli Abruzzi 24, 10129 Torino, Italy  Abstract  We study the dynamical equations of charge carriers in curved Dirac ma-  terials, in the presence of a local Fermi velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' An explicit parameterization  of the latter emerging quantity for a nanoscroll cylindrical geometry is also  provided, together with a discussion of related physical effects and observable  properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  Keywords: Dirac equation, graphene, local Fermi velocity, nanoscrolls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  1 Introduction  Dirac equation is one of the most relevant contributions in the history of quantum  mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Over the decades, its study has been conducted from different points of  view [1, 2], with countless applications in many areas of physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' The reformulation  of the Dirac formalism in curved backgrounds is an appealing field of research  due to its remarkable applications in high-energy physics, quantum field theory,  analogue gravity scenarios and condensed matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  A real, solid-state system where to observe the properties of Dirac spino-  rial quantum fields in a curved space is provided by graphene and other two-  dimensional materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' The latter have attracted great interest because of their  ∗bbagchi123@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='com  †antonio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='gallerati@polito.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='it  ‡rg928@snu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='in  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='12952v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='mes-hall]  30 Jan 2023  electronic, mechanical and optical characteristics [3–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' The analysis of the elec-  tronic structure of these special substrates has received further contributions from  theoretical studies on the dynamics of Dirac particles in curved spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' The  latter formulation, in particular, is a powerful tool to describe the motion of  graphene low-energy charge carriers [7–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Furthermore, from a practical per-  spective, stability, flexibility and near-ballistic transport at room temperature,  make Dirac materials an ideal framework for many nanoscale applications [18–  23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  It is well recognized that a gap formation in graphene-like systems [24, 25]  suggests the inclusion of a spatially-varying Fermi velocity [26–29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Confirmations  in this direction also come from spectroscopy experiments [30–33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' In addition,  the need for a local Fermi velocity (LFV) emerges from the electronic transport  in two-dimensional strained Dirac materials [34];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' the latter feature has triggered  a wide range of researches [35–41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  On the other hand, the interest in position-dependent mass (PDM) effects [42–  53] has led to a large amount of theoretical studies, including those on wave packet  dynamics and propagation in topological materials [54–57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  In the context of  PDM, we are dealing with an extended form of the Schr¨odinger equation involving  a set of ambiguity parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' In this regard, von Roos [58] showed that it is  possible to construct a Hermitian form for the governing Hamiltonian, whose  expression depends on the mentioned parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  In this work we focus on the dynamical equations of charge carriers in Dirac  materials, in the presence of a non-trivial local Fermi velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  The paper is  organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' In Section 2, we discuss the procedure we use to parameterize  the underlying curved spacetime, restricting on a set of suitable cylindrical for the  related Dirac equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' In Section 3, we analyze the influence of the LFV on the  Dirac equation, reformulating the problem in terms of modified γ-matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' In  Section 4, we take into account the impact of PDM in our formulation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' we also  discuss the explicit example of a nanoscroll parametrization, taking into account  the density of states as a simple observable and commenting on the measurement  issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Finally, in Section 5, we briefly summarize the obtained results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  2 Parameterizing the curved spacetime  We start recovering the coordinates of a conventional three-dimensional spacetime  in cylindrical coordinates as specified in [13, 59]:  xa = (t, x, y, z)  ⇒  xµ = (t, φ, z) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  (1)  2  We then express  x = r(φ) cos(φ) ,  y = r(φ) sin(φ) ,  z = z ,  (2)  where we have made explicit the φ–dependence of the radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Making use of the  Jacobian Ja  µ = ∂xa  ∂xµ , we find the explicit form of the metric gµν:  gµν = diag  �  1, −f(φ)2, −1  �  ,  (3)  where the quantity f(φ)2 reads  f(φ)2 = r(φ)2 + r′(φ)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  (4)  As a result, the line element for the curved space assumes the form  ds2 = gµν dxµ dxν = dt2 − f(φ)2 dφ2 − dz2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  (5)  The flat-space Dirac equation is defined in terms of flat γ-matrices that we can  parameterize in terms of Pauli matrices as  γa = {σ3, −i σ1, i σ2} ,  a = 0, 1, 2  (6)  and obey the Clifford algebra  {γa, γb} = 2 ηab 1 ,  ηab =  �  1  0  0  −1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  (7)  To derive the form of the Dirac equation in curved space, we consider a tetrad  basis at each spacetime point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  The γ-matrices for the curved background are  obtained by means of the vielbeins eµa:  γµ = eµa γa ,  (8)  where γµ = gµν γν satisfy the curved space Clifford algebra  {γµ, γν} = 2 gµν 1,  µ = 0, 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  (9)  The flat spacetime Dirac equation in the presence of an electromagnetic field  Aa(x),  �  iℏ γa �  ∂a + g  iℏAa  �  − m  �  Ψ = 0 ,  (10)  3  acquires, in the curved background, the form:  �  iℏ γµ �  ∂µ + Γµ + g  iℏAµ  �  − m  �  Ψ = 0 ,  (11)  where Γµ = 1  4 ωµab Mab is the spin affine connection, expressed in terms of the  spin connection ωµab and Lorentz generators Mab = 1  2[γa, γb] [30, 60, 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  Imposing the cylindrical symmetry of space, the spin affine connection Γµ  vanishes and (11) simplifies to  �  iℏ γµ �  ∂µ + g  iℏAµ  �  − m  �  Ψ = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  (12)  3 Impact of a local Fermi velocity  For heterostructures, the Fermi velocity becomes position-dependent [62]:  vf � vf(z, φ) ,  (13)  and this also affects the Dirac equation (10), that is modified as  �  iℏ ˜γa√vf  �  ∂a  √vf + g  iℏ  √vf Aa  �  − m  �  Ψ = 0 ,  (14)  where  ˜γa =  �  γ0  vf(z, φ), γ1, γ2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  (15)  We also introduce the curved ˜γµ matrices  ˜γµ =  �  1  vf(z, φ) e0a γa, e1a γa, e2a γa  �  ,  eµa = gµν eνa ,  (16)  so that a curved space Dirac equation in the presence of LFV takes the form  �  i˜γµ√vf  �  ∂µ  √vf + g  i  √vfAµ  �  − m  �  Ψ = 0 ,  (17)  where, from now on, we assume ℏ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  Let us now study more in detail eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' In order to simplify our analysis,  we apply the following transformation on Ψ(z, φ):  �  vf(z, φ) Ψ(z, φ) = χ(z, φ) ,  (18)  4  so that the above equation can be rewritten as  �  i˜γµvf  �  ∂µ + g  i Aµ  �  − m  �  χ = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  (19)  Explicitly, the components of γa and γµ read  γa = {σ3, −iσ1, iσ2} =  ��  1  0  0  −1  �  ,  �  0  −i  −i  0  �  ,  �  0  1  −1  0  ��  ,  (20)  γµ = eµa γa =  ��  1  0  0  −1  �  ,  �  0  if(φ)  if(φ)  0  �  ,  �  0  −1  1  0  ��  ,  (21)  where f(φ) is given by (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Note that the ˜γµ’s in our case are given by  ˜γµ = gµν ˜γν =  �  1  vf  �  1  0  0  −1  �  ,  �  0  −  i  f(φ)  −  i  f(φ)  0  �  ,  �  0  1  −1  0  ��  ,  (22)  where we have used the inverse of (3) for gµν and employed (15) and (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='1 Dirac Equation  Using the explicit form of the modified gamma matrices, we can turn equation  (19) to the form  �  E − m  vf  f  �  ∂φ + g  i Aφ  �  + ivf  �  ∂z + g  i Az  �  vf  f  �  ∂φ + g  i Aφ  �  − ivf  �  ∂z + g  i Az  �  −(E + m)  � �  χ1  χ2  �  = 0 ,  (23)  where we adopted the column representation for χ = (χ1, χ2) and restricted the  local Fermi velocity to be a function of z only, vf = vf(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  We now make the following ansatz for χ:  χ = e−i λ E t  �  χ1(φ, z)  χ2(φ, z)  �  = e−i λ E t ei kφ  � φ  f(φ′) dφ′  �  χ+(z)  χ−(z)  �  ,  (24)  were both χ+ and χ− are functions of z only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' We also assume the four vector Aµ  to be expressed as  Aµ =  �  0, 0, Aφ(φ, z), Az(φ, z)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  (25)  5  Expanding then (23), we obtain a pair of coupled equations of the form  �vf  f  �  ∂φ + g  i Aφ  �  + i vf  �  ∂z + g  i Az  ��  χ2 + (E − m) χ1 = 0 ,  (26)  �vf  f  �  ∂φ + g  i Aφ  �  − i vf  �  ∂z + g  i Az  ��  χ1 − (E + m) χ2 = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  (27)  Uncoupling the above system, we get two separate second-order differential equa-  tions, corresponding to separate conditions for the upper component χ+ and the  lower component χ− .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' In particular, we find for χ+  �  v2  f ∂2  z +  �  vf v′  f + 2 v2  f  g  i Az  �  ∂z +  �  g vf (vf Wφ)′ − kφ vf v′  f + 2 g kφ v2  f Wφ − g2 v2  f W 2  φ − k2  φ v2  f  �  +  +  �g  i vf (vf Az)′ − g2 v2  f A2  z  �  +  �  E2 − m2� �  χ+ = 0 ,  (28)  where we have also used the separation Aφ = f(φ) Wφ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  To obtain a real form for the above equation, we set Az = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' This reduces  (28) to the form  �  v2  f ∂2  z + vf v′  f ∂z +  �  g vf (vf Wφ)′ − kφ vf v′  f + 2 g kφ v2  f Wφ − g2 v2  f W 2  φ − k2  φ v2  f  �  +  +  �  E2 − m2� �  χ+ = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  (29)  Since we want an explicit analytic solution, we need to define the explicit form  of the LFV and the four vector components Aφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' In the following Section, we  exploit a typical choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' We also briefly comment PDM schemes characterized by  an extended Schr¨odinger equation, the latter in turn dependent on a generalized  class of potentials containing a set of ambiguity parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  4 An explicit model  The study of the position-dependent mass problems [58] has found a huge amount  of applications in the last two decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' These include the study of the electronic  properties of condensed-matter systems, such as compositionally-graded crystals,  quantum dots and liquid crystals [63, 64], quantum many-body systems focusing  on the nuclei, quantum liquids, helium and metal clusters [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Furthermore, PDM  has found relevance in different theoretical contexts that include supersymmetric  quantum mechanics [66–68], coherent states [69] and use of indefinite effective  mass [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Several techniques have been developed to tackle PDM problems like  the so-called point canonical transformation [71] and potential algebra formalism  6  (see, for instance, [40] and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  Motivated by our earlier works on the PDM issue [39, 40], let us propose the  following forms for the LFV and Wφ function:  vf(z) = v0 eαz ,  Wφ(z) = w0 e−αz ,  (30)  where α > 0, while v0 and w0 are real constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' We then see from (29) that the  upper component χ+ satisfies  �  v2  0 e2αz ∂2  z + α v2  0 e2αz ∂z −  �  α kφ v2  0 + k2  φ v2  0  �  e2αz + 2 B kφ v0 eαz − κ2�  χ+(z) = 0 ,  (31)  where  κ2 = m2 + B2 − E2 ,  B = g v0 w0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  (32)  If we now make the change of variable  y =  � z  1  vf(q) dq = −e−αz  v0 α ,  (33)  the equation can be recast as  �  ∂2  y − kφ  α  �  1 + kφ  α  � 1  y2 − 2 B kφ  α  1  y + κ2  �  χ+(y) = 0 ,  −∞ < y < 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  (34)  The solution of the above equation is given by  χ+(y) ∝ M  �  −B kφ  α κ , 2 kφ + α  2 α  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 2 y κ  �  ,  (35)  where M(a, b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' y) is the well known Whittaker function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='1 In terms of the variable  z, we then find  χ+(z) = C1 M  �  −B kφ  α κ , 2 kφ + α  2 α  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' − 2 κ  v0 α eαz  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  (36)  On the other hand, equation (26) can be used to calculate the lower component  χ− of the spinor wavefunction, giving  χ−(z) = C1 M  �B kφ  α κ , 2 kφ + α  2 α  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 2 κ  v0 α eαz  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  (37)  1 The function M(a, b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' z) is a solution of a Whittaker equation and is expressed in terms of the  Kummer confluent hypergeometric function as M(a, b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' z) = e− z  2 eb+ 1  2 1F1  �  b − a + 1  2, 1 + 2 b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' z  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  7  For the energy levels, we exploit the relation  κn = −  B kφ  kφ + α + n α ,  n = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  (38)  to determine  E2  n = m2 + B2  �  1 −  k2  φ  (kφ + α + n α)2  �  ,  n = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  (39)  the above expression ensuring that we always get real energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='1 Application: density of states  Once obtained an explicit solution χ to the modified Dirac equation (23), we  can consider the probability density P =  �  det(gµν) χ2 , in terms of which a  normalization condition can be written as  �  dΣ P(z, φ) =  �  dz dφ  �  r(φ)2 + r′(φ)2 |χ(z, φ)|2 = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  (40)  In order to write an expression for P, we need an explicit surface parameterization  r(φ) for the sample surface, together with the previous solutions eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' (36) and (37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='1 Example: nanoscroll parametrization  Graphene nanoscrolls [72–74] have received great attention due to their unusual  properties and potential applications [75–79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' They consist in carbon–based struc-  tures obtained by rolling a graphene layer into a cylindrical geometry [72], and  exhibit some exciting qualities due to their distinctively different conformation  [80, 81], making them conceptually interesting and experimentally relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  Unusual electronic and optical properties of carbon nanoscrolls are due to  their unique topology and peculiar structure [13, 59, 82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' From a geometrical point  of view, nanoscrolls geometry can be parameterized in terms of Archimedean–  type spirals in cylindrical coordinates [83–85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Let us then consider the following  parameterization for the cylindrical geometry of the layer surface [13, 59]  r(φ) = R  �  1 − 1  2π  φ  DN  �2  ,  (41)  where the coefficient D controls the distance between the layers and R is a typical  dimension for the average radius of the cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' The integer number N takes  into account the windings of the wrapped graphene layer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' from a physical point of  view, it acts as a momentum cutoff, correctly restricting the analysis to the long  8  wavelength continuum approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='2 The boundary conditions in the cylindri-  cal symmetry give rise to a quantization condition on the transverse component  of the charge carriers momentum kφ of the form [13, 59]  kφ = 2π  ζN  (42)  having defined the geometrical parameter ζN  ζN ≡  2πN  �  0  dφ f(φ) ,  (43)  roughly expressing a measure of the cylinder spiral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  In Figures 1–3 we plot the normalized probability density P as a function of  the coordinates (z, φ) for a nanoscroll geometry with different dimensions, number  of windings N and energy parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  Figure 1: Normalized probability density P as a function of the coordinates (z, φ) for a  nanoscroll geometry with parameters N = 4, R = 5, D = 100, g = 10−3,  α = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='5 × 10−2, v0 = 10, w0 = 10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  2 This requires that the number N be small when compared with the ratio of the cylinder  circumference to the graphene lattice dimension a  �  N !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 2πR  a  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  9  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' × 10-3  2 π  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' × 10-3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' × 10-3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  0  元  50  100  0Figure 2: Normalized probability density P as a function of the coordinates (z, φ) for a  nanoscroll geometry with parameters N = 5, R = 10, D = 100, g = 10−3,  α = 10−2, v0 = 10, w0 = 10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  Figure 3: Normalized probability density P as a function of the coordinates (z, φ) for a  nanoscroll geometry with parameters N = 3, R = 25, D = 100, g = 10−3,  α = 5 × 10−3, v0 = 10, w0 = 10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  10  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='6×10-3  2 爪  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='1× 10-3  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' × 10-4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  0  元  50  100  150  200  08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' × 10-4  2 元  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' × 10-4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' × 10-4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  0  100  2  200  04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='2 Local density of states  Given a location X on the layer surface and an energy value E, the local density of  states (LDOS) of the sample can be expressed in terms of the probabilty density  as [86]  ρs(E, X, 0) = 1  ε  E  �  E−ε  P(X) ,  (44)  for small values of ε, where the 0–coordinate states that we are considering pseu-  doparticles on a two-dimensional space (zero-distance from the substrate surface).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  The LDOS then expresses the number of charge carriers per unit surface and unit  energy range of size ε, at a given surface location X and energy E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' The sample  LDOS is not only an interesting direct physical observable, but also a substrate  feature of great importance for electronic applications, being the availability of  empty valence and conduction states (states below and above the Fermi level)  crucial for the transition rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  Measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  The sample LDOS can be mapped using a scanning tunnel-  ing microscope (STM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' The latter is an experimental device based on quantum  mechanical tunneling, where the wave-like properties of charge carries allow them  to penetrate, through a potential barrier, into regions that are forbidden to them  in the classical picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' STM spectroscopy provides insight into the surface elec-  tronic properties of the substrate, being the tunneling current strongly affected  by the local density of states ρs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' The latter is in turn related to the probability  density P through definition (44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  A typical STM device consists of sharp conductive tip, brought within tun-  neling distance (< nm) from a sample surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' A small voltage bias V is then  applied between the probe tip and the substrate, causing charge carriers to tunnel  across the gap, resulting in a tunneling current between the sample and the tip 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  In the STM–map setup, the density of states at some fixed energy is mapped  as a function of the position (z, φ) on the sample surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' If we assume ε = e V to  be very small with respect to the work function Φw (minimum energy required to  extract an electron from the surface), the sample states with energy lying between  Ef − ε and Ef are very close to the Fermi level and have non-zero probability of  tunneling into the tip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' The resulting tunneling current I is directly proportional  to the number of states on the substrate within our energy range of width ε,  this number depending on the local properties of the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Including all the  sample states in the chosen energy range, the measured tunneling current can be  3 To simplify our discussion, we are assuming that both materials have the same Fermi level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  11  modelled, in first approximation, as [86]  I ∝  Ef  �  Ef−e V  P e−2 κ d ,  (45)  where κ is some decay constant in the barrier separation depending on Φw .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' The  exponential function gives the suppression for charge carriers tunneling in the  classically forbidden region of width d (sample-tip separation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' The tunneling  current can be then measured, for constant separation d, at different X positions  and, for sufficiently small V , it can be conveniently expressed in terms of the  LDOS of the sample as [86]:  I(X) ∝ ρs(Ef, X, 0) e−2 κ d e V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  (46)  For finite bias voltage and different Fermi levels for the sample and tip, the tunnel-  ing current and its relation with the sample local density of states can be obtained  from Bardeen time-dependent perturbation approach [86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  5 Summary  The study of graphene-like systems is a captivating and multidisciplinary field of  research, combining notion and techniques from quantum mechanics, general rel-  ativity and condensed matter physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' These special materials realize the physics  of Dirac fermions in a real laboratory framework, the substrate acting as a lower-  dimensional curved spacetime for the charge carriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' This provides a direct con-  nection between condensed matter and theoretical quantum models, with the  possibility to explore the analogues of many high energy physics effects in a solid  state system [87–97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  In this paper we studied the governing equations of the charge carriers in Dirac  materials, in the presence of a local Fermi velocity and non-trivial energy-mass  parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' The curvature of the sample mimics a curved spacetime background  for the Dirac pseudoparticles, whose dynamics is then governed by a suitably  modified Dirac equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' The procedure led us to an analytic expression for the  wave functions of the quasiparticle modes, that was then applied to an explicitly  example involving a nanoscroll geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' We also discussed the impact on the  sample density of states as a simple and straightforward physical observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  12  Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  RG thanks Shiv Nadar University for the grant of senior  research fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' AG thanks Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Laviano that supported these studies  with his funds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  Data availability statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  All data supporting this study are included in  the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  Conflicts of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  The authors declare no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  13  References  [1] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Thaller, “The Dirac Equation”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Springer Verlag, Berlin, DE (1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 1]  [2] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Peskin and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Schroeder, “An Introduction to quantum field theory”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  Addison-Wesley, Reading, USA (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 1]  [3] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Novoselov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Geim, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Morozov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Jiang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Zhang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Dubonos,  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Grigorieva and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Firsov, “Electric Field Effect in Atomically Thin Carbon  Films”, Science 306 (2004), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 5696, 666–669.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 2]  [4] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Novoselov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Geim, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Morozov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Jiang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Katsnelson, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  Grigorieva, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Dubonos and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Firsov, “Two-dimensional gas of massless  Dirac fermions in graphene”, Nature 438 (2005) 197, [cond-mat/0509330].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 2]  [5] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Gusynin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Sharapov and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Carbotte, “Unusual microwave response  of Dirac quasiparticles in graphene”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 96 (2006) 256802,  [cond-mat/0603267].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 2]  [6] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Katsnelson and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Novoselov, “Graphene: New bridge between condensed  matter physics and quantum electrodynamics”, Solid State Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 143  (2007), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 1, 3–13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 2]  [7] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Castro Neto, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Guinea, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Peres, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Novoselov and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Geim,  “The electronic properties of graphene”, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 81 (2009) 109–162,  [arXiv:0709.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='1163].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 2]  [8] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Geim and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Novoselov, “The rise of graphene”, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 6 (2007),  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 3, 183.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 2]  [9] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Cortijo and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Vozmediano, “Effects of topological defects and local cur-  vature on the electronic properties of planar graphene”, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' B 763  (2007) 293–308, [cond-mat/0612374].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 2]  [10] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Geim, “Graphene: status and prospects”, Science 324 (2009), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 5934,  1530–1534.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 2]  [11] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Vozmediano, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Katsnelson and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Guinea, “Gauge fields in graphene”,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Rept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 496 (2010) 109–148, [arXiv:1003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='5179].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 2]  [12] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Downing and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Portnoi, “Massless Dirac fermions in two dimensions:  Confinement in nonuniform magnetic fields”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' B 94 (2016), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 16,  165407, [arXiv:1610.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='03032].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 2]  14  [13] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Gallerati, “Graphene properties from curved space Dirac equation”, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Plus 134 (2019), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 5, 202, [arXiv:1808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='01187].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 2, 8, 9]  [14] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Fillion-Gourdeau, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Lorin and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' MacLean, “Numerical quasiconformal  transformations for electron dynamics on strained graphene surfaces”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' E 103 (2021), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 1, 013312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 2]  [15] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Gallerati, “Negative-curvature spacetime solutions for graphene”, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Matt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 33 (2021), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 13, 135501, [arXiv:2101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='03010].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 2]  [16] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Hamil and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' L¨utf¨uo˘glu, “Dunkl graphene in constant magnetic field”,  Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Plus 137 (2022), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 11, 1241, [arXiv:2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='11729].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 2]  [17] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' De Oliveira, “Connecting the Dirac Equation in Flat and Curved Space-  times via Unitary Transformation”, Few-Body Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 63 (2022), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 2, 1–11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 2]  [18] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Wang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Zhang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Lee and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Dai, “Chemically derived, ultra-  smooth graphene nanoribbon semiconductors”, Science 319 (2008), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 5867,  1229–1232.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 2]  [19] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Stankovich, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Dikin, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Dommett, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Kohlhaas, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Zimney, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Stach,  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Piner, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Nguyen and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Ruoff, “Graphene-based composite materials”,  Nature 442 (2006), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 7100, 282–286.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 2]  [20] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Bonaccorso, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Sun, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Hasan and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Ferrari, “Graphene photonics and  optoelectronics”, Nature Phot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 4 (2010), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 9, 611–622.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 2]  [21] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Sun, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Hasan, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Torrisi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Popa, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Privitera, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Wang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Bonaccorso,  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Basko and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Ferrari, “Graphene mode-locked ultrafast laser”, ACS Nano  4 (2010), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 2, 803–810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 2]  [22] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Lui, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Mak, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Shan, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Heinz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=', “Ultrafast photoluminescence from  graphene”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 105 (2010), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 12, 127404.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 2]  [23] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Andrei, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Li and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Du, “Electronic properties of graphene: a per-  spective from scanning tunneling microscopy and magneto-transport”, Rept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 75 (2012) 056501, [arXiv:1204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='4532].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 2]  [24] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' de Juan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Cortijo and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Vozmediano, “Charge inhomogeneities due  to smooth ripples in graphene sheets”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' B 76 (2007), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 16, 165409.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 2]  15  [25] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Gui, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Li and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Zhong, “Band structure engineering of graphene by  strain: First-principles calculations”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' B 78 (2008), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 7, 075435.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 2]  [26] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' de Juan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Manes and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Vozmediano, “Gauge fields from strain in  graphene”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' B 87 (2013), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 16, 165131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 2]  [27] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Oliva-Leyva and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Naumis, “Generalizing the Fermi velocity of  strained graphene from uniform to nonuniform strain”, Physics Letters A  379 (2015), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 40-41, 2645–2651.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 2]  [28] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Lima and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Moraes, “Indirect band gap in graphene from modulation  of the Fermi velocity”, Solid State Communications 201 (2015) 82–87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 2]  [29] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Downing and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Portnoi, “Localization of massless Dirac particles via  spatial modulations of the Fermi velocity”, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Matt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 29 (2017),  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 31, 315301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 2]  [30] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' De Juan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Sturla and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Vozmediano, “Space dependent Fermi ve-  locity in strained graphene”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 108 (2012), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 22, 227205,  [arXiv:1201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='2656].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 2, 4]  [31] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Yan, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Chu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Yan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Liu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Meng, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Yang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Fan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Wang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Dou,  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=', “Superlattice Dirac points and space-dependent Fermi velocity  in a corrugated graphene monolayer”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' B 87 (2013), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 7, 075405.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 2]  [32] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Hwang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Siegel, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Mo, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Regan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Ismach, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Zhang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Zettl  and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Lanzara, “Fermi velocity engineering in graphene by substrate modi-  fication”, Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 2 (2012), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 1, 1–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 2]  [33] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Jang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Kim, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Shin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Wang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Jang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Kim, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Lee, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Kim,  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Song and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Kahng, “Observation of spatially-varying Fermi velocity in  strained-graphene directly grown on hexagonal boron nitride”, Carbon 74  (2014) 139–145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 2]  [34] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Phan and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Le, “Electronic transport in two-dimensional strained  Dirac materials under multi-step Fermi velocity barrier:  transfer matrix  method for supersymmetric systems”, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' B 94 (2021), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 8, 165,  [arXiv:2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='10902].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 2]  [35] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Mustafa, “(1+1)-Dirac bound states in one dimension, with position-  dependent Fermi velocity and mass”, Open Physics 11 (2013), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 4, 480–486.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 2]  16  [36] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Oliva-Leyva,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Barrios-Vargas and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Wang,  “Fingerprints of a  position-dependent Fermi velocity on scanning tunnelling spectra of strained  graphene”, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Matt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 30 (2018), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 8, 085702.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 2]  [37] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Oliva-Leyva, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Barrios-Vargas and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' De la Cruz, “Effective magnetic  field induced by inhomogeneous Fermi velocity in strained honeycomb struc-  tures”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' B 102 (2020), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 3, 035447.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 2]  [38] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Ishkhanyan and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Jakubsk´y, “Two-dimensional Dirac fermion in pres-  ence of an asymmetric vector potential”, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' A 51 (2018), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 49, 495205,  [arXiv:1801.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='05045].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 2]  [39] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Ghosh, “Position-dependent mass Dirac equation and local Fermi veloc-  ity”, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' A 55 (2022), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 1, 015307, [arXiv:2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='01668].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 2, 7]  [40] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Bagchi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Ghosh and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Quesne, “so(2, 1) algebra, local Fermi velocity,  and position-dependent mass Dirac equation”, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' A 55 (2022), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 37,  375204, [arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='02017].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 2, 7]  [41] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Valencia-Torres,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Avenda˜no,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Garc´ıa-Ravelo and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Chore˜no,  “Position-dependent mass with modulated velocity in 1-D heterostructures”,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Scripta 97 (2022), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 10, 105306.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 2]  [42] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Peres, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Neto and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Guinea, “Dirac fermion confinement in graphene”,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' B 73 (2006), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 24, 241403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 2]  [43] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Grassi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Poli, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Gnani, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Gnudi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Reggiani and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Baccarani, “Tight-  binding and effective mass modeling of armchair graphene nanoribbon FETs”,  Solid State Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 53 (2009), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 4, 462–467.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 2]  [44] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Yannouleas, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Romanovsky and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Landman, “Beyond the constant-mass  Dirac physics: Solitons, charge fractionization, and the emergence of topo-  logical insulators in graphene rings”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' B 89 (2014), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 3, 035432.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 2]  [45] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Reijnders, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Minenkov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Katsnelson and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Dobrokhotov, “Elec-  tronic optics in graphene in the semiclassical approximation”, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  397 (2018) 65–135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 2]  [46] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Oliveira, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Ara´ujo Filho, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Maluf and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Almeida, “The  relativistic Aharonov–Bohm–Coulomb system with position-dependent mass”,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' A 53 (2020), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 4, 045304, [arXiv:1812.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='07756].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 2]  17  [47] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Contreras-Astorga, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Fern´andez C and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Negro, “Solutions of the  Dirac equation in a magnetic field and intertwining operators”, Symm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Integr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 8 (2012) 082.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 2]  [48] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Downing and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Portnoi, “Trapping charge carriers in low-dimensional  Dirac materials”, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Nanosci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 18 (2019), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 03n04, 1940001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 2]  [49] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Serafim, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Santos, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Lima, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Filgueiras and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Moraes, “Position-  dependent mass effects in the electronic transport of two-dimensional quan-  tum systems: Applications to nanotubes”, Physica E 108 (2019) 139–146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 2]  [50] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Schulze-Halberg, “Arbitrary-order Darboux transformations for two-  dimensional Dirac equations with position-dependent mass”, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  Plus 135 (2020), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 3, 1–13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 2]  [51] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Schulze-Halberg, “Higher-order Darboux transformations for the Dirac  equation with position-dependent mass at nonvanishing energy”, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Plus 135 (2020), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 10, 863.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 2]  [52] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Schulze-Halberg, “Darboux transformations for Dirac equations in polar  coordinates with vector potential and position-dependent mass”, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Plus 137 (2022), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 7, 1–16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 2]  [53] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Tezcan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Sever and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Yesiltas, “A new approach to the exact solutions  of the effective mass Schrodinger equation”, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 47 (2008)  1713.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 2]  [54] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Raghu and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Haldane, “Analogs of quantum-Hall-effect edge states in  photonic crystals”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' A 78 (2008), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 3, 033834.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 2]  [55] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Bernevig and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Hughes, “Topological Insulators and Topological Super-  conductors”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Princeton University Press, Princeton, USA (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 2]  [56] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Xie and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Zhu, “Wave packet dynamics in slowly modulated photonic  graphene”, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Equ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 267 (2019), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 10, 5775–5808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 2]  [57] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Hu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Hong and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Zhu, “Linear and nonlinear electromagnetic waves in  modulated honeycomb media”, Stud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 144 (2020), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 1, 18–45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 2]  [58] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' von Roos, “Position-dependent effective masses in semiconductor theory”,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' B 27 (1983) 7547–7552.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 2, 6]  18  [59] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Gallerati, “Graphene, Dirac equation and analogue gravity”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Scripta  97 (2022), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 6, 064005, [arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='08843].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 2, 8, 9]  [60] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Pedernales, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Beau, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Pittman, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Egusquiza, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Lamata,  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Solano and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' del Campo, “Dirac Equation in (1+1)-Dimensional Curved  Spacetime and the Multiphoton Quantum Rabi Model”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 120  (2018), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 16, 160403, [arXiv:1707.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='07520].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 4]  [61] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' de Juan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Cortijo and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Vozmediano, “Charge inhomogeneities due to  smooth ripples in graphene sheets”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' B 76 (2007), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 16, 165409.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 4]  [62] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Peres, “Scattering in one-dimensional heterostructures described by  the Dirac equation”, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Matter 21 (2009) 095501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 4]  [63] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Bastard, “Wave mechanics applied to semiconductor heterostructures”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  John Wiley and Sons Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=', New York, USA (1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 6]  [64] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' de-la Huerta-Sainz, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Ballesteros and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Cordero, “Gaussian Curvature  Effects on Graphene Quantum Dots”, Nanomaterials 13 (2022), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 1, 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='6]  [65] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Ring and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Schuck, “The nuclear many-body problem”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Springer-Verlag,  Berlin, DE (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 6]  [66] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Bagchi and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Tanaka, “A generalized non-Hermitian oscillator Hamil-  tonian, N-fold supersymmetry and position-dependent mass models”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' A 372 (2008), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 33, 5390–5393.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 6]  [67] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Bagchi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Banerjee, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Quesne and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Tkachuk, “Deformed shape in-  variance and exactly solvable Hamiltonians with position-dependent effective  mass”, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' A Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 38 (2005), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 13, 2929.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 6]  [68] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Quesne, “Infinite families of position-dependent mass Schr¨odinger equa-  tions with known ground and first excited states”, Annals Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 399 (2018)  270–288, [arXiv:1808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='07367].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 6]  [69] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Cruz y Cruz and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Rosas-Ortiz, “Position-dependent mass oscillators and  coherent states”, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' A Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 42 (2009), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 18, 185205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 6]  [70] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Znojil and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Levai, “Schr¨odinger equations with indefinite effective  mass”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' A 376 (2012), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 45, 3000–3005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 6]  [71] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Quesne, “Point canonical transformation versus deformed shape invari-  ance for position-dependent mass Schr¨odinger equations”, Symm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Integr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 5 (2009) 046.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 6]  19  [72] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Xie, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Ju, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Feng, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Sun, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Zhou, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Liu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Fan, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Li and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Jiang,  “Controlled fabrication of high-quality carbon nanoscrolls from monolayer  graphene”, Nano Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 9 (2009), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 7, 2565–2570.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 8]  [73] Braga, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' and Coluci, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' and Legoas, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' and Giro, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' and Galv˜ao,  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' and Baughman, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=', “Structure and dynamics of carbon nanoscrolls”,  Nano Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 4 (2004), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 5, 881–884.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 8]  [74] Kim, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' and Lee, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' and Malone, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' and Chan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' and Alem´an, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' and  Regan, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' and Gannett, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' and Crommie, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' and Cohen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' and Zettl,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=', “Multiply folded graphene”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' B 83 (2011), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 24, 245433.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 8]  [75] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Chen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Lu and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Gao, “Structural and electronic study of nanoscrolls  rolled up by a single graphene sheet”, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' C 111 (2007), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 4,  1625–1630.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 8]  [76] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Mpourmpakis, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Tylianakis and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Froudakis, “Carbon nanoscrolls: a  promising material for hydrogen storage”, Nano Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 7 (2007), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 7, 1893–  1897.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 8]  [77] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Berman, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Deshmukh, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Sankaranarayanan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Erdemir and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Sumant,  “Macroscale superlubricity enabled by graphene nanoscroll formation”, Sci-  ence 348 (2015), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 6239, 1118–1122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 8]  [78] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Li, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Papadakis, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Jafri, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Thersleff, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Michler, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Ottosson and  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Leifer, “Superior adhesion of graphene nanoscrolls”, Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 1  (2018), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 1, 1–7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 8]  [79] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Saini, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Reshmi, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Gouda, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Kumar, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Sriram and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Bhattacharjee,  “Low reflectance of carbon nanotube and nanoscroll-based thin film coatings:  a case study”, Nanoscale Advances 3 (2021), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 11, 3184–3198.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 8]  [80] Zhang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' and Akatyeva, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' and Dumitric˘a, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=', “Bending ultrathin  graphene at the margins of continuum mechanics”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 106  (2011), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 25, 255503.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 8]  [81] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Xu and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Buehler, “Geometry controls conformation of graphene sheets:  membranes, ribbons, and scrolls”, ACS Nano 4 (2010), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 7, 3869–3876.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 8]  [82] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Fogler, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Neto and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Guinea, “Effect of external conditions on the  structure of scrolled graphene edges”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' B 81 (2010), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 16, 161408.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 8]  20  [83] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Chen, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Zhou, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Wang and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Chen, “Formation of Graphene Nano-  scrolls and Their Electronic Structures Based on Ab Initio Calculations”, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Lett 13 (2022) 2500–2506.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 8]  [84] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Trushin and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Neto, “Stability of a rolled-up conformation state for two-  dimensional materials in aqueous solutions”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 127 (2021),  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 15, 156101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 8]  [85] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Hassanzadazar, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Ahmadi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Ismail and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Goudarzi, “Electrical prop-  erty analytical prediction on Archimedes chiral carbon nanoscrolls”, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Elec-  tron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 45 (2016), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 10, 5404–5411.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 8]  [86] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Chen, “Introduction to scanning tunneling microscopy”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Oxford Univer-  sity Press, 1 ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 11, 12]  [87] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Cvetic and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Gibbons, “Graphene and the Zermelo Optical Metric of the  BTZ Black Hole”, Annals Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 327 (2012) 2617–2626, [arXiv:1202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='2938].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 12]  [88] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Stegmann and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Szpak, “Current flow paths in deformed graphene: from  quantum transport to classical trajectories in curved space”, New J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
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+page_content=' Pincak and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
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+page_content=' Capozziello, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Pincak and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
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+page_content=' Saridakis, “Constructing superconductors  by graphene Chern-Simons wormholes”, Annals Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
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+page_content=' 413 (2020) 168064, [arXiv:1907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
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+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Cerchiai, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' D’Auria, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Gallerati, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Noris, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Tri-  giante and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
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+page_content=' 12]  21  [95] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Gallerati, “Supersymmetric theories and graphene”, PoS ICHEP 2020  (2020) 662.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
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+page_content=' 12]  [96] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Morresi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Binosi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Simonucci, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Piergallini, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Roche, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Pugno and  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Simone, “Exploring event horizons and Hawking radiation through de-  formed graphene membranes”, 2D Materials 7 (2020), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 4, 041006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content='  [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' 12]  [97] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Capozziello, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Pinˇcak and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
+page_content=' Bartoˇs, “Chern-Simons Current of Left and  Right Chiral Superspace in Graphene Wormhole”, Symmetry 12 (2020), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FOT4oBgHgl3EQf5DRG/content/2301.12952v1.pdf'}
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+ON THE DIRECTIONAL DERIVATIVE OF THE
+HAUSDORFF DIMENSION OF QUADRATIC POLYNOMIAL
+JULIA SETS AT -2
+LUDWIK JAKSZTAS
+Abstract. Let d(δ) denote the Hausdorff dimension of the Julia set of
+the polynomial fδ(z) = z2 − 2 + δ.
+In this paper we will study the directional derivative of the function
+d along directions landing at the parameter 0, which corresponds to −2
+in the case of family pc(z) = z2 + c. We will consider all directions,
+except the one δ ∈ R+, which is inside the Mandelbrot set.
+We will prove asymptotic formula for the directional derivative of d.
+Moreover, we will see that the derivative is negative for all directions in
+the closed left half-plane. Computer calculations show that it is negative
+except a cone (with opening angle approximately 74◦) around R+.
+1. Introduction
+Let f be a polynomial in one complex variable of degree at least 2. The
+filled-in Julia set Kf we define as the set of all points that do not escape to
+infinity under iteration of f, i.e.
+Kf = {z ∈ C : fn(z) ↛ ∞}.
+It is a compact set whose boundary is called the Julia set. So, let us write
+Jf := ∂Kf.
+Instead of the classical quadratic family pc(z) = z2 + c, we will consider for
+technical reasons
+fδ(z) = z2 − 2 + δ,
+i.e. δ = c + 2. We will use the following abbreviations:
+Jδ := Jfδ,
+Kδ := Kfδ.
+Let d(δ) := HD(Jδ) denote the Hausdorff dimension of the Julia set. In this
+paper, we will deal with the function
+δ �→ d(δ).
+We define the Mandelbrot set (for the family fδ) as follows
+M := {δ ∈ C : fn
+δ (0) ↛ ∞}.
+Recall that a polynomial f : C → C is called hyperbolic (expanding) if
+there exists n ⩾ 1 such that |(fn)′(z)| > 1 for every z ∈ Jf.
+2020 Mathematics Subject Classification. Primary 37F44, Secondary 37F35.
+Key words and phrases. Hausdorff dimension, Julia sets, quadratic family.
+The
+author
+was
+partially
+supported
+by
+National
+Science
+Centre
+Grant
+2019/33/B/ST1/00275 (Poland).
+1
+arXiv:2301.04519v1  [math.DS]  11 Jan 2023
+
+2
+LUDWIK JAKSZTAS
+The function d(δ) is real-analytic on each hyperbolic component of Int M
+(consisting of parameters related to hyperbolic maps) as well as on the ex-
+terior of M (see [14]). For α ∈ [0, 2π), let us write
+R(α) := {z ∈ C∗ : α = arg z}.
+We will study properties of the function d(δ) when the parameter δ /∈ M
+tends to 0 ∈ ∂M along the ray R(α). So, we will consider all directions
+except R(0) which is inside M.
+Parameter δ = 0 corresponds to c = −2, so this is Misiurewicz’s pa-
+rameter. Note that a parameter is called Misiurewicz’s if the critical point
+is strictly preperiodic (that is preperiodic but not periodic). In our casse
+0 �→ −2 �→ 2, where 2 is a fixed point. Moreover the Julia set J0 is equal to
+the interval [−2, 2] (see for example [1]).
+J. Rivera-Letelier proved in [14] the following:
+Theorem R-L. There exists C0 > 0 such that if δn → 0 and
+Re(δn) ⩽ 0
+or
+| Im(δn)| > C0| Re(δn)|3/2,
+then δn /∈ M and d(δn) → d(0) = 1.
+An easy consequence of the above Theorem is that d(δ) converges to 1
+along any ray R(α), α ∈ (0, 2π).
+In [6], A. Fan, Y. Jiang and J. Wu, gave estimate of d restricted to the
+real line. They proved that:
+Theorem FJW. There exists constants K > 0 and δ1 < 0 such that
+1 − K−1�
+|δ| ⩽ d(δ) ⩽ 1 − K
+�
+|δ|,
+for all δ1 ⩽ δ < 0.
+So, one can expect that derivative d′(δ) tends to ∞ like 1/
+�
+|δ|, when
+δ → 0−. In this paper we prove much more general result. In order to state
+our main theorem we need two definitions.
+Let F be a real function defined on a domain U ∈ C. If z ∈ U and v ∈ C∗,
+then
+F ′
+v(z) := lim
+h→0
+F(z + hv) − F(z)
+h
+,
+if the limit exists.
+For α ∈ (0, π] we write
+Ω−2(α) :=
+1
+√
+6 π log 2
+�
+cos α − 1
+2
+√
+sin α
+π
+�
+α
+√
+sin x dx
+�
+.
+(1.1)
+If α ∈ (π, 2π) then we define Ω−2(α) := Ω−2(2π − α).
+The main Theorem in this paper is:
+Theorem 1.1. For every α ∈ (0, 2π) we have
+lim
+δ→0
+�
+|δ| · d′
+v(δ) = Ω−2(α),
+where α = arg δ and v = eiα.
+
+ON THE DIRECTIONAL DERIVATIVE OF THE HAUSDORFF DIMENSION
+3
+Figure 1. Graph of the function Ω−2(α) for α ∈ (0, π].
+Because of symmetry, it is enough to prove the Theorem for α ∈ (0, π].
+The function Ω−2(α) is obviously negative for α ∈ [π/2, π], whereas is pos-
+itive for small α.
+Moreover it is easy to check that
+∂
+∂αΩ−2(α) < 0 for
+α ∈ (0, π/2).
+Therefore there exists a unique α0 ∈ (0, π/2) such that
+Ω−2(α0) = 0.
+Thus we have d′
+v(δ) → +∞ for every α ∈ (0, α0), and
+d′
+v(δ) → −∞ for every α ∈ (α0, π].
+A numerically made picture of the graph of Ω(α) (see Figure 1.) suggests
+that it is a decreasing function on whole interval (0, π), whereas α0 is slightly
+greater than π/5 (close to 37◦).
+In the real case (i.e. α = π), we see that
+lim
+δ→0
+�
+|δ| · d′
+−1(δ) =
+−1
+√
+6 π log 2.
+Note that this is the directional derivative, whereas the derivative of d(δ)
+has opposite sign. So, integrating we obtain:
+Corollary 1.2. If δ ∈ R−, then for |δ| small we have
+d(δ) = 1 −
+√
+6
+3 π log 2
+�
+|δ| + o(
+�
+|δ|).
+Remark. Estimates along the ray R(0) (inside the Mandelbrot set).
+Recently N. Dobbs, J. Graczyk and N. Mihalache proved in [3] the follow-
+ing significant result:
+Theorem DGM. There exists constants κ∗ > 0 and δ0 > 0 such that for
+every δ ∈ (0, δ0]
+d(δ) ⩾ 1 + κ∗
+√
+δ.
+In that paper also an upper bound is proved, which holds on a "large" set
+of parameters.
+The fact that we can pass with Ω−2(α) to the limit as α → 0+, that is
+lim
+α→0+ Ω−2(α) =
+1
+√
+6 π log 2,
+leads to the following:
+
+0.15 
+0.10 E
+ s0'0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+0.05 
+-0.10 E
+-0.15 E4
+LUDWIK JAKSZTAS
+Conjecture.
+The supremum over all constants κ∗ > 0 for which there
+exists δ0 > 0 such that the statement of Theorem DGM holds, is equal to
+√
+6/(3 π log 2). In particular
+d(δ) ⩾ 1 +
+√
+6
+3 π log 2
+√
+δ − o(
+√
+δ),
+for small δ ∈ R+.
+Very recently A. Dudko, I. Gorbovickis and W. Tucker computed in [5]
+rigorous lower bounds of d(δ) for δ ∈ [0, 4] (in our parametrization). They
+estimated d(δ) on some intervals and also on some additional parameters.
+Now, using lower bounds of d(δ) for parameters (which are greater than
+estimates for intervals containing them) we make the following observation:
+For δn ≈ n · 0.004, where 1 ⩽ n ⩽ 25, precise values of estimates (which we
+received from the Authors) lead to
+d(δn) > 1 + 0.362
+�
+δn,
+whereas
+√
+6/(3 π log 2) ≈ 0.375.
+This is the first paper concerning the derivative of the dimension function
+close to a non-parabolic parameter. On the other hand this is the second
+paper about the directional derivative after [10], where the derivative close
+to the parameter c = 1/4 was studied (parabolic parameter with one petal).
+As in [10] we will us the formula for the derivative (quotient of two inte-
+grals, see Proposition 2.1). The integral from the denominator is equal to the
+Lyapunov exponent (if the measure is normalized) and tends to a positive
+constant. The main problem is to estimate integral from the numerator.
+Note that our situation is much different than in [10]. In our case a crucial
+role is played by a part of the Julia set that is close to 0, and we will have
+to rescale the set Jδ in order to study a location of Jδ and behaviour of
+conformal and invariant measures in detail.
+Notation: if z ∈ C \ R, then √z denotes square root with positive real
+part. If z ∈ R− then we assume that √z has positive imaginary part.
+2. Thermodynamic formalism
+The main goal of this section is to establish a formula for directional
+derivative of the Hausdorff dimension (see Proposition 2.1) (cf. [10, Section
+2]).
+Because of hyperbolicity, the Julia set moves holomorphically over every
+simply connected subset of C \ M. In particular, we can take the set U,
+which is connected component of B(0, 1/5) \ M containing δ0 = −1/10.
+Thus, there exists family of injections ϕδ : Jδ0 → ˆC, δ ∈ U such that
+ϕδ0 = idJδ0, ϕδ(Jδ0) = Jδ and
+ϕδ ◦ fδ0(s) = fδ ◦ ϕδ(s),
+for s ∈ Jδ0 (i.e. ϕδ conjugates fδ0 to fδ). Moreover, the map δ �→ ϕδ(s) is
+holomorphic, whereas for every δ ∈ U the map ϕδ extends to a quasiconfor-
+mal (so also Hölder) map of the sphere to itself (see [8]).
+
+ON THE DIRECTIONAL DERIVATIVE OF THE HAUSDORFF DIMENSION
+5
+Note that for δ = 0, there exists function ϕ0 that semiconjugates fδ0 to
+f0. A point z has two preimages under ϕ0 iff fn
+0 (z) = 0 for some n ⩾ 0 (see
+[2, Theorem 1.6]).
+Now we use the thermodynamic formalism, which holds for hyperbolic
+rational maps (see [12]). Let X = Jδ0, T = fδ0, and let φ : X → R be a
+potential function of the form φ = −τ log |f′
+δ(ϕδ)|, for δ ∈ U and τ ∈ R.
+The topological pressure can be defined as follows:
+P(T, φ) := lim
+n→∞
+1
+n log
+�
+s∈T −n(s)
+eSn(φ(s)),
+where Sn(φ) = �n−1
+k=0 φ ◦ T k.
+The limit exists and does not depend on
+s ∈ Jδ0 (see [12, Corollary 12.5.13]). If φ = −τ log |f′
+δ(ϕδ)| and ϕδ(s) = z,
+then eSn(φ(s)) = |(fn
+δ )′(z)|−τ, hence
+P(T, −τ log |f′
+δ(ϕδ)|) = lim
+n→∞
+1
+n log
+�
+z∈f−n
+δ
+(z)
+|(fn
+δ )′(z)|−τ.
+The function τ �→ P(T, −τ log |f′
+δ(ϕδ)|) is strictly decreasing from +∞ to
+−∞. So, there exists a unique τ0 such that P(T, −τ0 log |f′
+δ(ϕδ)|) = 0. By
+Bowen’s Theorem (see [12, Corollary 9.1.7] or [15, Theorem 5.12]) we obtain
+τ0 = d(δ).
+Thus, we have P(T, −d(δ) log |f′
+δ(ϕδ)|) = 0. Write φδ := −d(δ) log |f′
+δ(ϕδ)|.
+The Ruelle operator or the transfer operator Lφ : C0(X) → C0(X), is
+defined as
+Lφ(u)(s) :=
+�
+s∈T −1(s)
+u(s)eφ(s).
+(2.1)
+The Perron-Frobenius-Ruelle theorem [15, Theorem 4.1] asserts that β =
+eP(T,φ) is a single eigenvalue of Lφ associated to an eigenfunction ˜hφ > 0.
+Moreover there exists a unique probability measure ˜ωφ such that L∗
+φ(˜ωφ) =
+β˜ωφ, where L∗
+φ is dual to Lφ.
+For φ = φδ we have β = 1, and then ˜µφδ := ˜hφδ ˜ωφδ is a T-invariant
+measure called an equilibrium state (if it is normalized). We denote by ˜ωδ
+and ˜µδ the measures ˜ωφδ and ˜µφδ respectively (measures supported on Jδ0).
+Next, we take µδ := (ϕδ)∗ ˜µδ, and ωδ := (ϕδ)∗ ˜ωδ (measures supported on the
+Julia set Jδ).
+So, the measure µδ is fδ-invariant, whereas ωδ (after normalization) is
+called fδ-conformal measure with exponent d(δ), i.e. ωδ is a Borel probability
+measure such that for every Borel subset A ⊂ Jδ,
+ωδ(fδ(A)) =
+�
+A
+|f′
+δ|d(δ)dωδ,
+(2.2)
+provided fδ is injective on A.
+However we will not assume that µδ and ωδ are probability measures.
+Because J0 = [−2, 2] we will take µδ(Jδ) = ωδ(Jδ) = 4.
+
+6
+LUDWIK JAKSZTAS
+It follows from [15, Proposition 6.11] or [12, Theorem 5.6.5] that for every
+Hölder continuous functions ψ, ˆψ : X → R, at every t ∈ R, we have
+∂
+∂tP(T, ψ + t ˆψ) =
+�
+X
+ˆψ d˜µψ+t ˆψ.
+Let us consider parameters of the form δ = tv, where v = eiα, α = arg δ,
+and t > 0. Since τ = d(tv) is the unique zero of the pressure function, for
+the potential φ = −τ log |f′
+tv(ϕtv)|, the implicit function theorem combined
+with the above formula leads to (see [7, Proposition 2.1] or [9, Proposition
+2.1]):
+Proposition 2.1. If α ∈ (0, 2π) and v = eiα, then for every t > 0 such that
+tv /∈ M we have
+d′
+v(tv) = −d(tv)
+�
+Jδ0
+∂
+∂t log |f′
+tv(ϕtv)|d˜µtv
+�
+Jδ0 log |f′
+tv(ϕtv)|d˜µtv
+.
+(2.3)
+3. Conjugation close to the fixed point
+Suppose p is a repelling (or attracting) fixed point of a holomorphic func-
+tion f. It is well known fact that f is conjugated to z �→ f′(p)z in a neigh-
+borhood of p. Now we recall how to construct the conjugation.
+The polynomials fδ, δ ∈ U have two repelling fixed points. We will con-
+sider
+pδ := 1
+2 + 3
+2
+�
+1 − 4
+9δ = 2 − 1
+3 δ − 1
+27 δ2 + O(δ3),
+(3.1)
+which becomes p0 = 2 for δ = 0. So, the multiplier λδ at pδ is equal to
+λδ = f′
+δ(pδ) = 2pδ.
+In order to construct the conjugation Φδ, we first "move" pδ to 0. Put
+ˆfδ(z) := fδ(z + pδ) − pδ.
+Indeed, 0 is the fixed point of ˆfδ. We will write ˆJδ := J ˆfδ.
+The sequence ˆΦn,δ that converges to the function ˆΦδ, which conjugates ˆfδ
+to z �→ λδz (i.e. ˆΦδ ◦ ˆfδ = λδ ˆΦδ), we define as follows:
+ˆΦn,δ(z) := λn
+δ ˆf−n
+δ
+(z) = λn
+δ (f−n
+δ
+(pδ + z) − pδ).
+Note that
+ˆΦ−1
+n,δ(z) = ˆfn
+δ
+� z
+λn
+δ
+�
+= fn
+δ
+�
+pδ + z
+λn
+δ
+�
+− pδ.
+(3.2)
+We have
+ˆΦn,δ(z) ⇒ ˆΦδ(z),
+and the convergence is uniform independently of the parameter δ ∈ B(0, r△),
+where z ∈ B(0, rz), for some r△ > 0, rz > 0. Thus, ˆΦδ depends analytically
+on δ ∈ B(0, r△) (see for example [1, Chapter 2]).
+
+ON THE DIRECTIONAL DERIVATIVE OF THE HAUSDORFF DIMENSION
+7
+Possibly changing rz > 0, we can assume that both functions ˆΦδ and ˆΦ−1
+δ
+are defined on B(0, rz). Next, because ˆΦ′
+δ(0) = 1, we can also assume that
+these functions are not too far from the identity map, namely
+���
+ˆΦδ(z)
+z
+− 1
+��� < 1
+2,
+and
+���
+ˆΦ−1
+δ (z)
+z
+− 1
+��� < 1
+2,
+(3.3)
+for z ∈ B(0, rz) and δ ∈ B(0, r△).
+Finally, we take Φδ(z) := ˆΦδ(z − pδ), and then we have
+Φδ ◦ fδ = λδΦδ,
+(3.4)
+where the functions Φδ are defined on B(pδ, rz). Possibly changing rz and
+r△, we can also assume that Φδ are defined on B(2, rz) for δ ∈ B(0, r△).
+4. The Julia sets Jδ for δ close to 0
+4.1. The Hausdorff metric. If X and Y are two non-empty compact sub-
+sets of C, then their Hausdorff distance is defined as follows:
+dH(X, Y ) = max
+�
+sup
+x∈X
+dist(x, Y ), sup
+y∈Y
+dist(y, X)
+�
+.
+Now we state an important Theorem from [4, Section 5].
+Theorem 4.1. If Kˆδ has empty interior, that is Kˆδ = Jˆδ, then the function
+δ �→ Jδ,
+is continuous at ˆδ, as the function from C into the space of non-empty com-
+pact subsets of C equipped with the Hausdorff metric.
+Corollary 4.2. If δ → 0, then
+Jδ → J0 = [−2, 2],
+in the space of non-empty compact subsets of C equipped with the Hausdorff
+metric.
+4.2. Conjugations and location of Jδ. It is known fact that the function
+x �→ 2
+π arcsin x conjugates polynomial g(x) = −2x2 + 1 defined on [−1, 1],
+to the tent map T : [−1, 1] → [−1, 1] defined by T(x) = −2|x| + 1 (see for
+example [11, Chapter II]).
+Since x �→ −x/2 conjugates f0 to g, let us write
+ϕv(x) = 4
+π arcsin
+�x
+2
+�
+,
+ϕ−1
+v (x) = 2 sin
+�π
+4 x
+�
+,
+and next
+V (x) =
+�
+2x − 2
+if
+x ∈ [0, 2],
+−2x − 2
+if
+x ∈ [−2, 0).
+Thus, ϕv conjugates f0 to V . Hence
+fn
+0 (x) = ϕ−1
+v
+◦ V n ◦ ϕv(x),
+x ∈ [−2, 2], n ∈ N.
+Differentiating the above equality, and using the fact that (ϕ−1
+v )′(V ◦ ϕv) =
+(ϕ−1
+v )′(ϕv ◦ f0), we can get
+|(fn
+0 )′(x)| = 2n
+�
+4 − (fn
+0 (x))2
+4 − x2
+.
+(4.1)
+
+8
+LUDWIK JAKSZTAS
+The function F(z) = z + 1
+z is a semiconjugation between z �→ z2 and
+f0. Image of the unit circle under F is equal to the Julia set J0 = [−2, 2].
+Let us consider the family of circles S(0, r) where r > 1. Then, the images
+F(S(0, r)) =: ξr are ellipses, which can be parameterized as follows:
+ξr(t) : reit + 1
+re−it.
+For every r > 1, the focuses are ±2. Note that
+f0(ξr(t)) = ξr2(2t).
+If z = x + iy, then we get
+ξr(t) :
+�
+x(t) = (r + 1/r) cos t,
+y(t) = (r − 1/r) sin t.
+Thus the semi-minor and semi-major axes are equal to (r−1/r) and (r+1/r)
+respectively.
+Filled ellipses will be denoted by Er, i.e.
+Er :
+x2
+(r + 1/r)2 +
+y2
+(r − 1/r)2 ⩽ 1.
+Note that for r ⩾ 1, we have
+r + 1
+r ⩽ 2 + (r − 1)2,
+and
+r − 1
+r ⩽ 2(r − 1).
+(4.2)
+Now we give some estimates concerning precise location of the sets Jδ.
+Lemma 4.3. For every δ ̸= 0 we have
+Jδ ⊂ E1+√
+|δ| ⊂
+�
+z ∈ C : | Im z| ⩽ 2
+�
+|δ|
+�
+.
+Proof. Let r = 1 + κ
+�
+|δ|, where κ ⩾ 1. It is easy to prove that distance
+between ellipses ξr and ξr2 is equal to semi-major axis difference, namely
+�
+r2 + 1
+r2
+�
+−
+�
+r + 1
+r
+�
+=
+�
+r − 1
+�2�
+1 + 1
+r + 1
+r2
+�
+>
+�
+r − 1
+�2 = κ2|δ|.
+Because
+fδ(ξr(t)) = ξr2(2t) + δ,
+we see that Er is included in region bounded by fδ(ξr(t)), whereas exterior of
+Er is mapped onto exterior of fδ(ξr(t)). So, we conclude that if z /∈ E1+√
+|δ|,
+then fn
+δ (z) → ∞, hence
+Jδ ⊂ E1+√
+|δ|.
+Since Er is contained in {z ∈ C : | Im z| ⩽ r − 1/r}, and for r = 1 +
+�
+|δ|
+we have r − 1/r ⩽ 2
+�
+|δ| (cf. (4.2)), and the statement follows.
+□
+Lemma 4.4. There exists η > 0 such that if z ∈ Jδ, then
+(1)
+| Im z − Im pδ| < |δ|17/16
+provided
+Re z ⩾ Re pδ − |δ|15/16,
+| Im z + Im pδ| < |δ|17/16
+provided
+Re z ⩽ − Re pδ + |δ|15/16,
+(2)
+| Re z| < Re pδ + |δ|2,
+where 0 < |δ| < η.
+
+ON THE DIRECTIONAL DERIVATIVE OF THE HAUSDORFF DIMENSION
+9
+Proof. Proof will be carried out for ˆfδ and z ∈ ˆJδ (i.e. pδ is moved to 0).
+The curves
+γδ,β(t) = eiβet log λδ,
+where β ∈ R, are invariant under z �→ λδz, so the trajectories of ˆfδ lies on
+ˆΦ−1
+δ (γδ,β) (close to 0).
+The curve γδ,0(t) intersect circle S(0, |δ|h) for t = h log|λδ| |δ|.
+Let us
+denote the intersection point by bδ,h. Then, arg(bδ,h) = h log|λδ| |δ| · arg λδ.
+Since | arg λδ| < |δ|, we see that
+arg(bδ,h) = h log|λδ| |δ| · arg λδ → 0,
+uniformly, when δ → 0 and, 2 ⩾ h ⩾ 3/10. Moreover
+dist(bδ,h, R+) ⩽ |δ|h · h log|λδ| |δ| · arg λδ,
+thus for h ⩾ 3/10 we obtain
+dist(bδ,h, R+) < |δ|5/4,
+where 0 < |δ| < η, for suitably chosen η > 0.
+Next, if a line eiθR+ intersects γδ,0 at a point bδ,h′ where 2 ⩾ h′ ⩾ 3/10,
+we also have
+dist(bδ,h, eiθR+) < |δ|5/4,
+(4.3)
+for h ⩾ h′. Moreover this estimate will holds after rotation, so we can allow
+β ̸= 0.
+Let βδ and tδ be such that γδ,βδ(tδ) = −|δ|5/16 + 3i|δ|1/2 (then γδ,βδ(tδ)
+lies on a circle S(0, |δ|hδ), where hδ is close to 5/16 > 3/10).
+Because
+ˆΦ−1
+δ
+= z + aδz2 + O(z3), we see that
+Im(ˆΦ−1
+δ (γδ,βδ(tδ))) > 2|δ|1/2.
+The same estimate is also valid for parameters close to tδ, therefore we
+conclude from Lemma 4.3 that the Julia set lies below the curve ˆΦ−1
+δ (γδ,βδ).
+Let z = x + iy, then lδ : y = −3|δ|3/16x, is the line that intersects γδ,βδ(t)
+at tδ. We have lδ(−|δ|15/16) = 3|δ|9/8. Let t′
+δ be such that Re(γδ,βδ(t′
+δ)) =
+−|δ|15/16. Then, |γδ,βδ(t′
+δ) − 3|δ|9/8| is close to dist(γδ,βδ(t′
+δ), lδ), hence (4.3)
+gives us
+|γδ,βδ(t′
+δ) − 3|δ|9/8| < 2|δ|5/4.
+Since ˆΦ−1
+δ
+= z + aδz2 + O(z3), we see that
+Im(ˆΦ−1
+δ (γδ,βδ(t′
+δ))) < |δ|17/16.
+The same estimate are also valid for parameters less than t′
+δ. Analogously,
+considering a curve γδ,β−
+δ passing through the point −|δ|5/16 − 3i|δ|1/2, we
+can get lower estimate. So, the first statement follows from the symmetry.
+The second statement follows from the fact that ˆJδ lies between the curves
+ˆΦ−1
+δ (γδ,β±
+δ ), whereas arguments of the intersection points with S(0, |δ|2) are
+close to π.
+□
+
+10
+LUDWIK JAKSZTAS
+4.3. The uniform convergence. Now we are going to prove that ϕδ con-
+verges uniformly to ϕ0.
+Let Uδ, where δ ∈ U, be a simple connected neighborhood of Jδ containing
+0, disjoint from the postcritical set P(fδ), where
+P(fδ) :=
+�
+n⩾1
+fn
+δ (0).
+If δ = 0, then we assume that (−2, 2) ⊂ U0, and U0 does not contain {−2, 2}.
+Let f−n
+δ0,ν be an inverse branch of fn
+δ0 defined on Uδ0. Then we denote by
+f−n
+δ,ν related inverse branches of fn
+δ defined on Uδ, where δ ∈ U ∪ {0} (by
+related we mean that the function δ �→ f−n
+δ,ν (0) is continuous). Let Vn be the
+set of all possible choices of the inverse branches. For ν ∈ Vn, δ ∈ U we write
+Cδ,ν := f−n
+δ,ν (Jδ).
+If δ = 0, then we take C0,ν := f−n
+0,ν (−2, 2).
+Lemma 4.5. For every α ∈ (0, 2π) and n ⩾ 1 we have
+lim
+δ→0 sup
+ν∈Vn
+dH(Cδ,ν, C0,ν) = 0,
+where α = arg δ.
+Proof. Fix n ⩾ 1. If arg δ = π, then every two consecutive cylinders (that
+are included in R) are separated by an appropriate preimage of 0. Because
+f−n
+δ,ν (0) → f−n
+0,ν (0) and pδ → p0, the assertion follows. Of course cylinders
+are also separated by preimages of the imaginary axis Y , that are dinjoint
+from Jδ.
+Fix α ∈ (0, 2π) \ {π}. If we prove that f−n
+δ,ν (Y ) are disjoint from Jδ also
+for arg δ = α, the assertion will follow from Corollary 4.2.
+We have fδ(Y ) = −2 + δ + R−, so the imaginary part of every point from
+fδ(Y ) is equal to Im δ, whereas Im(−pδ) ≈ Im δ/3. Thus, we conclude from
+Lemma 4.4, that fδ(Y ) is disjoint from the Julia set, so also f−n
+δ,ν (Y ).
+□
+Since we have
+lim
+n→∞ sup
+ν∈Vn
+diam(C0,ν) = 0,
+Lemma 4.5 leads to:
+Proposition 4.6. If α ∈ (0, 2π) and δ → 0 where α = arg δ, then the
+convergence
+ϕδ ⇒ ϕ0,
+is uniform on the set Jδ0.
+5. Postcritical set
+Now we give a Proposition that helps us to control the postcritical set
+P(fδ). In particular we will see that P(fδ) is not to close to 0.
+Proposition 5.1. For every α ∈ (0, 2π) and θ > 0, there exist r > 0 and
+η > 0 such that
+P(fδ) ⊂
+�
+(C \ Er) ∪ B(−2, θ) ∪ B(2, θ)
+�
+,
+where 0 < |δ| < η and α = arg δ.
+
+ON THE DIRECTIONAL DERIVATIVE OF THE HAUSDORFF DIMENSION
+11
+Proof. Fix α ∈ (0, 2π) and θ > 0. Let δ1, where arg δ1 = α, be such that
+��δ1
+�� < 3
+8 min
+�
+rz, θ
+2
+�
+,
+(5.1)
+and
+3
+4 dist
+�
+− 8
+3t · δ1, R−�
+< dist
+�
+ˆΦ−1
+0
+�
+− 8
+3t · δ1
+�
+, R−�
+,
+(5.2)
+for t ∈ (0, 1].
+Write δt := t · δ1. We will consider sequences of parameters of the form
+δt/4n, for t ∈ (1/4, 1].
+Let vt,n := f2
+δt/4n(0) − pδt/4n. Then, using (3.2) we obtain
+fn+2
+δt/4n(0) = fn
+δt/4n(pδt/4n + vt,n)
+= fn
+δt/4n
+�
+pδt/4n +
+vt,nλn
+δt/4n
+λn
+δt/4n
+�
+= ˆΦ−1
+n,δt/4n(vt,nλn
+δt/4n) + pδt/4n,
+(5.3)
+provided ˆΦ−1
+n,δt/4n(vt,nλn
+δt/4n) is well defined, that is |vt,nλn
+δt/4n| < rz, for n
+large enough.
+Now we will compute the limit of vt,nλn
+δt/4n when n → ∞. First, note that
+pδt/4n + vt,n = f2
+δt/4n(0) = 2 − 3 δt
+4n + δ2
+t
+42n .
+Because
+pδt/4n = 2 − 1
+3
+δt
+4n + O
+� δ2
+t
+42n
+�
+,
+we get
+vt,n = −8
+3
+δt
+4n + O
+� δ2
+t
+42n
+�
+.
+Therefore
+vt,nλn
+δt/4n = −8
+3
+λn
+δt/4n
+4n
+δt + O
+�λn
+δt/4n
+42n δ2
+t
+�
+.
+(5.4)
+We have
+λδ = f′
+δ(pδ) = 4 + 3
+��
+1 − 4
+9δ − 1
+�
+,
+hence
+λn
+δt/4n
+4n
+=
+�
+1 + 3
+4
+��
+1 − 4
+9
+δt
+4n − 1
+��n
+.
+Since (
+�
+1 − 4
+9
+δt
+4n − 1) → 0, and
+��
+1 − 4
+9
+δt
+4n − 1
+�
+n → 0,
+we obtain
+λn
+δt/4n
+4n
+=
+�
+1 + 3
+4
+��
+1 − 4
+9
+δt
+4n − 1
+��n
+→ e0 = 1.
+Combining this with (5.4), we see that
+vt,nλn
+δt/4n → −8
+3δt.
+
+12
+LUDWIK JAKSZTAS
+So, for n large enough, the assumption (5.1) leads to |vt,nλn
+δt/4n| < rz.
+Therefore ˆΦ−1
+n,δt/4n(vt,nλn
+δt/4n) is well defined, and using (5.3) we obtain
+fn+2
+δt/4n(0) = ˆΦ−1
+n,δt/4n(vt,nλn
+δt/4n) + pδt/4n ⇒ ˆΦ−1
+0
+�
+− 8
+3δt
+�
++ 2,
+(5.5)
+where convergence is uniform with respect to t ∈ (1/4, 1]. Next, the assump-
+tion (5.2) gives us
+1
+2 dist
+�
+− 8
+3δt + 2, [−2, 2]
+�
+< dist
+�
+fn+2
+δt/4n(0), [−2, 2]
+�
+,
+for t ∈ (1/4, 1] and n > n0 for suitable n0 ∈ N. So, there exists r > 1 such
+that
+fn+2
+δt/4n(0) ∈ C \ Er,
+(5.6)
+for t ∈ (1/4, 1] and n > n0. Since f0(Er) = Er2 and distance between Er and
+Er2 is equal to (r2 + 1/r2) − (r + 1/r) (semi-major axis length difference),
+we conclude that
+fδ(C \ Er) ⊂ C \ Er,
+provided |δ| < (r2 + 1/r2) − (r + 1/r). Changing n0 if necessary, we can
+assume that δ1/4n0 < (r2 + 1/r2) − (r + 1/r) and then (5.6) gives us
+fn+2+k
+δt/4n (0) ∈ C \ Er,
+for t ∈ (1/4, 1], n > n0 and k ⩾ 0.
+Using (5.5), (3.3) and the assumption (5.1), we obtain fn+2
+δt/4n(0) ∈ B(2, θ).
+Since fδt/4n(0) ∈ B(−2, θ) and then fk+2
+δt/4n(0) ∈ B(2, θ), where 0 ⩽ k < n,
+n > n0 and t ∈ (1/4, 1], the statement holds for η = |δ1|/4n0.
+□
+6. Cylinders
+In this section we define cylinders C−2
+δ,n, C0
+δ,n, C+2
+δ,n, that give partitions of
+neighborhoods of the points −pδ, 0, pδ, respectively. However, first we will
+deal with inverse branches of fδ.
+Since δ0 ∈ R−, the Julia set Jδ0 is disjoint with the imaginary axis, so let
+us write
+f−1
+δ0,+ : Jδ0 → Jδ0,+ = Jδ0 ∩ {z ∈ C : Re z > 0},
+f−1
+δ0,− : Jδ0 → Jδ0,− = Jδ0 ∩ {z ∈ C : Re z < 0}.
+Using the holomorphic motion ϕδ, we can define f−1
+δ,± for parameters δ ∈ U,
+namely: f−1
+δ,± : Jδ → ϕδ(Jδ0,±).
+But, for δ close to 0 (i.e. 0 < |δ| < η(α)), the Julia sets are also disjoint
+from iR (see proof of Lemma 4.5), so we can define f−1
+δ,± analogously as for
+δ0:
+f−1
+δ,+ : Jδ → Jδ,+ = Jδ ∩ {z ∈ C : Re z > 0},
+f−1
+δ,− : Jδ → Jδ,− = Jδ ∩ {z ∈ C : Re z < 0}.
+For δ = 0 we take f−1
+0,+ : [−2, 2] → [0, 2], and f−1
+0,− : [−2, 2] → [−2, 0].
+Moreover
+f−n
+δ,+ := (f−1
+δ,+)n,
+f−n
+δ,− := (f−1
+δ,−)n.
+Let us now define partition of Jδ,+ onto cylinders C+2
+δ,n, n ⩾ 0, which can
+be used to describe neighborhoods of pδ ≈ 2.
+
+ON THE DIRECTIONAL DERIVATIVE OF THE HAUSDORFF DIMENSION
+13
+Figure 2. Partition Jδ onto cylinders, where δ = 1
+50 + 1
+50i.
+First, note that Jδ,+ = Jδ,+− ∪ Jδ,++, where Jδ,+− := f−1
+δ,+(Jδ,−), and
+Jδ,++ := f−1
+δ,+(Jδ,+). We take
+C+2
+δ,0 := Jδ,+−,
+and
+C+2
+δ,n := f−n
+δ,+(C+2
+δ,0 ).
+In particular we have Jδ,++ = �∞
+n=1 C+2
+δ,n ∪ {pδ} and
+Jδ,+ =
+∞
+�
+n=0
+C+2
+δ,n ∪ {pδ}.
+Cylinders C−2
+δ,n, n ⩾ 0, describing neighborhoods of −pδ ≈ −2, are placed
+symmetrically with respect to 0. So we obtain
+Jδ,− =
+∞
+�
+n=0
+C−2
+δ,n ∪ {−pδ}.
+In order to describe partition of Jδ close to 0, we take
+C0+
+δ,0 := Jδ,++,
+and
+C0+
+δ,n := f−1
+δ,+(C−2
+δ,n−1)
+for n ⩾ 1,
+thus Jδ,+ = �∞
+n=0 C0+
+δ,n∪{f−1
+δ,+(−pδ)}, whereas cylinders C0−
+δ,n, n ⩾ 0 are placed
+symmetrically with respect to 0. Finally
+C0
+δ,n := C0−
+δ,n ∪ C0+
+δ,n.
+Note that
+Jδ =
+∞
+�
+n=0
+C0
+δ,n ∪ f−1
+δ
+({−pδ}).
+We have the following important relations
+fδ(C0+
+δ,n) = fδ(C0−
+δ,n) = C−2
+δ,n−1,
+fδ(C−2
+δ,n) = C+2
+δ,n−1,
+for n ⩾ 1, hence f2
+δ (C0+
+δ,n) = f2
+δ (C0−
+δ,n) = C+2
+δ,n−2, for n ⩾ 2.
+We will also consider the sets:
+M0
+δ,N =
+�
+n⩾N
+C0
+δ,n.
+Next, we take C0
+n := ϕ−1
+δ (C0
+δ,n) and M0
+N := ϕ−1
+δ (M0
+δ,N) (subsets of Jδ0).
+Notation: Let C±2
+δ,n := C−2
+δ,n ∪ C+2
+δ,n, whereas C0±
+δ,n means "C0−
+δ,n or C0+
+δ,n".
+
+C
+Ce2
+Cei
+C1
+Cf2
+C
+Csi
+Cs2
+Cs
+Cg2
+C14
+LUDWIK JAKSZTAS
+Lemma 6.1. For every α ∈ (0, 2π) there exist K > 1 and η > 0 such that
+if z ∈ C0
+δ,n, n ⩾ 0, then
+(1) |z| > K−1|λδ|−n/2,
+(2) |z| < K|λδ|−n/2, provided |z| >
+�
+|δ|,
+(3) diam C0±
+δ,n < K|λδ|−n/2,
+where 0 < |δ| < η and α = arg δ.
+Proof. Fix α ∈ (0, 2π) and let α = arg δ.
+Let n0 ∈ N be the smallest
+number for which f2
+0 (C0
+0,n0) ⊂ B(2, rz). Then, there exists η > 0, such that
+f2
+δ (C0
+δ,n0) ⊂ B(2, rz), for 0 < |δ| < η.
+Step 1. If z ∈ C0
+δ,n, where n ⩾ n0, then
+|fδ(z) − fδ(0)| = |z|2.
+Because −pδ ≈ −2 + δ/3 (cf. (3.1)), whereas fδ(0) = −2 + δ, using Lemma
+4.4 we get
+K−1
+1 |fδ(z) − (−pδ)| < |fδ(z) − fδ(0)|,
+for some K1 > 0 (depending on α). If |z| >
+�
+|δ| then fδ(z) /∈ B(−2+δ, |δ|),
+so changing K1 if necessary, we obtain
+|fδ(z) − fδ(0)| < K1|fδ(z) − (−pδ)|.
+Therefore
+K−1
+1 |fδ(z) − (−pδ)| < |z|2 < K1|fδ(z) − (−pδ)|,
+(6.1)
+where right-hand side inequality holds under the assumption |z| >
+�
+|δ|.
+We have fn−n0+1
+δ
+(fδ(z)) ∈ f2
+δ (C0
+δ,n0) and of course fn−n0+1
+δ
+(−pδ) = pδ.
+Since fδ is conjugated to z �→ λδz on B(2, rz), we get
+|fn−n0+2
+δ
+(z) − pδ| ≍ |λδ|n−n0+1|fδ(z) − (−pδ)|.
+The fact that |fn−n0+2
+δ
+(z) − pδ| is separated from zero, leads to
+|λδ|n−n0+1|fδ(z) − (−pδ)| ≍ 1,
+hence
+|fδ(z) − (−pδ)| ≍ |λδ|−n.
+Thus, the first two statements for n ⩾ n0 follow from (6.1).
+Step 2. If z ∈ C0
+δ,n, n ⩾ n0 then the first statement gives us |f′
+δ(z)| >
+K−1
+2 |λδ|−n/2, so we conclude that
+diam fδ(C0
+δ,n) = diam fδ(C0±
+δ,n) > K−1
+3 |λδ|−n/2 diam C0±
+δ,n.
+Since fn−n0+1
+δ
+(fδ(C0
+δ,n)) = f2
+δ (C0
+δ,n0), we obtain
+|λδ|n−n0+1 diam fδ(C0
+δ,n) ≍ diam f2
+δ (C0
+δ,n0).
+Therefore
+diam C0±
+δ,n < K4|λδ|−n/2+n0−1 diam f2
+δ (C0
+δ,n0) < K5|λδ|−n/2,
+so all three statements hold for n ⩾ n0, and some constant K > 0.
+Next, possibly changing K, we can assume that the statements hold for
+n ⩾ 0.
+□
+
+ON THE DIRECTIONAL DERIVATIVE OF THE HAUSDORFF DIMENSION
+15
+7. The Julia set close to 0
+In order to compute the integral from the numerator of the formula (2.3),
+we need precise information on the location of the Julia set close to 0. Corol-
+lary 4.2 tells us nothing precise for δ ̸= 0, so we need to change scale in order
+to see the details. Because preimages of the point −pδ are close to ±
+√
+6
+3
+√
+−δ,
+we change scale by the factor 1/
+�
+|δ| in every direction. Thus, let us write
+J ⋇
+δ :=
+1
+�
+|δ|
+Jδ.
+Next, we define small neighborhoods of 0, and its rescaled version, as follows:
+Nδ,R := Jδ ∩ {z ∈ C : | Re z| ⩽ R
+�
+|δ|},
+N ⋇
+δ,R := J ⋇
+δ ∩ {z ∈ C : | Re z| ⩽ R},
+where R ⩾ 1. So, we have Nδ,R =
+�
+|δ|N ⋇
+δ,R. Write NR := ϕ−1
+δ (Nδ,R) ⊂ Jδ0.
+We denote by bδ one of the points being close to a preimage of −pδ, namely
+bδ :=
+√
+6
+3
+√
+−δ =
+√
+6
+3
+�
+|δ| sin α
+2 − i
+√
+6
+3
+�
+|δ| cos α
+2 .
+Indeed fδ(bδ) = −2 + 1
+3 δ ≈ −pδ (cf. (3.1)).
+We need to define the following sets:
+Hδ,R :
+� xy = − 1
+3 Im δ = − 1
+3|δ| sin α,
+√
+6
+3
+�
+|δ| sin α
+2 ⩽ |x| ⩽ R
+�
+|δ|,
+where x = Re z, y = Im z, δ ̸= 0, arg δ = α ∈ (0, 2π), and R ⩾ 1. Branches
+of Hδ,R included in left and right half-planes will be denoted by H−
+δ,R and
+H+
+δ,R respectively. The endpoints of H+
+δ are bδ, and the point which will be
+denoted by zδ,R (i.e. Re zδ,R = R
+�
+|δ|).
+After substitution
+�
+|δ|x,
+�
+|δ|y in place of x, y, for α ∈ (0, 2π) we obtain
+H⋇
+α,R :
+� xy = − 1
+3 sin α,
+√
+6
+3 sin α
+2 ⩽ |x| ⩽ R.
+Thus we have Hδ,R =
+�
+|δ|H⋇
+α,R, where α = arg δ. We define H⋇,−
+α,R and
+H⋇,+
+α,R analogously as H±
+δ,R. The endpoints of H⋇,+
+α,R are
+b⋇
+α :=
+1
+�
+|δ|
+bδ =
+√
+6
+3 sin α
+2 − i
+√
+6
+3 cos α
+2 ,
+z⋇
+α,R := R − i 1
+3R sin α.
+(7.1)
+Note that arg(z⋇
+α,R) = arg(zδ,R) = −γα,R, where
+γα,R = arctan
+� 1
+3R2 sin α
+�
+.
+(7.2)
+Hence, γα,R > 0 for α ∈ (0, π).
+Now we prove the following:
+Proposition 7.1. For every α ∈ (0, 2π) and R ⩾ 1, if δ → 0 where α =
+arg δ, then
+N ⋇
+δ,R → H⋇
+α,R,
+in the space of non-empty compact subsets of C equipped with the Hausdorff
+metric.
+
+16
+LUDWIK JAKSZTAS
+Figure 3. The Julia set Jδ for δ = 1
+25 +
+1
+100i, close to 0.
+Proof. Fix α ∈ (0, 2π) and R ⩾ 1.
+Step 1. First we will prove that
+lim
+δ→0 sup
+z∈N ⋇
+δ,R
+dist(z, H⋇
+α,R) = 0,
+(7.3)
+where α = arg δ.
+If z = x + iy, then we define
+l+
+δ : xy = −1
+3 Im δ + |δ|17/16,
+l−
+δ : xy = −1
+3 Im δ − |δ|17/16.
+The images of l±
+δ are horizontal lines such that
+Im(fδ(l±
+δ )
+�
+= 1
+3 Im δ ± 2|δ|17/16.
+The set | Re z| ⩽ R
+�
+|δ| is mapped into the half-plane Re z ⩽ −2+R2|δ|+
+Re δ. Since −pδ = −2 + δ/3 + O(δ2) (cf, (3.1)), we conclude from Lemma
+4.4 (1) that the set Nδ,R lies between the lines l±
+δ .
+Moreover Nδ,R is bounded by f−1
+δ
+(kδ) where kδ : Re z = − Re pδ −|δ|2 (see
+Lemma 4.4 (2)). Since fδ(bδ) = −2 + δ/3, (3.1) leads to | − pδ − fδ(bδ)| ≈
+|δ|2/27, whereas the critical value is equal to −2 + δ. Thus we see that the
+distance between f−1
+δ
+(kδ) and the points ±bδ is of order |δ|3/2.
+So, the set N ⋇
+δ,R lies between the curves
+xy = −1
+3 sin α ± |δ|1/16,
+and rescaled curves f−1
+δ
+(kδ) whose distance from ±b⋇
+α is of order |δ|. Thus
+(7.3) holds.
+Step 2. Now we will prove that
+lim
+δ→0 sup
+z∈H⋇
+α,R
+dist(z, N ⋇
+δ,R) = 0.
+(7.4)
+Suppose that, on the contrary, there is z0 ∈ H⋇
+α,R, r0 > 0, and a sequence
+δn → 0 such that B(z0, r)∩N ⋇
+δn,R = ∅. Note that the distances between ±b⋇
+α
+and rescaled preimages of f−1
+δ
+(−pδ) are of order |δ|, so the rescaled Julia set
+is located on ”both sides” of B(z0, r0). We can assume that 0 /∈ B(z0, r0).
+
+ON THE DIRECTIONAL DERIVATIVE OF THE HAUSDORFF DIMENSION
+17
+Next, we have B(
+�
+|δn|z0,
+�
+|δn|r0) ∩ Nδn,R = ∅, and then
+fδn
+�
+B(
+�
+|δn|z0,
+�
+|δn|r0)
+�
+∩ Jδn = ∅.
+Because f′
+δn(
+�
+|δn|z0) = κ
+�
+|δn|, for some κ ∈ C \ {0}, Koebe one-quarter
+Theorem leads to
+B
+�
+fδn(
+�
+|δn|z0), κr0|δn|/4
+�
+⊂ fδn
+�
+B(
+�
+|δn|z0,
+�
+|δn|r0)
+�
+.
+Since Im fδn(
+�
+|δn|z0) = Im δn/3, the point fδ(
+�
+|δn|z0) lies between the
+lines Im z = − Im pδn ± |δn|17/16. So, we conclude from Lemma 4.4 (1) that
+there exists 0 < r1 ⩽ κr0/4, such that
+Aα,n := {z ∈ C : | Re(z − fδn(
+�
+|δn|z0))| ⩽ r1|δn|} ∩ Jδn = ∅.
+The diameter of the set fδn(Nδn,R) is close to R2|δn|, whereas fδn(Nδn,R) can
+be mapped with bounded distortion onto a set of diameter grater than rz/5.
+So, the ”width” of the images of Aα,n is separated from 0. This contradicts
+Corollary 4.2. Therefore (7.4) holds, and the statement follows.
+□
+8. Conformal and Invariant measures
+Recall from Section 2, that ωδ (after normalization) is the conformal mea-
+sure with exponent d(δ) (cf. (2.2)), whereas µδ is the fδ-invariant measure,
+equivalent to ωδ. We assume that µδ(Jδ) = ωδ(Jδ) = 4.
+Let us define the rescaled measures as follows:
+ω⋇
+δ (A) :=
+1
+�
+|δ|
+d(δ) ωδ(
+�
+|δ|A),
+and
+µ⋇
+δ (A) :=
+1
+�
+|δ|
+d(δ) µδ(
+�
+|δ|A).
+Hence, ω⋇
+δ and µ⋇
+δ are supported on J ⋇
+δ .
+The Main result of this Section is Proposition 8.4, which concerns the
+limit of ω⋇
+δ (so also µ⋇
+δ ) when δ → 0. But first, we will deal with the limits
+of ωδ and µδ.
+For δ = 0 situation is as follows: The measure ω0 is simply the Lebesgue
+measure λ. Next, we know that the map ϕv(z) = ( 4
+π) arcsin( z
+2) conjugates
+f0 to V (see Section 4.2). Because the Lebesgue measure is V -invariant, the
+measure defined by ϕ∗
+vλ(A) = λ(ϕv(A)) is f0-invariant with density
+dµ0
+dω0
+(z) = 2
+π
+1
+�
+1 − (z/2)2 ,
+(8.1)
+(cf. [11, Chapter V]).
+First fact follows from [13, Section 5.1]:
+Proposition 8.1. For every α ∈ (0, 2π) the measure ω0 is equal to weak*
+limit of ωδ, where δ → 0 and α = arg δ.
+Now we prove:
+Proposition 8.2. For every α ∈ (0, 2π) the measure µ0 is equal to weak*
+limit of µδ, where δ → 0 and α = arg δ.
+Proof. Fix α ∈ (0, 2π).
+
+18
+LUDWIK JAKSZTAS
+Step 1. We see from [15, Theorems 4.1, 4.12]) that the density hδ :=
+dµδ/dωδ is equal to the limit Ln
+φδ(1), where φδ = −d(δ) log |f′
+δ(ϕδ)|, n → ∞.
+So, if z = ϕδ(s), then (2.1) leads to
+hδ(z) = lim
+n→∞ Ln
+φδ(1)(s) = lim
+n→∞
+�
+s∈T −n(s)
+|(fn
+δ )′(ϕδ(s))|−d(δ).
+(8.2)
+Let δk → 0, where arg δk = α, be a sequence such that µδk tends to a
+measure ˆµ0 in the weak* topology.
+If z0 ∈ (−2, 2), then we can find r such that for k large enough, and δ
+close to 0, B(z0, r) is separated from the postcritical set (cf. Proposition 5.1).
+Then, all inverse branches of fn
+δk, n ⩾ 1 have uniformly bounded distortion
+on B(z0, r). So, we see from (8.2) that there exists a constant K1 > 0 such
+that
+hδk
+��
+B(z0,r)∩Jδk < K1.
+(8.3)
+Passing to the limit (and possibly changing K1), we obtain
+K−1
+1
+< h0
+��
+(z0−r,z0+r) < K1,
+(8.4)
+or h0 = 0 on B(z0 − r, z0 + r). In the latter case we get h0 = 0 on (−2, 2).
+The limit measure ˆµ0 is f0-invariant. So, we see that if ˆµ0 has an atom,
+the only possibility is the fixed point p0 = 2. Therefore, it is enough to show
+that there are no atom at p0, because in that case (8.4) holds, and ˆµ0 is
+absolutely continuous with respect to ω0. Then, by uniqueness ˆµ0 = µ0, so
+µ0 in the only possible limit, and the statement follows.
+Step 2. So, now we prove that there are no atom at p0. If ϕδ(s) = z ∈ C+2
+δ,n,
+then using (8.2) we obtain
+hδ(z) =
+∞
+�
+k=0
+|(f−1
+δ,− ◦ f−k
+δ,+)′(z)|d(δ) · hδ((f−1
+δ,− ◦ f−k
+δ,+)(z)),
+(8.5)
+where (f−1
+δ,− ◦ f−k
+δ,+)(z) ∈ C−2
+δ,n+k+1. So, we must estimate values of hδ on the
+cylinders C−2
+δ,n+k+1. We have
+hδ(z) = |(f−1
+δ,−)′(z)|d(δ) · hδ(f−1
+δ,−(z)) + |(f−1
+δ,+)′(z)|d(δ) · hδ(f−1
+δ,+(z)).
+(8.6)
+If z ∈ C−2
+δ,n+k+1 then f−1
+δ,±(z) ∈ C0±
+δ,n+k+2.
+Let z0 = 0 and let r be such that (8.3) holds. Then, Lemma 6.1 combined
+with Proposition 7.1, implies that there exists n0 ∈ N such that C0±
+δ,n+k+2 ⊂
+B(0, r) where n ⩾ n0, k ∈ N.
+Again using Lemma 6.1 (1), we get
+|(f−1
+δ,±)′(z)|d(δ) = |f′
+δ(f−1
+δ,±(z))|−d(δ) < K2|λδ|d(δ)(n+k+2)/2,
+thus (8.3) and (8.6) leads to
+hδ(z) < K3|λδ|d(δ)(n+k+2)/2,
+where z ∈ C−2
+δ,n+k+1.
+
+ON THE DIRECTIONAL DERIVATIVE OF THE HAUSDORFF DIMENSION
+19
+If z ∈ C+2
+δ,n, then formula (8.5) combined with the above estimate, and the
+fact that |(f−1
+δ,− ◦ f−k
+δ,+)(z)|d(δ) ≍ |λδ|−d(δ)(k+1), gives us
+hδ(z) < K4
+∞
+�
+k=0
+|λδ|d(δ)(n−k)/2 < K5|λδ|d(δ)n/2,
+where n ⩾ n0.
+We have ωδ(C+2
+δ,n) ≍ (diam C+2
+δ,n)d(δ) ≍ |λδ|−d(δ)n. So, if N ⩾ n0 then
+µδ
+�
+∞
+�
+n=N
+C+2
+δ,n
+�
+=
+�
+�∞
+n=N C+2
+δ,n
+hδ dωδ < K6
+∞
+�
+n=N
+|λδ|d(δ)n/2|λδ|−d(δ)n
+< K6
+∞
+�
+n=N
+4−n/3 < 3K6
+�1
+4
+�N/3
+−−−−→
+N→∞ 0.
+Because the above estimate does not depend on δ, the statement follows.
+□
+Lemma 8.3. For every α ∈ (0, 2π) and ε > 0 there exist neighborhood U ∋ 0
+and η > 0 such that
+�
+1 − ε
+� 2
+π < dµδ
+dωδ
+���
+U∩Jδ
+<
+�
+1 + ε
+� 2
+π,
+where 0 < |δ| < η and α = arg δ or δ = 0.
+Proof. Fix α ∈ (0, 2π) and ε > 0. As before, we write hδ = dµδ/dωδ.
+We see from (8.1) that there exists an interval (−κ, κ) such that
+2
+π ⩽ h0(z) ⩽
+�
+1 + ε
+3
+� 2
+π,
+(8.7)
+where z ∈ (−κ, κ). So the statement holds for δ = 0.
+We can find (small) 0 < r < κ such that distortion of all inverse branches
+of fn
+δ , n ⩾ 1 on B(0, r) is as close to 1 as we need. So, taking into account
+formula (8.2), we can assume for every z1, z2 ∈ B(0, r) ∩ Jδ, we have
+��hδ(z1) − hδ(z2)
+�� < ε
+3
+2
+π.
+(8.8)
+Since h0 and hδ are close to constant functions on (−r, r) and B(0, r)∩Jδ
+respectively, we conclude from Propositions 8.1, 8.2 that there exist z3 ∈
+(−r, r) and z4 ∈ B(0, r) ∩ Jδ such that |h0(z3) − hδ(z4)| <
+ε
+3
+2
+π.
+So the
+statement follows from (8.7) and (8.8).
+□
+Let us denote by lα,R the arc length measure supported on H⋇
+α,R. Now we
+prove the main result of this Section:
+Proposition 8.4. For every α ∈ (0, 2π) and R ⩾ 1, we have
+ω⋇
+δ
+���
+N ⋇
+δ,R
+→ lα,R,
+in the weak∗ topology, where δ → 0 and α = arg δ.
+
+20
+LUDWIK JAKSZTAS
+Proof. Fix α ∈ (0, 2π), R ⩾ 1, and let α = arg δ. It is enough to consider
+the set N ⋇,+
+δ,R := N ⋇
+δ,R ∩ {z ∈ C : Re z > 0}. Write N +
+δ,R =
+�
+|δ|N ⋇,+
+δ,R .
+Step 1. Let U ′ be a small neighborhood of H⋇,+
+α,R separated from 0. Let
+U = U ′ ∩ {z ∈ C : 0 < Re z < R}. For any Borel set A ⊂ U, such that fn+1
+δ
+is injective on
+�
+|δ|A, we have
+ω⋇
+δ (A) =
+1
+�
+|δ|
+d(δ) ωδ(
+�
+|δ|A) =
+1
+�
+|δ|
+d(δ)
+�
+fn+1
+δ
+(√
+|δ|A)
+��(f−n−1
+δ,ν
+)′��d(δ)dωδ
+=
+�
+fn+1
+δ
+(√
+|δ|A)
+����
+(f−n−1
+δ,ν
+)′
+�
+|δ|
+����
+d(δ)
+dωδ,
+(8.9)
+where f−n−1
+δ,ν
+denotes the inverse branch that maps fn+1
+δ
+(
+�
+|δ|U) onto
+�
+|δ|U.
+Our aim is to show that if δ → 0, then we can find n(δ) such that for
+every Borel set A ⊂ U, satisfying lα,R(∂A) = 0, the above integral tends to
+lα,R(A).
+Step 2. Let rδ > 0 be the smallest number such that there exists n ∈ N
+for which
+fn+1
+δ
+(N +
+δ,R) ⊂ B(pδ, rδ),
+and fn+2
+δ
+(N +
+δ,R) /⊂ B(pδ, rz/2).
+(8.10)
+In particular we see that rδ ⩽ rz/2, and possible values of rδ are separated
+from 0. It follows from Proposition 7.1 that N +
+δ,R ⊂ U for δ small enough.
+Let (δk) be a sequence of parameters tending to 0 for which rδk → q when
+k → ∞. Let nk be such that (8.10) holds for δk.
+We have fnk+1
+δ
+(
+�
+|δk|U) ⊂ B(pδ, rz).
+If z ∈ fδ(
+�
+|δk|U), then −z ∈
+B(pδ, rz), so using (3.4) we obtain
+fnk
+δk (z) = fnk
+δk (−z) = Φ−1
+δk
+�
+λnk
+δk Φδk(−z)
+�
+.
+Therefore, if z ∈ U, then we have
+Fα,q,k(z) := fnk+1
+δk
+��
+|δk|z
+�
+= Φ−1
+δk
+�
+λnk
+δk Φδk(−|δk|z2 + 2 − δk)
+�
+.
+(8.11)
+Step 3. We will prove that the sequence Fα,q,k converges uniformly on U.
+Write
+gδ(z) := 1
+|δ|Φδ
+�
+− |δ|z2 + 2 − δ
+�
+= 1
+|δ|Φδ
+�
+pδ − |δ|z2 + 2 − δ − pδ
+�
+,
+so we have
+Fα,q,k(z) = Φ−1
+δk
+�
+|δk|λnk
+δk gδk(z)
+�
+.
+(8.12)
+Note that
+2 − δk − pδk = −2
+3δk + O(|δk|2).
+Therefore
+gδk(z) = Φδk
+�
+pδk + |δk|(−z2 − 2
+3v + O(|δk|))
+�
+|δk|
+,
+where v = eiα.
+Since Φ′
+δk(pδk) = 1, δk → 0 whereas U is bounded, we
+conclude that
+gδk(z) ⇒ −z2 − 2
+3v.
+(8.13)
+
+ON THE DIRECTIONAL DERIVATIVE OF THE HAUSDORFF DIMENSION
+21
+Proposition 7.1 and the fact that Φδ ⇒ Φ0, lead to
+diam(gδk(N ⋇,+
+δk,R)) → K1,
+for some constant K1 (which does not depend on q and is close to R2). On
+the other hand
+diam(Fα,q,k(N ⋇,+
+δk,R)) → q.
+So, (8.12) and the fact arg(λnk) = O(δk), lead to
+|δk|λnk
+δk → κ ∈ R.
+Thus, (8.12) combined with (8.13) and the above, gives us
+Fα,q,k(z) ⇒ Φ−1
+0
+�
+κ
+�
+− z2 − 2
+3v
+��
+:= Fα,q(z),
+where k → ∞.
+Step 4. Since (b⋇
+α)2 = − 2
+3v, we have Fα,q(b⋇
+α) = p0 = 2. Thus, we conclude
+from Proposition 7.1, and the choice of (δk), that
+Fα,q
+��
+H+
+α,R : H+
+α,R → [2 − q, 2] ⊂ J0.
+Moreover Fα,q
+��
+H+
+α,R is a bijection, and then
+�
+Fα,q(A)
+��(F −1
+α,q)′��dω0 = lα,R(A).
+(8.14)
+Hence, the assumption lα,R(∂A) = 0 leads to ω0(Fα,q(∂A)) = 0.
+We have (cf. (8.11))
+F −1
+α,q,k(z) =
+f−nk−1
+δk
+(z)
+�
+|δk|
+→ F −1
+α,q(z).
+Therefore (8.9) gives us
+ω⋇
+δk(A) =
+�
+Fα,q,k(A)
+��(F −1
+α,q,k)′��d(δk)dωδk.
+Since ω0(Fα,q(∂A)) = 0 and Fα,q,k converges uniformly to Fα,q, measure of
+symmetric difference ωδk(Fα,q,k(A)∆Fα,q(A)) tends to 0. So uniform conver-
+gence of |(F −1
+α,q,k)′|d(δk), Proposition 8.1 and (8.14) lead to
+�
+Fα,q,k(A)
+��(F −1
+α,q,k)′��d(δk)dωδk →
+�
+Fα,q(A)
+��(F −1
+α,q)′��dω0 = lα,R(A).
+Thus, the statement holds if r(δk) → q.
+But, lα,R is the only possible
+limit, because for every δk we can always find a subsequence δjk such that
+r(δjk) → q.
+□
+
+22
+LUDWIK JAKSZTAS
+9. Key integral
+9.1.
+The main problem in the proof of the Theorem 1.1, is to estimate
+integral from the numerator of the formula (2.3), namely:
+�
+Jδ0
+∂
+∂t log |f′
+tv(ϕtv)|d˜µtv.
+(9.1)
+Put ˙ϕδ := ∂
+∂δϕδ. We have
+∂
+∂t(f′
+tv(ϕtv)) = ∂
+∂t(2ϕtv) = 2v ˙ϕδ
+��
+δ=tv.
+Therefore, the integrand can be rewritten as follows:
+∂
+∂t log |f′
+tv(ϕtv)| = Re
+� ∂
+∂t(f′
+tv(ϕtv))
+f′
+tv(ϕtv)
+�
+= Re
+�
+v ˙ϕδ
+��
+δ=tv
+ϕtv
+�
+.
+(9.2)
+9.2.
+Now we derive the formula for ˙ϕδ. The function ϕδ conjugates fδ0 = T
+to fδ, so ϕδ(T(s)) = ϕ2
+δ(s) − 2 + δ. Differentiating both sides with respect
+to δ, we get
+˙ϕδ(T(s)) = 2ϕδ(s) ˙ϕδ(s) + 1,
+hence
+˙ϕδ(s) = −
+1
+2ϕδ(s) + ˙ϕδ(T(s))
+2ϕδ(s) .
+(9.3)
+Next, replacing s by T(s), T 2(s),..., T m−1(s), we obtain
+˙ϕδ(s) = −
+m−1
+�
+k=0
+1
+2ϕδ(s) · 2ϕδ(T(s)) · ... · 2ϕδ(T k(s))+
++
+˙ϕδ(T m(s))
+2ϕδ(s) · 2ϕδ(T(s)) · ... · 2ϕδ(T m−1(s)).
+Since 2ϕδ(s) = f′
+δ(ϕδ(s)), the chain rule leads to
+˙ϕδ(s) = −
+m
+�
+k=1
+1
+(fk
+δ )′(ϕδ(s)) +
+˙ϕδ(T m(s))
+(fm
+δ )′(ϕδ(s)).
+(9.4)
+9.3.
+Let us assume that z ∈ Jδ is close to 0, and z = ϕδ(s). Then the
+denominator of (9.2) is small, moreover we see from (9.3) that
+˙ϕδ(s) = − 1
+2z (1 − ˙ϕδ(T(s))).
+Since ϕδ(T(s)) is close to −2, it follows from the formula (9.4) that ˙ϕδ(T(s))
+should usually be close to 1/3. If this is so, we get (cf. (9.2))
+Re
+�
+v ˙ϕδ
+ϕδ
+�
+≈ 1
+3 Re
+�
+− v
+z2
+�
+.
+Thus, we can expect that the function d′
+v(tv), and the integral of Re(−v/z2)
+over a neighborhood of 0, with respect to µδ, have the same order.
+
+ON THE DIRECTIONAL DERIVATIVE OF THE HAUSDORFF DIMENSION
+23
+Indeed, integral of Re(−v/z2) has decisive influence on d′
+v(tv), and we will
+compute it over the sets Nδ,R in Proposition 9.1. But first, for α ∈ (0, π)
+and R ⩾ 1, let us define
+Iα,R :=
+√
+6 cos α
+��
+sin(2γα,R)
+sin α
+− 1
+�
++
+√
+6
+2
+√
+sin α
+π−2γα,R
+�
+α
+�
+sin β dβ,
+(9.5)
+where γα,R is given by the formula (7.2). For α = π we take
+Iπ,R =
+√
+6
+�
+1 −
+√
+6
+3
+1
+R
+�
+=
+√
+6 − 2
+R.
+(9.6)
+Note that
+�
+sin(2γα,R)
+sin α
+=
+√
+6
+3
+1
+R
+1
+�
+1 + ( sin α
+3R2 )2
+,
+therefore Iα,R → Iπ,R, when α → π−.
+Proposition 9.1. For every α ∈ (0, π] and R ⩾ 1 we have
+lim
+δ→0 |δ|1− 1
+2 d(δ)
+�
+Nδ,R
+Re
+�
+− v
+z2
+�
+dµδ(z) = 2
+πIα,R,
+where α = arg δ and v = eiα.
+Proof. Fix α ∈ (0, π) and R ⩾ 1. First, using Proposition 8.4, we obtain
+|δ|
+�
+|δ|
+d(δ)
+�
+Nδ,R
+Re
+�
+− v
+z2
+�
+dωδ(z) =
+1
+�
+|δ|
+d(δ)
+�
+Nδ,R
+Re
+�
+−
+v
+(z/
+�
+|δ|)2
+�
+dωδ(z)
+=
+�
+N ⋇
+δ,R
+Re
+�
+− v
+z2
+�
+dω⋇
+δ (z) →
+�
+H⋇
+α,R
+Re
+�
+− v
+z2
+�
+dlα,R(z).
+So, taking into account Lemma 8.3, we have to prove that the integral over
+H⋇
+α,R is equal to Iα,R. It is enough to consider the integral restricted to
+H⋇,+
+α,R , because value of the integral over H⋇,−
+α,R is the same.
+We have
+�
+H⋇,+
+α,R
+Re
+�
+− v
+z2
+�
+dlα,R(z) = −
+�
+H⋇,+
+α,R
+(x2 − y2) cos α + 2xy sin α
+(x2 + y2)2
+dlα,R(z),
+(9.7)
+where x = Re(z) and y = Im(z).
+If α ∈ (0, π), then H⋇,+
+α,R is the arc of hyperbola xy = − 1
+3 sin α contained
+in IV quadrant, which joins the points b⋇
+α and z⋇
+α,R (see (7.1)). So, in the
+polar coordinates H⋇,+
+α,R can be written as follows:
+H⋇,+
+α,R :
+�
+x(t) = hα(t) · cos t,
+y(t) = hα(t) · sin t.
+t ∈
+�
+arctan Im b⋇
+α
+Re b⋇α
+, arctan
+Im z⋇
+α,R
+Re z⋇
+α,R
+�
+,
+
+24
+LUDWIK JAKSZTAS
+where h2
+α(t) · 1
+2 sin(2t) = − 1
+3 sin α, and
+Im b⋇
+α
+Re b⋇α
+= − cot α
+2 = tan
+�α
+2 − π
+2
+�
+,
+Im z⋇
+α,R
+Re z⋇
+α,R
+= − 1
+3R2 sin α.
+Therefore, we obtain (cf. (7.2))
+H⋇,+
+α,R :
+�
+�
+�
+x(t) =
+�
+− 2
+3
+sin α
+sin(2t) · cos t,
+y(t) =
+�
+− 2
+3
+sin α
+sin(2t) · sin t,
+t ∈
+�1
+2(α − π), −γα,R
+�
+.
+Next, we can get
+dlα,R =
+�
+h2α(t) + (h′α(t))2 dt =
+�
+−2
+3
+sin α
+sin3(2t) dt.
+Since x(t)y(t) = − 1
+3 sin α, and x2(t) + y2(t) = h2
+α(t), moreover
+x2(t) − y2(t) = h2
+α(t)
+�
+cos2 t − sin2 t
+�
+= −2
+3 sin αcos(2t)
+sin(2t) ,
+the integrals from (9.7) are equal to
+−
+−γα,R
+�
+1
+2 (α−π)
+− 2
+3 sin α cos α cos(2t)
+sin(2t) − 2
+3 sin2 α
+4
+9 sin2 α
+1
+sin2(2t)
+�
+−2
+3
+sin α
+sin3(2t) dt.
+Simplifying we obtain
+√
+6
+2
+−γα,R
+�
+1
+2 (α−π)
+�cos α
+sin α
+cos(2t)
+sin(2t) + 1
+�
+sin2(2t)
+�
+− sin α
+sin3(2t) dt.
+Substitution β = 2t + π, and the above computations, lead to
+�
+H⋇,+
+α,R
+Re
+�
+− v
+z2
+�
+dlα,R(z) =
+√
+6
+4
+π−2γα,R
+�
+α
+�cos α
+sin α
+cos β
+sin β + 1
+�
+sin2 β
+�
+sin α
+sin3 β dβ.
+We divide the integrand into two parts. First we compute:
+√
+6
+4
+π−2γα,R
+�
+α
+cos α
+sin α
+cos β
+sin β sin2 β
+�
+sin α
+sin3 β dβ =
+√
+6
+4
+cos α
+√
+sin α
+π−2γα,R
+�
+α
+cos β
+√sin β dβ
+=
+√
+6
+4
+cos α
+√
+sin α
+�
+2
+�
+sin β
+����
+π−2γα,R
+α
+=
+√
+6
+2 cos α
+�
+sin 2γα,R
+sin α
+−
+√
+6
+2 cos α.
+Next we have
+√
+6
+4
+π−2γα,R
+�
+α
+sin2 β
+�
+sin α
+sin3 β dβ =
+√
+6
+4
+√
+sin α
+π−2γα,R
+�
+α
+�
+sin β dβ.
+Thus, the integral over H⋇,+
+α,R is equal 1
+2Iα,R (cf. (9.5)), so the statement
+holds for α ∈ (0, π).
+If α = π, then H⋇,+
+π,R :
+√
+6
+3 ⩽ x ⩽ R, and
+
+ON THE DIRECTIONAL DERIVATIVE OF THE HAUSDORFF DIMENSION
+25
+�
+H⋇,+
+π,R
+Re
+�
+− −1
+z2
+�
+dlπ,R(z) =
+R
+�
+√
+6
+3
+1
+x2 dx = −1
+x
+���
+R
+√
+6
+3
+=
+√
+6
+2 − 1
+R = 1
+2Iπ,R,
+(cf. (9.6)) thus the proof is finished.
+□
+Now we give two technical Corollaries.
+Corollary 9.2. For every α ∈ (0, π] and R ⩾ 1 there exists Kα,R ∈ R such
+that
+lim
+δ→0 |δ|1− 1
+2 d(δ)
+�
+Nδ,R
+1
+|z|2 dµδ(z) = Kα,R,
+where α = arg δ.
+Proof. Analogously as in the proof of Proposition 9.1, it is enough to compute
+integral
+�
+H⋇,+
+α,R |z|−2dlα,R(z). For α ∈ (0, π), proceeding as before, we obtain
+�
+H⋇,+
+α,R
+1
+|z|2 dlα,R(z) =
+−γα,R
+�
+1
+2 (α−π)
+−3
+2
+sin(2t)
+sin α
+�
+−2
+3
+sin α
+sin3(2t) dt
+=
+�
+3
+2
+1
+√
+sin α
+−γα,R
+�
+1
+2 (α−π)
+1
+�
+− sin(2t)
+dt := 1
+2Kα,R.
+If α = π, then
+R
+�
+√
+6
+3
+1
+x2 dx =
+√
+6
+2 − 1
+R := 1
+2Kπ,R,
+and the proof is finished.
+□
+Corollary 9.3. For every α ∈ (0, π] and R ⩾ 1 there exists Kα,R ∈ R such
+that
+lim
+δ→0 |δ|
+1
+2 (1−d(δ))
+�
+Nδ,R
+1
+|z| dµδ(z) = Kα,R,
+where α = arg δ.
+Proof. Using Proposition 8.4, analogously as in the proof of Proposition 9.1
+we get
+�
+|δ|
+�
+|δ|
+d(δ)
+�
+Nδ,R
+1
+|z| dωδ(z) =
+�
+N ⋇
+δ,R
+1
+|z| dω⋇
+δ (z) →
+�
+H⋇
+α,R
+1
+|z| dlα,R(z).
+
+26
+LUDWIK JAKSZTAS
+For α ∈ (0, π), proceeding as before, we obtain
+�
+H⋇,+
+α,R
+1
+|z| dlα,R(z) =
+−γα,R
+�
+1
+2 (α−π)
+�
+−3
+2
+sin(2t)
+sin α
+�
+−2
+3
+sin α
+sin3(2t) dt
+=
+−γα,R
+�
+1
+2 (α−π)
+1
+| sin(2t)| dt = −1
+2 log | tan t|
+����
+−γα,R
+1
+2 (α−π)
+= 1
+2 log 3
+2 + log
+R
+sin α
+2
+=: 1
+2Kα,R.
+If α = π, we get
+R
+�
+√
+6
+3
+1
+x dx = log R + 1
+2 log 3
+2 =: 1
+2Kπ,R,
+and the proof is finished.
+□
+10. Integral over Jδ0 \ NR
+The main result of this Section is Proposition 10.5, which shows us that
+the integral of (9.2) over Jδ0 \ NR is small (after dividing by |δ|1− 1
+2 d(δ)).
+This result, together with Proposition 11.1 (integral over NR) are the main
+ingredients in the proof of Theorem 1.1.
+But, we begin with a few facts about (fn
+δ )′, and Proposition 10.4, which
+will be used in the proofs of both mentioned Propositions.
+Because fδ are conjugated to z �→ λδz on the set B(2, rz) we conclude
+that:
+Lemma 10.1. For every α ∈ (0, 2π) there exists K > 1 such that if z ∈ C±2
+δ,n,
+n ⩾ 1 and 1 ⩽ j ⩽ n, then
+K−1|λδ|j < |(fj
+δ )′(z)| < K|λδ|j,
+where α = arg δ and 0 < |δ| < η.
+If U is a neighborhood of 0 and z ∈ Jδ \U, then f′
+δ(z) is close to f′
+0(Re z),
+where Re z ∈ J0 provided | Re z| ⩽ 2.
+The cylinders Cδ,ν ∋ z and C0,ν
+are close each other (cf. Lemma 4.5), so also finite trajectories fn
+δ (z) and
+fn
+0 (Re z). Thus the formula (4.1) leads to:
+Lemma 10.2. For every α ∈ (0, 2π), ε > 0, N ∈ N and open set U ∋ 0,
+there exists η > 0 such that if z ∈ Jδ, | Re(z)| ⩽ 2, {z, fδ(z), . . . , fn−1
+δ
+(z)} ∩
+U = ∅, n ⩽ N, then
+(1 − ε) 2n
+�
+4 − (Re fn
+0 (z))2
+4 − (Re z)2
+< |(fn
+δ )′(z)| < (1 + ε) 2n
+�
+4 − (Re fn
+0 (z))2
+4 − (Re z)2
+,
+where 0 < |δ| < η and α = arg δ.
+Lemma 10.2 for n = i, Lemma 6.1 (1), and Lemma 10.1 for n = j −1 lead
+to:
+
+ON THE DIRECTIONAL DERIVATIVE OF THE HAUSDORFF DIMENSION
+27
+Corollary 10.3. For every α ∈ (0, 2π), N ∈ N there exists K > 1 and η > 0
+such that if z ∈ f−i
+δ (C0
+δ,m), m ⩾ 0, then
+|(fi+j
+δ
+)′(z)| > K−12i|λδ|j−1− m
+2 ,
+where 0 ⩽ i ⩽ N, 0 ⩽ j ⩽ m, 0 < |δ| < η and α = arg δ.
+Proposition 10.4. For every α ∈ (0, π] and s > 0, we have
+lim
+δ→0 |δ|s
+�
+Jδ0
+�� ˙ϕδ
+�� d ˜µδ = 0,
+where α = arg δ.
+Proof. Fix α ∈ (0, π], s > 0 and N0 ⩾ 4 such that 2N0 ⩾ 8K, where K is a
+constant from Corollary 10.3.
+Step 1. First, we define the set Xδ,N0, on which it can be easily proven
+that |(fN0
+δ
+)′| > 10.
+If fN0
+δ
+(z) ∈ M0
+δ,1 (note that M0
+0,1 = [−
+√
+2,
+√
+2]), then Lemma 10.2 gives
+us
+|(fN0
+δ
+)′(z)| > 2N0 9
+10
+�
+2
+4 ⩾ 24 9
+10
+√
+2
+2
+> 10.
+The derivative will be also grater than 10 for z ∈ C±2
+δ,N0+n, n ⩾ 1 and z = ±pδ,
+therefore we define
+Xδ,N0 := f−N0
+δ
+(M0
+δ,1) ∪
+∞
+�
+n=1
+C±2
+δ,N0+n ∪ {−pδ, pδ}.
+Write XN0 := ϕ−1
+δ (Xδ,N0). Let s ∈ Xδ,N0, then (9.4) for m = N0, gives us:
+˙ϕδ(s) = −
+N0
+�
+k=1
+1
+(fk
+δ )′(ϕδ(s)) + ˙ϕδ(T N0(s))
+(fN0
+δ
+)′(z)
+.
+Since fN0
+δ
+(z) ∈ M0
+δ,1, the trajectory {z, fδ(z), . . . , fN0−1
+δ
+(z)} is disjoint from
+M0
+δ,N0, hence is separated from 0. Thus, the derivatives (fk
+δ )′(ϕδ(s)) are also
+separated from 0 (by a constant depending on N0). So, the fact that the
+measure ˜µδ is T-invariant, leads to
+�
+XN0
+| ˙ϕδ| d ˜µδ < K1(N0) + 1
+10
+�
+XN0
+| ˙ϕδ(T N0)| d ˜µδ
+< K1(N0) + 1
+10
+�
+T N0(XN0)
+| ˙ϕδ| d ˜µδ < K1(N0) + 1
+10
+�
+Jδ0
+| ˙ϕδ| d ˜µδ.
+(10.1)
+Step 2. Now, we will deal with the set
+Jδ \ Xδ,N0 =
+∞
+�
+n=1
+(f−N0
+δ
+(C±2
+δ,n) \ C±2
+δ,n+N0) =
+∞
+�
+n=1
+G−N0
+δ,n ,
+where G−N0
+δ,n
+:= f−N0
+δ
+(C±2
+δ,n) \ C±2
+δ,n+N0. Let G−N0
+δ,n
+:= ϕ−1
+δ (G−N0
+δ,n ).
+
+28
+LUDWIK JAKSZTAS
+If ϕδ(s) = z ∈ G−N0
+δ,n , then we will use the formula (9.4) for m = N0 + n:
+˙ϕδ(s) = −
+N0+n
+�
+j=1
+1
+(fj
+δ )′(ϕδ(s))
++ ˙ϕδ(T N0+n(s))
+(fN0+n
+δ
+)′(z)
+.
+(10.2)
+Before we pass to estimations, let us rewrite the set G−N0
+δ,n . We have
+f−N0
+δ
+(C−2
+δ,n) = f−(N0−1)
+δ
+(f−1
+δ
+(C−2
+δ,n)) = f−(N0−1)
+δ
+(C0
+δ,n+1),
+and next
+f−N0
+δ
+(C+2
+δ,n) = C+2
+δ,n+N0 ∪
+N0−1
+�
+k=0
+f−(N0−k−1)
+δ
+(f−1
+δ,−(C+2
+δ,n+k))
+= C+2
+δ,n+N0 ∪ C−2
+δ,n+N0 ∪
+N0−2
+�
+k=0
+f−(N0−k−2)
+δ
+(C0
+δ,n+k+2).
+Therefore
+G−N0
+δ,n
+=
+N0−2
+�
+k=−1
+f−(N0−k−2)
+δ
+(C0
+δ,n+k+2) =
+N0
+�
+k=1
+f−(N0−k)
+δ
+(C0
+δ,n+k).
+(10.3)
+Step 3. Now we will estimate integral of the "tail" of (10.2). If z ∈ G−N0
+δ,n ,
+then we see that there exists k ∈ {1, . . . , N0} such that fN0−k
+δ
+(z) ∈ Cδ,n+k.
+Thus, Corollary 10.3 for N = N0 − 1, i = N0 − k, m = n + k and j = n + k,
+leads to
+|(fN0+n
+δ
+)′(z)| > K−12N0−k|λδ|
+1
+2 (n+k)−1
+= K−12N0|λδ|
+n
+2 −1��
+|λδ|
+2
+�k
+> 9
+40K−12N0�19
+10
+�n
+.
+Therefore
+�
+G−N0
+n
+����
+˙ϕδ(T N0+n)
+(fN0+n
+δ
+)′(ϕδ)
+���� d ˜µδ < 40
+9 K2−N0�10
+19
+�n �
+G−N0
+n
+| ˙ϕδ(T N0+n)| d ˜µδ
+< 40
+9 K2−N0�10
+19
+�n �
+Jδ0
+| ˙ϕδ| d ˜µδ.
+By assumption, 2N0 > 8K, we get
+40
+9 K2−N0
+∞
+�
+n=1
+�10
+19
+�n
+= 4K2−N0�10
+9
+�2
+< 1
+2
+�10
+9
+�2
+< 5
+8.
+Thus we obtain
+∞
+�
+n=1
+�
+G−N0
+n
+����
+˙ϕδ(T N0+n)
+(fN0+n
+δ
+)′(ϕδ)
+���� d ˜µδ < 5
+8
+�
+Jδ0
+| ˙ϕδ| d ˜µδ.
+(10.4)
+
+ON THE DIRECTIONAL DERIVATIVE OF THE HAUSDORFF DIMENSION
+29
+Step 4. We have to estimate finite sum from (10.2). If ϕδ(s) = z ∈ G−N0
+δ,n
+and fN0−k
+δ
+(z) ∈ C0
+δ,n+k (cf. (10.3)), then we can write
+−
+N0+n
+�
+j=1
+1
+(fj
+δ )′(z)
+= −
+N0−k
+�
+j=1
+1
+(fj
+δ )′(z)
+−
+N0+n
+�
+j=N0−k+1
+1
+(fj
+δ )′(z)
+.
+(10.5)
+Now we will estimate first sum on the right. If k = N0, then we will
+assume that it is equal to 0.
+Since fN0−k
+δ
+(z) ∈ C0
+δ,n+k, the trajectory {z, fδ(z), . . . , fN0−k−1
+δ
+(z)} is dis-
+joint from M0
+δ,N0, hence is separated from 0. Therefore the derivatives are
+separated from 0, and the sum is bounded by a constant K2(N0)/4. Thus
+we have
+∞
+�
+n=1
+N0
+�
+k=1
+�
+f−N0+k
+δ
+(C0
+δ,n+k)
+N0−k
+�
+j=1
+����
+1
+(fj
+δ )′(z)
+���� dµδ(z) < K2(N0).
+(10.6)
+Step 5. The rightmost sum from (10.5) can be rewritten in the form
+−
+N0+n
+�
+j=N0−k+1
+1
+(fj
+δ )′(z)
+= −
+n+k
+�
+j=1
+1
+(fj
+δ )′(fN0−k
+δ
+(z)) · (fN0−k
+δ
+)′(z)
+= −
+1
+(fN0−k
+δ
+)′(z)
+·
+1
+f′
+δ(fN0−k
+δ
+(z))
+�
+1 +
+n+k−1
+�
+j=1
+1
+(fj
+δ )′(fN0−k+1
+δ
+(z))
+�
+.
+If fN0−k
+δ
+(z) ∈ C0
+δ,n+k then |(fN0−k
+δ
+)′(z)| > 1 (cf. Lemma 10.2). Moreover
+fN0−k+1
+δ
+(z) ∈ C−2
+δ,n+k−1, so using Lemma 10.1, we can find K3 > 1 such that
+����
+N0+n
+�
+j=N0−k+1
+1
+(fj
+δ )′(z)
+���� < 2K3
+1
+|f′
+δ(fN0−k
+δ
+(z))|
+= K3
+1
+|fN0−k
+δ
+(z)|
+.
+(10.7)
+Step 6. In order to estimate the above expression, we will consider two
+cases. First, we will deal with the sets G−N0
+δ,n
+such that if z ∈ G−N0
+δ,n , then
+the trajectory {z, . . . , fN0−1
+δ
+(z)} is disjoint from Nδ,1. It is enough to check
+if z ∈ C0
+δ,n+N0 then | Re(z)| >
+�
+|δ|. Since |z| < K|λδ|− 1
+2 (n+N0) (see Lemma
+6.1 (2)), we get
+�
+|δ| < K|λδ|− 1
+2 (n+N0), and then 1 ⩽ n < log|λδ|
+� K2
+|δ|
+�
+− N0.
+We have
+K3
+�
+f−N0+k
+δ
+(C0
+δ,n+k)
+1
+|fN0−k
+δ
+(z)|
+dµδ(z) = K3
+�
+C0
+δ,n+k
+1
+|z| dµδ(z).
+(10.8)
+
+30
+LUDWIK JAKSZTAS
+So, using (10.7), (10.8), the fact that µδ(C0
+δ,n) ≍ ωδ(C0
+δ,n) ≍ (diam(C0
+δ,n))d(δ)
+and Lemma 6.1 we obtain
+N0
+�
+k=1
+�
+f−N0+k
+δ
+(C0
+δ,n+k)
+����
+N0+n
+�
+j=N0−k+1
+1
+(fj
+δ )′(z)
+����dµδ(z) < K3
+N0
+�
+k=1
+�
+C0
+δ,n+k
+1
+|z|dµδ(z)
+< K4
+N0
+�
+k=1
+|λδ|
+1
+2 (n+k)|λδ|− 1
+2 d(δ)(n+k) < K5(N0)|λδ|
+1
+2 (1−d(δ))n.
+We can assume that 1 − d(δ) ⩽ |1 − d(δ)| < s/2, thus the integral over all
+sets G−N0
+δ,n , where 1 ⩽ n ⩽ log|λδ|
+� K2
+|δ|
+�
+− N0 can be estimated by
+K5(N0)
+log|λδ|
+�
+K2
+|δ|
+�
+−N0
+�
+n=1
+|λδ|
+1
+2 |1−d(δ)|n < K5(N0)
+�K2
+|δ|
+� s
+4 log|λδ|
+�K2
+|δ|
+�
+< |δ|− s
+4 |δ|− s
+4 = |δ|− s
+2 .
+(10.9)
+Step 7. Let nδ be the smallest number such that there exists z ∈ G−N0
+δ,nδ for
+which {z, . . . , fN0−1
+δ
+(z)} ∩ Nδ,1 ̸= ∅, hence C0
+δ,nδ+N0 ∩ Nδ,1 ̸= ∅. Thus, there
+exists R > 1 (R ≍ 2N0) such that M0
+δ,nδ ⊂ Nδ,R. So, using (10.3), we obtain
+∞
+�
+n=nδ
+G−N0
+δ,n
+⊂
+N0
+�
+k=1
+f−N0+k
+δ
+(M0
+δ,nδ) ⊂
+N0
+�
+k=1
+f−N0+k
+δ
+(Nδ,R).
+Using (10.7), (10.8) (analogously as in the Step 5), and next Corollary
+9.3, we obtain
+N0
+�
+k=1
+�
+f−N0+k
+δ
+(Nδ,R)
+����
+N0+n
+�
+j=N0−k+1
+1
+(fj
+δ )′(z)
+����dµδ(z)
+< K3
+N0
+�
+k=1
+�
+f−N0+k
+δ
+(Nδ,R)
+1
+|fN0−k
+δ
+(z)|
+dµδ(z) = K3 ·N0
+�
+Nδ,R
+1
+|z| dµδ(z) < |δ|− s
+2 .
+(10.10)
+Step 8. Estimates (10.1), (10.4), (10.6), (10.9) and (10.10) give us
+�
+Jδ0
+�� ˙ϕδ
+�� d ˜µδ < K1(N0) + K2(N0) +
+� 1
+10 + 5
+8
+� �
+Jδ0
+�� ˙ϕδ
+�� d ˜µδ + 2|δ|− s
+2 .
+Since
+1
+10 + 5
+8 < 3
+4, we obtain
+1
+4
+�
+Jδ0
+�� ˙ϕδ
+�� d ˜µδ < K1(N0) + K2(N0) + 2|δ|− s
+2 ,
+and the statement follows.
+□
+
+ON THE DIRECTIONAL DERIVATIVE OF THE HAUSDORFF DIMENSION
+31
+Proposition 10.5. For every α ∈ (0, π], ε > 0 there exist R ⩾ 1 and η > 0
+such that
+|δ|1− 1
+2 d(δ)
+�
+Jδ0\NR
+��� ˙ϕδ
+ϕδ
+��� d ˜µδ < ε,
+where 0 < |δ| < η and α = arg δ.
+Proof. We conclude from Proposition 10.4 that it is enough to estimate the
+integral over the set M0
+1 \ NR.
+Fix α ∈ (0, π] and ε > 0. If ϕδ(s) = z ∈ C0
+δ,n, then formula (9.4) for m = n
+leads to
+˙ϕδ(s)
+ϕδ(s) = −
+1
+z · f′
+δ(z)
+�
+1 +
+n−1
+�
+k=1
+1
+(fk
+δ )′(fδ(z))
+�
++
+˙ϕδ(T n(s))
+ϕδ(s) · (fn
+δ )′(ϕδ(s)).
+where α = arg δ.
+We have |z| > R
+�
+|δ| for some R large enough, on the other hand Lemma
+6.1 gives us |z| < K1|λδ|− 1
+2 n. So we will consider 1 ⩽ n ⩽ log|λδ|
+� K2
+1
+R2|δ|
+�
+.
+Since fδ(z) ∈ C−2
+δ,n−1, using Lemma 10.1 and again Lemma 6.1, we get
+|z · (fn
+δ )′(z)| = |z · f′
+δ(z) · (fn−1
+δ
+)′(fδ(z))| > K2|λδ|−n|λδ|n−1 > K2.
+So, the above inequality, and the fact that T n(C0
+n) = M0
+1 lead to
+log|λδ|
+� K2
+1
+R2|δ|
+�
+�
+n=1
+�
+C0n
+����
+˙ϕδ(T n)
+ϕδ · (fn
+δ )′(ϕδ)
+���� d ˜µδ < K3 log|λδ|
+� K2
+1
+R2|δ|
+� �
+M0
+1
+�� ˙ϕδ
+�� d ˜µδ.
+Because 1 − d(δ)/2 is close to 1/2, Proposition 10.4 gives us
+lim
+δ→0 |δ|1− 1
+2 d(δ)K3 log|λδ|
+� K2
+1
+R2|δ|
+� �
+M0
+1
+�� ˙ϕδ
+�� d ˜µδ = 0.
+Since fδ(z) ∈ C−2
+δ,n−1, using Lemma 10.1, next Lemma 6.1, and the fact
+that µδ(C0
+δ,n) ≍ ωδ(C0
+δ,n) ≍ (diam(C0
+δ,n))d(δ), we obtain
+log|λδ|
+� K2
+1
+R2|δ|
+�
+�
+n=1
+�
+C0
+δ,n
+����
+1
+z · f′
+δ(z)
+�
+1 +
+n−1
+�
+k=1
+1
+(fk
+δ )′(fδ(z))
+����� dµδ
+< K4
+log|λδ|
+� K2
+1
+R2|δ|
+�
+�
+n=1
+�
+C0
+δ,n
+1
+|z|2 dµδ < K5
+log|λδ|
+� K2
+1
+R2|δ|
+�
+�
+n=1
+|λδ|(1− 1
+2 d(δ))n.
+The rightmost sum can be estimated by the integral:
+K6
+log|λδ|
+� K2
+1
+R2|δ|
+�
+�
+1
+|λδ|(1− 1
+2 d(δ))xdx < K7
+� K2
+1
+R2|δ|
+�1− 1
+2 d(δ)
+< ε|δ|−1+ 1
+2 d(δ).
+
+32
+LUDWIK JAKSZTAS
+The last inequality holds for R large enough, since K1 and K7 does not
+depend on R. Thus the assertion follows.
+□
+11. Integral over NR and Proof of the main Theorem
+Now, using Proposition 9.1, we estimate integral of (9.2) over NR, which
+has decisive influence on (9.1). Next we will prove Theorem 1.1.
+Proposition 11.1. For every α ∈ (0, π] and R ⩾ 1 we have
+lim
+δ→0 |δ|1− 1
+2 d(δ)
+�
+NR
+Re
+�
+v ˙ϕδ
+ϕδ
+�
+d ˜µδ = 1
+3
+2
+πIα,R,
+where α = arg δ and v = eiα.
+Proof. Fix α ∈ (0, π], R ⩾ 1 and ε1 > 0. Let ε = ε1/Kα,R where Kα,R is the
+constant from Corollary 9.2.
+Step 1. There exists 0 < rε ⩽ rz such that
+����
+(Φ−1
+δ )′(z)
+(Φ−1
+δ )′(w) − 1
+���� < ε,
+(11.1)
+for z, w ∈ B(0, rε) and δ ∈ B(0, r△).
+If z ∈ Nδ,R, then −fδ(z) is close to pδ. Let nδ be the largest number
+such that λnδ
+δ Φδ(−fδ(z)) ⊂ B(0, rε) for all points z ∈ Nδ,R, where α = arg δ.
+Since diam(Nδ,R) → 0 if δ → 0, we conclude that nδ → ∞.
+Formula (9.4) for m = nδ + 1 leads to
+˙ϕδ(s) = −
+1
+f′
+δ(ϕδ(s))
+�
+1 +
+nδ
+�
+k=1
+1
+(fk
+δ )′(fδ(ϕδ(s)))
+�
++
+˙ϕδ(T nδ+1(s))
+(fnδ+1
+δ
+)′(ϕδ(s))
+. (11.2)
+Now we estimate finite sum from the above formula. If Φ(z) ∈ B(0, rε/λnδ
+δ )
+and 1 ⩽ k ⩽ nδ, then we have (cf. (3.4))
+fk
+δ (z) = Φ−1
+δ (λk
+δΦδ(z)),
+and
+(fk
+δ )′(z) = (Φ−1
+δ )′(λk
+δΦδ(z)) · λk
+δ · Φ′
+δ(z).
+Therefore, (11.1) gives us
+����
+λk
+δ
+(fk
+δ )′(z) − 1
+���� < ε
+and
+����
+1
+(fk
+δ )′(z) − 1
+λk
+δ
+���� <
+ε
+|λk
+δ|.
+(11.3)
+Since nδ → ∞ and λδ → 4, we get
+����
+nδ
+�
+k=1
+1
+λk
+δ
+− 1
+3
+���� < ε.
+Thus, (11.3) and the above estimate, lead to
+����
+nδ
+�
+k=1
+1
+(fk
+δ )′(z) − 1
+3
+���� < ε +
+nδ
+�
+k=1
+ε
+|λk
+δ| < 2ε.
+
+ON THE DIRECTIONAL DERIVATIVE OF THE HAUSDORFF DIMENSION
+33
+If z ∈ Nδ,R, then Φ(−fδ(z)) ∈ B(0, rε/λnδ
+δ ). Moreover (fk
+δ )′(−fδ(z)) =
+−(fk
+δ )′(fδ(z)), so we have
+����
+nδ
+�
+k=1
+1
+(fk
+δ )′(fδ(z)) + 1
+3
+���� < 2ε.
+(11.4)
+Step 2.
+Now we estimate integral of a "tail" from the formula (11.2).
+Distortion of fnδ+1
+δ
+on Nδ,R is bounded by a constant depending on α and
+R. Since diam(Nδ,R) < K1
+�
+|δ| and diam(fnδ+1
+δ
+(Nδ,R)) > K2ε, we get
+��(fnδ+1
+δ
+)′(z)
+�� > K3(α, R, ε)
+1
+�
+|δ|
+,
+for z ∈ Nδ,R. On the other hand |z| > K4
+�
+|δ| (cf. Proposition 7.1), so if
+ϕδ(s) = z ∈ Nδ,R, we obtain
+���� Re
+�
+v
+ϕδ(s)
+˙ϕδ(T nδ+1(s))
+(fnδ+1
+δ
+)′(ϕδ(s))
+����� < K5(α, R, ε)
+�� ˙ϕδ(T nδ+1(s))
+��.
+Therefore, Proposition 10.4 leads to
+|δ|1− 1
+2 d(δ)
+�
+NR
+���� Re
+�
+v
+ϕδ(s)
+˙ϕδ(T nδ+1(s))
+(fnδ+1
+δ
+)′(ϕδ(s))
+�����d ˜µδ → 0.
+(11.5)
+Step 3. We have f′
+δ(z) = 2z, so (11.4) leads to
+���� Re
+�v
+z · −1
+f′
+δ(z)
+�
+1 +
+nδ
+�
+k=1
+1
+(fk
+δ )′(fδ(z))
+��
+− Re
+�
+− v
+2z2
+2
+3
+����� <
+ε
+|z|2 .
+Next, we integrate both sides over NR (where z = ϕδ(s)) with respect to
+˜µδ, and multiply by |δ|1− 1
+2 d(δ).
+So, formula (11.2) combined with (11.5),
+Proposition 9.1 and Corollary 9.2, gives us
+���� lim
+δ→0 |δ|1− 1
+2 d(δ)
+�
+NR
+Re
+�
+v ˙ϕδ
+ϕδ
+�
+d ˜µδ − 1
+3
+2
+πIα,R
+���� < εKα,R = ε1.
+Because ε1 was arbitrary, the assertion follows.
+□
+Proof of the Theorem 1.1. Fix α ∈ (0, π]. We have (cf. (9.5), (9.6))
+lim
+R→∞ Iα,R = −
+√
+6 cos α +
+√
+6
+2
+√
+sin α
+π
+�
+α
+�
+sin β dβ := Iα.
+Let α = arg δ and let v = eiα. Then, Propositions 11.1 and 10.5 lead to
+lim
+δ→0 |δ|1− 1
+2 d(δ)
+�
+Jδ0
+Re
+�
+v ˙ϕδ
+ϕδ
+�
+d ˜µδ = 1
+3
+2
+πIα.
+(11.6)
+Because f0 is conjugated to V , the entropy hµ0(f0) is equal to log 2. Next,
+we conclude from [12, Theorem 11.4.1] that 1 = HD(µ0) = hµ0(f0)/χµ0(f0)
+(where χµ0(f0) is the Lyapunov exponent). Therefore χµ0(f0) = log 2, hence
+�
+J0
+log |f′
+0|dµ0 = 4χµ0(f0) = 4 log 2.
+
+34
+LUDWIK JAKSZTAS
+Since f′
+δ(ϕδ) = 2ϕδ, Propositions 4.6, 8.2, give us
+lim
+δ→0
+�
+Jδ0
+log |f′
+δ(ϕδ)|d˜µδ = 4 log 2.
+Thus, formula (2.3) combined with (9.2), (11.6) and the above limit, leads
+to
+lim
+δ→0 |δ|1− 1
+2 d(δ) · d′
+v(δ) = −
+1
+6π log 2Iα,
+(11.7)
+where α = arg δ and let v = eiα.
+In order to finish the proof, we have to replace 1 − d(δ)/2 by 1/2 in the
+exponent. Because Iα is bounded, the above gives us
+|d′
+v(tv)| < K1 · t
+1
+2 d(tv)−1.
+Thus we have
+��d(tv) − 1
+�� ⩽
+t
+�
+0
+��d′
+v(sv)
+�� ds < K1
+t
+�
+0
+s
+1
+2 d(sv)−1ds < K1
+t
+�
+0
+s−2/3ds
+= 3K1 · t1/3 < t1/4,
+and then
+1 ⩾ t|d(tv)−1| ⩾ tt1/4 = et1/4 log t → e0 = 1.
+So, we obtain
+lim
+t→0+
+t1− 1
+2 d(tv)
+t
+1
+2
+= lim
+t→0+ t
+1
+2 (1−d(tv)) = 1,
+and finally (11.7) and (1.1) lead to
+lim
+δ→0 |δ|
+1
+2 d′
+v(δ) = −
+1
+6π log 2Iα = Ω−2(α),
+where α = arg δ and let v = eiα.
+□
+References
+[1] L. Carleson, T.W. Gamelin, Complex Dynamics, Springer-Verlag, New York, 1993.
+[2] Y.-C.
+Chen,
+T.
+Kawahira,
+From
+Cantor
+to
+semi-hyperbolic
+parameter
+along
+external
+rays,
+Trans.
+Amer.
+Math.
+Soc.
+372
+(2019)
+7959–7992.
+https://doi.org/10.1090/tran/7839.
+[3] N. Dobbs, J. Graczyk, N. Mihalache, Hausdorff Dimension of Julia sets in the
+logistic family, arXiv:2007.07661.
+[4] A. Douady, Does a Julia set depend continuously on the polynomial?, in: R. De-
+vaney (ed.), Complex dynamical systems, Proc. Sympos. Appl. Math. 49, Amer.
+Math. Soc., 1994, pp. 91–135. https://doi.org/10.1090/psapm/049/1315535.
+[5] A. Dudko, I. Gorbovickis, W. Tucker, Lower bounds on the Hausdorff dimension
+of some Julia sets, arXiv:2204.07880.
+[6] A. Fan, Y. Jiang, J. Wu, Asymptotic Hausdorff dimensions of Cantor sets associ-
+ated with an asymptotically non-hyperbolic family, Ergod. Th. & Dynam. Sys. 25
+(2005) 1799-1808. https://doi.org/10.1017/S014338570500009X.
+[7] G. Havard, M. Zinsmeister, Thermodynamic formalism and variations of the Haus-
+dorff dimension of quadratic Julia sets, Commun. Math. Phys. 210 (2000) 225–247.
+https://doi.org/10.1007/s002200050778.
+[8] J. H. Hubbard, Teichmüller Theory and Applications to Geometry, Topology and
+Dynamics. vol. 1. Teichmüller Theory, Matrix Editions, Ithaca, NY. 2006.
+
+ON THE DIRECTIONAL DERIVATIVE OF THE HAUSDORFF DIMENSION
+35
+[9] L.
+Jaksztas,
+On
+the
+derivative
+of
+the
+Hausdorff
+dimension
+of
+the
+qua-
+dratic
+Julia
+sets,
+Trans.
+Amer.
+Math.
+Soc.
+363
+(2011)
+5251–5291.
+https://doi.org/10.1090/S0002-9947-2011-05208-6.
+[10] L.
+Jaksztas,
+On
+the
+directonal
+derivative
+of
+the
+Hausdorff
+dimension
+of
+quadratic polynomial Julia sets at 1/4, Nonlinearity 33 (2020) 5919–5960.
+https://doi.org/10.1088/1361-6544/ab9a1a.
+[11] W. de Melo, S. van Strien, One-Dimensional Dynamics, Springer-Verlag, Berlin,
+1993.
+[12] F. Przytycki, M. Urbanski, Fractals in the Plane, Ergodic Theory Methods, Cam-
+bridge University press, vol. 371, 2010.
+[13] J. Rivera-Letelier, On the continuity of Hausdorff dimension of Julia sets and simi-
+larity between the Mandelbrot set and Julia sets, Fund. Math. 170 (2001) 287–317.
+https://doi.org/10.4064/fm170-3-6.
+[14] D. Ruelle, Repellers for real analytic maps, Ergod. Th. & Dynam. Sys. 2 (1982)
+99–107. https://doi.org/10.1017/S0143385700009603.
+[15] M. Zinsmeister, Thermodynamic Formalism and Holomorphic Dynamical Systems,
+SMF/AMS Texts and Monographs, vol. 2, 1996.
+Faculty of Mathematics and Information Science, Warsaw University of
+Technology, ul. Koszykowa 75, 00-662 Warsaw, Poland, ORCID: 0000-0002-
+9283-8841
+Email address: jaksztas@impan.pl
+
diff --git a/XdE3T4oBgHgl3EQfbwqk/content/tmp_files/load_file.txt b/XdE3T4oBgHgl3EQfbwqk/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..b522ed10726d7ea1b6fc1ad831b44caf76cddcc0
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@@ -0,0 +1,1072 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf,len=1071
+page_content='ON THE DIRECTIONAL DERIVATIVE OF THE  HAUSDORFF DIMENSION OF QUADRATIC POLYNOMIAL  JULIA SETS AT -2  LUDWIK JAKSZTAS  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Let d(δ) denote the Hausdorff dimension of the Julia set of  the polynomial fδ(z) = z2 − 2 + δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  In this paper we will study the directional derivative of the function  d along directions landing at the parameter 0, which corresponds to −2  in the case of family pc(z) = z2 + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' We will consider all directions,  except the one δ ∈ R+, which is inside the Mandelbrot set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  We will prove asymptotic formula for the directional derivative of d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Moreover, we will see that the derivative is negative for all directions in  the closed left half-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Computer calculations show that it is negative  except a cone (with opening angle approximately 74◦) around R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Introduction  Let f be a polynomial in one complex variable of degree at least 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' The  filled-in Julia set Kf we define as the set of all points that do not escape to  infinity under iteration of f, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Kf = {z ∈ C : fn(z) ↛ ∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  It is a compact set whose boundary is called the Julia set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' So, let us write  Jf := ∂Kf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Instead of the classical quadratic family pc(z) = z2 + c, we will consider for  technical reasons  fδ(z) = z2 − 2 + δ,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' δ = c + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' We will use the following abbreviations:  Jδ := Jfδ,  Kδ := Kfδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Let d(δ) := HD(Jδ) denote the Hausdorff dimension of the Julia set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' In this  paper, we will deal with the function  δ �→ d(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  We define the Mandelbrot set (for the family fδ) as follows  M := {δ ∈ C : fn  δ (0) ↛ ∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Recall that a polynomial f : C → C is called hyperbolic (expanding) if  there exists n ⩾ 1 such that |(fn)′(z)| > 1 for every z ∈ Jf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Primary 37F44, Secondary 37F35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Hausdorff dimension, Julia sets, quadratic family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  The  author  was  partially  supported  by  National  Science  Centre  Grant  2019/33/B/ST1/00275 (Poland).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='04519v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='DS]  11 Jan 2023  2  LUDWIK JAKSZTAS  The function d(δ) is real-analytic on each hyperbolic component of Int M  (consisting of parameters related to hyperbolic maps) as well as on the ex-  terior of M (see [14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' For α ∈ [0, 2π), let us write  R(α) := {z ∈ C∗ : α = arg z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  We will study properties of the function d(δ) when the parameter δ /∈ M  tends to 0 ∈ ∂M along the ray R(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' So, we will consider all directions  except R(0) which is inside M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Parameter δ = 0 corresponds to c = −2, so this is Misiurewicz’s pa-  rameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Note that a parameter is called Misiurewicz’s if the critical point  is strictly preperiodic (that is preperiodic but not periodic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' In our casse  0 �→ −2 �→ 2, where 2 is a fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Moreover the Julia set J0 is equal to  the interval [−2, 2] (see for example [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Rivera-Letelier proved in [14] the following:  Theorem R-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' There exists C0 > 0 such that if δn → 0 and  Re(δn) ⩽ 0  or  | Im(δn)| > C0| Re(δn)|3/2,  then δn /∈ M and d(δn) → d(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  An easy consequence of the above Theorem is that d(δ) converges to 1  along any ray R(α), α ∈ (0, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  In [6], A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Fan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Jiang and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Wu, gave estimate of d restricted to the  real line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' They proved that:  Theorem FJW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' There exists constants K > 0 and δ1 < 0 such that  1 − K−1�  |δ| ⩽ d(δ) ⩽ 1 − K  �  |δ|,  for all δ1 ⩽ δ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  So, one can expect that derivative d′(δ) tends to ∞ like 1/  �  |δ|, when  δ → 0−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' In this paper we prove much more general result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' In order to state  our main theorem we need two definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Let F be a real function defined on a domain U ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' If z ∈ U and v ∈ C∗,  then  F ′  v(z) := lim  h→0  F(z + hv) − F(z)  h  ,  if the limit exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  For α ∈ (0, π] we write  Ω−2(α) :=  1  √  6 π log 2  �  cos α − 1  2  √  sin α  π  �  α  √  sin x dx  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1)  If α ∈ (π, 2π) then we define Ω−2(α) := Ω−2(2π − α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  The main Theorem in this paper is:  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' For every α ∈ (0, 2π) we have  lim  δ→0  �  |δ| · d′  v(δ) = Ω−2(α),  where α = arg δ and v = eiα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  ON THE DIRECTIONAL DERIVATIVE OF THE HAUSDORFF DIMENSION  3  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Graph of the function Ω−2(α) for α ∈ (0, π].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Because of symmetry, it is enough to prove the Theorem for α ∈ (0, π].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  The function Ω−2(α) is obviously negative for α ∈ [π/2, π], whereas is pos-  itive for small α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Moreover it is easy to check that  ∂  ∂αΩ−2(α) < 0 for  α ∈ (0, π/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Therefore there exists a unique α0 ∈ (0, π/2) such that  Ω−2(α0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Thus we have d′  v(δ) → +∞ for every α ∈ (0, α0), and  d′  v(δ) → −∞ for every α ∈ (α0, π].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  A numerically made picture of the graph of Ω(α) (see Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=') suggests  that it is a decreasing function on whole interval (0, π), whereas α0 is slightly  greater than π/5 (close to 37◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  In the real case (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' α = π), we see that  lim  δ→0  �  |δ| · d′  −1(δ) =  −1  √  6 π log 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Note that this is the directional derivative, whereas the derivative of d(δ)  has opposite sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' So, integrating we obtain:  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' If δ ∈ R−, then for |δ| small we have  d(δ) = 1 −  √  6  3 π log 2  �  |δ| + o(  �  |δ|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Estimates along the ray R(0) (inside the Mandelbrot set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Recently N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Dobbs, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Graczyk and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Mihalache proved in [3] the follow-  ing significant result:  Theorem DGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' There exists constants κ∗ > 0 and δ0 > 0 such that for  every δ ∈ (0, δ0]  d(δ) ⩾ 1 + κ∗  √  δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  In that paper also an upper bound is proved, which holds on a "large" set  of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  The fact that we can pass with Ω−2(α) to the limit as α → 0+, that is  lim  α→0+ Ω−2(α) =  1  √  6 π log 2,  leads to the following:  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content="10 E  s0'0  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='10 E  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='15 E4  LUDWIK JAKSZTAS  Conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  The supremum over all constants κ∗ > 0 for which there  exists δ0 > 0 such that the statement of Theorem DGM holds, is equal to  √  6/(3 π log 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' In particular  d(δ) ⩾ 1 +  √  6  3 π log 2  √  δ − o(  √  δ),  for small δ ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Very recently A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Dudko, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Gorbovickis and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Tucker computed in [5]  rigorous lower bounds of d(δ) for δ ∈ [0, 4] (in our parametrization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' They  estimated d(δ) on some intervals and also on some additional parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Now, using lower bounds of d(δ) for parameters (which are greater than  estimates for intervals containing them) we make the following observation:  For δn ≈ n · 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='004, where 1 ⩽ n ⩽ 25, precise values of estimates (which we  received from the Authors) lead to  d(δn) > 1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='362  �  δn,  whereas  √  6/(3 π log 2) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='375.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  This is the first paper concerning the derivative of the dimension function  close to a non-parabolic parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' On the other hand this is the second  paper about the directional derivative after [10], where the derivative close  to the parameter c = 1/4 was studied (parabolic parameter with one petal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  As in [10] we will us the formula for the derivative (quotient of two inte-  grals, see Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' The integral from the denominator is equal to the  Lyapunov exponent (if the measure is normalized) and tends to a positive  constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' The main problem is to estimate integral from the numerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Note that our situation is much different than in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' In our case a crucial  role is played by a part of the Julia set that is close to 0, and we will have  to rescale the set Jδ in order to study a location of Jδ and behaviour of  conformal and invariant measures in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Notation: if z ∈ C \\ R, then √z denotes square root with positive real  part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' If z ∈ R− then we assume that √z has positive imaginary part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Thermodynamic formalism  The main goal of this section is to establish a formula for directional  derivative of the Hausdorff dimension (see Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' [10, Section  2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Because of hyperbolicity, the Julia set moves holomorphically over every  simply connected subset of C \\ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' In particular, we can take the set U,  which is connected component of B(0, 1/5) \\ M containing δ0 = −1/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Thus, there exists family of injections ϕδ : Jδ0 → ˆC, δ ∈ U such that  ϕδ0 = idJδ0, ϕδ(Jδ0) = Jδ and  ϕδ ◦ fδ0(s) = fδ ◦ ϕδ(s),  for s ∈ Jδ0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' ϕδ conjugates fδ0 to fδ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Moreover, the map δ �→ ϕδ(s) is  holomorphic, whereas for every δ ∈ U the map ϕδ extends to a quasiconfor-  mal (so also Hölder) map of the sphere to itself (see [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  ON THE DIRECTIONAL DERIVATIVE OF THE HAUSDORFF DIMENSION  5  Note that for δ = 0, there exists function ϕ0 that semiconjugates fδ0 to  f0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' A point z has two preimages under ϕ0 iff fn  0 (z) = 0 for some n ⩾ 0 (see  [2, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Now we use the thermodynamic formalism, which holds for hyperbolic  rational maps (see [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Let X = Jδ0, T = fδ0, and let φ : X → R be a  potential function of the form φ = −τ log |f′  δ(ϕδ)|, for δ ∈ U and τ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  The topological pressure can be defined as follows:  P(T, φ) := lim  n→∞  1  n log  �  s∈T −n(s)  eSn(φ(s)),  where Sn(φ) = �n−1  k=0 φ ◦ T k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  The limit exists and does not depend on  s ∈ Jδ0 (see [12, Corollary 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' If φ = −τ log |f′  δ(ϕδ)| and ϕδ(s) = z,  then eSn(φ(s)) = |(fn  δ )′(z)|−τ, hence  P(T, −τ log |f′  δ(ϕδ)|) = lim  n→∞  1  n log  �  z∈f−n  δ  (z)  |(fn  δ )′(z)|−τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  The function τ �→ P(T, −τ log |f′  δ(ϕδ)|) is strictly decreasing from +∞ to  −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' So, there exists a unique τ0 such that P(T, −τ0 log |f′  δ(ϕδ)|) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' By  Bowen’s Theorem (see [12, Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='7] or [15, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='12]) we obtain  τ0 = d(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Thus, we have P(T, −d(δ) log |f′  δ(ϕδ)|) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Write φδ := −d(δ) log |f′  δ(ϕδ)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  The Ruelle operator or the transfer operator Lφ : C0(X) → C0(X), is  defined as  Lφ(u)(s) :=  �  s∈T −1(s)  u(s)eφ(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1)  The Perron-Frobenius-Ruelle theorem [15, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1] asserts that β =  eP(T,φ) is a single eigenvalue of Lφ associated to an eigenfunction ˜hφ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Moreover there exists a unique probability measure ˜ωφ such that L∗  φ(˜ωφ) =  β˜ωφ, where L∗  φ is dual to Lφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  For φ = φδ we have β = 1, and then ˜µφδ := ˜hφδ ˜ωφδ is a T-invariant  measure called an equilibrium state (if it is normalized).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' We denote by ˜ωδ  and ˜µδ the measures ˜ωφδ and ˜µφδ respectively (measures supported on Jδ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Next, we take µδ := (ϕδ)∗ ˜µδ, and ωδ := (ϕδ)∗ ˜ωδ (measures supported on the  Julia set Jδ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  So, the measure µδ is fδ-invariant, whereas ωδ (after normalization) is  called fδ-conformal measure with exponent d(δ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' ωδ is a Borel probability  measure such that for every Borel subset A ⊂ Jδ,  ωδ(fδ(A)) =  �  A  |f′  δ|d(δ)dωδ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='2)  provided fδ is injective on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  However we will not assume that µδ and ωδ are probability measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Because J0 = [−2, 2] we will take µδ(Jδ) = ωδ(Jδ) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  6  LUDWIK JAKSZTAS  It follows from [15, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='11] or [12, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='5] that for every  Hölder continuous functions ψ, ˆψ : X → R, at every t ∈ R, we have  ∂  ∂tP(T, ψ + t ˆψ) =  �  X  ˆψ d˜µψ+t ˆψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Let us consider parameters of the form δ = tv, where v = eiα, α = arg δ,  and t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Since τ = d(tv) is the unique zero of the pressure function, for  the potential φ = −τ log |f′  tv(ϕtv)|, the implicit function theorem combined  with the above formula leads to (see [7, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1] or [9, Proposition  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1]):  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' If α ∈ (0, 2π) and v = eiα, then for every t > 0 such that  tv /∈ M we have  d′  v(tv) = −d(tv)  �  Jδ0  ∂  ∂t log |f′  tv(ϕtv)|d˜µtv  �  Jδ0 log |f′  tv(ϕtv)|d˜µtv  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='3)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Conjugation close to the fixed point  Suppose p is a repelling (or attracting) fixed point of a holomorphic func-  tion f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' It is well known fact that f is conjugated to z �→ f′(p)z in a neigh-  borhood of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Now we recall how to construct the conjugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  The polynomials fδ, δ ∈ U have two repelling fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' We will con-  sider  pδ := 1  2 + 3  2  �  1 − 4  9δ = 2 − 1  3 δ − 1  27 δ2 + O(δ3),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1)  which becomes p0 = 2 for δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' So, the multiplier λδ at pδ is equal to  λδ = f′  δ(pδ) = 2pδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  In order to construct the conjugation Φδ, we first "move" pδ to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Put  ˆfδ(z) := fδ(z + pδ) − pδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Indeed, 0 is the fixed point of ˆfδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' We will write ˆJδ := J ˆfδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  The sequence ˆΦn,δ that converges to the function ˆΦδ, which conjugates ˆfδ  to z �→ λδz (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' ˆΦδ ◦ ˆfδ = λδ ˆΦδ), we define as follows:  ˆΦn,δ(z) := λn  δ ˆf−n  δ  (z) = λn  δ (f−n  δ  (pδ + z) − pδ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Note that  ˆΦ−1  n,δ(z) = ˆfn  δ  � z  λn  δ  �  = fn  δ  �  pδ + z  λn  δ  �  − pδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='2)  We have  ˆΦn,δ(z) ⇒ ˆΦδ(z),  and the convergence is uniform independently of the parameter δ ∈ B(0, r△),  where z ∈ B(0, rz), for some r△ > 0, rz > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Thus, ˆΦδ depends analytically  on δ ∈ B(0, r△) (see for example [1, Chapter 2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  ON THE DIRECTIONAL DERIVATIVE OF THE HAUSDORFF DIMENSION  7  Possibly changing rz > 0, we can assume that both functions ˆΦδ and ˆΦ−1  δ  are defined on B(0, rz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Next, because ˆΦ′  δ(0) = 1, we can also assume that  these functions are not too far from the identity map, namely  ���  ˆΦδ(z)  z  − 1  ��� < 1  2,  and  ���  ˆΦ−1  δ (z)  z  − 1  ��� < 1  2,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='3)  for z ∈ B(0, rz) and δ ∈ B(0, r△).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Finally, we take Φδ(z) := ˆΦδ(z − pδ), and then we have  Φδ ◦ fδ = λδΦδ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='4)  where the functions Φδ are defined on B(pδ, rz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Possibly changing rz and  r△, we can also assume that Φδ are defined on B(2, rz) for δ ∈ B(0, r△).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' The Julia sets Jδ for δ close to 0  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' The Hausdorff metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' If X and Y are two non-empty compact sub-  sets of C, then their Hausdorff distance is defined as follows:  dH(X, Y ) = max  �  sup  x∈X  dist(x, Y ), sup  y∈Y  dist(y, X)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Now we state an important Theorem from [4, Section 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' If Kˆδ has empty interior, that is Kˆδ = Jˆδ, then the function  δ �→ Jδ,  is continuous at ˆδ, as the function from C into the space of non-empty com-  pact subsets of C equipped with the Hausdorff metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' If δ → 0, then  Jδ → J0 = [−2, 2],  in the space of non-empty compact subsets of C equipped with the Hausdorff  metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Conjugations and location of Jδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' It is known fact that the function  x �→ 2  π arcsin x conjugates polynomial g(x) = −2x2 + 1 defined on [−1, 1],  to the tent map T : [−1, 1] → [−1, 1] defined by T(x) = −2|x| + 1 (see for  example [11, Chapter II]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Since x �→ −x/2 conjugates f0 to g, let us write  ϕv(x) = 4  π arcsin  �x  2  �  ,  ϕ−1  v (x) = 2 sin  �π  4 x  �  ,  and next  V (x) =  �  2x − 2  if  x ∈ [0, 2],  −2x − 2  if  x ∈ [−2, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Thus, ϕv conjugates f0 to V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Hence  fn  0 (x) = ϕ−1  v  V n ◦ ϕv(x),  x ∈ [−2, 2], n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Differentiating the above equality, and using the fact that (ϕ−1  v )′(V ◦ ϕv) =  (ϕ−1  v )′(ϕv ◦ f0), we can get  |(fn  0 )′(x)| = 2n  �  4 − (fn  0 (x))2  4 − x2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1)  8  LUDWIK JAKSZTAS  The function F(z) = z + 1  z is a semiconjugation between z �→ z2 and  f0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Image of the unit circle under F is equal to the Julia set J0 = [−2, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Let us consider the family of circles S(0, r) where r > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Then, the images  F(S(0, r)) =: ξr are ellipses, which can be parameterized as follows:  ξr(t) : reit + 1  re−it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  For every r > 1, the focuses are ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Note that  f0(ξr(t)) = ξr2(2t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  If z = x + iy, then we get  ξr(t) :  �  x(t) = (r + 1/r) cos t,  y(t) = (r − 1/r) sin t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Thus the semi-minor and semi-major axes are equal to (r−1/r) and (r+1/r)  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Filled ellipses will be denoted by Er, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Er :  x2  (r + 1/r)2 +  y2  (r − 1/r)2 ⩽ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Note that for r ⩾ 1, we have  r + 1  r ⩽ 2 + (r − 1)2,  and  r − 1  r ⩽ 2(r − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='2)  Now we give some estimates concerning precise location of the sets Jδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' For every δ ̸= 0 we have  Jδ ⊂ E1+√  |δ| ⊂  �  z ∈ C : | Im z| ⩽ 2  �  |δ|  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Let r = 1 + κ  �  |δ|, where κ ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' It is easy to prove that distance  between ellipses ξr and ξr2 is equal to semi-major axis difference, namely  �  r2 + 1  r2  �  −  �  r + 1  r  �  =  �  r − 1  �2�  1 + 1  r + 1  r2  �  >  �  r − 1  �2 = κ2|δ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Because  fδ(ξr(t)) = ξr2(2t) + δ,  we see that Er is included in region bounded by fδ(ξr(t)), whereas exterior of  Er is mapped onto exterior of fδ(ξr(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' So, we conclude that if z /∈ E1+√  |δ|,  then fn  δ (z) → ∞, hence  Jδ ⊂ E1+√  |δ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Since Er is contained in {z ∈ C : | Im z| ⩽ r − 1/r}, and for r = 1 +  �  |δ|  we have r − 1/r ⩽ 2  �  |δ| (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='2)), and the statement follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  □  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' There exists η > 0 such that if z ∈ Jδ, then  (1)  | Im z − Im pδ| < |δ|17/16  provided  Re z ⩾ Re pδ − |δ|15/16,  | Im z + Im pδ| < |δ|17/16  provided  Re z ⩽ − Re pδ + |δ|15/16,  (2)  | Re z| < Re pδ + |δ|2,  where 0 < |δ| < η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  ON THE DIRECTIONAL DERIVATIVE OF THE HAUSDORFF DIMENSION  9  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Proof will be carried out for ˆfδ and z ∈ ˆJδ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' pδ is moved to 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  The curves  γδ,β(t) = eiβet log λδ,  where β ∈ R, are invariant under z �→ λδz, so the trajectories of ˆfδ lies on  ˆΦ−1  δ (γδ,β) (close to 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  The curve γδ,0(t) intersect circle S(0, |δ|h) for t = h log|λδ| |δ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Let us  denote the intersection point by bδ,h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Then, arg(bδ,h) = h log|λδ| |δ| · arg λδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Since | arg λδ| < |δ|, we see that  arg(bδ,h) = h log|λδ| |δ| · arg λδ → 0,  uniformly, when δ → 0 and, 2 ⩾ h ⩾ 3/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Moreover  dist(bδ,h, R+) ⩽ |δ|h · h log|λδ| |δ| · arg λδ,  thus for h ⩾ 3/10 we obtain  dist(bδ,h, R+) < |δ|5/4,  where 0 < |δ| < η, for suitably chosen η > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Next, if a line eiθR+ intersects γδ,0 at a point bδ,h′ where 2 ⩾ h′ ⩾ 3/10,  we also have  dist(bδ,h, eiθR+) < |δ|5/4,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='3)  for h ⩾ h′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Moreover this estimate will holds after rotation, so we can allow  β ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Let βδ and tδ be such that γδ,βδ(tδ) = −|δ|5/16 + 3i|δ|1/2 (then γδ,βδ(tδ)  lies on a circle S(0, |δ|hδ), where hδ is close to 5/16 > 3/10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Because  ˆΦ−1  δ  = z + aδz2 + O(z3), we see that  Im(ˆΦ−1  δ (γδ,βδ(tδ))) > 2|δ|1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  The same estimate is also valid for parameters close to tδ, therefore we  conclude from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='3 that the Julia set lies below the curve ˆΦ−1  δ (γδ,βδ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Let z = x + iy, then lδ : y = −3|δ|3/16x, is the line that intersects γδ,βδ(t)  at tδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' We have lδ(−|δ|15/16) = 3|δ|9/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Let t′  δ be such that Re(γδ,βδ(t′  δ)) =  −|δ|15/16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Then, |γδ,βδ(t′  δ) − 3|δ|9/8| is close to dist(γδ,βδ(t′  δ), lδ), hence (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='3)  gives us  |γδ,βδ(t′  δ) − 3|δ|9/8| < 2|δ|5/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Since ˆΦ−1  δ  = z + aδz2 + O(z3), we see that  Im(ˆΦ−1  δ (γδ,βδ(t′  δ))) < |δ|17/16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  The same estimate are also valid for parameters less than t′  δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Analogously,  considering a curve γδ,β−  δ passing through the point −|δ|5/16 − 3i|δ|1/2, we  can get lower estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' So, the first statement follows from the symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  The second statement follows from the fact that ˆJδ lies between the curves  ˆΦ−1  δ (γδ,β±  δ ), whereas arguments of the intersection points with S(0, |δ|2) are  close to π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  □  10  LUDWIK JAKSZTAS  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' The uniform convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Now we are going to prove that ϕδ con-  verges uniformly to ϕ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Let Uδ, where δ ∈ U, be a simple connected neighborhood of Jδ containing  0, disjoint from the postcritical set P(fδ), where  P(fδ) :=  �  n⩾1  fn  δ (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  If δ = 0, then we assume that (−2, 2) ⊂ U0, and U0 does not contain {−2, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Let f−n  δ0,ν be an inverse branch of fn  δ0 defined on Uδ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Then we denote by  f−n  δ,ν related inverse branches of fn  δ defined on Uδ, where δ ∈ U ∪ {0} (by  related we mean that the function δ �→ f−n  δ,ν (0) is continuous).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Let Vn be the  set of all possible choices of the inverse branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' For ν ∈ Vn, δ ∈ U we write  Cδ,ν := f−n  δ,ν (Jδ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  If δ = 0, then we take C0,ν := f−n  0,ν (−2, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' For every α ∈ (0, 2π) and n ⩾ 1 we have  lim  δ→0 sup  ν∈Vn  dH(Cδ,ν, C0,ν) = 0,  where α = arg δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Fix n ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' If arg δ = π, then every two consecutive cylinders (that  are included in R) are separated by an appropriate preimage of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Because  f−n  δ,ν (0) → f−n  0,ν (0) and pδ → p0, the assertion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Of course cylinders  are also separated by preimages of the imaginary axis Y , that are dinjoint  from Jδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Fix α ∈ (0, 2π) \\ {π}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' If we prove that f−n  δ,ν (Y ) are disjoint from Jδ also  for arg δ = α, the assertion will follow from Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  We have fδ(Y ) = −2 + δ + R−, so the imaginary part of every point from  fδ(Y ) is equal to Im δ, whereas Im(−pδ) ≈ Im δ/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Thus, we conclude from  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='4, that fδ(Y ) is disjoint from the Julia set, so also f−n  δ,ν (Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  □  Since we have  lim  n→∞ sup  ν∈Vn  diam(C0,ν) = 0,  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='5 leads to:  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' If α ∈ (0, 2π) and δ → 0 where α = arg δ, then the  convergence  ϕδ ⇒ ϕ0,  is uniform on the set Jδ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Postcritical set  Now we give a Proposition that helps us to control the postcritical set  P(fδ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' In particular we will see that P(fδ) is not to close to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' For every α ∈ (0, 2π) and θ > 0, there exist r > 0 and  η > 0 such that  P(fδ) ⊂  �  (C \\ Er) ∪ B(−2, θ) ∪ B(2, θ)  �  ,  where 0 < |δ| < η and α = arg δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  ON THE DIRECTIONAL DERIVATIVE OF THE HAUSDORFF DIMENSION  11  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Fix α ∈ (0, 2π) and θ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Let δ1, where arg δ1 = α, be such that  ��δ1  �� < 3  8 min  �  rz, θ  2  �  ,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1)  and  3  4 dist  �  − 8  3t · δ1, R−�  < dist  �  ˆΦ−1  0  �  − 8  3t · δ1  �  , R−�  ,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='2)  for t ∈ (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Write δt := t · δ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' We will consider sequences of parameters of the form  δt/4n, for t ∈ (1/4, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Let vt,n := f2  δt/4n(0) − pδt/4n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Then, using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='2) we obtain  fn+2  δt/4n(0) = fn  δt/4n(pδt/4n + vt,n)  = fn  δt/4n  �  pδt/4n +  vt,nλn  δt/4n  λn  δt/4n  �  = ˆΦ−1  n,δt/4n(vt,nλn  δt/4n) + pδt/4n,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='3)  provided ˆΦ−1  n,δt/4n(vt,nλn  δt/4n) is well defined, that is |vt,nλn  δt/4n| < rz, for n  large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Now we will compute the limit of vt,nλn  δt/4n when n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' First, note that  pδt/4n + vt,n = f2  δt/4n(0) = 2 − 3 δt  4n + δ2  t  42n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Because  pδt/4n = 2 − 1  3  δt  4n + O  � δ2  t  42n  �  ,  we get  vt,n = −8  3  δt  4n + O  � δ2  t  42n  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Therefore  vt,nλn  δt/4n = −8  3  λn  δt/4n  4n  δt + O  �λn  δt/4n  42n δ2  t  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='4)  We have  λδ = f′  δ(pδ) = 4 + 3  ��  1 − 4  9δ − 1  �  ,  hence  λn  δt/4n  4n  =  �  1 + 3  4  ��  1 − 4  9  δt  4n − 1  ��n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Since (  �  1 − 4  9  δt  4n − 1) → 0, and  ��  1 − 4  9  δt  4n − 1  �  n → 0,  we obtain  λn  δt/4n  4n  =  �  1 + 3  4  ��  1 − 4  9  δt  4n − 1  ��n  → e0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Combining this with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='4), we see that  vt,nλn  δt/4n → −8  3δt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  12  LUDWIK JAKSZTAS  So, for n large enough, the assumption (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1) leads to |vt,nλn  δt/4n| < rz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Therefore ˆΦ−1  n,δt/4n(vt,nλn  δt/4n) is well defined, and using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='3) we obtain  fn+2  δt/4n(0) = ˆΦ−1  n,δt/4n(vt,nλn  δt/4n) + pδt/4n ⇒ ˆΦ−1  0  �  − 8  3δt  �  + 2,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='5)  where convergence is uniform with respect to t ∈ (1/4, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Next, the assump-  tion (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='2) gives us  1  2 dist  �  − 8  3δt + 2, [−2, 2]  �  < dist  �  fn+2  δt/4n(0), [−2, 2]  �  ,  for t ∈ (1/4, 1] and n > n0 for suitable n0 ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' So, there exists r > 1 such  that  fn+2  δt/4n(0) ∈ C \\ Er,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='6)  for t ∈ (1/4, 1] and n > n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Since f0(Er) = Er2 and distance between Er and  Er2 is equal to (r2 + 1/r2) − (r + 1/r) (semi-major axis length difference),  we conclude that  fδ(C \\ Er) ⊂ C \\ Er,  provided |δ| < (r2 + 1/r2) − (r + 1/r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Changing n0 if necessary, we can  assume that δ1/4n0 < (r2 + 1/r2) − (r + 1/r) and then (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='6) gives us  fn+2+k  δt/4n (0) ∈ C \\ Er,  for t ∈ (1/4, 1], n > n0 and k ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='5), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='3) and the assumption (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1), we obtain fn+2  δt/4n(0) ∈ B(2, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Since fδt/4n(0) ∈ B(−2, θ) and then fk+2  δt/4n(0) ∈ B(2, θ), where 0 ⩽ k < n,  n > n0 and t ∈ (1/4, 1], the statement holds for η = |δ1|/4n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  □  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Cylinders  In this section we define cylinders C−2  δ,n, C0  δ,n, C+2  δ,n, that give partitions of  neighborhoods of the points −pδ, 0, pδ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' However, first we will  deal with inverse branches of fδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Since δ0 ∈ R−, the Julia set Jδ0 is disjoint with the imaginary axis, so let  us write  f−1  δ0,+ : Jδ0 → Jδ0,+ = Jδ0 ∩ {z ∈ C : Re z > 0},  f−1  δ0,− : Jδ0 → Jδ0,− = Jδ0 ∩ {z ∈ C : Re z < 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Using the holomorphic motion ϕδ, we can define f−1  δ,± for parameters δ ∈ U,  namely: f−1  δ,± : Jδ → ϕδ(Jδ0,±).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  But, for δ close to 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' 0 < |δ| < η(α)), the Julia sets are also disjoint  from iR (see proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='5), so we can define f−1  δ,± analogously as for  δ0:  f−1  δ,+ : Jδ → Jδ,+ = Jδ ∩ {z ∈ C : Re z > 0},  f−1  δ,− : Jδ → Jδ,− = Jδ ∩ {z ∈ C : Re z < 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  For δ = 0 we take f−1  0,+ : [−2, 2] → [0, 2], and f−1  0,− : [−2, 2] → [−2, 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Moreover  f−n  δ,+ := (f−1  δ,+)n,  f−n  δ,− := (f−1  δ,−)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Let us now define partition of Jδ,+ onto cylinders C+2  δ,n, n ⩾ 0, which can  be used to describe neighborhoods of pδ ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  ON THE DIRECTIONAL DERIVATIVE OF THE HAUSDORFF DIMENSION  13  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Partition Jδ onto cylinders, where δ = 1  50 + 1  50i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  First, note that Jδ,+ = Jδ,+− ∪ Jδ,++, where Jδ,+− := f−1  δ,+(Jδ,−), and  Jδ,++ := f−1  δ,+(Jδ,+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' We take  C+2  δ,0 := Jδ,+−,  and  C+2  δ,n := f−n  δ,+(C+2  δ,0 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  In particular we have Jδ,++ = �∞  n=1 C+2  δ,n ∪ {pδ} and  Jδ,+ =  ∞  �  n=0  C+2  δ,n ∪ {pδ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Cylinders C−2  δ,n, n ⩾ 0, describing neighborhoods of −pδ ≈ −2, are placed  symmetrically with respect to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' So we obtain  Jδ,− =  ∞  �  n=0  C−2  δ,n ∪ {−pδ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  In order to describe partition of Jδ close to 0, we take  C0+  δ,0 := Jδ,++,  and  C0+  δ,n := f−1  δ,+(C−2  δ,n−1)  for n ⩾ 1,  thus Jδ,+ = �∞  n=0 C0+  δ,n∪{f−1  δ,+(−pδ)}, whereas cylinders C0−  δ,n, n ⩾ 0 are placed  symmetrically with respect to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Finally  C0  δ,n := C0−  δ,n ∪ C0+  δ,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Note that  Jδ =  ∞  �  n=0  C0  δ,n ∪ f−1  δ  ({−pδ}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  We have the following important relations  fδ(C0+  δ,n) = fδ(C0−  δ,n) = C−2  δ,n−1,  fδ(C−2  δ,n) = C+2  δ,n−1,  for n ⩾ 1, hence f2  δ (C0+  δ,n) = f2  δ (C0−  δ,n) = C+2  δ,n−2, for n ⩾ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  We will also consider the sets:  M0  δ,N =  �  n⩾N  C0  δ,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Next, we take C0  n := ϕ−1  δ (C0  δ,n) and M0  N := ϕ−1  δ (M0  δ,N) (subsets of Jδ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Notation: Let C±2  δ,n := C−2  δ,n ∪ C+2  δ,n, whereas C0±  δ,n means "C0−  δ,n or C0+  δ,n".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  C  Ce2  Cei  C1  Cf2  C  Csi  Cs2  Cs  Cg2  C14  LUDWIK JAKSZTAS  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' For every α ∈ (0, 2π) there exist K > 1 and η > 0 such that  if z ∈ C0  δ,n, n ⩾ 0, then  (1) |z| > K−1|λδ|−n/2,  (2) |z| < K|λδ|−n/2, provided |z| >  �  |δ|,  (3) diam C0±  δ,n < K|λδ|−n/2,  where 0 < |δ| < η and α = arg δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Fix α ∈ (0, 2π) and let α = arg δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Let n0 ∈ N be the smallest  number for which f2  0 (C0  0,n0) ⊂ B(2, rz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Then, there exists η > 0, such that  f2  δ (C0  δ,n0) ⊂ B(2, rz), for 0 < |δ| < η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' If z ∈ C0  δ,n, where n ⩾ n0, then  |fδ(z) − fδ(0)| = |z|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Because −pδ ≈ −2 + δ/3 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1)), whereas fδ(0) = −2 + δ, using Lemma  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='4 we get  K−1  1 |fδ(z) − (−pδ)| < |fδ(z) − fδ(0)|,  for some K1 > 0 (depending on α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' If |z| >  �  |δ| then fδ(z) /∈ B(−2+δ, |δ|),  so changing K1 if necessary, we obtain  |fδ(z) − fδ(0)| < K1|fδ(z) − (−pδ)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Therefore  K−1  1 |fδ(z) − (−pδ)| < |z|2 < K1|fδ(z) − (−pδ)|,  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1)  where right-hand side inequality holds under the assumption |z| >  �  |δ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  We have fn−n0+1  δ  (fδ(z)) ∈ f2  δ (C0  δ,n0) and of course fn−n0+1  δ  (−pδ) = pδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Since fδ is conjugated to z �→ λδz on B(2, rz), we get  |fn−n0+2  δ  (z) − pδ| ≍ |λδ|n−n0+1|fδ(z) − (−pδ)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  The fact that |fn−n0+2  δ  (z) − pδ| is separated from zero, leads to  |λδ|n−n0+1|fδ(z) − (−pδ)| ≍ 1,  hence  |fδ(z) − (−pδ)| ≍ |λδ|−n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Thus, the first two statements for n ⩾ n0 follow from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' If z ∈ C0  δ,n, n ⩾ n0 then the first statement gives us |f′  δ(z)| >  K−1  2 |λδ|−n/2, so we conclude that  diam fδ(C0  δ,n) = diam fδ(C0±  δ,n) > K−1  3 |λδ|−n/2 diam C0±  δ,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Since fn−n0+1  δ  (fδ(C0  δ,n)) = f2  δ (C0  δ,n0), we obtain  |λδ|n−n0+1 diam fδ(C0  δ,n) ≍ diam f2  δ (C0  δ,n0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Therefore  diam C0±  δ,n < K4|λδ|−n/2+n0−1 diam f2  δ (C0  δ,n0) < K5|λδ|−n/2,  so all three statements hold for n ⩾ n0, and some constant K > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Next, possibly changing K, we can assume that the statements hold for  n ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  □  ON THE DIRECTIONAL DERIVATIVE OF THE HAUSDORFF DIMENSION  15  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' The Julia set close to 0  In order to compute the integral from the numerator of the formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='3),  we need precise information on the location of the Julia set close to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Corol-  lary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='2 tells us nothing precise for δ ̸= 0, so we need to change scale in order  to see the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Because preimages of the point −pδ are close to ±  √  6  3  √  −δ,  we change scale by the factor 1/  �  |δ| in every direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Thus, let us write  J ⋇  δ :=  1  �  |δ|  Jδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Next, we define small neighborhoods of 0, and its rescaled version, as follows:  Nδ,R := Jδ ∩ {z ∈ C : | Re z| ⩽ R  �  |δ|},  N ⋇  δ,R := J ⋇  δ ∩ {z ∈ C : | Re z| ⩽ R},  where R ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' So, we have Nδ,R =  �  |δ|N ⋇  δ,R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Write NR := ϕ−1  δ (Nδ,R) ⊂ Jδ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  We denote by bδ one of the points being close to a preimage of −pδ, namely  bδ :=  √  6  3  √  −δ =  √  6  3  �  |δ| sin α  2 − i  √  6  3  �  |δ| cos α  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Indeed fδ(bδ) = −2 + 1  3 δ ≈ −pδ (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  We need to define the following sets:  Hδ,R :  � xy = − 1  3 Im δ = − 1  3|δ| sin α,  √  6  3  �  |δ| sin α  2 ⩽ |x| ⩽ R  �  |δ|,  where x = Re z, y = Im z, δ ̸= 0, arg δ = α ∈ (0, 2π), and R ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Branches  of Hδ,R included in left and right half-planes will be denoted by H−  δ,R and  H+  δ,R respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' The endpoints of H+  δ are bδ, and the point which will be  denoted by zδ,R (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Re zδ,R = R  �  |δ|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  After substitution  �  |δ|x,  �  |δ|y in place of x, y, for α ∈ (0, 2π) we obtain  H⋇  α,R :  � xy = − 1  3 sin α,  √  6  3 sin α  2 ⩽ |x| ⩽ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Thus we have Hδ,R =  �  |δ|H⋇  α,R, where α = arg δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' We define H⋇,−  α,R and  H⋇,+  α,R analogously as H±  δ,R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' The endpoints of H⋇,+  α,R are  b⋇  α :=  1  �  |δ|  bδ =  √  6  3 sin α  2 − i  √  6  3 cos α  2 ,  z⋇  α,R := R − i 1  3R sin α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1)  Note that arg(z⋇  α,R) = arg(zδ,R) = −γα,R, where  γα,R = arctan  � 1  3R2 sin α  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='2)  Hence, γα,R > 0 for α ∈ (0, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Now we prove the following:  Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' For every α ∈ (0, 2π) and R ⩾ 1, if δ → 0 where α =  arg δ, then  N ⋇  δ,R → H⋇  α,R,  in the space of non-empty compact subsets of C equipped with the Hausdorff  metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  16  LUDWIK JAKSZTAS  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' The Julia set Jδ for δ = 1  25 +  1  100i, close to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Fix α ∈ (0, 2π) and R ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' First we will prove that  lim  δ→0 sup  z∈N ⋇  δ,R  dist(z, H⋇  α,R) = 0,  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='3)  where α = arg δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  If z = x + iy, then we define  l+  δ : xy = −1  3 Im δ + |δ|17/16,  l−  δ : xy = −1  3 Im δ − |δ|17/16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  The images of l±  δ are horizontal lines such that  Im(fδ(l±  δ )  �  = 1  3 Im δ ± 2|δ|17/16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  The set | Re z| ⩽ R  �  |δ| is mapped into the half-plane Re z ⩽ −2+R2|δ|+  Re δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Since −pδ = −2 + δ/3 + O(δ2) (cf, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1)), we conclude from Lemma  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='4 (1) that the set Nδ,R lies between the lines l±  δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Moreover Nδ,R is bounded by f−1  δ  (kδ) where kδ : Re z = − Re pδ −|δ|2 (see  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='4 (2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Since fδ(bδ) = −2 + δ/3, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1) leads to | − pδ − fδ(bδ)| ≈  |δ|2/27, whereas the critical value is equal to −2 + δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Thus we see that the  distance between f−1  δ  (kδ) and the points ±bδ is of order |δ|3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  So, the set N ⋇  δ,R lies between the curves  xy = −1  3 sin α ± |δ|1/16,  and rescaled curves f−1  δ  (kδ) whose distance from ±b⋇  α is of order |δ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Thus  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='3) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Now we will prove that  lim  δ→0 sup  z∈H⋇  α,R  dist(z, N ⋇  δ,R) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='4)  Suppose that, on the contrary, there is z0 ∈ H⋇  α,R, r0 > 0, and a sequence  δn → 0 such that B(z0, r)∩N ⋇  δn,R = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Note that the distances between ±b⋇  α  and rescaled preimages of f−1  δ  (−pδ) are of order |δ|, so the rescaled Julia set  is located on ”both sides” of B(z0, r0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' We can assume that 0 /∈ B(z0, r0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  ON THE DIRECTIONAL DERIVATIVE OF THE HAUSDORFF DIMENSION  17  Next, we have B(  �  |δn|z0,  �  |δn|r0) ∩ Nδn,R = ∅, and then  fδn  �  B(  �  |δn|z0,  �  |δn|r0)  �  ∩ Jδn = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Because f′  δn(  �  |δn|z0) = κ  �  |δn|, for some κ ∈ C \\ {0}, Koebe one-quarter  Theorem leads to  B  �  fδn(  �  |δn|z0), κr0|δn|/4  �  ⊂ fδn  �  B(  �  |δn|z0,  �  |δn|r0)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Since Im fδn(  �  |δn|z0) = Im δn/3, the point fδ(  �  |δn|z0) lies between the  lines Im z = − Im pδn ± |δn|17/16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' So, we conclude from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='4 (1) that  there exists 0 < r1 ⩽ κr0/4, such that  Aα,n := {z ∈ C : | Re(z − fδn(  �  |δn|z0))| ⩽ r1|δn|} ∩ Jδn = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  The diameter of the set fδn(Nδn,R) is close to R2|δn|, whereas fδn(Nδn,R) can  be mapped with bounded distortion onto a set of diameter grater than rz/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  So, the ”width” of the images of Aα,n is separated from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' This contradicts  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Therefore (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='4) holds, and the statement follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  □  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Conformal and Invariant measures  Recall from Section 2, that ωδ (after normalization) is the conformal mea-  sure with exponent d(δ) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='2)), whereas µδ is the fδ-invariant measure,  equivalent to ωδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' We assume that µδ(Jδ) = ωδ(Jδ) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Let us define the rescaled measures as follows:  ω⋇  δ (A) :=  1  �  |δ|  d(δ) ωδ(  �  |δ|A),  and  µ⋇  δ (A) :=  1  �  |δ|  d(δ) µδ(  �  |δ|A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Hence, ω⋇  δ and µ⋇  δ are supported on J ⋇  δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  The Main result of this Section is Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='4, which concerns the  limit of ω⋇  δ (so also µ⋇  δ ) when δ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' But first, we will deal with the limits  of ωδ and µδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  For δ = 0 situation is as follows: The measure ω0 is simply the Lebesgue  measure λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Next, we know that the map ϕv(z) = ( 4  π) arcsin( z  2) conjugates  f0 to V (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Because the Lebesgue measure is V -invariant, the  measure defined by ϕ∗  vλ(A) = λ(ϕv(A)) is f0-invariant with density  dµ0  dω0  (z) = 2  π  1  �  1 − (z/2)2 ,  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1)  (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' [11, Chapter V]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  First fact follows from [13, Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1]:  Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' For every α ∈ (0, 2π) the measure ω0 is equal to weak*  limit of ωδ, where δ → 0 and α = arg δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Now we prove:  Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' For every α ∈ (0, 2π) the measure µ0 is equal to weak*  limit of µδ, where δ → 0 and α = arg δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Fix α ∈ (0, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  18  LUDWIK JAKSZTAS  Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' We see from [15, Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='12]) that the density hδ :=  dµδ/dωδ is equal to the limit Ln  φδ(1), where φδ = −d(δ) log |f′  δ(ϕδ)|, n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  So, if z = ϕδ(s), then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1) leads to  hδ(z) = lim  n→∞ Ln  φδ(1)(s) = lim  n→∞  �  s∈T −n(s)  |(fn  δ )′(ϕδ(s))|−d(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='2)  Let δk → 0, where arg δk = α, be a sequence such that µδk tends to a  measure ˆµ0 in the weak* topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  If z0 ∈ (−2, 2), then we can find r such that for k large enough, and δ  close to 0, B(z0, r) is separated from the postcritical set (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Then, all inverse branches of fn  δk, n ⩾ 1 have uniformly bounded distortion  on B(z0, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' So, we see from (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='2) that there exists a constant K1 > 0 such  that  hδk  ��  B(z0,r)∩Jδk < K1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='3)  Passing to the limit (and possibly changing K1), we obtain  K−1  1  < h0  ��  (z0−r,z0+r) < K1,  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='4)  or h0 = 0 on B(z0 − r, z0 + r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' In the latter case we get h0 = 0 on (−2, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  The limit measure ˆµ0 is f0-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' So, we see that if ˆµ0 has an atom,  the only possibility is the fixed point p0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Therefore, it is enough to show  that there are no atom at p0, because in that case (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='4) holds, and ˆµ0 is  absolutely continuous with respect to ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Then, by uniqueness ˆµ0 = µ0, so  µ0 in the only possible limit, and the statement follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' So, now we prove that there are no atom at p0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' If ϕδ(s) = z ∈ C+2  δ,n,  then using (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='2) we obtain  hδ(z) =  ∞  �  k=0  |(f−1  δ,− ◦ f−k  δ,+)′(z)|d(δ) · hδ((f−1  δ,− ◦ f−k  δ,+)(z)),  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='5)  where (f−1  δ,− ◦ f−k  δ,+)(z) ∈ C−2  δ,n+k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' So, we must estimate values of hδ on the  cylinders C−2  δ,n+k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' We have  hδ(z) = |(f−1  δ,−)′(z)|d(δ) · hδ(f−1  δ,−(z)) + |(f−1  δ,+)′(z)|d(δ) · hδ(f−1  δ,+(z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='6)  If z ∈ C−2  δ,n+k+1 then f−1  δ,±(z) ∈ C0±  δ,n+k+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Let z0 = 0 and let r be such that (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='3) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Then, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1 combined  with Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1, implies that there exists n0 ∈ N such that C0±  δ,n+k+2 ⊂  B(0, r) where n ⩾ n0, k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Again using Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1 (1), we get  |(f−1  δ,±)′(z)|d(δ) = |f′  δ(f−1  δ,±(z))|−d(δ) < K2|λδ|d(δ)(n+k+2)/2,  thus (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='3) and (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='6) leads to  hδ(z) < K3|λδ|d(δ)(n+k+2)/2,  where z ∈ C−2  δ,n+k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  ON THE DIRECTIONAL DERIVATIVE OF THE HAUSDORFF DIMENSION  19  If z ∈ C+2  δ,n, then formula (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='5) combined with the above estimate, and the  fact that |(f−1  δ,− ◦ f−k  δ,+)(z)|d(δ) ≍ |λδ|−d(δ)(k+1), gives us  hδ(z) < K4  ∞  �  k=0  |λδ|d(δ)(n−k)/2 < K5|λδ|d(δ)n/2,  where n ⩾ n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  We have ωδ(C+2  δ,n) ≍ (diam C+2  δ,n)d(δ) ≍ |λδ|−d(δ)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' So, if N ⩾ n0 then  µδ  �  ∞  �  n=N  C+2  δ,n  �  =  �  �∞  n=N C+2  δ,n  hδ dωδ < K6  ∞  �  n=N  |λδ|d(δ)n/2|λδ|−d(δ)n  < K6  ∞  �  n=N  4−n/3 < 3K6  �1  4  �N/3  −−−−→  N→∞ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Because the above estimate does not depend on δ, the statement follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  □  Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' For every α ∈ (0, 2π) and ε > 0 there exist neighborhood U ∋ 0  and η > 0 such that  �  1 − ε  � 2  π < dµδ  dωδ  ���  U∩Jδ  <  �  1 + ε  � 2  π,  where 0 < |δ| < η and α = arg δ or δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Fix α ∈ (0, 2π) and ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' As before, we write hδ = dµδ/dωδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  We see from (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1) that there exists an interval (−κ, κ) such that  2  π ⩽ h0(z) ⩽  �  1 + ε  3  � 2  π,  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='7)  where z ∈ (−κ, κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' So the statement holds for δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  We can find (small) 0 < r < κ such that distortion of all inverse branches  of fn  δ , n ⩾ 1 on B(0, r) is as close to 1 as we need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' So, taking into account  formula (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='2), we can assume for every z1, z2 ∈ B(0, r) ∩ Jδ, we have  ��hδ(z1) − hδ(z2)  �� < ε  3  2  π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='8)  Since h0 and hδ are close to constant functions on (−r, r) and B(0, r)∩Jδ  respectively, we conclude from Propositions 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='2 that there exist z3 ∈  (−r, r) and z4 ∈ B(0, r) ∩ Jδ such that |h0(z3) − hδ(z4)| <  ε  3  2  π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  So the  statement follows from (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='7) and (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  □  Let us denote by lα,R the arc length measure supported on H⋇  α,R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Now we  prove the main result of this Section:  Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' For every α ∈ (0, 2π) and R ⩾ 1, we have  ω⋇  δ  ���  N ⋇  δ,R  → lα,R,  in the weak∗ topology, where δ → 0 and α = arg δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  20  LUDWIK JAKSZTAS  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Fix α ∈ (0, 2π), R ⩾ 1, and let α = arg δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' It is enough to consider  the set N ⋇,+  δ,R := N ⋇  δ,R ∩ {z ∈ C : Re z > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Write N +  δ,R =  �  |δ|N ⋇,+  δ,R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Let U ′ be a small neighborhood of H⋇,+  α,R separated from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Let  U = U ′ ∩ {z ∈ C : 0 < Re z < R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' For any Borel set A ⊂ U, such that fn+1  δ  is injective on  �  |δ|A, we have  ω⋇  δ (A) =  1  �  |δ|  d(δ) ωδ(  �  |δ|A) =  1  �  |δ|  d(δ)  �  fn+1  δ  (√  |δ|A)  ��(f−n−1  δ,ν  )′��d(δ)dωδ  =  �  fn+1  δ  (√  |δ|A)  ����  (f−n−1  δ,ν  )′  �  |δ|  ����  d(δ)  dωδ,  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='9)  where f−n−1  δ,ν  denotes the inverse branch that maps fn+1  δ  (  �  |δ|U) onto  �  |δ|U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Our aim is to show that if δ → 0, then we can find n(δ) such that for  every Borel set A ⊂ U, satisfying lα,R(∂A) = 0, the above integral tends to  lα,R(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Let rδ > 0 be the smallest number such that there exists n ∈ N  for which  fn+1  δ  (N +  δ,R) ⊂ B(pδ, rδ),  and fn+2  δ  (N +  δ,R) /⊂ B(pδ, rz/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='10)  In particular we see that rδ ⩽ rz/2, and possible values of rδ are separated  from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' It follows from Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1 that N +  δ,R ⊂ U for δ small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Let (δk) be a sequence of parameters tending to 0 for which rδk → q when  k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Let nk be such that (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='10) holds for δk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  We have fnk+1  δ  (  �  |δk|U) ⊂ B(pδ, rz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  If z ∈ fδ(  �  |δk|U), then −z ∈  B(pδ, rz), so using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='4) we obtain  fnk  δk (z) = fnk  δk (−z) = Φ−1  δk  �  λnk  δk Φδk(−z)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Therefore, if z ∈ U, then we have  Fα,q,k(z) := fnk+1  δk  ��  |δk|z  �  = Φ−1  δk  �  λnk  δk Φδk(−|δk|z2 + 2 − δk)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='11)  Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' We will prove that the sequence Fα,q,k converges uniformly on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Write  gδ(z) := 1  |δ|Φδ  �  − |δ|z2 + 2 − δ  �  = 1  |δ|Φδ  �  pδ − |δ|z2 + 2 − δ − pδ  �  ,  so we have  Fα,q,k(z) = Φ−1  δk  �  |δk|λnk  δk gδk(z)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='12)  Note that  2 − δk − pδk = −2  3δk + O(|δk|2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Therefore  gδk(z) = Φδk  �  pδk + |δk|(−z2 − 2  3v + O(|δk|))  �  |δk|  ,  where v = eiα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Since Φ′  δk(pδk) = 1, δk → 0 whereas U is bounded, we  conclude that  gδk(z) ⇒ −z2 − 2  3v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='13)  ON THE DIRECTIONAL DERIVATIVE OF THE HAUSDORFF DIMENSION  21  Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1 and the fact that Φδ ⇒ Φ0, lead to  diam(gδk(N ⋇,+  δk,R)) → K1,  for some constant K1 (which does not depend on q and is close to R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' On  the other hand  diam(Fα,q,k(N ⋇,+  δk,R)) → q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  So, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='12) and the fact arg(λnk) = O(δk), lead to  |δk|λnk  δk → κ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Thus, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='12) combined with (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='13) and the above, gives us  Fα,q,k(z) ⇒ Φ−1  0  �  κ  �  − z2 − 2  3v  ��  := Fα,q(z),  where k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Since (b⋇  α)2 = − 2  3v, we have Fα,q(b⋇  α) = p0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Thus, we conclude  from Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1, and the choice of (δk), that  Fα,q  ��  H+  α,R : H+  α,R → [2 − q, 2] ⊂ J0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Moreover Fα,q  ��  H+  α,R is a bijection, and then  �  Fα,q(A)  ��(F −1  α,q)′��dω0 = lα,R(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='14)  Hence, the assumption lα,R(∂A) = 0 leads to ω0(Fα,q(∂A)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  We have (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='11))  F −1  α,q,k(z) =  f−nk−1  δk  (z)  �  |δk|  → F −1  α,q(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Therefore (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='9) gives us  ω⋇  δk(A) =  �  Fα,q,k(A)  ��(F −1  α,q,k)′��d(δk)dωδk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Since ω0(Fα,q(∂A)) = 0 and Fα,q,k converges uniformly to Fα,q, measure of  symmetric difference ωδk(Fα,q,k(A)∆Fα,q(A)) tends to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' So uniform conver-  gence of |(F −1  α,q,k)′|d(δk), Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1 and (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='14) lead to  �  Fα,q,k(A)  ��(F −1  α,q,k)′��d(δk)dωδk →  �  Fα,q(A)  ��(F −1  α,q)′��dω0 = lα,R(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Thus, the statement holds if r(δk) → q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  But, lα,R is the only possible  limit, because for every δk we can always find a subsequence δjk such that  r(δjk) → q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  □  22  LUDWIK JAKSZTAS  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Key integral  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  The main problem in the proof of the Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1, is to estimate  integral from the numerator of the formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='3), namely:  �  Jδ0  ∂  ∂t log |f′  tv(ϕtv)|d˜µtv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1)  Put ˙ϕδ := ∂  ∂δϕδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' We have  ∂  ∂t(f′  tv(ϕtv)) = ∂  ∂t(2ϕtv) = 2v ˙ϕδ  ��  δ=tv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Therefore, the integrand can be rewritten as follows:  ∂  ∂t log |f′  tv(ϕtv)| = Re  � ∂  ∂t(f′  tv(ϕtv))  f′  tv(ϕtv)  �  = Re  �  v ˙ϕδ  ��  δ=tv  ϕtv  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='2)  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Now we derive the formula for ˙ϕδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' The function ϕδ conjugates fδ0 = T  to fδ, so ϕδ(T(s)) = ϕ2  δ(s) − 2 + δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Differentiating both sides with respect  to δ, we get  ˙ϕδ(T(s)) = 2ϕδ(s) ˙ϕδ(s) + 1,  hence  ˙ϕδ(s) = −  1  2ϕδ(s) + ˙ϕδ(T(s))  2ϕδ(s) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='3)  Next, replacing s by T(s), T 2(s),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=', T m−1(s), we obtain  ˙ϕδ(s) = −  m−1  �  k=0  1  2ϕδ(s) · 2ϕδ(T(s)) · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' · 2ϕδ(T k(s))+  +  ˙ϕδ(T m(s))  2ϕδ(s) · 2ϕδ(T(s)) · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' · 2ϕδ(T m−1(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Since 2ϕδ(s) = f′  δ(ϕδ(s)), the chain rule leads to  ˙ϕδ(s) = −  m  �  k=1  1  (fk  δ )′(ϕδ(s)) +  ˙ϕδ(T m(s))  (fm  δ )′(ϕδ(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='4)  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Let us assume that z ∈ Jδ is close to 0, and z = ϕδ(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Then the  denominator of (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='2) is small, moreover we see from (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='3) that  ˙ϕδ(s) = − 1  2z (1 − ˙ϕδ(T(s))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Since ϕδ(T(s)) is close to −2, it follows from the formula (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='4) that ˙ϕδ(T(s))  should usually be close to 1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' If this is so, we get (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='2))  Re  �  v ˙ϕδ  ϕδ  �  ≈ 1  3 Re  �  − v  z2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Thus, we can expect that the function d′  v(tv), and the integral of Re(−v/z2)  over a neighborhood of 0, with respect to µδ, have the same order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  ON THE DIRECTIONAL DERIVATIVE OF THE HAUSDORFF DIMENSION  23  Indeed, integral of Re(−v/z2) has decisive influence on d′  v(tv), and we will  compute it over the sets Nδ,R in Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' But first, for α ∈ (0, π)  and R ⩾ 1, let us define  Iα,R :=  √  6 cos α  ��  sin(2γα,R)  sin α  − 1  �  +  √  6  2  √  sin α  π−2γα,R  �  α  �  sin β dβ,  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='5)  where γα,R is given by the formula (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' For α = π we take  Iπ,R =  √  6  �  1 −  √  6  3  1  R  �  =  √  6 − 2  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='6)  Note that  �  sin(2γα,R)  sin α  =  √  6  3  1  R  1  �  1 + ( sin α  3R2 )2  ,  therefore Iα,R → Iπ,R, when α → π−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' For every α ∈ (0, π] and R ⩾ 1 we have  lim  δ→0 |δ|1− 1  2 d(δ)  �  Nδ,R  Re  �  − v  z2  �  dµδ(z) = 2  πIα,R,  where α = arg δ and v = eiα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Fix α ∈ (0, π) and R ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' First, using Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='4, we obtain  |δ|  �  |δ|  d(δ)  �  Nδ,R  Re  �  − v  z2  �  dωδ(z) =  1  �  |δ|  d(δ)  �  Nδ,R  Re  �  −  v  (z/  �  |δ|)2  �  dωδ(z)  =  �  N ⋇  δ,R  Re  �  − v  z2  �  dω⋇  δ (z) →  �  H⋇  α,R  Re  �  − v  z2  �  dlα,R(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  So, taking into account Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='3, we have to prove that the integral over  H⋇  α,R is equal to Iα,R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' It is enough to consider the integral restricted to  H⋇,+  α,R , because value of the integral over H⋇,−  α,R is the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  We have  �  H⋇,+  α,R  Re  �  − v  z2  �  dlα,R(z) = −  �  H⋇,+  α,R  (x2 − y2) cos α + 2xy sin α  (x2 + y2)2  dlα,R(z),  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='7)  where x = Re(z) and y = Im(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  If α ∈ (0, π), then H⋇,+  α,R is the arc of hyperbola xy = − 1  3 sin α contained  in IV quadrant, which joins the points b⋇  α and z⋇  α,R (see (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' So, in the  polar coordinates H⋇,+  α,R can be written as follows:  H⋇,+  α,R :  �  x(t) = hα(t) · cos t,  y(t) = hα(t) · sin t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  t ∈  �  arctan Im b⋇  α  Re b⋇α  , arctan  Im z⋇  α,R  Re z⋇  α,R  �  ,  24  LUDWIK JAKSZTAS  where h2  α(t) · 1  2 sin(2t) = − 1  3 sin α, and  Im b⋇  α  Re b⋇α  = − cot α  2 = tan  �α  2 − π  2  �  ,  Im z⋇  α,R  Re z⋇  α,R  = − 1  3R2 sin α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Therefore, we obtain (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='2))  H⋇,+  α,R :  �  �  �  x(t) =  �  − 2  3  sin α  sin(2t) · cos t,  y(t) =  �  − 2  3  sin α  sin(2t) · sin t,  t ∈  �1  2(α − π), −γα,R  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Next, we can get  dlα,R =  �  h2α(t) + (h′α(t))2 dt =  �  −2  3  sin α  sin3(2t) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Since x(t)y(t) = − 1  3 sin α, and x2(t) + y2(t) = h2  α(t), moreover  x2(t) − y2(t) = h2  α(t)  �  cos2 t − sin2 t  �  = −2  3 sin αcos(2t)  sin(2t) ,  the integrals from (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='7) are equal to  −  −γα,R  �  1  2 (α−π)  − 2  3 sin α cos α cos(2t)  sin(2t) − 2  3 sin2 α  4  9 sin2 α  1  sin2(2t)  �  −2  3  sin α  sin3(2t) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Simplifying we obtain  √  6  2  −γα,R  �  1  2 (α−π)  �cos α  sin α  cos(2t)  sin(2t) + 1  �  sin2(2t)  �  − sin α  sin3(2t) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Substitution β = 2t + π, and the above computations, lead to  �  H⋇,+  α,R  Re  �  − v  z2  �  dlα,R(z) =  √  6  4  π−2γα,R  �  α  �cos α  sin α  cos β  sin β + 1  �  sin2 β  �  sin α  sin3 β dβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  We divide the integrand into two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' First we compute:  √  6  4  π−2γα,R  �  α  cos α  sin α  cos β  sin β sin2 β  �  sin α  sin3 β dβ =  √  6  4  cos α  √  sin α  π−2γα,R  �  α  cos β  √sin β dβ  =  √  6  4  cos α  √  sin α  �  2  �  sin β  ����  π−2γα,R  α  =  √  6  2 cos α  �  sin 2γα,R  sin α  −  √  6  2 cos α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Next we have  √  6  4  π−2γα,R  �  α  sin2 β  �  sin α  sin3 β dβ =  √  6  4  √  sin α  π−2γα,R  �  α  �  sin β dβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Thus, the integral over H⋇,+  α,R is equal 1  2Iα,R (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='5)), so the statement  holds for α ∈ (0, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  If α = π, then H⋇,+  π,R :  √  6  3 ⩽ x ⩽ R, and  ON THE DIRECTIONAL DERIVATIVE OF THE HAUSDORFF DIMENSION  25  �  H⋇,+  π,R  Re  �  − −1  z2  �  dlπ,R(z) =  R  �  √  6  3  1  x2 dx = −1  x  ���  R  √  6  3  =  √  6  2 − 1  R = 1  2Iπ,R,  (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='6)) thus the proof is finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  □  Now we give two technical Corollaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' For every α ∈ (0, π] and R ⩾ 1 there exists Kα,R ∈ R such  that  lim  δ→0 |δ|1− 1  2 d(δ)  �  Nδ,R  1  |z|2 dµδ(z) = Kα,R,  where α = arg δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Analogously as in the proof of Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1, it is enough to compute  integral  �  H⋇,+  α,R |z|−2dlα,R(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' For α ∈ (0, π), proceeding as before, we obtain  �  H⋇,+  α,R  1  |z|2 dlα,R(z) =  −γα,R  �  1  2 (α−π)  −3  2  sin(2t)  sin α  �  −2  3  sin α  sin3(2t) dt  =  �  3  2  1  √  sin α  −γα,R  �  1  2 (α−π)  1  �  − sin(2t)  dt := 1  2Kα,R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  If α = π, then  R  �  √  6  3  1  x2 dx =  √  6  2 − 1  R := 1  2Kπ,R,  and the proof is finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  □  Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' For every α ∈ (0, π] and R ⩾ 1 there exists Kα,R ∈ R such  that  lim  δ→0 |δ|  1  2 (1−d(δ))  �  Nδ,R  1  |z| dµδ(z) = Kα,R,  where α = arg δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Using Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='4, analogously as in the proof of Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1  we get  �  |δ|  �  |δ|  d(δ)  �  Nδ,R  1  |z| dωδ(z) =  �  N ⋇  δ,R  1  |z| dω⋇  δ (z) →  �  H⋇  α,R  1  |z| dlα,R(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  26  LUDWIK JAKSZTAS  For α ∈ (0, π), proceeding as before, we obtain  �  H⋇,+  α,R  1  |z| dlα,R(z) =  −γα,R  �  1  2 (α−π)  �  −3  2  sin(2t)  sin α  �  −2  3  sin α  sin3(2t) dt  =  −γα,R  �  1  2 (α−π)  1  | sin(2t)| dt = −1  2 log | tan t|  ����  −γα,R  1  2 (α−π)  = 1  2 log 3  2 + log  R  sin α  2  =: 1  2Kα,R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  If α = π, we get  R  �  √  6  3  1  x dx = log R + 1  2 log 3  2 =: 1  2Kπ,R,  and the proof is finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  □  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Integral over Jδ0 \\ NR  The main result of this Section is Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='5, which shows us that  the integral of (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='2) over Jδ0 \\ NR is small (after dividing by |δ|1− 1  2 d(δ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  This result, together with Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1 (integral over NR) are the main  ingredients in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  But, we begin with a few facts about (fn  δ )′, and Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='4, which  will be used in the proofs of both mentioned Propositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Because fδ are conjugated to z �→ λδz on the set B(2, rz) we conclude  that:  Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' For every α ∈ (0, 2π) there exists K > 1 such that if z ∈ C±2  δ,n,  n ⩾ 1 and 1 ⩽ j ⩽ n, then  K−1|λδ|j < |(fj  δ )′(z)| < K|λδ|j,  where α = arg δ and 0 < |δ| < η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  If U is a neighborhood of 0 and z ∈ Jδ \\U, then f′  δ(z) is close to f′  0(Re z),  where Re z ∈ J0 provided | Re z| ⩽ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  The cylinders Cδ,ν ∋ z and C0,ν  are close each other (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='5), so also finite trajectories fn  δ (z) and  fn  0 (Re z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Thus the formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1) leads to:  Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' For every α ∈ (0, 2π), ε > 0, N ∈ N and open set U ∋ 0,  there exists η > 0 such that if z ∈ Jδ, | Re(z)| ⩽ 2, {z, fδ(z), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' , fn−1  δ  (z)} ∩  U = ∅, n ⩽ N, then  (1 − ε) 2n  �  4 − (Re fn  0 (z))2  4 − (Re z)2  < |(fn  δ )′(z)| < (1 + ε) 2n  �  4 − (Re fn  0 (z))2  4 − (Re z)2  ,  where 0 < |δ| < η and α = arg δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='2 for n = i, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1 (1), and Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1 for n = j −1 lead  to:  ON THE DIRECTIONAL DERIVATIVE OF THE HAUSDORFF DIMENSION  27  Corollary 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' For every α ∈ (0, 2π), N ∈ N there exists K > 1 and η > 0  such that if z ∈ f−i  δ (C0  δ,m), m ⩾ 0, then  |(fi+j  δ  )′(z)| > K−12i|λδ|j−1− m  2 ,  where 0 ⩽ i ⩽ N, 0 ⩽ j ⩽ m, 0 < |δ| < η and α = arg δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' For every α ∈ (0, π] and s > 0, we have  lim  δ→0 |δ|s  �  Jδ0  �� ˙ϕδ  �� d ˜µδ = 0,  where α = arg δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Fix α ∈ (0, π], s > 0 and N0 ⩾ 4 such that 2N0 ⩾ 8K, where K is a  constant from Corollary 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' First, we define the set Xδ,N0, on which it can be easily proven  that |(fN0  δ  )′| > 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  If fN0  δ  (z) ∈ M0  δ,1 (note that M0  0,1 = [−  √  2,  √  2]), then Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='2 gives  us  |(fN0  δ  )′(z)| > 2N0 9  10  �  2  4 ⩾ 24 9  10  √  2  2  > 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  The derivative will be also grater than 10 for z ∈ C±2  δ,N0+n, n ⩾ 1 and z = ±pδ,  therefore we define  Xδ,N0 := f−N0  δ  (M0  δ,1) ∪  ∞  �  n=1  C±2  δ,N0+n ∪ {−pδ, pδ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Write XN0 := ϕ−1  δ (Xδ,N0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Let s ∈ Xδ,N0, then (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='4) for m = N0, gives us:  ˙ϕδ(s) = −  N0  �  k=1  1  (fk  δ )′(ϕδ(s)) + ˙ϕδ(T N0(s))  (fN0  δ  )′(z)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Since fN0  δ  (z) ∈ M0  δ,1, the trajectory {z, fδ(z), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' , fN0−1  δ  (z)} is disjoint from  M0  δ,N0, hence is separated from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Thus, the derivatives (fk  δ )′(ϕδ(s)) are also  separated from 0 (by a constant depending on N0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' So, the fact that the  measure ˜µδ is T-invariant, leads to  �  XN0  | ˙ϕδ| d ˜µδ < K1(N0) + 1  10  �  XN0  | ˙ϕδ(T N0)| d ˜µδ  < K1(N0) + 1  10  �  T N0(XN0)  | ˙ϕδ| d ˜µδ < K1(N0) + 1  10  �  Jδ0  | ˙ϕδ| d ˜µδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1)  Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Now, we will deal with the set  Jδ \\ Xδ,N0 =  ∞  �  n=1  (f−N0  δ  (C±2  δ,n) \\ C±2  δ,n+N0) =  ∞  �  n=1  G−N0  δ,n ,  where G−N0  δ,n  := f−N0  δ  (C±2  δ,n) \\ C±2  δ,n+N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Let G−N0  δ,n  := ϕ−1  δ (G−N0  δ,n ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  28  LUDWIK JAKSZTAS  If ϕδ(s) = z ∈ G−N0  δ,n , then we will use the formula (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='4) for m = N0 + n:  ˙ϕδ(s) = −  N0+n  �  j=1  1  (fj  δ )′(ϕδ(s))  + ˙ϕδ(T N0+n(s))  (fN0+n  δ  )′(z)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='2)  Before we pass to estimations, let us rewrite the set G−N0  δ,n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' We have  f−N0  δ  (C−2  δ,n) = f−(N0−1)  δ  (f−1  δ  (C−2  δ,n)) = f−(N0−1)  δ  (C0  δ,n+1),  and next  f−N0  δ  (C+2  δ,n) = C+2  δ,n+N0 ∪  N0−1  �  k=0  f−(N0−k−1)  δ  (f−1  δ,−(C+2  δ,n+k))  = C+2  δ,n+N0 ∪ C−2  δ,n+N0 ∪  N0−2  �  k=0  f−(N0−k−2)  δ  (C0  δ,n+k+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Therefore  G−N0  δ,n  =  N0−2  �  k=−1  f−(N0−k−2)  δ  (C0  δ,n+k+2) =  N0  �  k=1  f−(N0−k)  δ  (C0  δ,n+k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='3)  Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Now we will estimate integral of the "tail" of (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' If z ∈ G−N0  δ,n ,  then we see that there exists k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' , N0} such that fN0−k  δ  (z) ∈ Cδ,n+k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Thus, Corollary 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='3 for N = N0 − 1, i = N0 − k, m = n + k and j = n + k,  leads to  |(fN0+n  δ  )′(z)| > K−12N0−k|λδ|  1  2 (n+k)−1  = K−12N0|λδ|  n  2 −1��  |λδ|  2  �k  > 9  40K−12N0�19  10  �n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Therefore  �  G−N0  n  ����  ˙ϕδ(T N0+n)  (fN0+n  δ  )′(ϕδ)  ���� d ˜µδ < 40  9 K2−N0�10  19  �n �  G−N0  n  | ˙ϕδ(T N0+n)| d ˜µδ  < 40  9 K2−N0�10  19  �n �  Jδ0  | ˙ϕδ| d ˜µδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  By assumption, 2N0 > 8K, we get  40  9 K2−N0  ∞  �  n=1  �10  19  �n  = 4K2−N0�10  9  �2  < 1  2  �10  9  �2  < 5  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Thus we obtain  ∞  �  n=1  �  G−N0  n  ����  ˙ϕδ(T N0+n)  (fN0+n  δ  )′(ϕδ)  ���� d ˜µδ < 5  8  �  Jδ0  | ˙ϕδ| d ˜µδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='4)  ON THE DIRECTIONAL DERIVATIVE OF THE HAUSDORFF DIMENSION  29  Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' We have to estimate finite sum from (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' If ϕδ(s) = z ∈ G−N0  δ,n  and fN0−k  δ  (z) ∈ C0  δ,n+k (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='3)), then we can write  −  N0+n  �  j=1  1  (fj  δ )′(z)  = −  N0−k  �  j=1  1  (fj  δ )′(z)  −  N0+n  �  j=N0−k+1  1  (fj  δ )′(z)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='5)  Now we will estimate first sum on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' If k = N0, then we will  assume that it is equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Since fN0−k  δ  (z) ∈ C0  δ,n+k, the trajectory {z, fδ(z), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' , fN0−k−1  δ  (z)} is dis-  joint from M0  δ,N0, hence is separated from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Therefore the derivatives are  separated from 0, and the sum is bounded by a constant K2(N0)/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Thus  we have  ∞  �  n=1  N0  �  k=1  �  f−N0+k  δ  (C0  δ,n+k)  N0−k  �  j=1  ����  1  (fj  δ )′(z)  ���� dµδ(z) < K2(N0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='6)  Step 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' The rightmost sum from (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='5) can be rewritten in the form  −  N0+n  �  j=N0−k+1  1  (fj  δ )′(z)  = −  n+k  �  j=1  1  (fj  δ )′(fN0−k  δ  (z)) · (fN0−k  δ  )′(z)  = −  1  (fN0−k  δ  )′(z) 1  f′  δ(fN0−k  δ  (z))  �  1 +  n+k−1  �  j=1  1  (fj  δ )′(fN0−k+1  δ  (z))  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  If fN0−k  δ  (z) ∈ C0  δ,n+k then |(fN0−k  δ  )′(z)| > 1 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Moreover  fN0−k+1  δ  (z) ∈ C−2  δ,n+k−1, so using Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1, we can find K3 > 1 such that  ����  N0+n  �  j=N0−k+1  1  (fj  δ )′(z)  ���� < 2K3  1  |f′  δ(fN0−k  δ  (z))|  = K3  1  |fN0−k  δ  (z)|  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='7)  Step 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' In order to estimate the above expression, we will consider two  cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' First, we will deal with the sets G−N0  δ,n  such that if z ∈ G−N0  δ,n , then  the trajectory {z, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' , fN0−1  δ  (z)} is disjoint from Nδ,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' It is enough to check  if z ∈ C0  δ,n+N0 then | Re(z)| >  �  |δ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Since |z| < K|λδ|− 1  2 (n+N0) (see Lemma  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1 (2)), we get  �  |δ| < K|λδ|− 1  2 (n+N0), and then 1 ⩽ n < log|λδ|  � K2  |δ|  �  − N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  We have  K3  �  f−N0+k  δ  (C0  δ,n+k)  1  |fN0−k  δ  (z)|  dµδ(z) = K3  �  C0  δ,n+k  1  |z| dµδ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='8)  30  LUDWIK JAKSZTAS  So, using (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='7), (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='8), the fact that µδ(C0  δ,n) ≍ ωδ(C0  δ,n) ≍ (diam(C0  δ,n))d(δ)  and Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1 we obtain  N0  �  k=1  �  f−N0+k  δ  (C0  δ,n+k)  ����  N0+n  �  j=N0−k+1  1  (fj  δ )′(z)  ����dµδ(z) < K3  N0  �  k=1  �  C0  δ,n+k  1  |z|dµδ(z)  < K4  N0  �  k=1  |λδ|  1  2 (n+k)|λδ|− 1  2 d(δ)(n+k) < K5(N0)|λδ|  1  2 (1−d(δ))n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  We can assume that 1 − d(δ) ⩽ |1 − d(δ)| < s/2, thus the integral over all  sets G−N0  δ,n , where 1 ⩽ n ⩽ log|λδ|  � K2  |δ|  �  − N0 can be estimated by  K5(N0)  log|λδ|  �  K2  |δ|  �  −N0  �  n=1  |λδ|  1  2 |1−d(δ)|n < K5(N0)  �K2  |δ|  � s  4 log|λδ|  �K2  |δ|  �  < |δ|− s  4 |δ|− s  4 = |δ|− s  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='9)  Step 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Let nδ be the smallest number such that there exists z ∈ G−N0  δ,nδ for  which {z, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' , fN0−1  δ  (z)} ∩ Nδ,1 ̸= ∅, hence C0  δ,nδ+N0 ∩ Nδ,1 ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Thus, there  exists R > 1 (R ≍ 2N0) such that M0  δ,nδ ⊂ Nδ,R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' So, using (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='3), we obtain  ∞  �  n=nδ  G−N0  δ,n  ⊂  N0  �  k=1  f−N0+k  δ  (M0  δ,nδ) ⊂  N0  �  k=1  f−N0+k  δ  (Nδ,R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Using (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='7), (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='8) (analogously as in the Step 5), and next Corollary  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='3, we obtain  N0  �  k=1  �  f−N0+k  δ  (Nδ,R)  ����  N0+n  �  j=N0−k+1  1  (fj  δ )′(z)  ����dµδ(z)  < K3  N0  �  k=1  �  f−N0+k  δ  (Nδ,R)  1  |fN0−k  δ  (z)|  dµδ(z) = K3 ·N0  �  Nδ,R  1  |z| dµδ(z) < |δ|− s  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='10)  Step 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Estimates (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1), (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='4), (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='6), (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='9) and (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='10) give us  �  Jδ0  �� ˙ϕδ  �� d ˜µδ < K1(N0) + K2(N0) +  � 1  10 + 5  8  � �  Jδ0  �� ˙ϕδ  �� d ˜µδ + 2|δ|− s  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Since  1  10 + 5  8 < 3  4, we obtain  1  4  �  Jδ0  �� ˙ϕδ  �� d ˜µδ < K1(N0) + K2(N0) + 2|δ|− s  2 ,  and the statement follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  □  ON THE DIRECTIONAL DERIVATIVE OF THE HAUSDORFF DIMENSION  31  Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' For every α ∈ (0, π], ε > 0 there exist R ⩾ 1 and η > 0  such that  |δ|1− 1  2 d(δ)  �  Jδ0\\NR  ��� ˙ϕδ  ϕδ  ��� d ˜µδ < ε,  where 0 < |δ| < η and α = arg δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' We conclude from Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='4 that it is enough to estimate the  integral over the set M0  1 \\ NR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Fix α ∈ (0, π] and ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' If ϕδ(s) = z ∈ C0  δ,n, then formula (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='4) for m = n  leads to  ˙ϕδ(s)  ϕδ(s) = −  1  z · f′  δ(z)  �  1 +  n−1  �  k=1  1  (fk  δ )′(fδ(z))  �  +  ˙ϕδ(T n(s))  ϕδ(s) · (fn  δ )′(ϕδ(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  where α = arg δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  We have |z| > R  �  |δ| for some R large enough, on the other hand Lemma  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1 gives us |z| < K1|λδ|− 1  2 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' So we will consider 1 ⩽ n ⩽ log|λδ|  � K2  1  R2|δ|  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Since fδ(z) ∈ C−2  δ,n−1, using Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1 and again Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1, we get  |z · (fn  δ )′(z)| = |z · f′  δ(z) · (fn−1  δ  )′(fδ(z))| > K2|λδ|−n|λδ|n−1 > K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  So, the above inequality, and the fact that T n(C0  n) = M0  1 lead to  log|λδ|  � K2  1  R2|δ|  �  �  n=1  �  C0n  ����  ˙ϕδ(T n)  ϕδ · (fn  δ )′(ϕδ)  ���� d ˜µδ < K3 log|λδ|  � K2  1  R2|δ|  � �  M0  1  �� ˙ϕδ  �� d ˜µδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Because 1 − d(δ)/2 is close to 1/2, Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='4 gives us  lim  δ→0 |δ|1− 1  2 d(δ)K3 log|λδ|  � K2  1  R2|δ|  � �  M0  1  �� ˙ϕδ  �� d ˜µδ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Since fδ(z) ∈ C−2  δ,n−1, using Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1, next Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1, and the fact  that µδ(C0  δ,n) ≍ ωδ(C0  δ,n) ≍ (diam(C0  δ,n))d(δ), we obtain  log|λδ|  � K2  1  R2|δ|  �  �  n=1  �  C0  δ,n  ����  1  z · f′  δ(z)  �  1 +  n−1  �  k=1  1  (fk  δ )′(fδ(z))  ����� dµδ  < K4  log|λδ|  � K2  1  R2|δ|  �  �  n=1  �  C0  δ,n  1  |z|2 dµδ < K5  log|λδ|  � K2  1  R2|δ|  �  �  n=1  |λδ|(1− 1  2 d(δ))n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  The rightmost sum can be estimated by the integral:  K6  log|λδ|  � K2  1  R2|δ|  �  �  1  |λδ|(1− 1  2 d(δ))xdx < K7  � K2  1  R2|δ|  �1− 1  2 d(δ)  < ε|δ|−1+ 1  2 d(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  32  LUDWIK JAKSZTAS  The last inequality holds for R large enough, since K1 and K7 does not  depend on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Thus the assertion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  □  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Integral over NR and Proof of the main Theorem  Now, using Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1, we estimate integral of (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='2) over NR, which  has decisive influence on (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Next we will prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' For every α ∈ (0, π] and R ⩾ 1 we have  lim  δ→0 |δ|1− 1  2 d(δ)  �  NR  Re  �  v ˙ϕδ  ϕδ  �  d ˜µδ = 1  3  2  πIα,R,  where α = arg δ and v = eiα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Fix α ∈ (0, π], R ⩾ 1 and ε1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Let ε = ε1/Kα,R where Kα,R is the  constant from Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' There exists 0 < rε ⩽ rz such that  ����  (Φ−1  δ )′(z)  (Φ−1  δ )′(w) − 1  ���� < ε,  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1)  for z, w ∈ B(0, rε) and δ ∈ B(0, r△).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  If z ∈ Nδ,R, then −fδ(z) is close to pδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Let nδ be the largest number  such that λnδ  δ Φδ(−fδ(z)) ⊂ B(0, rε) for all points z ∈ Nδ,R, where α = arg δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Since diam(Nδ,R) → 0 if δ → 0, we conclude that nδ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Formula (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='4) for m = nδ + 1 leads to  ˙ϕδ(s) = −  1  f′  δ(ϕδ(s))  �  1 +  nδ  �  k=1  1  (fk  δ )′(fδ(ϕδ(s)))  �  +  ˙ϕδ(T nδ+1(s))  (fnδ+1  δ  )′(ϕδ(s))  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='2)  Now we estimate finite sum from the above formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' If Φ(z) ∈ B(0, rε/λnδ  δ )  and 1 ⩽ k ⩽ nδ, then we have (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='4))  fk  δ (z) = Φ−1  δ (λk  δΦδ(z)),  and  (fk  δ )′(z) = (Φ−1  δ )′(λk  δΦδ(z)) · λk  δ · Φ′  δ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Therefore, (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1) gives us  ����  λk  δ  (fk  δ )′(z) − 1  ���� < ε  and  ����  1  (fk  δ )′(z) − 1  λk  δ  ���� <  ε  |λk  δ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='3)  Since nδ → ∞ and λδ → 4, we get  ����  nδ  �  k=1  1  λk  δ  − 1  3  ���� < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Thus, (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='3) and the above estimate, lead to  ����  nδ  �  k=1  1  (fk  δ )′(z) − 1  3  ���� < ε +  nδ  �  k=1  ε  |λk  δ| < 2ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  ON THE DIRECTIONAL DERIVATIVE OF THE HAUSDORFF DIMENSION  33  If z ∈ Nδ,R, then Φ(−fδ(z)) ∈ B(0, rε/λnδ  δ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Moreover (fk  δ )′(−fδ(z)) =  −(fk  δ )′(fδ(z)), so we have  ����  nδ  �  k=1  1  (fk  δ )′(fδ(z)) + 1  3  ���� < 2ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='4)  Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Now we estimate integral of a "tail" from the formula (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Distortion of fnδ+1  δ  on Nδ,R is bounded by a constant depending on α and  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Since diam(Nδ,R) < K1  �  |δ| and diam(fnδ+1  δ  (Nδ,R)) > K2ε, we get  ��(fnδ+1  δ  )′(z)  �� > K3(α, R, ε)  1  �  |δ|  ,  for z ∈ Nδ,R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' On the other hand |z| > K4  �  |δ| (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1), so if  ϕδ(s) = z ∈ Nδ,R, we obtain  ���� Re  �  v  ϕδ(s)  ˙ϕδ(T nδ+1(s))  (fnδ+1  δ  )′(ϕδ(s))  ����� < K5(α, R, ε)  �� ˙ϕδ(T nδ+1(s))  ��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Therefore, Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='4 leads to  |δ|1− 1  2 d(δ)  �  NR  ���� Re  �  v  ϕδ(s)  ˙ϕδ(T nδ+1(s))  (fnδ+1  δ  )′(ϕδ(s))  �����d ˜µδ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='5)  Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' We have f′  δ(z) = 2z, so (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='4) leads to  ���� Re  �v  z · −1  f′  δ(z)  �  1 +  nδ  �  k=1  1  (fk  δ )′(fδ(z))  ��  − Re  �  − v  2z2  2  3  ����� <  ε  |z|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Next, we integrate both sides over NR (where z = ϕδ(s)) with respect to  ˜µδ, and multiply by |δ|1− 1  2 d(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  So, formula (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='2) combined with (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='5),  Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1 and Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='2, gives us  ���� lim  δ→0 |δ|1− 1  2 d(δ)  �  NR  Re  �  v ˙ϕδ  ϕδ  �  d ˜µδ − 1  3  2  πIα,R  ���� < εKα,R = ε1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Because ε1 was arbitrary, the assertion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  □  Proof of the Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Fix α ∈ (0, π].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' We have (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='5), (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='6))  lim  R→∞ Iα,R = −  √  6 cos α +  √  6  2  √  sin α  π  �  α  �  sin β dβ := Iα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Let α = arg δ and let v = eiα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Then, Propositions 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='5 lead to  lim  δ→0 |δ|1− 1  2 d(δ)  �  Jδ0  Re  �  v ˙ϕδ  ϕδ  �  d ˜µδ = 1  3  2  πIα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='6)  Because f0 is conjugated to V , the entropy hµ0(f0) is equal to log 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Next,  we conclude from [12, Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1] that 1 = HD(µ0) = hµ0(f0)/χµ0(f0)  (where χµ0(f0) is the Lyapunov exponent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Therefore χµ0(f0) = log 2, hence  �  J0  log |f′  0|dµ0 = 4χµ0(f0) = 4 log 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  34  LUDWIK JAKSZTAS  Since f′  δ(ϕδ) = 2ϕδ, Propositions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='6, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='2, give us  lim  δ→0  �  Jδ0  log |f′  δ(ϕδ)|d˜µδ = 4 log 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Thus, formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='3) combined with (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='2), (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='6) and the above limit, leads  to  lim  δ→0 |δ|1− 1  2 d(δ) · d′  v(δ) = −  1  6π log 2Iα,  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='7)  where α = arg δ and let v = eiα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  In order to finish the proof, we have to replace 1 − d(δ)/2 by 1/2 in the  exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Because Iα is bounded, the above gives us  |d′  v(tv)| < K1 · t  1  2 d(tv)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Thus we have  ��d(tv) − 1  �� ⩽  t  �  0  ��d′  v(sv)  �� ds < K1  t  �  0  s  1  2 d(sv)−1ds < K1  t  �  0  s−2/3ds  = 3K1 · t1/3 < t1/4,  and then  1 ⩾ t|d(tv)−1| ⩾ tt1/4 = et1/4 log t → e0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  So, we obtain  lim  t→0+  t1− 1  2 d(tv)  t  1  2  = lim  t→0+ t  1  2 (1−d(tv)) = 1,  and finally (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='7) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1) lead to  lim  δ→0 |δ|  1  2 d′  v(δ) = −  1  6π log 2Iα = Ω−2(α),  where α = arg δ and let v = eiα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  □  References  [1] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Carleson, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Gamelin, Complex Dynamics, Springer-Verlag, New York, 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  [2] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Chen,  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Kawahira,  From  Cantor  to  semi-hyperbolic  parameter  along  external  rays,  Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  372  (2019)  7959–7992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1090/tran/7839.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  [3] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Dobbs, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Graczyk, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Mihalache, Hausdorff Dimension of Julia sets in the  logistic family, arXiv:2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='07661.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  [4] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Douady, Does a Julia set depend continuously on the polynomial?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=', in: R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' De-  vaney (ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' ), Complex dynamical systems, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Sympos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' 49, Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=', 1994, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' 91–135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1090/psapm/049/1315535.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  [5] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Dudko, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Gorbovickis, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Tucker, Lower bounds on the Hausdorff dimension  of some Julia sets, arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='07880.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  [6] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Fan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Jiang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Wu, Asymptotic Hausdorff dimensions of Cantor sets associ-  ated with an asymptotically non-hyperbolic family, Ergod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' & Dynam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Sys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' 25  (2005) 1799-1808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1017/S014338570500009X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  [7] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Havard, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Zinsmeister, Thermodynamic formalism and variations of the Haus-  dorff dimension of quadratic Julia sets, Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' 210 (2000) 225–247.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1007/s002200050778.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  [8] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Hubbard, Teichmüller Theory and Applications to Geometry, Topology and  Dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Teichmüller Theory, Matrix Editions, Ithaca, NY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  ON THE DIRECTIONAL DERIVATIVE OF THE HAUSDORFF DIMENSION  35  [9] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Jaksztas,  On  the  derivative  of  the  Hausdorff  dimension  of  the  qua-  dratic  Julia  sets,  Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  363  (2011)  5251–5291.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1090/S0002-9947-2011-05208-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  [10] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Jaksztas,  On  the  directonal  derivative  of  the  Hausdorff  dimension  of  quadratic polynomial Julia sets at 1/4, Nonlinearity 33 (2020) 5919–5960.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1088/1361-6544/ab9a1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  [11] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' de Melo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' van Strien, One-Dimensional Dynamics, Springer-Verlag, Berlin,  1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  [12] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Przytycki, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Urbanski, Fractals in the Plane, Ergodic Theory Methods, Cam-  bridge University press, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' 371, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  [13] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Rivera-Letelier, On the continuity of Hausdorff dimension of Julia sets and simi-  larity between the Mandelbrot set and Julia sets, Fund.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' 170 (2001) 287–317.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='4064/fm170-3-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  [14] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Ruelle, Repellers for real analytic maps, Ergod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' & Dynam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Sys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' 2 (1982)  99–107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='1017/S0143385700009603.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  [15] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Zinsmeister, Thermodynamic Formalism and Holomorphic Dynamical Systems,  SMF/AMS Texts and Monographs, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' 2, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='  Faculty of Mathematics and Information Science, Warsaw University of  Technology, ul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content=' Koszykowa 75, 00-662 Warsaw, Poland, ORCID: 0000-0002-  9283-8841  Email address: jaksztas@impan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
+page_content='pl' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfbwqk/content/2301.04519v1.pdf'}
diff --git a/Y9FRT4oBgHgl3EQfPDf9/content/tmp_files/2301.13516v1.pdf.txt b/Y9FRT4oBgHgl3EQfPDf9/content/tmp_files/2301.13516v1.pdf.txt
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@@ -0,0 +1,1173 @@
+Recurrences reveal shared causal drivers of complex time series
+William Gilpin∗
+Department of Physics, The University of Texas at Austin, Austin, Texas 78712, USA
+and
+Oden Institute for Computational Engineering and Sciences,
+The University of Texas at Austin, Austin, Texas 78712, USA
+(Dated: February 1, 2023)
+Many experimental time series measurements share an unobserved causal driver. Examples in-
+clude genes targeted by transcription factors, ocean flows influenced by large-scale atmospheric
+currents, and motor circuits steered by descending neurons. Reliably inferring this unseen driving
+force is necessary to understand the intermittent nature of top-down control schemes in diverse
+biological and engineered systems. Here, we introduce a new unsupervised learning algorithm that
+uses recurrences in time series measurements to gradually reconstruct an unobserved driving signal.
+Drawing on the mathematical theory of skew-product dynamical systems, we identify recurrence
+events shared across response time series, which implicitly define a recurrence graph with glass-like
+structure. As the amount or quality of observed data improves, this recurrence graph undergoes
+a percolation transition manifesting as weak ergodicity breaking for random walks on the induced
+landscape—revealing the shared driver’s dynamics, even in the presence of strongly corrupted or
+noisy measurements. Across several thousand random dynamical systems, we empirically quantify
+the dependence of reconstruction accuracy on the rate of information transfer from a chaotic driver
+to the response systems, and we find that effective reconstruction proceeds through gradual approx-
+imation of the driver’s dominant unstable periodic orbits. Through extensive benchmarks against
+classical and neural-network-based signal processing techniques, we demonstrate our method’s strong
+ability to extract causal driving signals from diverse real-world datasets spanning neuroscience, ge-
+nomics, fluid dynamics, and physiology.
+I.
+INTRODUCTION
+Diverse biological and engineered systems exhibit
+descending control,
+in which a central driving sig-
+nal causally influences the state of multiple response
+variables—which cannot, in turn, influence the driver.
+In ecological food webs, environmental changes trigger
+complex cascades of trophic interactions [1], while in an-
+imal nervous systems, descending neurons convey cogni-
+tive imperatives to motor circuits that implement behav-
+ior [2]. In systems biology, master regulators such as the
+Hox gene family orchestrate numerous complex gene reg-
+ulatory networks during development [3]. The ubiquity
+of top-down control may arise from the robustness and
+parsimony it confers relative to the number of dynamical
+interactions [4]; however, it also introduces experimen-
+tal challenges when the driving variable is not known a
+priori, or is not directly observable.
+From a computational perspective, the driving signal
+encodes information about itself into the states of the re-
+sponding subsystems. The coupling between the driver
+and responses acts as a lossy transmission channel—
+motivating recent methods for detection of causal interac-
+tions based on mutual information and transfer entropy
+[5–8]. Alternatively, causal relationships can be inferred
+based on properties of the dynamical system that gen-
+erates a time series: theorems from nonlinear dynam-
+ics suggest that partial information present in observ-
+able subsystems can reveal subregimes within the driv-
+∗ wgilpin@utexas.edu
+ing signal [9], motivating recent methods that determine
+causality based on attractor reconstruction or unsuper-
+vised learning of latent representations [10–14], which
+draw inspiration from classical results regarding the ob-
+servability of dynamical systems [15–17].
+However, while many existing methods detect the av-
+erage strength and directionality of causal interactions
+among known variables, resolving their temporal dynam-
+ics has received less attention. Yet understanding how
+causal forces change over time is necessary to under-
+stand control schemes, particularly in biological systems
+in which control signals are sparse and infrequent [18].
+Time-resolved driving signals may be detected based on
+signal denoising or assimilation techniques, or by general-
+izing statistical causality measures using time-dependent
+kernels [9, 19, 20]. However, these methods fail to lever-
+age existing theoretical works on skew-product dynami-
+cal systems and chaotic synchronization [21–23], which
+provide a strong inductive bias that may help obtain
+robust, physically-motivated representations even from
+highly corrupted measurements.
+Here, we introduce a general unsupervised learning
+method that reconstructs an unobserved driver time se-
+ries, given many partial or noisy observations of response
+time series. Based on the intuition that recurrences in re-
+sponse systems imply recurrences in the driver state, we
+traverse a network representation of the observed time
+series in order to identify unique driver states, allowing
+our method to identify temporal progressions of driver
+states even from highly nonlinear or noisy response mea-
+surements.
+Our method applies to both discrete- and
+continuous-time signals with either oscillatory or aperi-
+odic driving, and we show that its underlying accuracy
+arXiv:2301.13516v1  [cs.LG]  31 Jan 2023
+
+2
+corresponds to structural properties of the observed time
+series in the form of an ergodicity-breaking percolation
+transition in the time series adjacency graph. We exten-
+sively compare our technique against existing methods
+based on mutual information, artificial neural networks,
+and spectral transformations, and obtain strong results
+on diverse datasets spanning electrophysiology, hydrody-
+namics, and systems biology.
+Moreover, we find that
+our method’s accuracy arises from its ability to identify
+and approximate unstable periodic orbits that the driv-
+ing signal transiently shadows—linking our algorithm’s
+performance to topological properties of the underlying
+systems.
+II.
+APPROACH
+A.
+Recurrence structure of skew-product systems
+Suppose an unknown signal z(t) drives N response
+dynamical systems
+˙yk(t)
+=
+gk(z(t), yk(t), t),
+k
+∈
+{1, 2, ..., N}.
+Given a set of response signals {xk(t)}N
+comprising nonlinear measurements xk(t) = Mk ◦ yk(t)
+we seek a reconstruction of the original driver ˆz(t). In
+principle, this problem is underdetermined, because the
+measurement operator Mk may be noninvertible, or the
+internal dynamics of each response variable yk may only
+very weakly couple to the driving signal.
+However, if the driver state lies on an attractor Z,
+while the kth response signal lies on an attractor Yk,
+then the overall dynamics have skew product structure
+with combined attractor Z × Y1 × Y2 × ... × YN. While
+individual Yk are not directly observable, the skew prod-
+uct reconstruction theorem of Stark suggests that a one-
+to-one mapping ˆYk ∼ Yk may be constructed from the
+observed signal xk(t) by applying a lift φ(.) to the sig-
+nal using time delays ˆyk(t) = φ(xk(t)) = [xk(t), xk(t −
+τ), xk(t − 2τ), ..., xk(t − (D − 1)τ)]⊤, where D and τ are
+hyperparameters chosen using a variety of heuristics [15–
+17]. Nonlinear generalizations of the lifting map φ may
+be obtained with attractor reconstruction techniques like
+singular value decomposition, or even autoencoder neural
+networks [9, 24–26].
+Recent work by Sauer notes that the one-to-one map-
+ping between Yk and ˆYk requires that a recurrence in
+the kth reconstructed response ˆyk(t) ≈ ˆyk(t′), t ̸= t′
+implies a recurrence in the driving signal z(t) ≈ z(t′);
+however, the converse does not hold true due to the skew
+product structure [23]. Thus, tracking recurrence events
+in a time series driven by an ergodic dynamical system
+gradually reveals information about the driver attractor
+Z [14]. Observing more response signals provides more
+opportunities to observe recurrences.
+Sauer distills this insight into an exact algorithm
+for reconstructing a discrete-time periodic driver given
+noiseless response dynamics. Given a reconstructed re-
+sponse time series ˆyk(n), n ∈ 1, 2, ..., T, then near-
+recurrences correspond to two timepoints n and n′ where
+||ˆyk(n) − ˆyk(n′)|| ≤ ϵ, where ϵ is a small constant act-
+ing as an adjustable hyperparameter for the method.
+If a given response series ˆyk recurs at a set of times
+Rk = {t(k)
+1 , t(k)
+2 , ...}, and another response series ˆyℓ re-
+curs at a set of times Rℓ = {t(ℓ)
+1 , t(ℓ)
+2 , ...}, then if any
+timepoints are shared between Rk and Rℓ, then all time-
+points in either set belong to the same equivalence class
+and thus correspond to the same unique driver state. As
+a result, each one of the T timepoints observed across the
+N response time series can be labelled by their equiva-
+lence class, resulting in a time series of driver states.
+However, while this algorithm is motived by fundamen-
+tal properties of skew-product systems, it cannot readily
+be applied to most experimental time series because it
+requires discrete-time, low-period, and noise-free signals
+where each observed timepoint can unambiguously be as-
+signed to a single equivalence class.
+B.
+Generalizing recurrence detection for real-world
+time series
+We start by observing that the classical skew prod-
+uct reconstruction method can be interpreted as discov-
+ering disjoint sets within an aggregated time series ad-
+jacency graph: each lifted response time series ˆyk(n) ∈
+RT ×D has a distance graph d(ˆyk, ˆyk) ∈ RT ×T , where
+dm,n = ||ˆyk(m) − ˆyk(n)||. We can merge these N re-
+sponse time series graphs into a single, aggregated adja-
+cency graph d∗ ∈ RT ×T in which d∗
+ij = infk d(ˆyk, ˆyk)ij
+which can be discretized into a binary adjacency matrix
+Aij = Θ(ϵ − d∗
+ij) by applying step function centered at a
+thresholding hyperparameter ϵ. Identifying driver states
+therefore represents a graph partitioning problem on A,
+which can be solved in ≲ O(T 2) operations by finding
+the leading eigenvector of the associated graph Laplacian
+matrix.
+Our key insight is that skew product reconstruction
+exactly describes the the long-time behavior of an en-
+semble of random walkers navigating the time series ad-
+jacency graph. Distinct driver states induce partitions
+corresponding to discrete basins of attraction, or equiv-
+alence classes. The exact dynamics of skew product re-
+construction therefore represents a low-temperature limit
+of a more general set of algorithms for analyzing time
+series. From a dynamical systems perspective, the ag-
+gregated adjacency graph A defines the state transi-
+tion matrix for a Markov chain, which acts as a dis-
+crete Perron-Frobenius operator propagating distribu-
+tions of random walkers initialized at different nodes.
+Under this operator, the walker distribution asymptoti-
+cally evolves towards a stationary distribution associated
+with the largest eigenvalue of A, with sub-leading eigen-
+values describing transients. If the graph partitions into
+fully-disconnected subgraphs, walkers remain kinetically
+trapped within their basin, revealing an invariant mea-
+sure associated with the dynamics.
+However,
+for most real-world datasets containing
+
+3
+Causal Driver (unknown)
+Reconstructed Driver
+Observe random nonlinear dynamics 
+with missing values
+Distance Matrix Ensemble
+distance
+time
+Graph representation and 
+topological node ordering
+Figure 1.
+An example application of the shared dynamics reconstruction method.
+An unobserved driver influences an
+ensemble of dynamical systems, which each contain unique internal dynamics and random measurement filters. Here, we use
+the R¨ossler dynamical system to drive an ensemble of Lorenz systems with random parameters that have been filtered with
+random Gaussian response functions. Timepoint-wise recurrence graphs are separately calculated for each response, and then
+aggregated to produce a consensus adjacency graph. This graph is traversed using a diffusion process, the longest transient of
+which reconstructs the driver dynamics.
+noise or continuous-time dynamics, the transition matrix
+fails to exhibit fully disjoint structure due to spurious
+recurrences—leading to false adjacencies among nodes
+that prevent classical skew product reconstruction from
+succeeding. The leading eigenvector of the transition ma-
+trix instead approaches the uniform distribution at long
+times.
+However, some structure of the driving signal
+may remain encoded in transients associated with sub-
+leading eigenvectors—however, this information disap-
+pears at long times because ergodicity is not fully broken.
+These weaker partitions are known as “almost invariant
+sets” in fluid dynamics, and they correspond to subre-
+gions within the graph where random walkers become
+trapped for extended durations before slowly escaping
+into other subregions [27–30].
+We therefore calculate almost-invariant sets as a gener-
+alized shared dynamics reconstruction algorithm for ar-
+bitrary signals. In our implementation, we first embed
+each response time series separately, and then aggregate
+them into a timepoint-wise connectivity graph using a
+norm that balances the bias-variance tradeoff between
+sensitivity and spurious recurrences.
+We then identify
+almost-invariant sets using either spectral eigenmaps for
+a continuous time driver, or graph community detection
+for discrete time [31, 32]. We describe our implementa-
+tion in further detail in the supplementary appendix.
+Generalizations of time-delay embeddings for non-
+autonomous systems have previous been leveraged to in-
+fer causal structure using likelihood-based methods, par-
+ticularly in the context of ecological time series [33, 34].
+Our particular algorithm conceptually resembles prior
+works that aggregate time series graphs, and then iden-
+tify contiguous paths through the resulting network
+[35, 36]; as well as similar methods for spatiotemporal
+measurements based on Laplacian eigenmaps [31]. More
+broadly, our reconstruction process conceptually resem-
+bles pseudotime, a topological graph ordering method
+used in systems biology that infers developmental trajec-
+tories given gene expression measurements of unsorted
+cell populations [37].
+Our method differs from prior
+works on time series recurrence by explicitly treating re-
+sponse variables as distinct dynamical systems, which
+yield information about the driver only through recur-
+rences that gradually reveal the structure of the driver’s
+own recurrence network [38–40].
+In contrast, recent
+works on temporal Bayes filtration and neural data as-
+similation treat response observations as instantaneous
+nonlinear samples from a latent process [41–46].
+Our
+emphasis on incremental information gain from recur-
+rences distinguishes our approach from multiview and
+cross-embeddings [10–14, 47, 48], as well as generaliza-
+tions of static causal network inference to temporal net-
+works [9, 19, 20, 49, 50].
+We demonstrate the key steps of our method by us-
+ing the dynamics of the chaotic R¨ossler attractor to
+drive multiple realizations of the Lorenz equations with
+random dynamical parameters (Figure 1). In order to
+make the reconstruction task more difficult, we filter the
+Lorenz response dynamics through random Gaussian ker-
+nels, resulting in sparse, heavily-corrupted measurements
+that confound linear reconstruction methods like princi-
+pal components analysis.
+
+生9800088904
+A
+B
+Time
+Turbulence
+Electrocardiogram
+Gene Expression
+Neurons
+Accuracy
+Figure 2.
+The shared dynamics method discovers causal drivers in diverse datasets. (A) Example driver signals
+(blue) and their reconstructions (black) from response variables for four different experimental time series.
+(B) Spearman
+correlation scores for multiple possible reconstruction methods.
+Error bars denote bootstrapped one-sided standard errors
+around random model initializations trained on random subsets of the original time series.
+III.
+RESULTS
+A.
+The shared dynamics algorithm identifies causal
+drivers in diverse datasets
+In order to understand our method’s properties, we
+perform extensive benchmarks comparing our approach
+to similar methods. Beyond demonstrating our method’s
+strong performance as an off-the-shelf method for ex-
+ploratory data analysis, we aim to highlight how viewing
+time series through the lens of recurrences potentially
+represents a physically-motivated method for represent-
+ing arbitrary sequential datasets.
+For our experiments, we apply our method and others
+to a diverse datasets in which a known driving signal in-
+fluences a large, diverse set of response variables: (1) The
+turbulent motion of a set of diffusing tracer particles in
+a double gyre flow. The driver is long-wavelength sinu-
+soidal forcing, while the response variables comprise the
+radial coordinates of individual particles, as would be ob-
+served via Doppler velocimetry. (2) A fetal electrocardio-
+gram time series. The driver is the maternal respiration
+signal, while the response variables comprise multi-point
+electrocardiogram recordings placed across the fetal body
+[51]. (3) Stochastic gene expression dynamics in a subset
+of the yeast metabolic gene regulatory network, as gen-
+erated by an experiment design suite used in cell biology
+[52]. The driver corresponds to the expression level of a
+known transcriptional regulator in response to a knock-
+down perturbation, and the response states are the ex-
+pression levels of downstream genes in the network. (4)
+Spiking activities of neurons in the monkey primary mo-
+tor cortex, as the animal performs a reaching task [53].
+The driver is the reaching velocity, while the response
+variables comprise firing rates of individual neurons.
+We train unsupervised reconstruction models on only
+the response time series, and then compare the result-
+ing outputs to the true driving signal. We compare our
+method to a wide variety of alternative models: prin-
+cipal components analysis (PCA), independent compo-
+nents analysis (ICA), canonical correlation analysis be-
+tween past and future values (CCA), simple averaging
+
+1.0
+0.0
+gpfa
+ec
+nan
+shi
+m
+e1.0
+0.0
+ec
+Mean
+nan
+Kalm
+bo1.0
+0.0
+Mean
+Kalman1.0
+0.0
+Kalman
+Mean
+LFA5
+(Mean), a Kalman filter (Kalman), and a surrogate-
+weighted ensemble Fourier transform (fPCA)[9]. We also
+consider several newer dynamical decomposition meth-
+ods: dynamical components analysis (DCA), which finds
+low-dimensional modes that maximize predictive infor-
+mation [54]; slow-feature analysis (SFA), which seeks lin-
+ear combinations of nonlinear kernels that minimize time
+variation [55]; Gaussian-process factor analysis (GPFA),
+which finds latent factors parametrizing the mean of a
+one-step correlated Gaussian process [56]. We also con-
+sider artificial neural networks in the form of a causal con-
+volutional network (cCNN), an established architecture
+for time series representation learning [57]; and Latent
+Factor Analysis via Dynamical Systems (LFADS), a re-
+current variational autoencoder [58]. For all benchmark
+models we tune hyperparameters using cross-validation
+and grid search on a separate training dataset than the
+held-out testing data. We do not vary any hyperparam-
+eters for our own method.
+We use multiple accuracy
+metrics to compare the reconstructed driver ˆz(t) to the
+true driver z(t), including: mean-squared error, Spear-
+man correlation, covariance, symmetric mean absolute
+error, mean absolute scaled error, dynamic time warp-
+ing distance, mutual information, and nonlinear Granger
+causality; all yield comparable results, and so we report
+Spearman correlations here for clarity. Additional infor-
+mation about the benchmark experiment design is in-
+cluded in the supplementary material.
+Across all methods, we observe consistent strong per-
+formance of our approach relative to other methods—
+both conceptually and computationally simpler methods
+such as PCA or SFA, but also sophisticated deep-neural-
+network based time series models like LFADS (Figure 2).
+The strongest-performing benchmark model, the causal
+autoencoder neural network, takes the hierarchical and
+sequential structure of time series data into account [57],
+but otherwise does not specifically exploit potential skew
+product structure of the underlying time series. Addi-
+tionally, because our method requires only matrix de-
+composition and does not require extensive hyperparam-
+eter tuning, it is among the fastest to compute of the
+benchmark methods.
+Across the experimental datasets, we find that ape-
+riodic driving signals are consistently harder to recon-
+struct, with all tested methods performing weakly on
+the neuroscience and gene expression network datasets.
+We speculate that the prescence of multiple significant
+but widely-spaced timescales presents a more challenging
+problem setting, motivating our exploration of unstable
+periodic orbits below. Interestingly, while the turbulent
+flow dataset contains a monochromatic periodic driver,
+the stochastic component of the response systems masks
+the driving signal from detection by linear methods like
+the Fourier transform or PCA.
+We note that many of our example datasets represent
+cases in which the reconstructed driving signal ˆz(t) ex-
+erts nearly-instantaneous influence on the response dy-
+namics, allowing our method to reconstruct drivers that
+Figure 3.
+Correlation between chaoticity and ac-
+curacy across thousands of dynamical systems.
+(A)
+Thousands of random skew-product dynamical systems pro-
+duced by sampling pairs from a set of 132 named chaotic
+systems (e.g. Lorenz, Rossler, etc) and using one system as a
+driver for replicates of the other system. Linear fits with 95%
+bootstrapped confidence intervals highlight significant Spear-
+man correlations between the reconstruction accuracy and the
+Lyapunov exponents of the driver (red, ρ = 0.16 ± 0.03) and
+response (blue, ρ = −0.20 ± 0.03). (B) For each individual
+dynamical system, the average accuracy of a reconstruction
+in which it appears as a response variable, versus one in which
+it appears as a driver.
+qualitatively resemble the known driver z(t).
+In prin-
+ciple, however, our method can be applied to any dy-
+namical system where the response dynamics have the
+form ˙yk(t) = gk(yk(t), z(t), t). However, we focus on the
+synchronized regime z(t) ≈ ˆz(t) because a nonlinearly-
+transformed or time-delayed driver introduces ambiguity
+regarding the appropriate definition of the true driver.
+For example, for the system ˙x = x + h(z(t − τ)), the
+true driving signal for x(t) could be defined as z(t), h(t),
+or h(t − τ). As a result, it is challenging to construct a
+fair benchmark experiment for very weak, nonlinear, and
+time-delayed driving, particularly because methods such
+as PCA will tend to find driving terms that linearly and
+instantaneously influence the dynamics.
+B.
+Chaotic drivers hinder reconstruction, but
+chaotic responses improve it
+In order to further understand which systems are more
+difficult to reconstruct, we perform an additional bench-
+mark using our recently-introduced database of 132 low-
+dimensional ordinary differential equations [59]. Initially
+curated from published literature to include common sys-
+tems like the Lorenz, R¨ossler, and Chua attractors, the
+dataset has grown through crowdsourced contributions
+to contain diverse systems spanning domains such as as-
+trophysics, climatology, and biochemistry. Each system
+is annotated with estimates of descriptive mathematical
+properties such as maximum Lyapunov exponents, frac-
+tal dimensions, and entropy rates, which quantify differ-
+ent aspects of each system’s underlying complexity.
+Using
+this
+database,
+we
+create
+random
+skew-
+
+A
+B
+1.0
+max
+Response
+10-3
+10-1
+101
+e
+0.8
+Accuracy
+0.6
+Driver
+0.2
+0.1
+0.0
+0.0
+0.2
+0.4
+0.6
+1.0
+10.0
+0.8
+0.1
+1.0
+Lyapunov Exponent >
+Accuracy as Driver
+max6
+product systems in which one randomly-selected system
+drives multiple randomly-initialized versions of another
+randomly-selected system.
+The Lorenz-Rossler system
+featured in Figure 1 represents one such skew-product
+system from the ∼ 105 possible pairs.
+We scale the
+coupling strength to a constant multiple of the aver-
+age amplitude of both systems, and we scale integration
+timesteps and trajectory lengths based on precomputed
+estimates of Lipschitz constants and dominant timescales
+annotated with each system within our database.
+We
+sample all possible pairs, but retain time series samples
+only from skew systems that still exhibit chaotic dynam-
+ics. We apply our reconstruction method to each random
+skew product time series, and correlate reconstruction ac-
+curacy with the separate mathematical properties of the
+driver and response systems.
+We observe weak but consistent negative correlation
+between the driver Lyapunov exponent and reconstruc-
+tion accuracy, implying that more chaotic systems are
+more difficult to reconstruct (Figure 3A). Surprisingly, we
+observe the opposite for response systems: more chaotic
+response systems improve reconstruction quality. As a
+result, a given dynamical system that proves easier to
+reconstruct will tend to weaken reconstruction of other
+systems (Figure 3B).
+This discrepancy arises from the dual roles of the driver
+and response systems: more chaotic response dynamics
+produce more diverse and thus informative time series,
+leading to more unique recurrence events. Conversely, a
+nearly non-chaotic response system (λmax ≈ 0) yields
+only a finite amount of information over an extended
+duration, or across many replicates.
+Less chaotic sys-
+tems thus hinder reconstruction because they discover
+new information about the driver at a slower rate. Con-
+versely, more chaotic drivers transmit information at a
+faster rate, thus requiring more information to be gained
+from the response time series. This concept of the driver
+as an information source, and response systems as re-
+ceivers, mirrors classical formulations of skew-product
+systems [60]. The entropy production rate of a chaotic
+system is bounded from below by the largest Lyapunov
+exponent [61], which acts as a proxy for the information
+transmission rate between the driver and response. In
+the limit of a perfectly lossless transmission from driver
+to response system, the systems would exhibit general-
+ized synchronization, in which the response becomes a
+function of solely the driver state [62].
+C.
+Reconstruction accuracy requires a percolation
+transition in the time series adjacency graph.
+Our reconstruction method implicitly defines a diffu-
+sion process on the time series adjacency graph. In or-
+der to understand how this dynamical process couples
+to the dynamics of the driver itself, we apply our al-
+gorithm to response systems driven by the logistic map
+as it undergoes a series of period-doubling bifurcations
+Figure 4.
+A period-doubling bifurcation reveals that
+percolation loss precedes accurate reconstruction. (A)
+Driver reconstruction accuracy (adjusted Rand index) for re-
+sponses driven by the logistic map in its period-2 (blue),
+period-4 (turquoise), period-8 (red), and fully-chaotic (ma-
+genta) regimes; plotted from left to right as a function of the
+noise level, driver to response coupling strength, and number
+of response time series. In the last plot, dashed guides indi-
+cate null hypotheses for scaling of accuracy with dataset size,
+as predicted by the β-distribution. (Lower) The size of the
+largest connected component (a measure of percolation) in
+the reconstructed time series adjacency graph. All plots show
+median and interquartile range across 60 replicates with ran-
+dom initial conditions and algorithm seeds. (B) Changes in
+inferred adjacency graph of the period-8 sequence as the data
+noise is varied; frames correspond to points on the red trace
+marked with dots in the first panel.
+leading to chaos (Figure 4A). We track the point-wise
+accuracy of the reconstructed discrete driver using the
+adjusted Rand accuracy, which accounts for the permu-
+tation invariance of labels: an adjusted Rand accuracy
+of 0 implies a driver state labelling with no greater ac-
+curacy than random chance, while a score of one implies
+that every driver state is correctly identified for each of
+the 3 × 103 timepoints tested. We evaluate our method’s
+properties by recording its accuracy as a function of three
+control parameters: the relative amount of stochasticity
+present in the response dynamics (signal-to-noise ratio),
+the strength of coupling between the response dynam-
+ics and driver, and the number of response time series
+used for reconstruction N. In the latter case, we naively
+expect that reconstruction accuracy will increase propor-
+tionally to the total number of recurrences observed and
+thus N, resulting in an accuracy scaling given by the β-
+distribution (which describes the distribution of the ex-
+pected value of the mean of a binomial process, if driver
+recurrence states are discovered at random).
+We observe that shorter-period drivers are generally
+easier to reconstruct. Interestingly, at higher-period driv-
+ing rates, we observe a staircase pattern in the accu-
+
+A
+1.0
+Accuracy
+0.0
+Component
+Largest
+0.5
+0
+20
+10-2
+10-1
+101
+C
+102
+Signal-to-noise (dB)
+Coupling Strength (k)
+Number of time series (N)
+B7
+racy as reconstruction quality improves. This occurs be-
+cause, at lower accuracies and longer periods, our ap-
+proach fails to differentiate all driver states, resulting in
+groups of states merging together into a coarse-grained,
+lower-period driving signal. These clusters split into dis-
+tinct driver substates as reconstruction quality improves
+(Figure 4B).
+In order to understand how changes in method accu-
+racy depend on structural properties of the underlying re-
+currence graph, for each condition we measure the order
+parameter TLCC/T ∈ [0, 1], where T is the total number
+of nodes and TLCC is the number of nodes comprising the
+largest connected component. This order parameter de-
+tects percolation in finite-size undirected networks [63],
+and in our system it reveals a first-order phase transi-
+tion preceding an increase in accuracy when any of the
+three control parameters is varied. Intuitively, a highly-
+connected time series network contains many equivalent
+paths between two nodes, producing ambiguity regard-
+ing the correct assignment of recurrence events to driver
+states. This degeneracy below the percolation transition
+produces an underdetermined reconstruction, which be-
+comes determined as the amount or quality of response
+datasets increases.
+Considering a diffusion process on
+the recurrence graph, increasing any of the three con-
+trol parameters increases the duration of transients that
+reveal almost-invariant structure. The percolation tran-
+sition therefore indicates the onset of weak ergodicity
+breaking, in which the dynamics transition from having
+a short mixing time to a much longer time dominated
+by gradual leakage between highly-connected groups of
+nodes sharing the same driver state.
+Our observations mirror recent results reporting equiv-
+alency between the fitting accuracy of deep artificial neu-
+ral networks and the jamming transition [64].
+By de-
+creasing the ratio of data to model size, a deep neural
+network passes from underfitting to overfitting regimes
+mirroring a dilute-to-jammed transition. In our system,
+increasing training data size or effective size (by decreas-
+ing noise or increasing coupling) increases final accuracy
+by making the time series adjacency graph more dilute—
+thereby ensuring that well-defined distinctions emerge
+among different driver states. Our observation that high-
+period driver states initially appear as low-period states
+at the onset of the percolation transition is reminiscent
+of transformations of dynamical systems under the action
+of a renormalization operator [65].
+D.
+Reconstruction prioritizes dominant unstable
+periodic orbits.
+We next consider the process by which our method
+learns to reconstruct diverse signals. We return to our
+original demonstration, in which the R¨ossler dynamical
+equations drive many randomly-corrupted measurements
+of the Lorenz system. We repeat the reconstruction while
+varying the number of response series N, and thus the
+reconstruction accuracy.
+Given our method’s emphasis on inferring driver re-
+currence states, we speculate that its accuracy implicitly
+depends on the unstable periodic orbit spectrum of the
+driver [22]. Classical work on nonlinear dynamical sys-
+tems has shown that any chaotic system can be decom-
+posed into an infinite set of unstable periodic orbits [66],
+which trajectories shadow for extended durations propor-
+tional to each orbit’s relative stability. While chaotic sys-
+tems have continuous power spectra, the set of valid un-
+stable periodic orbits bears basic topological restrictions
+due to attractor geometry—making it a countable, al-
+beit infinite, hierarchical measure for the dynamical sys-
+tem. We compute several unstable periodic orbits of the
+R¨ossler driving system using the method of closest re-
+currences [67], and select the three most dominant orbits
+based on the relative duration they are shadowed by the
+driver dynamics.
+Comparing the pairwise distance matrix of the recon-
+structed signal with the distance matrix of individual un-
+stable periodic orbits (Figure 5) suggests that, as the
+amount of training data increases, the reconstructed dis-
+tance matrix initially approximates only the most dom-
+inant orbit, but then gradually approximates finer-scale
+details of the driver attractor that are encoded in sec-
+ondary orbits. We hypothesize that the recurrence struc-
+ture of our method biases it towards shorter-period and
+more stable orbits that dominate the graph partition.
+This phenomenon provides some context for our ear-
+lier observation that reconstruction accuracy depends in-
+versely on the driver’s Lyapunov exponent: systems with
+larger Lyapunov exponents tend to transition among a
+larger set of orbits, thus requiring more recurrence infor-
+mation to produce faithful reconstructions [68].
+IV.
+DISCUSSION
+We have presented a theoretically-motivated unsuper-
+vised learning method that reconstructs an unobserved
+causal driving signal, given observations of downstream
+response variables. Across extensive benchmarks against
+methods drawn from diverse domains, we find that our
+method quickly and robustly reconstructs diverse driv-
+ing signals in time series datasets ranging from fluid dy-
+namics to systems biology. Beyond introducing a new
+general-purpose data analysis method, our work provides
+empirical insight into the nature of information trans-
+mission in coupled dissipative systems: the topology of
+the driver’s characteristic unstable orbits manifests as
+structure within the recurrence graph of the response
+time series, with the efficiency of this linkage mediated
+by both the chaoticity of the driver, and the diversity
+of the responses.
+A diffusion process on the resulting
+time series graph reveals driver structure in its longest-
+lived transient modes, and methods that increase the life-
+time of these transients—such as more response datasets,
+increased driver-response coupling, or reduced measure-
+
+8
+Figure 5.
+Reconstruction of complex signals follows
+dominant unstable periodic orbits. (A) The three most
+stable unstable periodic orbits for the R¨ossler system. (B)
+Elementwise Pearson correlation between the distance matrix
+of the reconstructed R¨ossler driver and the three orbits, as
+the number of response time series (and thus overall driver
+reconstruction accuracy) increases. Ranges denote 95% con-
+fidence intervals around random model initializations trained
+on random subsets of available response series.(C) The over-
+laid true distance matrices of the three leading orbits, with
+RGB channels of the color image encoding each orbit’s indi-
+vidual distance matrix. (D) The distance matrices of the re-
+construction as the number of response subsystems increases.
+ment noise—produce a corresponding increase in recon-
+struction accuracy.
+Our diffusion process represents a
+heuristic solution to the difficult combinatorial optimiza-
+tion problem of graph traversal and topological node
+ordering, and we note that our method reduces to a
+previously-known exact algorithm in the limit of a noise-
+less and discrete periodic driver [23].
+Key limitations of our approach therefore stem from
+our use of diffusive dynamics for graph traversal. Certain
+measurement types may induce false recurrence events
+with systematic structure, such as sensor saturation or
+memory effects, that impose fixed structural barriers to
+diffusion on the graph adjacency matrix—akin to inho-
+mogenous diffusivity.
+Mitigating such effects requires
+sufficient information about the measurement dynamics
+to reconstruct and correct for response bias; within our
+framework, this might require preconditioning the graph
+Laplacian [69]. Additionally, non-stationary driving sig-
+nals can complicate reconstruction; for example if the
+attractor is time-dependent (e.g. energy gradually dissi-
+pates due to drag), then the driver may exhibit transient
+chaos before settling into a state of quiescence or oscil-
+lation [70]. In this case, a random walker’s transitions
+among nodes will no longer heed detailed balance and the
+transition operator will exhibit non-Hermitian structure,
+resulting in long-lived nonequilibrium steady-states that
+complicate driver state assignment [71–73]. These and
+other, more challenging data types may require nonlin-
+ear alternatives to diffusion processes on the recurrence
+graph. Our use of diffusion therefore represents a form of
+inductive bias in the time series representations learned
+by our model.
+However, the problem of discovering nonlinear pro-
+cesses from an empirical recurrence graph introduces a
+variety of potential generalizations grounded in recent
+results from statistical learning. Instead of directly fit-
+ting a linear operator, future work could leverage recent
+methods for data-driven inference of dynamical propaga-
+tors [74–77], which can even produce reduced-order ana-
+lytical models [78]. Generalizations may also incorporate
+new graph representation or traversal methods (such as
+message passing)[50, 79], particularly graph neural net-
+works [80, 81]. Thus, beyond our specific algorithm, we
+argue that our work motivates general parametrization of
+time series models in terms of recurrences, and the latent
+orbit structure that they encode.
+V.
+ACKNOWLEDGMENTS
+We
+thank
+Anthony
+Bao
+for
+feedback
+on
+the
+manuscript. W. G. was supported by the University of
+Texas at Austin.
+VI.
+CODE AVAILABILITY
+All code used in this study is available online at https:
+//github.com/williamgilpin/shrec
+[1] Rosenblatt, A. E. & Schmitz, O. J. Climate change, nu-
+trition, and bottom-up and top-down food web processes.
+Trends in Ecology & Evolution 31, 965–975 (2016).
+[2] Cande, J. et al.
+Optogenetic dissection of descending
+behavioral control in drosophila. Elife 7 (2018).
+[3] Leisner, M., Stingl, K., Frey, E. & Maier, B. Stochastic
+switching to competence. Current opinion in microbiol-
+ogy 11, 553–559 (2008).
+[4] Pezzulo, G. & Levin, M. Top-down models in biology:
+explanation and control of complex living systems above
+
+A
+C
+D
+N=
+人
+B
+upo 1
+upo 2
+Distance Matrix correlation
+upo3
+0.5
+N=300
+0.0
+3
+101
+102
+Number of response systems9
+the molecular level. Journal of The Royal Society Inter-
+face 13, 20160555 (2016).
+[5] Schwarze, A. C., Ichinaga, S. M. & Brunton, B. W.
+Network inference via process motifs for lagged cor-
+relation in linear stochastic processes.
+arXiv preprint
+arXiv:2208.08871 (2022).
+[6] Runge, J., Heitzig, J., Petoukhov, V. & Kurths, J. Escap-
+ing the curse of dimensionality in estimating multivari-
+ate transfer entropy. Physical review letters 108, 258701
+(2012).
+[7] Sugihara, G. et al. Detecting causality in complex ecosys-
+tems. science 338, 496–500 (2012).
+[8] Runge, J., Nowack, P., Kretschmer, M., Flaxman, S. &
+Sejdinovic, D. Detecting and quantifying causal associ-
+ations in large nonlinear time series datasets.
+Science
+Advances 5, eaau4996 (2019).
+[9] Kantz, H. & Schreiber, T. Nonlinear time series analysis,
+vol. 7 (Cambridge University Press, 2004).
+[10] Munch, S. B., Rogers, T. L. & Sugihara, G. Recent de-
+velopments in empirical dynamic modelling. Methods in
+Ecology and Evolution (2022).
+[11] Runge, J. et al. Inferring causation from time series in
+earth system sciences. Nature communications 10, 1–13
+(2019).
+[12] De Brouwer, E., Arany, A., Simm, J. & Moreau, Y. La-
+tent convergent cross mapping. In International Confer-
+ence on Learning Representations (2020).
+[13] Bennett, S. & Yu, R. Rethinking neural relational in-
+ference for granger causal discovery.
+In NeurIPS 2022
+Workshop on Causality for Real-world Impact (2022).
+[14] Quiroga, R. Q., Arnhold, J. & Grassberger, P. Learning
+driver-response relationships from synchronization pat-
+terns. Physical Review E 61, 5142 (2000).
+[15] Takens, F. Detecting strange attractors in turbulence. In
+Dynamical systems and turbulence, Warwick 1980, 366–
+381 (Springer, 1981).
+[16] Packard, N. H., Crutchfield, J. P., Farmer, J. D. & Shaw,
+R. S.
+Geometry from a time series.
+Physical Review
+Letters 45, 712 (1980).
+[17] Stark, J. Delay embeddings for forced systems. i. deter-
+ministic forcing. Journal of Nonlinear Science 9, 255–332
+(1999).
+[18] Sober, S. J., Sponberg, S., Nemenman, I. & Ting, L. H.
+Millisecond spike timing codes for motor control. Trends
+in neurosciences 41, 644–648 (2018).
+[19] Dhamala, M., Rangarajan, G. & Ding, M. Estimating
+granger causality from fourier and wavelet transforms
+of time series data. Physical review letters 100, 018701
+(2008).
+[20] Mastakouri, A. A., Sch¨olkopf, B. & Janzing, D.
+Nec-
+essary and sufficient conditions for causal feature selec-
+tion in time series with latent common causes.
+In In-
+ternational Conference on Machine Learning, 7502–7511
+(PMLR, 2021).
+[21] Pecora, L. M., Carroll, T. L., Johnson, G. A., Mar,
+D. J. & Heagy, J. F. Fundamentals of synchronization in
+chaotic systems, concepts, and applications. Chaos: An
+Interdisciplinary Journal of Nonlinear Science 7, 520–
+543 (1997).
+[22] Hunt, B. R., Ott, E. & Yorke, J. A. Differentiable gener-
+alized synchronization of chaos. Physical Review E 55,
+4029 (1997).
+[23] Sauer, T. D.
+Reconstruction of shared nonlinear dy-
+namics in a network. Physical review letters 93, 198701
+(2004).
+[24] Wehmeyer, C. & No´e, F.
+Time-lagged autoencoders:
+Deep learning of slow collective variables for molecular
+kinetics. The Journal of chemical physics 148, 241703
+(2018).
+[25] Gilpin, W.
+Deep reconstruction of strange attractors
+from time series. Advances in Neural Information Pro-
+cessing Systems 33 (2020).
+[26] Bakarji, J., Champion, K., Kutz, J. N. & Brunton, S. L.
+Discovering governing equations from partial measure-
+ments with deep delay autoencoders.
+arXiv preprint
+arXiv:2201.05136 (2022).
+[27] Froyland, G. & Padberg, K. Almost-invariant sets and in-
+variant manifolds—connecting probabilistic and geomet-
+ric descriptions of coherent structures in flows. Physica
+D: Nonlinear Phenomena 238, 1507–1523 (2009).
+[28] Brunton, S. L., Brunton, B. W., Proctor, J. L., Kaiser,
+E. & Kutz, J. N. Chaos as an intermittently forced linear
+system. Nature communications 8, 1–9 (2017).
+[29] Mezi´c, I.
+Spectral properties of dynamical systems,
+model reduction and decompositions. Nonlinear Dynam-
+ics 41, 309–325 (2005).
+[30] Costa, A. C., Ahamed, T., Jordan, D. & Stephens,
+G. Maximally predictive ensemble dynamics from data.
+arXiv preprint arXiv:2105.12811 (2021).
+[31] Giannakis, D. & Majda, A. J.
+Nonlinear laplacian
+spectral analysis for time series with intermittency and
+low-frequency variability.
+Proceedings of the National
+Academy of Sciences 109, 2222–2227 (2012).
+[32] Girvan, M. & Newman, M. E. Community structure in
+social and biological networks.
+Proceedings of the na-
+tional academy of sciences 99, 7821–7826 (2002).
+[33] Rogers, T. L. & Munch, S. B. Hidden similarities in the
+dynamics of a weakly synchronous marine metapopula-
+tion. Proceedings of the National Academy of Sciences
+117, 479–485 (2020).
+[34] Deyle, E. R., Maher, M. C., Hernandez, R. D., Basu, S. &
+Sugihara, G. Global environmental drivers of influenza.
+Proceedings of the National Academy of Sciences 113,
+13081–13086 (2016).
+[35] Hirata, Y., Horai, S. & Aihara, K.
+Reproduction of
+distance matrices and original time series from recur-
+rence plots and their applications. The European Physical
+Journal Special Topics 164, 13–22 (2008).
+[36] Hirata, Y. & Aihara, K.
+Identifying hidden common
+causes from bivariate time series: A method using re-
+currence plots. Physical Review E 81, 016203 (2010).
+[37] Haghverdi, L., B¨uttner, M., Wolf, F. A., Buettner, F. &
+Theis, F. J. Diffusion pseudotime robustly reconstructs
+lineage branching. Nature methods 13, 845–848 (2016).
+[38] Donner, R. V., Zou, Y., Donges, J. F., Marwan, N. &
+Kurths, J. Recurrence networks—a novel paradigm for
+nonlinear time series analysis. New Journal of Physics
+12, 033025 (2010).
+[39] Davidsen, J., Grassberger, P. & Paczuski, M. Networks of
+recurrent events, a theory of records, and an application
+to finding causal signatures in seismicity. Physical Review
+E 77, 066104 (2008).
+[40] Myers, A., Munch, E. & Khasawneh, F. A. Persistent ho-
+mology of complex networks for dynamic state detection.
+Physical Review E 100, 022314 (2019).
+[41] McCabe, M. & Brown, J.
+Learning to assimilate in
+chaotic dynamical systems.
+Advances in Neural Infor-
+mation Processing Systems 34, 12237–12250 (2021).
+
+10
+[42] Frerix, T. et al.
+Variational data assimilation with a
+learned inverse observation operator.
+In International
+Conference on Machine Learning, 3449–3458 (PMLR,
+2021).
+[43] Karl, M., Soelch, M., Bayer, J. & Van der Smagt, P.
+Deep variational bayes filters:
+Unsupervised learning
+of state space models from raw data.
+arXiv preprint
+arXiv:1605.06432 (2016).
+[44] Krishnan, R. G., Shalit, U. & Sontag, D. Deep kalman
+filters. arXiv preprint arXiv:1511.05121 (2015).
+[45] Koppe, G., Toutounji, H., Kirsch, P., Lis, S. & Durste-
+witz, D.
+Identifying nonlinear dynamical systems via
+generative recurrent neural networks with applications to
+fmri. PLoS computational biology 15, e1007263 (2019).
+[46] Hamilton, F., Berry, T. & Sauer, T. Ensemble kalman
+filtering without a model. Physical Review X 6, 011021
+(2016).
+[47] Ye, H. & Sugihara, G. Information leverage in intercon-
+nected ecosystems: Overcoming the curse of dimension-
+ality. Science 353, 922–925 (2016).
+[48] Leng, S. et al.
+Partial cross mapping eliminates indi-
+rect causal influences. Nature communications 11, 1–9
+(2020).
+[49] Alaa, A. & Van Der Schaar, M.
+Validating causal in-
+ference models via influence functions. In International
+Conference on Machine Learning,
+191–201 (PMLR,
+2019).
+[50] Williams, O. E., Lacasa, L., Mill´an, A. P. & Latora, V.
+The shape of memory in temporal networks. Nature com-
+munications 13, 1–8 (2022).
+[51] Sulas, E. et al. A non-invasive multimodal foetal ecg–
+doppler dataset for antenatal cardiology research. Scien-
+tific Data 8, 1–19 (2021).
+[52] Schaffter,
+T.,
+Marbach,
+D.
+&
+Floreano,
+D.
+Genenetweaver:
+in silico benchmark generation and
+performance profiling of network inference methods.
+Bioinformatics 27, 2263–2270 (2011).
+[53] Pei, F. et al. Neural latents benchmark’21: Evaluating la-
+tent variable models of neural population activity. arXiv
+preprint arXiv:2109.04463 (2021).
+[54] Clark, D., Livezey, J. & Bouchard, K. Unsupervised dis-
+covery of temporal structure in noisy data with dynami-
+cal components analysis. Advances in Neural Information
+Processing Systems 32 (2019).
+[55] Wiskott, L. & Sejnowski, T. J.
+Slow feature analysis:
+Unsupervised learning of invariances. Neural computa-
+tion 14, 715–770 (2002).
+[56] Yu, B. M. et al. Gaussian-process factor analysis for low-
+dimensional single-trial analysis of neural population ac-
+tivity. Advances in neural information processing systems
+21 (2008).
+[57] Franceschi, J.-Y., Dieuleveut, A. & Jaggi, M.
+Unsu-
+pervised scalable representation learning for multivariate
+time series. Advances in neural information processing
+systems 32 (2019).
+[58] Pandarinath, C. et al. Inferring single-trial neural popu-
+lation dynamics using sequential auto-encoders. Nature
+methods 15, 805–815 (2018).
+[59] Gilpin, W. Chaos as an interpretable benchmark for fore-
+casting and data-driven modelling. Advances in Neural
+Information Processing Systems 1 (2021).
+[60] Adler, R., Coppersmith, D. & Hassner, M. Algorithms
+for sliding block codes-an application of symbolic dynam-
+ics to information theory. IEEE Transactions on Infor-
+mation Theory 29, 5–22 (1983).
+[61] Pesin, Y. Characteristic lyapunov exponents and smooth
+ergodic theory.
+Russian Mathematical Surveys 32, 55
+(1977).
+[62] Pecora, L. M. & Carroll, T. L. Synchronization in chaotic
+systems. Physical Review Letters 64, 821 (1990).
+[63] H´ebert-Dufresne, L. & Allard, A. Smeared phase transi-
+tions in percolation on real complex networks. Physical
+Review Research 1, 013009 (2019).
+[64] Geiger, M. et al. Jamming transition as a paradigm to
+understand the loss landscape of deep neural networks.
+Physical Review E 100, 012115 (2019).
+[65] Feigenbaum, M. J.
+The universal metric properties of
+nonlinear transformations. Journal of Statistical Physics
+21, 669–706 (1979).
+[66] Cvitanovi´c, P. Invariant measurement of strange sets in
+terms of cycles. Physical Review Letters 61, 2729 (1988).
+[67] Lathrop, D. P. & Kostelich, E. J. Characterization of an
+experimental strange attractor by periodic orbits. Phys-
+ical Review A 40, 4028 (1989).
+[68] Ott, E. Chaos in dynamical systems (Cambridge Univer-
+sity Press, 2002).
+[69] Krishnan, D., Fattal, R. & Szeliski, R. Efficient precondi-
+tioning of laplacian matrices for computer graphics. ACM
+transactions on Graphics (tOG) 32, 1–15 (2013).
+[70] Motter, A. E., Gruiz, M., K´arolyi, G. & T´el, T. Doubly
+transient chaos: Generic form of chaos in autonomous
+dissipative systems. Physical review letters 111, 194101
+(2013).
+[71] Fruchart, M., Hanai, R., Littlewood, P. B. & Vitelli, V.
+Non-reciprocal phase transitions. Nature 592, 363–369
+(2021).
+[72] Battle, C. et al. Broken detailed balance at mesoscopic
+scales in active biological systems. Science 352, 604–607
+(2016).
+[73] Weis, C. et al.
+Coalescence of attractors: exceptional
+points in non-linear dynamical systems. arXiv preprint
+arXiv:2207.11667 (2022).
+[74] Bramburger, J. J., Kutz, J. N. & Brunton, S. L. Data-
+driven stabilization of periodic orbits. IEEE Access 9,
+43504–43521 (2021).
+[75] Raissi, M. & Karniadakis, G. E. Hidden physics models:
+Machine learning of nonlinear partial differential equa-
+tions. Journal of Computational Physics 357, 125–141
+(2018).
+[76] Chen, R. T., Rubanova, Y., Bettencourt, J. & Duvenaud,
+D. K. Neural ordinary differential equations. Advances
+in neural information processing systems 31 (2018).
+[77] Choudhary, A. et al. Physics-enhanced neural networks
+learn order and chaos. Physical Review E 101, 062207
+(2020).
+[78] Champion, K., Lusch, B., Kutz, J. N. & Brunton, S. L.
+Data-driven discovery of coordinates and governing equa-
+tions. Proceedings of the National Academy of Sciences
+116, 22445–22451 (2019).
+[79] Gilmer, J., Schoenholz, S. S., Riley, P. F., Vinyals, O. &
+Dahl, G. E. Neural message passing for quantum chem-
+istry. In International conference on machine learning,
+1263–1272 (PMLR, 2017).
+[80] Donoho, D. L., Maleki, A. & Montanari, A. Message-
+passing algorithms for compressed sensing. Proceedings
+of the National Academy of Sciences 106, 18914–18919
+(2009).
+
+11
+[81] Rossi, E. et al. Temporal graph networks for deep learn-
+ing on dynamic graphs. arXiv preprint arXiv:2006.10637
+(2020).
+
diff --git a/Y9FRT4oBgHgl3EQfPDf9/content/tmp_files/load_file.txt b/Y9FRT4oBgHgl3EQfPDf9/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf,len=725
+page_content='Recurrences reveal shared causal drivers of complex time series  William Gilpin∗  Department of Physics, The University of Texas at Austin, Austin, Texas 78712, USA  and  Oden Institute for Computational Engineering and Sciences,  The University of Texas at Austin, Austin, Texas 78712, USA  (Dated: February 1, 2023)  Many experimental time series measurements share an unobserved causal driver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Examples in-  clude genes targeted by transcription factors, ocean flows influenced by large-scale atmospheric  currents, and motor circuits steered by descending neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Reliably inferring this unseen driving  force is necessary to understand the intermittent nature of top-down control schemes in diverse  biological and engineered systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Here, we introduce a new unsupervised learning algorithm that  uses recurrences in time series measurements to gradually reconstruct an unobserved driving signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Drawing on the mathematical theory of skew-product dynamical systems, we identify recurrence  events shared across response time series, which implicitly define a recurrence graph with glass-like  structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' As the amount or quality of observed data improves, this recurrence graph undergoes  a percolation transition manifesting as weak ergodicity breaking for random walks on the induced  landscape—revealing the shared driver’s dynamics, even in the presence of strongly corrupted or  noisy measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Across several thousand random dynamical systems, we empirically quantify  the dependence of reconstruction accuracy on the rate of information transfer from a chaotic driver  to the response systems, and we find that effective reconstruction proceeds through gradual approx-  imation of the driver’s dominant unstable periodic orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Through extensive benchmarks against  classical and neural-network-based signal processing techniques, we demonstrate our method’s strong  ability to extract causal driving signals from diverse real-world datasets spanning neuroscience, ge-  nomics, fluid dynamics, and physiology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  INTRODUCTION  Diverse biological and engineered systems exhibit  descending control,  in which a central driving sig-  nal causally influences the state of multiple response  variables—which cannot, in turn, influence the driver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  In ecological food webs, environmental changes trigger  complex cascades of trophic interactions [1], while in an-  imal nervous systems, descending neurons convey cogni-  tive imperatives to motor circuits that implement behav-  ior [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' In systems biology, master regulators such as the  Hox gene family orchestrate numerous complex gene reg-  ulatory networks during development [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' The ubiquity  of top-down control may arise from the robustness and  parsimony it confers relative to the number of dynamical  interactions [4];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' however, it also introduces experimen-  tal challenges when the driving variable is not known a  priori, or is not directly observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  From a computational perspective, the driving signal  encodes information about itself into the states of the re-  sponding subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' The coupling between the driver  and responses acts as a lossy transmission channel—  motivating recent methods for detection of causal interac-  tions based on mutual information and transfer entropy  [5–8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Alternatively, causal relationships can be inferred  based on properties of the dynamical system that gen-  erates a time series: theorems from nonlinear dynam-  ics suggest that partial information present in observ-  able subsystems can reveal subregimes within the driv-  ∗ wgilpin@utexas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='edu  ing signal [9], motivating recent methods that determine  causality based on attractor reconstruction or unsuper-  vised learning of latent representations [10–14], which  draw inspiration from classical results regarding the ob-  servability of dynamical systems [15–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  However, while many existing methods detect the av-  erage strength and directionality of causal interactions  among known variables, resolving their temporal dynam-  ics has received less attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Yet understanding how  causal forces change over time is necessary to under-  stand control schemes, particularly in biological systems  in which control signals are sparse and infrequent [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Time-resolved driving signals may be detected based on  signal denoising or assimilation techniques, or by general-  izing statistical causality measures using time-dependent  kernels [9, 19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' However, these methods fail to lever-  age existing theoretical works on skew-product dynami-  cal systems and chaotic synchronization [21–23], which  provide a strong inductive bias that may help obtain  robust, physically-motivated representations even from  highly corrupted measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Here, we introduce a general unsupervised learning  method that reconstructs an unobserved driver time se-  ries, given many partial or noisy observations of response  time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Based on the intuition that recurrences in re-  sponse systems imply recurrences in the driver state, we  traverse a network representation of the observed time  series in order to identify unique driver states, allowing  our method to identify temporal progressions of driver  states even from highly nonlinear or noisy response mea-  surements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Our method applies to both discrete- and  continuous-time signals with either oscillatory or aperi-  odic driving, and we show that its underlying accuracy  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='13516v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='LG]  31 Jan 2023  2  corresponds to structural properties of the observed time  series in the form of an ergodicity-breaking percolation  transition in the time series adjacency graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' We exten-  sively compare our technique against existing methods  based on mutual information, artificial neural networks,  and spectral transformations, and obtain strong results  on diverse datasets spanning electrophysiology, hydrody-  namics, and systems biology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Moreover, we find that  our method’s accuracy arises from its ability to identify  and approximate unstable periodic orbits that the driv-  ing signal transiently shadows—linking our algorithm’s  performance to topological properties of the underlying  systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  APPROACH  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Recurrence structure of skew-product systems  Suppose an unknown signal z(t) drives N response  dynamical systems  ˙yk(t)  =  gk(z(t), yk(t), t),  k  ∈  {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Given a set of response signals {xk(t)}N  comprising nonlinear measurements xk(t) = Mk ◦ yk(t)  we seek a reconstruction of the original driver ˆz(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' In  principle, this problem is underdetermined, because the  measurement operator Mk may be noninvertible, or the  internal dynamics of each response variable yk may only  very weakly couple to the driving signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  However, if the driver state lies on an attractor Z,  while the kth response signal lies on an attractor Yk,  then the overall dynamics have skew product structure  with combined attractor Z × Y1 × Y2 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' × YN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' While  individual Yk are not directly observable, the skew prod-  uct reconstruction theorem of Stark suggests that a one-  to-one mapping ˆYk ∼ Yk may be constructed from the  observed signal xk(t) by applying a lift φ(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=') to the sig-  nal using time delays ˆyk(t) = φ(xk(t)) = [xk(t), xk(t −  τ), xk(t − 2τ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', xk(t − (D − 1)τ)]⊤, where D and τ are  hyperparameters chosen using a variety of heuristics [15–  17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Nonlinear generalizations of the lifting map φ may  be obtained with attractor reconstruction techniques like  singular value decomposition, or even autoencoder neural  networks [9, 24–26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Recent work by Sauer notes that the one-to-one map-  ping between Yk and ˆYk requires that a recurrence in  the kth reconstructed response ˆyk(t) ≈ ˆyk(t′), t ̸= t′  implies a recurrence in the driving signal z(t) ≈ z(t′);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  however, the converse does not hold true due to the skew  product structure [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Thus, tracking recurrence events  in a time series driven by an ergodic dynamical system  gradually reveals information about the driver attractor  Z [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Observing more response signals provides more  opportunities to observe recurrences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Sauer distills this insight into an exact algorithm  for reconstructing a discrete-time periodic driver given  noiseless response dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Given a reconstructed re-  sponse time series ˆyk(n), n ∈ 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', T, then near-  recurrences correspond to two timepoints n and n′ where  ||ˆyk(n) − ˆyk(n′)|| ≤ ϵ, where ϵ is a small constant act-  ing as an adjustable hyperparameter for the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  If a given response series ˆyk recurs at a set of times  Rk = {t(k)  1 , t(k)  2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='}, and another response series ˆyℓ re-  curs at a set of times Rℓ = {t(ℓ)  1 , t(ℓ)  2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='}, then if any  timepoints are shared between Rk and Rℓ, then all time-  points in either set belong to the same equivalence class  and thus correspond to the same unique driver state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' As  a result, each one of the T timepoints observed across the  N response time series can be labelled by their equiva-  lence class, resulting in a time series of driver states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  However, while this algorithm is motived by fundamen-  tal properties of skew-product systems, it cannot readily  be applied to most experimental time series because it  requires discrete-time, low-period, and noise-free signals  where each observed timepoint can unambiguously be as-  signed to a single equivalence class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Generalizing recurrence detection for real-world  time series  We start by observing that the classical skew prod-  uct reconstruction method can be interpreted as discov-  ering disjoint sets within an aggregated time series ad-  jacency graph: each lifted response time series ˆyk(n) ∈  RT ×D has a distance graph d(ˆyk, ˆyk) ∈ RT ×T , where  dm,n = ||ˆyk(m) − ˆyk(n)||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' We can merge these N re-  sponse time series graphs into a single, aggregated adja-  cency graph d∗ ∈ RT ×T in which d∗  ij = infk d(ˆyk, ˆyk)ij  which can be discretized into a binary adjacency matrix  Aij = Θ(ϵ − d∗  ij) by applying step function centered at a  thresholding hyperparameter ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Identifying driver states  therefore represents a graph partitioning problem on A,  which can be solved in ≲ O(T 2) operations by finding  the leading eigenvector of the associated graph Laplacian  matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Our key insight is that skew product reconstruction  exactly describes the the long-time behavior of an en-  semble of random walkers navigating the time series ad-  jacency graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Distinct driver states induce partitions  corresponding to discrete basins of attraction, or equiv-  alence classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' The exact dynamics of skew product re-  construction therefore represents a low-temperature limit  of a more general set of algorithms for analyzing time  series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' From a dynamical systems perspective, the ag-  gregated adjacency graph A defines the state transi-  tion matrix for a Markov chain, which acts as a dis-  crete Perron-Frobenius operator propagating distribu-  tions of random walkers initialized at different nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Under this operator, the walker distribution asymptoti-  cally evolves towards a stationary distribution associated  with the largest eigenvalue of A, with sub-leading eigen-  values describing transients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' If the graph partitions into  fully-disconnected subgraphs, walkers remain kinetically  trapped within their basin, revealing an invariant mea-  sure associated with the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  However,  for most real-world datasets containing  3  Causal Driver (unknown)  Reconstructed Driver  Observe random nonlinear dynamics  with missing values  Distance Matrix Ensemble  distance  time  Graph representation and  topological node ordering  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  An example application of the shared dynamics reconstruction method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  An unobserved driver influences an  ensemble of dynamical systems, which each contain unique internal dynamics and random measurement filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Here, we use  the R¨ossler dynamical system to drive an ensemble of Lorenz systems with random parameters that have been filtered with  random Gaussian response functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Timepoint-wise recurrence graphs are separately calculated for each response, and then  aggregated to produce a consensus adjacency graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' This graph is traversed using a diffusion process, the longest transient of  which reconstructs the driver dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  noise or continuous-time dynamics, the transition matrix  fails to exhibit fully disjoint structure due to spurious  recurrences—leading to false adjacencies among nodes  that prevent classical skew product reconstruction from  succeeding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' The leading eigenvector of the transition ma-  trix instead approaches the uniform distribution at long  times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  However, some structure of the driving signal  may remain encoded in transients associated with sub-  leading eigenvectors—however, this information disap-  pears at long times because ergodicity is not fully broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  These weaker partitions are known as “almost invariant  sets” in fluid dynamics, and they correspond to subre-  gions within the graph where random walkers become  trapped for extended durations before slowly escaping  into other subregions [27–30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  We therefore calculate almost-invariant sets as a gener-  alized shared dynamics reconstruction algorithm for ar-  bitrary signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' In our implementation, we first embed  each response time series separately, and then aggregate  them into a timepoint-wise connectivity graph using a  norm that balances the bias-variance tradeoff between  sensitivity and spurious recurrences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  We then identify  almost-invariant sets using either spectral eigenmaps for  a continuous time driver, or graph community detection  for discrete time [31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' We describe our implementa-  tion in further detail in the supplementary appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Generalizations of time-delay embeddings for non-  autonomous systems have previous been leveraged to in-  fer causal structure using likelihood-based methods, par-  ticularly in the context of ecological time series [33, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Our particular algorithm conceptually resembles prior  works that aggregate time series graphs, and then iden-  tify contiguous paths through the resulting network  [35, 36];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' as well as similar methods for spatiotemporal  measurements based on Laplacian eigenmaps [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' More  broadly, our reconstruction process conceptually resem-  bles pseudotime, a topological graph ordering method  used in systems biology that infers developmental trajec-  tories given gene expression measurements of unsorted  cell populations [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Our method differs from prior  works on time series recurrence by explicitly treating re-  sponse variables as distinct dynamical systems, which  yield information about the driver only through recur-  rences that gradually reveal the structure of the driver’s  own recurrence network [38–40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  In contrast, recent  works on temporal Bayes filtration and neural data as-  similation treat response observations as instantaneous  nonlinear samples from a latent process [41–46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Our  emphasis on incremental information gain from recur-  rences distinguishes our approach from multiview and  cross-embeddings [10–14, 47, 48], as well as generaliza-  tions of static causal network inference to temporal net-  works [9, 19, 20, 49, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  We demonstrate the key steps of our method by us-  ing the dynamics of the chaotic R¨ossler attractor to  drive multiple realizations of the Lorenz equations with  random dynamical parameters (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' In order to  make the reconstruction task more difficult, we filter the  Lorenz response dynamics through random Gaussian ker-  nels, resulting in sparse, heavily-corrupted measurements  that confound linear reconstruction methods like princi-  pal components analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  生9800088904  A  B  Time  Turbulence  Electrocardiogram  Gene Expression  Neurons  Accuracy  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  The shared dynamics method discovers causal drivers in diverse datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' (A) Example driver signals  (blue) and their reconstructions (black) from response variables for four different experimental time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  (B) Spearman  correlation scores for multiple possible reconstruction methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Error bars denote bootstrapped one-sided standard errors  around random model initializations trained on random subsets of the original time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  RESULTS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  The shared dynamics algorithm identifies causal  drivers in diverse datasets  In order to understand our method’s properties, we  perform extensive benchmarks comparing our approach  to similar methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Beyond demonstrating our method’s  strong performance as an off-the-shelf method for ex-  ploratory data analysis, we aim to highlight how viewing  time series through the lens of recurrences potentially  represents a physically-motivated method for represent-  ing arbitrary sequential datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  For our experiments, we apply our method and others  to a diverse datasets in which a known driving signal in-  fluences a large, diverse set of response variables: (1) The  turbulent motion of a set of diffusing tracer particles in  a double gyre flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' The driver is long-wavelength sinu-  soidal forcing, while the response variables comprise the  radial coordinates of individual particles, as would be ob-  served via Doppler velocimetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' (2) A fetal electrocardio-  gram time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' The driver is the maternal respiration  signal, while the response variables comprise multi-point  electrocardiogram recordings placed across the fetal body  [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' (3) Stochastic gene expression dynamics in a subset  of the yeast metabolic gene regulatory network, as gen-  erated by an experiment design suite used in cell biology  [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' The driver corresponds to the expression level of a  known transcriptional regulator in response to a knock-  down perturbation, and the response states are the ex-  pression levels of downstream genes in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' (4)  Spiking activities of neurons in the monkey primary mo-  tor cortex, as the animal performs a reaching task [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  The driver is the reaching velocity, while the response  variables comprise firing rates of individual neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  We train unsupervised reconstruction models on only  the response time series, and then compare the result-  ing outputs to the true driving signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' We compare our  method to a wide variety of alternative models: prin-  cipal components analysis (PCA), independent compo-  nents analysis (ICA), canonical correlation analysis be-  tween past and future values (CCA), simple averaging  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='0  gpfa  ec  nan  shi  m  e1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='0  ec  Mean  nan  Kalm  bo1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='0  Mean  Kalman1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='0  Kalman  Mean  LFA5  (Mean), a Kalman filter (Kalman), and a surrogate-  weighted ensemble Fourier transform (fPCA)[9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' We also  consider several newer dynamical decomposition meth-  ods: dynamical components analysis (DCA), which finds  low-dimensional modes that maximize predictive infor-  mation [54];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' slow-feature analysis (SFA), which seeks lin-  ear combinations of nonlinear kernels that minimize time  variation [55];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Gaussian-process factor analysis (GPFA),  which finds latent factors parametrizing the mean of a  one-step correlated Gaussian process [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' We also con-  sider artificial neural networks in the form of a causal con-  volutional network (cCNN), an established architecture  for time series representation learning [57];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' and Latent  Factor Analysis via Dynamical Systems (LFADS), a re-  current variational autoencoder [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' For all benchmark  models we tune hyperparameters using cross-validation  and grid search on a separate training dataset than the  held-out testing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' We do not vary any hyperparam-  eters for our own method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  We use multiple accuracy  metrics to compare the reconstructed driver ˆz(t) to the  true driver z(t), including: mean-squared error, Spear-  man correlation, covariance, symmetric mean absolute  error, mean absolute scaled error, dynamic time warp-  ing distance, mutual information, and nonlinear Granger  causality;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' all yield comparable results, and so we report  Spearman correlations here for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Additional infor-  mation about the benchmark experiment design is in-  cluded in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Across all methods, we observe consistent strong per-  formance of our approach relative to other methods—  both conceptually and computationally simpler methods  such as PCA or SFA, but also sophisticated deep-neural-  network based time series models like LFADS (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  The strongest-performing benchmark model, the causal  autoencoder neural network, takes the hierarchical and  sequential structure of time series data into account [57],  but otherwise does not specifically exploit potential skew  product structure of the underlying time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Addi-  tionally, because our method requires only matrix de-  composition and does not require extensive hyperparam-  eter tuning, it is among the fastest to compute of the  benchmark methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Across the experimental datasets, we find that ape-  riodic driving signals are consistently harder to recon-  struct, with all tested methods performing weakly on  the neuroscience and gene expression network datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  We speculate that the prescence of multiple significant  but widely-spaced timescales presents a more challenging  problem setting, motivating our exploration of unstable  periodic orbits below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Interestingly, while the turbulent  flow dataset contains a monochromatic periodic driver,  the stochastic component of the response systems masks  the driving signal from detection by linear methods like  the Fourier transform or PCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  We note that many of our example datasets represent  cases in which the reconstructed driving signal ˆz(t) ex-  erts nearly-instantaneous influence on the response dy-  namics, allowing our method to reconstruct drivers that  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Correlation between chaoticity and ac-  curacy across thousands of dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  (A)  Thousands of random skew-product dynamical systems pro-  duced by sampling pairs from a set of 132 named chaotic  systems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Lorenz, Rossler, etc) and using one system as a  driver for replicates of the other system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Linear fits with 95%  bootstrapped confidence intervals highlight significant Spear-  man correlations between the reconstruction accuracy and the  Lyapunov exponents of the driver (red, ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='16 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='03) and  response (blue, ρ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='03).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' (B) For each individual  dynamical system, the average accuracy of a reconstruction  in which it appears as a response variable, versus one in which  it appears as a driver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  qualitatively resemble the known driver z(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  In prin-  ciple, however, our method can be applied to any dy-  namical system where the response dynamics have the  form ˙yk(t) = gk(yk(t), z(t), t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' However, we focus on the  synchronized regime z(t) ≈ ˆz(t) because a nonlinearly-  transformed or time-delayed driver introduces ambiguity  regarding the appropriate definition of the true driver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  For example, for the system ˙x = x + h(z(t − τ)), the  true driving signal for x(t) could be defined as z(t), h(t),  or h(t − τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' As a result, it is challenging to construct a  fair benchmark experiment for very weak, nonlinear, and  time-delayed driving, particularly because methods such  as PCA will tend to find driving terms that linearly and  instantaneously influence the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Chaotic drivers hinder reconstruction, but  chaotic responses improve it  In order to further understand which systems are more  difficult to reconstruct, we perform an additional bench-  mark using our recently-introduced database of 132 low-  dimensional ordinary differential equations [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Initially  curated from published literature to include common sys-  tems like the Lorenz, R¨ossler, and Chua attractors, the  dataset has grown through crowdsourced contributions  to contain diverse systems spanning domains such as as-  trophysics, climatology, and biochemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Each system  is annotated with estimates of descriptive mathematical  properties such as maximum Lyapunov exponents, frac-  tal dimensions, and entropy rates, which quantify differ-  ent aspects of each system’s underlying complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Using  this  database,  we  create  random  skew-  A  B  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='0  max  Response  10-3  10-1  101  e  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='8  Accuracy  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='6  Driver  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='0  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='0  Lyapunov Exponent >  Accuracy as Driver  max6  product systems in which one randomly-selected system  drives multiple randomly-initialized versions of another  randomly-selected system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  The Lorenz-Rossler system  featured in Figure 1 represents one such skew-product  system from the ∼ 105 possible pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  We scale the  coupling strength to a constant multiple of the aver-  age amplitude of both systems, and we scale integration  timesteps and trajectory lengths based on precomputed  estimates of Lipschitz constants and dominant timescales  annotated with each system within our database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  We  sample all possible pairs, but retain time series samples  only from skew systems that still exhibit chaotic dynam-  ics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' We apply our reconstruction method to each random  skew product time series, and correlate reconstruction ac-  curacy with the separate mathematical properties of the  driver and response systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  We observe weak but consistent negative correlation  between the driver Lyapunov exponent and reconstruc-  tion accuracy, implying that more chaotic systems are  more difficult to reconstruct (Figure 3A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Surprisingly, we  observe the opposite for response systems: more chaotic  response systems improve reconstruction quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' As a  result, a given dynamical system that proves easier to  reconstruct will tend to weaken reconstruction of other  systems (Figure 3B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  This discrepancy arises from the dual roles of the driver  and response systems: more chaotic response dynamics  produce more diverse and thus informative time series,  leading to more unique recurrence events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Conversely, a  nearly non-chaotic response system (λmax ≈ 0) yields  only a finite amount of information over an extended  duration, or across many replicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Less chaotic sys-  tems thus hinder reconstruction because they discover  new information about the driver at a slower rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Con-  versely, more chaotic drivers transmit information at a  faster rate, thus requiring more information to be gained  from the response time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' This concept of the driver  as an information source, and response systems as re-  ceivers, mirrors classical formulations of skew-product  systems [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' The entropy production rate of a chaotic  system is bounded from below by the largest Lyapunov  exponent [61], which acts as a proxy for the information  transmission rate between the driver and response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' In  the limit of a perfectly lossless transmission from driver  to response system, the systems would exhibit general-  ized synchronization, in which the response becomes a  function of solely the driver state [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Reconstruction accuracy requires a percolation  transition in the time series adjacency graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Our reconstruction method implicitly defines a diffu-  sion process on the time series adjacency graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' In or-  der to understand how this dynamical process couples  to the dynamics of the driver itself, we apply our al-  gorithm to response systems driven by the logistic map  as it undergoes a series of period-doubling bifurcations  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  A period-doubling bifurcation reveals that  percolation loss precedes accurate reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' (A)  Driver reconstruction accuracy (adjusted Rand index) for re-  sponses driven by the logistic map in its period-2 (blue),  period-4 (turquoise), period-8 (red), and fully-chaotic (ma-  genta) regimes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' plotted from left to right as a function of the  noise level, driver to response coupling strength, and number  of response time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' In the last plot, dashed guides indi-  cate null hypotheses for scaling of accuracy with dataset size,  as predicted by the β-distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' (Lower) The size of the  largest connected component (a measure of percolation) in  the reconstructed time series adjacency graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' All plots show  median and interquartile range across 60 replicates with ran-  dom initial conditions and algorithm seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' (B) Changes in  inferred adjacency graph of the period-8 sequence as the data  noise is varied;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' frames correspond to points on the red trace  marked with dots in the first panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  leading to chaos (Figure 4A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' We track the point-wise  accuracy of the reconstructed discrete driver using the  adjusted Rand accuracy, which accounts for the permu-  tation invariance of labels: an adjusted Rand accuracy  of 0 implies a driver state labelling with no greater ac-  curacy than random chance, while a score of one implies  that every driver state is correctly identified for each of  the 3 × 103 timepoints tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' We evaluate our method’s  properties by recording its accuracy as a function of three  control parameters: the relative amount of stochasticity  present in the response dynamics (signal-to-noise ratio),  the strength of coupling between the response dynam-  ics and driver, and the number of response time series  used for reconstruction N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' In the latter case, we naively  expect that reconstruction accuracy will increase propor-  tionally to the total number of recurrences observed and  thus N, resulting in an accuracy scaling given by the β-  distribution (which describes the distribution of the ex-  pected value of the mean of a binomial process, if driver  recurrence states are discovered at random).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  We observe that shorter-period drivers are generally  easier to reconstruct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Interestingly, at higher-period driv-  ing rates, we observe a staircase pattern in the accu-  A  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='0  Accuracy  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='0  Component  Largest  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='5  0  20  10-2  10-1  101  C  102  Signal-to-noise (dB)  Coupling Strength (k)  Number of time series (N)  B7  racy as reconstruction quality improves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' This occurs be-  cause, at lower accuracies and longer periods, our ap-  proach fails to differentiate all driver states, resulting in  groups of states merging together into a coarse-grained,  lower-period driving signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' These clusters split into dis-  tinct driver substates as reconstruction quality improves  (Figure 4B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  In order to understand how changes in method accu-  racy depend on structural properties of the underlying re-  currence graph, for each condition we measure the order  parameter TLCC/T ∈ [0, 1], where T is the total number  of nodes and TLCC is the number of nodes comprising the  largest connected component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' This order parameter de-  tects percolation in finite-size undirected networks [63],  and in our system it reveals a first-order phase transi-  tion preceding an increase in accuracy when any of the  three control parameters is varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Intuitively, a highly-  connected time series network contains many equivalent  paths between two nodes, producing ambiguity regard-  ing the correct assignment of recurrence events to driver  states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' This degeneracy below the percolation transition  produces an underdetermined reconstruction, which be-  comes determined as the amount or quality of response  datasets increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Considering a diffusion process on  the recurrence graph, increasing any of the three con-  trol parameters increases the duration of transients that  reveal almost-invariant structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' The percolation tran-  sition therefore indicates the onset of weak ergodicity  breaking, in which the dynamics transition from having  a short mixing time to a much longer time dominated  by gradual leakage between highly-connected groups of  nodes sharing the same driver state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Our observations mirror recent results reporting equiv-  alency between the fitting accuracy of deep artificial neu-  ral networks and the jamming transition [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  By de-  creasing the ratio of data to model size, a deep neural  network passes from underfitting to overfitting regimes  mirroring a dilute-to-jammed transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' In our system,  increasing training data size or effective size (by decreas-  ing noise or increasing coupling) increases final accuracy  by making the time series adjacency graph more dilute—  thereby ensuring that well-defined distinctions emerge  among different driver states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Our observation that high-  period driver states initially appear as low-period states  at the onset of the percolation transition is reminiscent  of transformations of dynamical systems under the action  of a renormalization operator [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Reconstruction prioritizes dominant unstable  periodic orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  We next consider the process by which our method  learns to reconstruct diverse signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' We return to our  original demonstration, in which the R¨ossler dynamical  equations drive many randomly-corrupted measurements  of the Lorenz system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' We repeat the reconstruction while  varying the number of response series N, and thus the  reconstruction accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Given our method’s emphasis on inferring driver re-  currence states, we speculate that its accuracy implicitly  depends on the unstable periodic orbit spectrum of the  driver [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Classical work on nonlinear dynamical sys-  tems has shown that any chaotic system can be decom-  posed into an infinite set of unstable periodic orbits [66],  which trajectories shadow for extended durations propor-  tional to each orbit’s relative stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' While chaotic sys-  tems have continuous power spectra, the set of valid un-  stable periodic orbits bears basic topological restrictions  due to attractor geometry—making it a countable, al-  beit infinite, hierarchical measure for the dynamical sys-  tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' We compute several unstable periodic orbits of the  R¨ossler driving system using the method of closest re-  currences [67], and select the three most dominant orbits  based on the relative duration they are shadowed by the  driver dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Comparing the pairwise distance matrix of the recon-  structed signal with the distance matrix of individual un-  stable periodic orbits (Figure 5) suggests that, as the  amount of training data increases, the reconstructed dis-  tance matrix initially approximates only the most dom-  inant orbit, but then gradually approximates finer-scale  details of the driver attractor that are encoded in sec-  ondary orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' We hypothesize that the recurrence struc-  ture of our method biases it towards shorter-period and  more stable orbits that dominate the graph partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  This phenomenon provides some context for our ear-  lier observation that reconstruction accuracy depends in-  versely on the driver’s Lyapunov exponent: systems with  larger Lyapunov exponents tend to transition among a  larger set of orbits, thus requiring more recurrence infor-  mation to produce faithful reconstructions [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  DISCUSSION  We have presented a theoretically-motivated unsuper-  vised learning method that reconstructs an unobserved  causal driving signal, given observations of downstream  response variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Across extensive benchmarks against  methods drawn from diverse domains, we find that our  method quickly and robustly reconstructs diverse driv-  ing signals in time series datasets ranging from fluid dy-  namics to systems biology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Beyond introducing a new  general-purpose data analysis method, our work provides  empirical insight into the nature of information trans-  mission in coupled dissipative systems: the topology of  the driver’s characteristic unstable orbits manifests as  structure within the recurrence graph of the response  time series, with the efficiency of this linkage mediated  by both the chaoticity of the driver, and the diversity  of the responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  A diffusion process on the resulting  time series graph reveals driver structure in its longest-  lived transient modes, and methods that increase the life-  time of these transients—such as more response datasets,  increased driver-response coupling, or reduced measure-  8  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Reconstruction of complex signals follows  dominant unstable periodic orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' (A) The three most  stable unstable periodic orbits for the R¨ossler system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' (B)  Elementwise Pearson correlation between the distance matrix  of the reconstructed R¨ossler driver and the three orbits, as  the number of response time series (and thus overall driver  reconstruction accuracy) increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Ranges denote 95% con-  fidence intervals around random model initializations trained  on random subsets of available response series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' (C) The over-  laid true distance matrices of the three leading orbits, with  RGB channels of the color image encoding each orbit’s indi-  vidual distance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' (D) The distance matrices of the re-  construction as the number of response subsystems increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  ment noise—produce a corresponding increase in recon-  struction accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Our diffusion process represents a  heuristic solution to the difficult combinatorial optimiza-  tion problem of graph traversal and topological node  ordering, and we note that our method reduces to a  previously-known exact algorithm in the limit of a noise-  less and discrete periodic driver [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Key limitations of our approach therefore stem from  our use of diffusive dynamics for graph traversal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Certain  measurement types may induce false recurrence events  with systematic structure, such as sensor saturation or  memory effects, that impose fixed structural barriers to  diffusion on the graph adjacency matrix—akin to inho-  mogenous diffusivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Mitigating such effects requires  sufficient information about the measurement dynamics  to reconstruct and correct for response bias;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' within our  framework, this might require preconditioning the graph  Laplacian [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Additionally, non-stationary driving sig-  nals can complicate reconstruction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' for example if the  attractor is time-dependent (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' energy gradually dissi-  pates due to drag), then the driver may exhibit transient  chaos before settling into a state of quiescence or oscil-  lation [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' In this case, a random walker’s transitions  among nodes will no longer heed detailed balance and the  transition operator will exhibit non-Hermitian structure,  resulting in long-lived nonequilibrium steady-states that  complicate driver state assignment [71–73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' These and  other, more challenging data types may require nonlin-  ear alternatives to diffusion processes on the recurrence  graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Our use of diffusion therefore represents a form of  inductive bias in the time series representations learned  by our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  However, the problem of discovering nonlinear pro-  cesses from an empirical recurrence graph introduces a  variety of potential generalizations grounded in recent  results from statistical learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Instead of directly fit-  ting a linear operator, future work could leverage recent  methods for data-driven inference of dynamical propaga-  tors [74–77], which can even produce reduced-order ana-  lytical models [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Generalizations may also incorporate  new graph representation or traversal methods (such as  message passing)[50, 79], particularly graph neural net-  works [80, 81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Thus, beyond our specific algorithm, we  argue that our work motivates general parametrization of  time series models in terms of recurrences, and the latent  orbit structure that they encode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  We  thank  Anthony  Bao  for  feedback  on  the  manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' was supported by the University of  Texas at Austin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  CODE AVAILABILITY  All code used in this study is available online at https:  //github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='com/williamgilpin/shrec  [1] Rosenblatt, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' & Schmitz, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Climate change, nu-  trition, and bottom-up and top-down food web processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Trends in Ecology & Evolution 31, 965–975 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [2] Cande, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Optogenetic dissection of descending  behavioral control in drosophila.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Elife 7 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [3] Leisner, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Stingl, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Frey, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' & Maier, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Stochastic  switching to competence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Current opinion in microbiol-  ogy 11, 553–559 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [4] Pezzulo, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' & Levin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Top-down models in biology:  explanation and control of complex living systems above  A  C  D  N=  人  B  upo 1  upo 2  Distance Matrix correlation  upo3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='5  N=300  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='0  3  101  102  Number of response systems9  the molecular level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Journal of The Royal Society Inter-  face 13, 20160555 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [5] Schwarze, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Ichinaga, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' & Brunton, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Network inference via process motifs for lagged cor-  relation in linear stochastic processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  arXiv preprint  arXiv:2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='08871 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [6] Runge, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Heitzig, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Petoukhov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' & Kurths, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Escap-  ing the curse of dimensionality in estimating multivari-  ate transfer entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Physical review letters 108, 258701  (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [7] Sugihara, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Detecting causality in complex ecosys-  tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' science 338, 496–500 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [8] Runge, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Nowack, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Kretschmer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Flaxman, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' &  Sejdinovic, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Detecting and quantifying causal associ-  ations in large nonlinear time series datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Science  Advances 5, eaau4996 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [9] Kantz, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' & Schreiber, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Nonlinear time series analysis,  vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' 7 (Cambridge University Press, 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [10] Munch, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Rogers, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' & Sugihara, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Recent de-  velopments in empirical dynamic modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Methods in  Ecology and Evolution (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [11] Runge, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Inferring causation from time series in  earth system sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Nature communications 10, 1–13  (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [12] De Brouwer, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Arany, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Simm, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' & Moreau, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' La-  tent convergent cross mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' In International Confer-  ence on Learning Representations (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [13] Bennett, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' & Yu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Rethinking neural relational in-  ference for granger causal discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  In NeurIPS 2022  Workshop on Causality for Real-world Impact (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [14] Quiroga, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Arnhold, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' & Grassberger, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Learning  driver-response relationships from synchronization pat-  terns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Physical Review E 61, 5142 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [15] Takens, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Detecting strange attractors in turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' In  Dynamical systems and turbulence, Warwick 1980, 366–  381 (Springer, 1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [16] Packard, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Crutchfield, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Farmer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' & Shaw,  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Geometry from a time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Physical Review  Letters 45, 712 (1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [17] Stark, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Delay embeddings for forced systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' deter-  ministic forcing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Journal of Nonlinear Science 9, 255–332  (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [18] Sober, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Sponberg, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Nemenman, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' & Ting, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Millisecond spike timing codes for motor control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Trends  in neurosciences 41, 644–648 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [19] Dhamala, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Rangarajan, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' & Ding, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Estimating  granger causality from fourier and wavelet transforms  of time series data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Physical review letters 100, 018701  (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [20] Mastakouri, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Sch¨olkopf, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' & Janzing, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Nec-  essary and sufficient conditions for causal feature selec-  tion in time series with latent common causes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  In In-  ternational Conference on Machine Learning, 7502–7511  (PMLR, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [21] Pecora, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Carroll, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Johnson, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Mar,  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' & Heagy, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Fundamentals of synchronization in  chaotic systems, concepts, and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Chaos: An  Interdisciplinary Journal of Nonlinear Science 7, 520–  543 (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [22] Hunt, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Ott, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' & Yorke, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Differentiable gener-  alized synchronization of chaos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Physical Review E 55,  4029 (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [23] Sauer, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Reconstruction of shared nonlinear dy-  namics in a network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Physical review letters 93, 198701  (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [24] Wehmeyer, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' & No´e, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Time-lagged autoencoders:  Deep learning of slow collective variables for molecular  kinetics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' The Journal of chemical physics 148, 241703  (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [25] Gilpin, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Deep reconstruction of strange attractors  from time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Advances in Neural Information Pro-  cessing Systems 33 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [26] Bakarji, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Champion, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Kutz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' & Brunton, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Discovering governing equations from partial measure-  ments with deep delay autoencoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  arXiv preprint  arXiv:2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='05136 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [27] Froyland, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' & Padberg, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Almost-invariant sets and in-  variant manifolds—connecting probabilistic and geomet-  ric descriptions of coherent structures in flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Physica  D: Nonlinear Phenomena 238, 1507–1523 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [28] Brunton, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Brunton, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Proctor, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Kaiser,  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' & Kutz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Chaos as an intermittently forced linear  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Nature communications 8, 1–9 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [29] Mezi´c, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Spectral properties of dynamical systems,  model reduction and decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Nonlinear Dynam-  ics 41, 309–325 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [30] Costa, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Ahamed, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Jordan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' & Stephens,  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Maximally predictive ensemble dynamics from data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  arXiv preprint arXiv:2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='12811 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [31] Giannakis, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' & Majda, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Nonlinear laplacian  spectral analysis for time series with intermittency and  low-frequency variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Proceedings of the National  Academy of Sciences 109, 2222–2227 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [32] Girvan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' & Newman, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Community structure in  social and biological networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Proceedings of the na-  tional academy of sciences 99, 7821–7826 (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [33] Rogers, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' & Munch, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Hidden similarities in the  dynamics of a weakly synchronous marine metapopula-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Proceedings of the National Academy of Sciences  117, 479–485 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [34] Deyle, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Maher, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Hernandez, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Basu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' &  Sugihara, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Global environmental drivers of influenza.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Proceedings of the National Academy of Sciences 113,  13081–13086 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [35] Hirata, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Horai, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' & Aihara, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Reproduction of  distance matrices and original time series from recur-  rence plots and their applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' The European Physical  Journal Special Topics 164, 13–22 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [36] Hirata, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' & Aihara, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Identifying hidden common  causes from bivariate time series: A method using re-  currence plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Physical Review E 81, 016203 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [37] Haghverdi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', B¨uttner, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Wolf, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Buettner, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' &  Theis, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Diffusion pseudotime robustly reconstructs  lineage branching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Nature methods 13, 845–848 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [38] Donner, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Zou, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Donges, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Marwan, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' &  Kurths, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Recurrence networks—a novel paradigm for  nonlinear time series analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' New Journal of Physics  12, 033025 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [39] Davidsen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Grassberger, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' & Paczuski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Networks of  recurrent events, a theory of records, and an application  to finding causal signatures in seismicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Physical Review  E 77, 066104 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [40] Myers, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Munch, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' & Khasawneh, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Persistent ho-  mology of complex networks for dynamic state detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Physical Review E 100, 022314 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [41] McCabe, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' & Brown, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Learning to assimilate in  chaotic dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Advances in Neural Infor-  mation Processing Systems 34, 12237–12250 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  10  [42] Frerix, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Variational data assimilation with a  learned inverse observation operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  In International  Conference on Machine Learning, 3449–3458 (PMLR,  2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [43] Karl, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Soelch, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Bayer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' & Van der Smagt, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Deep variational bayes filters:  Unsupervised learning  of state space models from raw data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  arXiv preprint  arXiv:1605.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='06432 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [44] Krishnan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Shalit, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' & Sontag, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Deep kalman  filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' arXiv preprint arXiv:1511.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='05121 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [45] Koppe, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Toutounji, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Kirsch, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Lis, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' & Durste-  witz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Identifying nonlinear dynamical systems via  generative recurrent neural networks with applications to  fmri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' PLoS computational biology 15, e1007263 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [46] Hamilton, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Berry, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' & Sauer, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Ensemble kalman  filtering without a model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Physical Review X 6, 011021  (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [47] Ye, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' & Sugihara, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Information leverage in intercon-  nected ecosystems: Overcoming the curse of dimension-  ality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Science 353, 922–925 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [48] Leng, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Partial cross mapping eliminates indi-  rect causal influences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Nature communications 11, 1–9  (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [49] Alaa, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' & Van Der Schaar, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Validating causal in-  ference models via influence functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' In International  Conference on Machine Learning,  191–201 (PMLR,  2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [50] Williams, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Lacasa, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Mill´an, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' & Latora, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  The shape of memory in temporal networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Nature com-  munications 13, 1–8 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [51] Sulas, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' A non-invasive multimodal foetal ecg–  doppler dataset for antenatal cardiology research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Scien-  tific Data 8, 1–19 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [52] Schaffter,  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=',  Marbach,  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  &  Floreano,  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Genenetweaver:  in silico benchmark generation and  performance profiling of network inference methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Bioinformatics 27, 2263–2270 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [53] Pei, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Neural latents benchmark’21: Evaluating la-  tent variable models of neural population activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' arXiv  preprint arXiv:2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='04463 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [54] Clark, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Livezey, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' & Bouchard, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Unsupervised dis-  covery of temporal structure in noisy data with dynami-  cal components analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Advances in Neural Information  Processing Systems 32 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [55] Wiskott, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' & Sejnowski, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Slow feature analysis:  Unsupervised learning of invariances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Neural computa-  tion 14, 715–770 (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [56] Yu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Gaussian-process factor analysis for low-  dimensional single-trial analysis of neural population ac-  tivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Advances in neural information processing systems  21 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [57] Franceschi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Dieuleveut, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' & Jaggi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Unsu-  pervised scalable representation learning for multivariate  time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Advances in neural information processing  systems 32 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [58] Pandarinath, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Inferring single-trial neural popu-  lation dynamics using sequential auto-encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Nature  methods 15, 805–815 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [59] Gilpin, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Chaos as an interpretable benchmark for fore-  casting and data-driven modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Advances in Neural  Information Processing Systems 1 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [60] Adler, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Coppersmith, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' & Hassner, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Algorithms  for sliding block codes-an application of symbolic dynam-  ics to information theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' IEEE Transactions on Infor-  mation Theory 29, 5–22 (1983).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [61] Pesin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Characteristic lyapunov exponents and smooth  ergodic theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Russian Mathematical Surveys 32, 55  (1977).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [62] Pecora, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' & Carroll, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Synchronization in chaotic  systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Physical Review Letters 64, 821 (1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [63] H´ebert-Dufresne, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' & Allard, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Smeared phase transi-  tions in percolation on real complex networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Physical  Review Research 1, 013009 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [64] Geiger, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Jamming transition as a paradigm to  understand the loss landscape of deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Physical Review E 100, 012115 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [65] Feigenbaum, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  The universal metric properties of  nonlinear transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Journal of Statistical Physics  21, 669–706 (1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [66] Cvitanovi´c, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Invariant measurement of strange sets in  terms of cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Physical Review Letters 61, 2729 (1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [67] Lathrop, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' & Kostelich, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Characterization of an  experimental strange attractor by periodic orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Phys-  ical Review A 40, 4028 (1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [68] Ott, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Chaos in dynamical systems (Cambridge Univer-  sity Press, 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [69] Krishnan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Fattal, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' & Szeliski, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Efficient precondi-  tioning of laplacian matrices for computer graphics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' ACM  transactions on Graphics (tOG) 32, 1–15 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [70] Motter, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Gruiz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', K´arolyi, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' & T´el, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Doubly  transient chaos: Generic form of chaos in autonomous  dissipative systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Physical review letters 111, 194101  (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [71] Fruchart, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Hanai, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Littlewood, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' & Vitelli, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Non-reciprocal phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Nature 592, 363–369  (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [72] Battle, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Broken detailed balance at mesoscopic  scales in active biological systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Science 352, 604–607  (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [73] Weis, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Coalescence of attractors: exceptional  points in non-linear dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' arXiv preprint  arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='11667 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [74] Bramburger, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Kutz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' & Brunton, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Data-  driven stabilization of periodic orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' IEEE Access 9,  43504–43521 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [75] Raissi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' & Karniadakis, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Hidden physics models:  Machine learning of nonlinear partial differential equa-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Journal of Computational Physics 357, 125–141  (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [76] Chen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Rubanova, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Bettencourt, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' & Duvenaud,  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Neural ordinary differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Advances  in neural information processing systems 31 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [77] Choudhary, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Physics-enhanced neural networks  learn order and chaos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Physical Review E 101, 062207  (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [78] Champion, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Lusch, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Kutz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' & Brunton, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  Data-driven discovery of coordinates and governing equa-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Proceedings of the National Academy of Sciences  116, 22445–22451 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [79] Gilmer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Schoenholz, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Riley, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Vinyals, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' &  Dahl, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Neural message passing for quantum chem-  istry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' In International conference on machine learning,  1263–1272 (PMLR, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  [80] Donoho, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=', Maleki, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' & Montanari, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Message-  passing algorithms for compressed sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Proceedings  of the National Academy of Sciences 106, 18914–18919  (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='  11  [81] Rossi, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' Temporal graph networks for deep learn-  ing on dynamic graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content=' arXiv preprint arXiv:2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
+page_content='10637  (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FRT4oBgHgl3EQfPDf9/content/2301.13516v1.pdf'}
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+arXiv:2301.00166v1  [math.AP]  31 Dec 2022
+HOMOGENIZATION OF ACTIVE SUSPENSIONS
+AND REDUCTION OF EFFECTIVE VISCOSITY
+ARMAND BERNOU, MITIA DUERINCKX, AND ANTOINE GLORIA
+Abstract. We consider a suspension of active rigid particles (swimmers) in a steady
+Stokes flow, using for simplicity a steady-state model where particles are distributed ac-
+cording to a stationary ergodic random process, and we study its homogenization in the
+macroscopic limit. A key point in the model is that swimmers are allowed to adapt their
+propulsion to the surrounding fluid deformation: swimming forces are not prescribed
+a priori, but are rather obtained through the retroaction of the fluid. Qualitative ho-
+mogenization of this nonlinear model requires an unusual proof that crucially relies on
+a semi-quantitative two-scale analysis.
+Thanks to the introduction of new correctors
+that accurately capture spatial oscillations created by swimming forces, we identify the
+contribution of the activity to the effective viscosity. In agreement with the physics liter-
+ature, an analysis in the dilute regime shows that the activity of the particles can either
+increase or decrease the effective viscosity (depending on the swimming mechanism), in
+a way that strongly differs from the well-known effect of passive suspensions.
+Contents
+1.
+Introduction and main results
+1
+1.1.
+General overview
+1
+1.2.
+Reminder on passive suspensions
+3
+1.3.
+Hydrodynamic model for active suspensions
+5
+1.4.
+Heuristics and relation to the physics literature
+8
+1.5.
+Main results: well-posedness, homogenization, and dilute regime
+10
+1.6.
+Viscosity reduction
+14
+2.
+Corrector problems
+15
+2.1.
+Passive corrector problems
+15
+2.2.
+Active corrector problems
+17
+3.
+Proof of the homogenization result
+26
+3.1.
+Well-posedness of hydrodynamic model
+26
+3.2.
+Qualitative homogenization at fixed δ
+29
+3.3.
+Quantitative homogenization and limit δ ↓ 0
+48
+4.
+Dilute expansion of the effective viscosity
+51
+Acknowledgements
+56
+References
+56
+1. Introduction and main results
+1.1. General overview. This work is devoted to the large-scale rheology of suspensions
+of active particles in viscous fluids, where active particles are devices that can propel
+1
+
+2
+A. BERNOU, M. DUERINCKX, AND A. GLORIA
+themselves in the fluid (in a direction that can depend on the surrounding fluid flow itself).
+Our main purpose is to establish rigorously, in a simple (yet somewhat realistic) physical
+model, that the presence of active particles in a fluid can strongly reduce its viscosity.
+Important examples include suspensions of bacteria [36], micro-algae [37], nanomo-
+tors [29], etc.
+Compared to passive systems, active suspensions exhibit a particularly
+rich phenomenology, with the experimental observation of pattern formations [1] and un-
+steady whirls and jets [28]. Due to this complexity, the response to an external forcing can
+defy intuition, with rheological measurements displaying in some settings a transition to a
+superfluid-like behavior [27].
+Our contribution is threefold: we formulate a suitable steady-state model for active
+suspensions based on the physics literature, we prove homogenization for this model and
+identify the effective rheology on large scales, and we finally analyze the latter in the dilute
+regime — then recovering standard explicit results of the physics literature.
+As reviewed in Section 1.2 below, Einstein’s celebrated viscosity formula [12] shows that
+the presence of passive rigid particles can only increase the plain fluid viscosity on large
+scales as those particles hinder the fluid flow. The case of active suspensions is radically
+different: the activity of micro-swimmers may overcome viscous dissipation and reduce
+the effective viscosity. As first pointed out in [24], this is related to the emergence of a
+state of collective orientation of the particles under a given external force: the preferred
+orientation happens to determine the increase or decrease in viscosity and was predicted to
+depend on the type of swimming mechanism. In 2009, Sokolov and Aranson [35] observed
+experimentally a striking reduction of the effective viscosity of a suspension of Bacillus
+subtilis in a dilute regime.
+In contrast, for motile microalgae such as Chlamydomonas
+reinhardtii, a drastic increase in the viscosity was observed experimentally in [31] — drastic
+even compared to suspensions with the same amount of dead cells! This confirmed that
+the swimming mechanism plays a key role in the modification of the viscosity on large
+scales and subsequent studies have classified these into two categories:
+— “pusher” particles, such as Bacillus subtilis, move into the fluid via an extensile motion,
+e.g. using a flagella, and tend to decrease the effective viscosity;
+— “puller” particles, such as Chlamydomonas reinhardtii, are contractile swimmers, and
+their activity can increase the effective viscosity.
+Those experimental results have attracted much attention in the physics community over
+the past decade and several models have been introduced to explain the effect of the
+swimming mechanism on the effective viscosity. We refer in particular to the heuristic
+analysis in [22, 23, 32], where the preferred orientation of the particles is formally computed
+depending on the external shear flow, and is shown to affect the effective viscosity in the
+dilute regime, in agreement with experiments. In these works, the swimming mechanism
+is usually modeled for simplicity by point dipole forces at particle locations.
+We refer
+to [30, 33, 34] and references therein for the physical background.
+From a modeling point of view, a key question in the study of suspensions is how to
+describe the positions and orientations of the suspended particles and how they are coupled
+to the fluid flow. There are two natural ways to proceed:
+(I) Dynamical model: Particles follow the fluid flow and propel themselves, so their
+positions and orientations evolve in time accordingly. The geometry of the array
+of particles can then adapt dynamically to external forces, leading to a nonlinear
+response (non-Newtonian effects).
+
+HOMOGENIZATION FOR ACTIVE SUSPENSIONS
+3
+(II) Nonlinear steady-state model: The statistics of the positions are assumed to be
+given, as well as the statistics of the orientations, but the latter are allowed to depend
+on the surrounding fluid flow. The problem is then reduced to understanding the
+retroaction of the fluid on the self-propulsion of the particles as their orientations
+adapt to the surrounding fluid deformation.
+From the rigorous mathematical perspective, the literature is particularly scarce, with the
+exception of the recent work [19] by Girodroux-Lavigne. In this work, the author considers
+a dilute steady-state model, consisting of a steady Stokes fluid with a dilute suspension
+of active particles for which swimming forces are fully prescribed in advance (hence, are
+part of the data). This reduces the analysis to a combination of homogenization (due to
+the presence of rigid particles, via a dilute analysis) with averaging of the active part (in
+form of a source term). This can be viewed as a very first step towards both situations (I)
+and (II): it remains to be combined with the retroaction of the fluid, which can be modeled
+either by the particle dynamics (I), or by the dependence of particle orientations on the
+surrounding fluid in a nonlinear steady-state model (II). Rigorous results in a realistic
+dynamical setting (I) seem out of reach at the moment beyond dilute regimes [3], and
+the present contribution is rather devoted to the steady-state setting (II) in a general
+non-dilute regime.
+The sequel of this introduction is organized as follows: In Section 1.2, we recall the
+steady-state model for a steady Stokes fluid with a suspension of passive rigid particles,
+and we state the associated homogenization result previously obtained by the last two
+authors. In Section 1.3, we introduce a corresponding model for active suspensions in a
+steady Stokes flow, as inspired by the review articles [33, 34]. In Section 1.4, we relate this
+model to the physics literature and show how it can be used to recover standard results
+of the physics literature on a heuristic level. Section 1.5 is dedicated to the main results
+of this paper: the rigorous homogenization of this nonlinear model, and the analysis of
+the effective rheology in the dilute regime. Last, in Section 1.6, based on these results,
+we investigate the contribution of active particles to the effective viscosity, and rigorously
+establish that a significant reduction can take place in the case of pushers.
+1.2. Reminder on passive suspensions. Given an underlying probability space (Ω, P)
+(with expectation E [·]), let {xn}n be a random point process on the ambient space Rd,
+consider an associated collection of random shapes {I◦
+n}n, where each I◦
+n is a connected
+random open subset of the unit ball B centered at the origin (in the sense that
+´
+I◦n y dy = 0),
+and then define the corresponding inclusions
+In := xn + I◦
+n.
+Note that random shapes are not required to be independent of the point process {xn}n.
+We then consider the random set
+I :=
+�
+n
+In,
+which we assume to satisfy the following for some ϑ > 0.
+Hypothesis 1.1 (Particle suspension).
+(a) Stationarity and ergodicity: The random set I is stationary and ergodic.
+(b) Uniform C2 regularity: The random shapes {I◦
+n}n satisfy interior and exterior ball
+conditions with radius ϑ almost surely.
+
+4
+A. BERNOU, M. DUERINCKX, AND A. GLORIA
+(c) Uniform hardcore condition: For some ℓ ≥ ϑ, there holds (In + ℓB) ∩ (Im + ℓB) = ∅
+almost surely for all n ̸= m.
+We let ℓ be the largest such value, that is, half the
+interparticle distance
+ℓ := 1
+2 inf
+n̸=m dist(In, Im).
+(1.1)
+♦
+Now consider a tank, described as a bounded Lipschitz domain U ⊂ Rd, and assume
+that it is filled with a steady Stokes fluid with a suspension of particles of size ε, described
+as the ε-rescaling of I. More precisely, we only consider particles that are included in U
+and remove those close to the boundary: define Nε(U) as the set of indices n such that
+ε(In + ℓB) ⊂ U, and set
+Iε(U) :=
+�
+n∈Nε(U)
+εIn.
+We write uε for the fluid velocity, Pε for the corresponding pressure. We assume Dirichlet
+conditions uε = 0 on ∂U, and we extend the fluid velocity inside particles with the rigidity
+constraint
+D(uε) := 1
+2(∇uε + (∇uε)T ) = 0,
+in Iε(U).
+Recall the definition of the Cauchy stress tensor
+σ(uε, Pε) := 2 D(uε) − Pε Id,
+where Id denotes the identity matrix. Given an internal force h, the fluid velocity uε ∈
+H1
+0(U)d and the associated pressure Pε ∈ L2(U \ Iε(U)) are then given as the solutions of
+the Stokes system
+
+
+
+
+
+
+
+
+
+
+
+−△uε + ∇Pε = h
+in U \ Iε(U),
+div(uε) = 0,
+in U \ Iε(U),
+D(uε) = 0,
+in Iε(U),
+´
+ε∂In σ(uε, Pε)ν = 0,
+∀n ∈ Nε(U),
+´
+ε∂In Θ(x − εxn) · σ(uε, Pε)ν = 0,
+∀n ∈ Nε(U), Θ ∈ Mskew,
+(1.2)
+where ν stands for the outward normal to ∂In, where Mskew is the set of skew-symmetric
+matrices, and where we assume the additional anchoring condition
+ˆ
+U\Iε(U)
+Pε = 0,
+which we shall abbreviate as choosing Pε ∈ L2(U \ Iε(U))/R.
+The homogenization of
+the Stokes system (1.2) was the object of [2, 4, 8], where the last two authors proved
+that (uε, Pε1U\Iε(U)) converges almost surely weakly in H1
+0(U) × L2(U)/R to the unique
+solution (¯u, ¯P) of the homogenized Stokes system
+�
+−div(2 ¯Bpas D(¯u)) + ∇ ¯P = (1 − λ)f
+in U,
+div(¯u) = 0,
+in U,
+(1.3)
+where λ stands for the particle volume fraction
+λ := E [1I] ,
+(1.4)
+and where the effective viscosity ¯Bpas is a symmetric linear map on the set Msym
+0
+of
+symmetric trace-free matrices. We recall that the latter satisfies E : ¯BpasE > |E|2 for all
+E ∈ Msym
+0
+as soon as λ > 0, meaning that the presence of (passive) rigid particles always
+increases the effective viscosity. We refer to [9] for a review of the topic.
+
+HOMOGENIZATION FOR ACTIVE SUSPENSIONS
+5
+In view of the quantitative homogenization results that we shall need later, we occa-
+sionally make quantitative ergodicity assumptions in form of the validity of the following
+multiscale variance inequality introduced by the last two authors in [6, 7]. This assumption
+holds for instance for hardcore Poisson point process with exponentially decaying π.
+Hypothesis 1.2 (Quantitative mixing assumption). There exists a non-increasing weight
+function π : R+ → R+ with superalgebraic decay (that is, π(t) ≤ Cp⟨t⟩−p for all p < ∞)
+such that the random set I satisfies, for all σ(I)-measurable random variables Y (I),
+Var [Y (I)] ≤ E
+� ˆ ∞
+0
+ˆ
+Rd
+�
+∂osc
+I,Bt(x)Y (I)
+�2
+dx ⟨t⟩−dπ(t) dt
+�
+,
+where the “oscillation” ∂osc of the random variable Y (I) is defined by
+∂osc
+I,Bt(x)Y (I) := sup ess
+�
+Y (I′) : I′ ∩ (Rd \ Bt(x)) = I ∩ (Rd \ Bt(x))
+�
+− inf ess
+�
+Y (I′) : I′ ∩ (Rd \ Bt(x)) = I ∩ (Rd \ Bt(x))
+�
+.
+♦
+1.3. Hydrodynamic model for active suspensions. As opposed to passive particles,
+active particles propel themselves by applying a force on the surrounding fluid. In a steady-
+state perspective, we assume that we are given a random ensemble of particle positions and
+swimming directions, and we aim to evaluate the associated large-scale rheology. Swim-
+ming directions should not be taken as uniformly distributed, but should depend on the
+surrounding fluid deformation, which leads to a nontrivial interaction between the fluid
+flow and particles’ swimming forces. More precisely, our model is based on the following
+assumption: if the fluid is locally deformed, then the distribution of orientations depends
+on some local average of the symmetrized velocity gradient of the surrounding fluid around
+each particle. Although this steady-state perspective is certainly simplistic, our model does
+not prescribe the retroaction of the fluid on the particles a priori, but leaves it as part of
+the problem. We start by modeling the swimming mechanism for a single particle, before
+combining it with (1.2) into a model for the whole active suspension.
+1.3.1. Single-particle swimming mechanism. Let us place ourselves at the scale of an iso-
+lated particle I, and denote by u the fluid velocity outside I. Given a nonnegative smooth
+kernel χ with unit mass
+´
+Rd χ = 1, the locally averaged fluid deformation felt by the particle
+is taken of the form
+EI(u) :=
+ 
+I
+χ ∗ D(u).
+(1.5)
+The precise choice of this operator does not matter in our analysis, provided that it is a
+compact operator applied to a restriction of D(u) around I. Given a value EI(u) = E of
+this averaged fluid deformation, the particle adapts its random swimming direction: we
+denote by ¯f(E) ∈ Rd the resulting propulsion force and by ˜f(E) ∈ Mskew the resulting
+torque on the particle. By the action-reaction principle, this force and torque must result
+from an action of the particle on the surrounding fluid. The detail of this action depends
+on the details of the swimming mechanism (flagella, cilia, etc.). The force field exerted by
+the particle on the fluid is denoted by f(E) = f(·, E), depending on the deformation E,
+and is taken to be supported in the immediate neighborhood (I + B) \ I of the particle.
+
+6
+A. BERNOU, M. DUERINCKX, AND A. GLORIA
+By the action-reaction principle for forces and torques, we must have
+¯f(E) +
+ˆ
+(I+B)\I
+f(E)
+=
+0,
+(1.6)
+Θ : ˜f(E) +
+ˆ
+(I+B)\I
+Θx · f(E)
+=
+0,
+∀Θ ∈ Mskew,
+since the barycenter of particle I is
+´
+I y dy = 0. The relation (1.6) reads as a local neutrality
+condition that actually entails that swimming forces act as dipoles in the fluid equations.
+We emphasize that this is fundamentally different from the sedimentation problem studied
+in [11], for which the force on particles originates from gravity and is not compensated
+locally by opposite forces in the surrounding fluid — the backflow is then uniform and
+leads to more important large-scale effects.
+We further make the following assumptions on the regularity of the swimming force with
+respect to the fluid deformation.
+Hypothesis 1.3 (Swimming mechanism). The random force field defines almost surely a
+smooth map Msym
+0
+→ L∞((I + B) \ I)d : E �→ f(E) such that, for all k ≥ 1,
+∥f(E)∥L∞((I+B)\I) ≤ C⟨E⟩,
+∥∂kf(E)∥L∞((I+B)\I) ≤ Ck.
+♦
+In view of quantitative homogenization results, we shall occasionally need to further as-
+sume that for large strain rates E the swimming direction becomes a deterministic function
+of E. This technical assumption is physically reasonable.
+Hypothesis 1.4 (Swimming in large strain rate). There exist a deterministic direction field
+f ∞ : Msym
+0
+→ Rd, a random strength field S ∈ L∞((I + B) \ I), and an exponent γ > 0,
+such that for all E ∈ Msym
+0
+and k ≥ 1 we have almost surely
+∥f(E) − Sf ∞(E)∥L∞((I+B)\I) ≤ C⟨E⟩1−γ,
+∥∂kf(E) − S∂kf ∞(E)∥L∞((I+B)\I) ≤ Ck⟨E⟩−γ.
+♦
+1.3.2. Resulting system for many particles. Before including the above single-particle swim-
+ming mechanism into the passive suspension model (1.2), we start by making an assumption
+on the joint law of particles’ swimming forces.
+Hypothesis 1.5 (Joint swimming forces). Let {fn(E)}n be a sequence of random maps,
+such that fn satisfies Hypothesis 1.3 with I = In for all n, and such that I and �
+n fn(E)
+are jointly stationary for all E.
+♦
+In order to include these swimming forces into the model (1.2) for a suspension of
+small particles {εIn}n, they need to be properly rescaled. The natural scaling happens to
+be O(1
+ε), which is indeed the only scaling giving rise to a nontrivial and finite contribution
+in the macroscopic limit ε ↓ 0. We add a coupling parameter κ, which stands for the
+activity strength and will need to be chosen small enough to perform the analysis. In
+this ε-rescaling, the kernel χ defining the local averaged fluid deformations felt by the
+particles (1.5) should naturally be replaced by χε := ε−dχ( ·
+ε).
+This however leads to
+important difficulties due to the highly oscillatory local behavior of the fluid flow. Instead,
+we need to replace it by χδ := δ−dχ( ·
+δ), for some intermediate averaging scale ε ≪ δ ≪ 1.
+
+HOMOGENIZATION FOR ACTIVE SUSPENSIONS
+7
+The resulting hydrodynamic model takes on the following guise,
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+−△uε + ∇Pε
+= h + κ
+ε
+�
+n∈Nε(U) fn,ε(
+ffl
+εIn χδ ∗ D(uε)),
+in U \ Iε(U),
+div(uε) = 0,
+in U \ Iε(U),
+uε = 0,
+on ∂U,
+D(uε) = 0,
+in Iε(U),
+´
+ε∂In σ(uε, Pε)ν + κ
+ε ¯fn(
+ffl
+εIn χδ ∗ D(uε)) = 0,
+∀n ∈ Nε(U),
+´
+ε∂In Θ(· − εxn) · σ(uε, Pε)ν
++ κ
+ε Θ : ˜fn(
+ffl
+εIn χδ ∗ D(uε)) = 0,
+∀n ∈ Nε(U), Θ ∈ Mskew,
+(1.7)
+where we have set
+fn,ε(E) := fn,ε(·, E) := ε−dfn
+� ·
+ε, E
+�
+.
+As above, the pressure is anchored via
+´
+U\Iε(U) Pε = 0.
+In what follows, it will be convenient to use an equivalent formulation of swimming
+forces. While each force field fn is supported in the particle neighborhood (In + B) \ In in
+the fluid domain, we may naturally extend fn inside the particle domain In to match its
+propulsion force and torque. More precisely, we can uniquely define fn in In as an affine
+function such that
+¯fn(E) =
+ˆ
+In
+fn(E),
+˜fn(E) =
+ˆ
+In
+fn(E) ⊗ (x − xn).
+(1.8)
+In terms of these extensions, the neutrality condition (1.6) takes the simpler form
+ˆ
+In+B
+fn(E)
+=
+0,
+(1.9)
+ˆ
+In+B
+Θ(x − xn) · fn(E)
+=
+0,
+∀Θ ∈ Mskew,
+and the system (1.7) then becomes
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+−△uε + ∇Pε
+= h + κ
+ε
+�
+n∈Nε(U) fn,ε(
+ffl
+εIn χδ ∗ D(uε)),
+in U \ Iε(U),
+div(uε) = 0,
+in U \ Iε(U),
+uε = 0,
+on ∂U,
+D(uε) = 0,
+in Iε(U),
+´
+ε∂In σ(uε, Pε)ν + κ
+ε
+´
+εIn fn,ε(
+ffl
+εIn χδ ∗ D(uε)) = 0,
+∀n ∈ Nε(U),
+´
+ε∂In Θ(x − εxn) · σ(uε, Pε)ν
++ κ
+ε
+´
+εIn Θ(x − εxn) · fn,ε(
+ffl
+εIn χδ ∗ D(uε)) = 0,
+∀n ∈ Nε(U), Θ ∈ Mskew.
+(1.10)
+With this reformulation, we may readily check that the solution uε can be viewed as the
+orthogonal projection on {u ∈ H1
+0(U)d : D(u)|Iε(U) = 0, div(u) = 0} of the solution of
+
+
+
+−△vε + ∇Rε = h + κ
+ε
+�
+n∈Nε(U) fn,ε(
+ffl
+εIn χδ ∗ D(uε)),
+in U,
+div(vε) = 0,
+in U,
+vε = 0,
+on ∂U.
+(1.11)
+This observation is not used in the sequel.
+
+8
+A. BERNOU, M. DUERINCKX, AND A. GLORIA
+1.4. Heuristics and relation to the physics literature. In the physics literature, one
+usually considers a forcing term in form of an imposed strain rate E ∈ Msym
+0
+at infinity —
+in which case it is natural to replace (1.5) by E itself. The velocity field of the suspension
+on microscopic scale is then given by uE + Ex, where uE is a suitable solution of the
+following infinite-volume problem,
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+−△uE + ∇PE = κ �
+n fn(E),
+in Rd \ I,
+div(uE) = 0,
+in Rd \ I,
+D(uE + Ex) = 0,
+in I,
+´
+∂In σ(uE + Ex, PE)ν + κ
+´
+In fn(E) = 0,
+∀n,
+´
+∂In Θ(x − xn) · σ(uE + Ex, PE)ν
++κ
+´
+In Θ(x − xn) · fn(E) = 0,
+∀n, Θ ∈ Mskew.
+(1.12)
+In these terms, the effective viscosity ¯Btot(E) of the suspension in direction E is obtained
+as the associated ensemble-averaged stress. Splitting the contributions of the stress in the
+fluid domain and in the particles, and taking into account swimming forces, we get for
+all E′ ∈ Msym
+0
+,
+E′ : 2 ¯Btot(E) = E′ : E
+�
+σ(uE + Ex, PE)1Rd\I
+�
++ E′ : E
+�
+σE1I
+�
+− κE
+� �
+n
+E′(x − xn) · fn(E)
+�
+,
+(1.13)
+where σE stands for the stress inside the particles, which we shall define in the proof of
+Lemma 2.5 below via the extension problem
+�
+−div(σE) = κfn(E),
+in In,
+σEν = σ(uE + Ex, PE)ν,
+on ∂In.
+(1.14)
+Noting that rigidity constraints D(uE + Ex) = 0 in I yield
+σ(uE + Ex, PE)1Rd\I = 2 D(uE + Ex) − PE Id 1Rd\I,
+and recalling that E [D(uE)] = 0 (as the average of a gradient), the first contribution
+in (1.13) takes the form
+E′ : E
+�
+σ(uE + Ex, PE)1Rd\I
+�
+= 2E′ : E.
+Next, using (1.14), integrating by parts, and using (1.8), and the skew-symmetry of ˜fn(E),
+one can reformulate the second contribution in (1.13) as the ensemble-averaged stresslet
+on the particles
+E
+�
+σE1I
+�
+=
+E
+� �
+n
+1In
+|In|
+ˆ
+In
+σE
+�
+=
+E
+� �
+n
+1In
+|In|
+ˆ
+∂In
+σ(uE + Ex, PE)ν ⊗s (x − xn)
+�
+.
+The effective viscosity (1.13) thus takes the form
+E′ : 2 ¯Btot(E) = 2E′ : E
++ E
+� �
+n
+1In
+|In|
+ˆ
+∂In
+E′(x − xn) · σ(uE + Ex, PE)ν
+�
+− κE
+� �
+n
+E′(x − xn) · fn(E)
+�
+.
+(1.15)
+
+HOMOGENIZATION FOR ACTIVE SUSPENSIONS
+9
+For convenience, we shall distinguish the passive from the active contributions in this
+expression, and we decompose the solution uE as uE = ψE + κφE in terms of the so-called
+passive and active correctors ψE and φE, defined as suitable solutions of
+
+
+
+
+
+
+
+
+
+
+
+−△ψE + ∇ΣE = 0,
+in Rd \ I,
+div(ψE) = 0,
+in Rd \ I,
+D(ψE) + E = 0,
+in I,
+´
+∂In σ(Ex + ψE, ΣE)ν = 0,
+∀n,
+´
+∂In Θ(x − xn) · σ(Ex + ψE, ΣE)ν = 0,
+∀Θ ∈ Mskew, ∀n.
+and
+
+
+
+
+
+
+
+
+
+
+
+−△φE + ∇ΠE = �
+n fn(E),
+in Rd \ I,
+div(φE) = 0,
+in Rd \ I,
+D(φE) = 0,
+in I,
+´
+∂In σ(φE, ΠE)ν +
+´
+In fn(E) = 0,
+∀n,
+´
+∂In Θ(x − xn) · σ(φE, ΠE)ν +
+´
+In Θ(x − xn) · fn(E) = 0,
+∀Θ ∈ Mskew, ∀n.
+Precise definitions of these correctors are postponed to Section 2.
+In these terms, the
+effective viscosity takes the form
+¯Btot(E) = ¯BpasE + κ ¯Bact(E),
+(1.16)
+where the passive and active contributions are given by
+E′ : 2 ¯BpasE
+:=
+2E′ : E + E
+� �
+n
+1In
+|In|
+ˆ
+∂In
+E′(x − xn) · σ(ψE + Ex, ΣE)ν
+�
+,
+E′ : 2 ¯Bact(E)
+:=
+E
+� �
+n
+1In
+|In|
+ˆ
+∂In
+E′(x − xn) · σ(φE, ΠE)ν
+�
+−E
+� �
+n
+E′(x − xn) · fn(E)
+�
+.
+Using equations for correctors, these expressions are equivalently given by
+E : 2 ¯BpasE
+=
+E
+�
+2|D(ψE) + E|2�
+,
+E′ : 2 ¯Bact(E)
+=
+−E
+� �
+n
+(ψE′ + E′(x − xn)) · fn(E)
+�
+.
+(1.17)
+As one could have expected, the active contribution E′ : ¯Bact(E) coincides with the aver-
+aged swimming force along the passive corrector in direction E′. Note in particular that the
+active corrector φE does not appear in that formulation. The first main contribution of the
+present article is to properly justify these effective viscosity formulas using homogenization
+theory, cf. Theorem 1.7.
+While these general formulas are difficult to analyze in practice without resorting to
+numerical simulations, it is a classical problem in the physics community to derive simpler
+approximate formulas in the dilute regime, which are easier to interpret and provide a
+useful grasp at the physical behavior of suspensions. This is made possible by replacing
+correctors by explicit solutions of single-particle problems: we refer to Theorem 1.9 and
+Section 1.6 below for justification of such dilute approximations based on the methods
+introduced by the last two authors in [5].
+
+10
+A. BERNOU, M. DUERINCKX, AND A. GLORIA
+1.5. Main results: well-posedness, homogenization, and dilute regime. We turn
+to the statement of our main results and we start with the well-posedness of the hy-
+drodynamic model (1.7). It requires either the coupling constant κ to be small enough
+or the interparticle distance ℓ to be large enough. Note that condition (1.18) below is
+nearly almost optimal in general: the same condition with η = 0 is required to ensure
+the perturbative well-posedness of the homogenized equation (1.20), see first paragraph of
+Section 3.2.
+Proposition 1.6 (Well-posedness of the hydrodynamic model). Let Hypotheses 1.1, 1.3,
+and 1.5 hold, and assume εℓ ≤ δ. Provided for some η > 0 we have that
+κℓη−d ≪ 1
+(1.18)
+is small enough (only depending on η, the dimension d, the domain U, and the unscaled
+kernel χ), the system (1.7) is well-posed almost surely for any h ∈ L2(U)d: there exists
+a unique almost sure weak solution (uε, Pε) ∈ L2(Ω; H1
+0(U)d × L2(U \ Iε(U))/R) and it
+satisfies almost surely
+ˆ
+U
+|∇uε|2 +
+ˆ
+U\Iε(U)
+(Pε)2 ≲η (1 + κ2)
+�
+κ2ℓ−d +
+ˆ
+U\Iε(U)
+|h|2�
+.
+(1.19)
+♦
+We now state the homogenization result for this model in the macroscopic limit ε ↓ 0,
+in the simplified situation when the mesoscopic averaging scale δ is fixed (see the proof for
+the associated corrector result).
+Theorem 1.7 (Homogenization at fixed δ > 0). Let Hypotheses 1.1, 1.3, and 1.5 hold, as
+well as the smallness condition (1.18) to ensure well-posedness. For any h ∈ W 1,∞(U)d,
+as ε ↓ 0 with δ > 0 fixed, the almost sure weak solution (uε, Pε) of (1.7) satisfies almost
+surely
+uε
+⇀
+¯uδ,
+in H1
+0(U),
+Pε1U\Iε(U)
+⇀
+(1 − λ) ¯Pδ + (1 − λ)¯b : D(¯uδ)
++(1 − λ)κ
+�
+¯c(χδ ∗ D(¯uδ)) −
+ffl
+U ¯c(χδ ∗ D(¯uδ))
+�
+,
+in L2(U),
+where (¯uδ, ¯Pδ) ∈ H1
+0(U)d × L2(U)/R is the unique solution of the well-posed macroscopic
+system
+
+
+
+−div(2 ¯Bpas D(¯uδ)) − div(2κ ¯Bact(χδ ∗ D(¯uδ))) + ∇ ¯Pδ = (1 − λ)h,
+in U,
+div(¯uδ) = 0,
+in U,
+¯uδ = 0,
+on ∂U,
+(1.20)
+where λ := E [1I] is the particle volume fraction, and where the effective tensors ¯Bpas,
+¯Bact, ¯b, ¯c are defined as follows, in terms of the correctors (ψ, Σ) and (φ, Π) given in (2.1)
+and (2.4) below,
+• The passive effective viscosity ¯Bpas is a positive definite symmetric linear map on the
+space of symmetric trace-free matrices Msym
+0
+: together with the associated symmetric
+trace-free matrix ¯b ∈ Msym
+0
+, it is defined for all E ∈ Msym
+0
+by
+2( ¯Bpas − Id)E + (¯b : E) Id := E
+� �
+n
+1In
+|In|
+ˆ
+∂In
+σ(ψE + Ex, ΣE)ν ⊗s (x − xn)
+�
+,
+
+HOMOGENIZATION FOR ACTIVE SUSPENSIONS
+11
+or equivalently,
+E : 2 ¯BpasE
+:=
+E
+�
+2|D(ψE) + E|2�
+,
+(1.21)
+¯b : E
+:=
+1
+d E
+� �
+n
+1In
+|In|
+ˆ
+∂In
+(x − xn) · σ(ψE + Ex, ΣE)ν
+�
+.
+(1.22)
+• The active effective viscosity ¯Bact is given by
+¯Bact := ¯C + 1
+2 ¯F ,
+(1.23)
+where the map ¯C : Msym
+0
+→ Msym
+0
+, together with the associated map ¯c : Msym
+0
+→ R, is
+defined for all E ∈ Msym
+0
+by
+2 ¯C(E) + ¯c(E) Id := E
+� �
+n
+1In
+|In|
+ˆ
+∂In
+σ(φE, ΠE)ν ⊗s (x − xn)
+�
+,
+(1.24)
+and where the map ¯F : Msym
+0
+→ Msym
+0
+is defined for all E, E′ ∈ Msym
+0
+by
+E′ : ¯F (E) := −E
+� �
+n
+1In
+|In|
+ˆ
+In+B
+E′(x − xn) · fn(E)
+�
+,
+(1.25)
+or equivalently for all E, E′ ∈ Msym
+0
+,
+E′ : 2 ¯Bact(E)
+=
+−E
+� �
+n
+1In
+|In|
+ˆ
+In+B
+(ψE′ + E′(x − xn)) · fn(E)
+�
+,
+(1.26)
+¯c(E)
+=
+1
+dE
+� �
+n
+1In
+|In|
+� ˆ
+∂In
+(x − xn) · σ(φE, ΠE)ν
+��
+.
+♦
+Next, we combine the above (nonlocal) homogenization limit with the (local) limit δ ↓ 0.
+In order to get beyond a purely diagonal regime, cf. (3.55) below, we need to appeal to the
+quantitative homogenization techniques developed in [10] by the last two authors in the
+context of the Stokes equation with rigid inclusions. This requires a quantitative mixing
+assumption such as Hypothesis 1.2 (which holds in particular for hardcore Poisson point
+processes). Although from the modeling viewpoint the choice δ ∼ ε could be more natural,
+it is not accessible to our analysis, and we are restricted to (1.27) below.
+Theorem 1.8 (Homogenization as δ ↓ 0). Let Hypotheses 1.1, 1.3, and 1.5 hold, as well
+as the smallness condition (1.18) to ensure well-posedness, and further let the quantitative
+mixing Hypothesis 1.2 and the technical Hypothesis 1.4 hold. Then, for any h ∈ W 1,∞(U)d,
+as ε, δ ↓ 0, in the regime
+δ−sε → 0,
+for some s > d + 1,
+(1.27)
+the almost sure weak solution (uε, Pε) of (1.7) satisfies almost surely
+uε
+⇀
+¯u,
+in H1
+0(U)d,
+Pε1U\Iε(U)
+⇀
+(1 − λ) ¯P + (1 − λ)¯b : D(¯u)
++(1 − λ)κ
+�
+¯c(D(¯u)) −
+ffl
+U ¯c(D(¯u))
+�
+,
+in L2(U),
+where (¯u, ¯P) ∈ H1
+0(U)d × L2(U)/R is the unique solution of
+
+
+
+−div(2 ¯Btot(D(¯u))) + ∇ ¯P = (1 − λ)h,
+in U,
+div(¯u) = 0,
+in U,
+¯u = 0,
+on ∂U.
+(1.28)
+
+12
+A. BERNOU, M. DUERINCKX, AND A. GLORIA
+in terms of the total effective viscosity
+¯Btot(E) := ¯BpasE + κ ¯Bact(E),
+where we recall that λ, ¯Bpas, ¯Bact, ¯b, ¯c are defined in Theorem 1.7 above.
+♦
+The above shows that the effective stress-strain constitutive relation E �→ 2 ¯BpasE is
+replaced by the nonlinear (non-Newtonian) relation E �→ 2 ¯BpasE + 2 ¯Bact(E) due to the
+effect of particle activity.
+Our last result concerns the analysis of the latter in the dilute regime, and we establish
+the active counterpart to Einstein’s effective viscosity formula. Before stating the result,
+we need to recall some notation from [5]: Denote by Qr(x) = x + [− r
+2, r
+2)d the cube of
+sidelength r centered at x, and set Q(x) = Q1(x), Qr = Qr(0), and Q = Q1(0). The
+intensity of the point process {xn}n is
+λ1 := E
+�
+♯{n : xn ∈ Q}
+�
+,
+and we further define the two- and three-point intensities as
+λ2
+:=
+ℓ−2d sup
+x∈Rd E
+�
+♯
+�
+(n, m) : n ̸= m, xn ∈ Qℓ, xm ∈ Qℓ(x)
+��
+,
+λ3
+:=
+ℓ−3d
+sup
+x1,x2∈Rd E
+�
+♯
+�
+(n0, n1, n2) : n0, n1, n2 pairwise distinct,
+xn0 ∈ Qℓ, xn1 ∈ Qℓ(x1), xn2 ∈ Qℓ(x2)
+��
+,
+where we recall that ℓ stands for (half) the interparticle distance, cf. (1.1). Note that by
+definition the intensity can be compared to the particle volume fraction λ, cf. (1.4), and
+we have for k = 2, 3,
+λ1 ≃ λ ≲ ℓ−d
+and
+λk ≲ ℓ−kd.
+We further recall that the two- and three-point densities g2, g3 of the point process are
+defined by the following relations, for all ζ2 ∈ C∞
+c ((Rd)2) and ζ3 ∈ C∞
+c ((Rd)3),
+¨
+(Rd)2 ζ2(x, y) g2(x, y) dxdy
+=
+E
+� �
+n̸=m
+ζ2(xn, xm)
+�
+,
+(1.29)
+¨
+(Rd)3 ζ3(x, y, z) g3(x, y, z) dxdydz
+=
+E
+�
+�
+n0,n1,n2
+distinct
+ζ3(xn0, xn1, xn2)
+�
+.
+In these terms, the above definitions of the two- and three-point intensities are equivalently
+written as
+λ2
+=
+sup
+x∈Rd
+ 
+Qℓ
+ 
+Qℓ(x)
+g2(y, z) dydz,
+(1.30)
+λ3
+=
+sup
+x1,x2∈Rd
+ 
+Qℓ
+ 
+Qℓ(x1)
+ 
+Qℓ(x2)
+g3(x, y, z) dxdydz.
+With this notation at hand, we may now state the following result on the first-order dilute
+expansion of the effective viscosity. The expansion of the passive contribution ¯Bpas was
+already established in [5], and we extend it here to the active setting.
+Theorem 1.9 (Dilute expansion of the effective viscosity). Let Hypotheses 1.1, 1.3, and 1.5
+hold, and further assume
+
+HOMOGENIZATION FOR ACTIVE SUSPENSIONS
+13
+(a) Independence condition: The random shapes and swimming forces {I◦
+n, fn}n are iid
+copies of a given random open subset I◦ and of a random map f ◦, independently of the
+point process {xn}n.
+(b) Decay of correlations: The point process {xn}n is strongly mixing, and the two- and
+three-point correlation functions
+h2(x, y)
+:=
+g2(x, y) − λ2
+1,
+h3(x, y, z)
+:=
+g3(x, y, z) − λ3
+1 − λ1(h2(x, y) + h2(x, z) + h2(y, z)),
+have algebraic decay: there exists γ > 0 such that for all x, y, z,
+|h2(x, y)| ≲ ⟨x − y⟩−γ,
+|h3(x, y, z)| ≲ ⟨x − y⟩−γ ∧ ⟨x − z⟩−γ ∧ ⟨y − z⟩−γ.
+(1.31)
+Then, we have
+��� ¯Btot(E) −
+�
+E + λ1 ¯B(1)
+pasE + κλ1 ¯B(1)
+act(E)
+����
+≲ ⟨E⟩
+�
+λ2|log λ1| + (λ1λ2 + λ1λ3|log λ1|)
+1
+2
+�
+,
+(1.32)
+where we have set
+E′ : 2 ¯B(1)
+pasE
+:=
+E
+� ˆ
+∂I◦ E′x · σ(ψ◦
+E + Ex, Σ◦
+E)ν
+�
+,
+E′ : 2 ¯B(1)
+act(E)
+:=
+−E
+� ˆ
+2B
+(ψ◦
+E′ + E′x) · f ◦(E)
+�
+,
+in terms of the solution (ψ◦
+E, Σ◦
+E) of the single-particle problem
+
+
+
+
+
+
+
+
+
+
+
+−△ψ◦
+E + ∇Σ◦
+E = 0,
+in Rd \ I◦,
+div(ψ◦
+E) = 0,
+in Rd \ I◦,
+D(ψ◦
+E + Ex) = 0,
+in I◦,
+´
+∂I◦ σ(ψ◦
+E + Ex, Σ◦
+E)ν = 0,
+´
+∂I◦ Θx · σ(ψ◦
+E + Ex, Σ◦
+E)ν = 0,
+∀Θ ∈ Mskew.
+(1.33)
+In particular, in case of spherical particles I◦ = B, these expressions take the explicit forms
+E′ : 2 ¯B(1)
+pasE := (d + 2)|B|E′ : E,
+E′ : 2 ¯B(1)
+act(E) := −
+ˆ
+Rd\B
+�
+1 −
+1
+|x|d+2
+�
+E′x · E [f ◦(E)]
+(1.34)
++ d+2
+2
+ˆ
+Rd\B
+�
+1 −
+1
+|x|2
+� (x·E′x)x
+|x|d+2 · E [f ◦(E)] .
+♦
+Note that the error bound in (1.32) is ≪ λ1 provided that λ2|log λ1| ≪ λ1, and is in
+particular bounded by ℓ−3d/2 ≪ ℓ−d in the regime ℓ ≫ 1.
+Alternatively, arguing similarly as for the reformulation (1.26) of (1.24)–(1.25), the dilute
+active effective viscosity can be written as
+E′ : 2 ¯B(1)
+act(E) = E′ : E
+� �
+n
+1In
+|In|
+� ˆ
+∂In
+σ(φn
+E, Πn
+E)ν⊗s(x−xn)−
+ˆ
+In+B
+fn(E)⊗s(x−xn)
+��
+.
+
+14
+A. BERNOU, M. DUERINCKX, AND A. GLORIA
+In particular, if fn(E) is rotationally symmetric around the direction ¯fn(E), we find by
+symmetry,
+2 ¯B(1)
+act(E) = λ1E
+�
+α(E)
+� ¯fn(E)⊗ ¯fn(E)
+| ¯fn(E)|2
+− 1
+d Id
+��
+,
+(1.35)
+for some random prefactor α(E) ∈ R, the sign of which is actually of critical interest and
+depends on the shape of the swimming mechanism.
+1.6. Viscosity reduction. The main motivation of this work is to introduce a nontrivial
+(and hopefully somewhat realistic) model for active suspensions and rigorously establish
+a reduction of the viscosity of the plain fluid due to the activity of the particles. In the
+dilute regime, the above formulas provide a rigorous contribution to this celebrated topic.
+Although the question can be reduced to understanding the sign of α(E) in (1.35), we take
+a shorter path here based on (1.34). To make computations explicit, we restrict ourselves
+to the following simplified model for the swimming mechanism, see e.g. [19, 22],
+f ◦(E) := ¯f(E)(|B|−11B − δx(E)),
+where the force exerted by the particle on the surrounding fluid is reduced to a Dirac force
+at a surrounding point x(E) ∈ Rd\B. Consider a shear deformation E = s
+2(e1⊗e2+e2⊗e1)
+for some s ∈ R. In these terms, the formula (1.34) reads
+E : ¯B(1)
+act(E) = s
+2
+�
+1 −
+1
+|x(E)|d+2
+��
+x(E)1 ¯f(E)2 + x(E)2 ¯f(E)1
+�
+− s d+2
+2
+�
+1 −
+1
+|x(E)|2
+�x(E)1x(E)2
+|x(E)|d+2 x(E) · ¯f(E).
+For a spherical particle with its swimming device viewed as a rigid elongated particle, the
+motion in shear flow has been well-studied on the formal level: in the limit of a strong
+angular diffusion, a standard heuristic computation shows that the preferred orientation
+of the particle is e :=
+1
+√
+2(e1 + e2) or its opposite; see e.g. [14, Section V.8]. This leads us
+to choosing
+¯f(E) = ±| ¯f(E)|e
+and
+x(E) = ±γ|x(E)|e,
+where γ = 1 in case when the Dirac swimming force is ahead of the particle (“puller”
+particle) and γ = −1 in case when it is behind the particle (“pusher” particle). Hence, we
+get
+E : ¯B(1)
+act(E) = γ s
+2|x(E)|| ¯f(E)|
+�
+1 − d+2
+2
+1
+|x(E)|d + d
+2
+1
+|x(E)|d+2
+�
+,
+which is negative (resp. positive) in case of a pusher (resp. puller). This is in full agreement
+with well-known experiments and predictions of [35, 36, 27].
+To conclude on the extent of the possible viscosity reduction, we come back to Theo-
+rem 1.9 in case ℓ ≫ 1. As stated, the error bound in the result is then of order O(ℓ−3d/2),
+¯Btot(E) =
+�
+1 + d+2
+2 |B|λ1
+�
+E + κλ1 ¯B(1)
+act(E) + O(ℓ−3d/2).
+(1.36)
+If ¯B(1)
+act(E) < 0, we infer that the total effective viscosity is smaller than the viscosity of
+the plain fluid,
+E : ¯Btot(E) < |E|2,
+provided that the activity is strong enough κ ≫ 1 in such a way that the active contribution
+exceeds the passive one. Moreover, as λ1 = O(ℓ−d), we see that the viscosity reduction
+could become of order 1 if κ = O(ℓd).
+This drastic reduction of viscosity is however
+
+HOMOGENIZATION FOR ACTIVE SUSPENSIONS
+15
+prohibited by the (only nearly optimal) condition (1.18), which is used to ensure the well-
+posedness of the microscopic model. In some sense, the above analysis can be compared
+to the enhancement of elastostriction by active charges analyzed in [13].
+Outline of the article. The article is organized as follows. Section 2 is dedicated to
+the introduction of correctors, the key quantities in homogenization. The proofs of the
+homogenization results of Theorems 1.7 and 1.8 are the object of Section 3, while the
+dilute analysis and the proof of Theorem 1.9 are postponed to Section 4.
+Notation.
+— For vector fields u, u′, and matrix fields T, T ′, we set (∇u)ij = ∇jui, div(T)i = ∇jTij,
+T : T ′ = TijT ′
+ij, (u ⊗ u′)ij = uiu′
+j, (T s)ij = 1
+2(Tij + Tji), D(u) = (∇u)s, (u ⊗s u′) =
+(u ⊗ u′)s. For a 3-tensor field S, the matrix div(S) is defined by div(S)ij = ∇kSijk.
+For a matrix E and a vector field u, we write ∂Eu = E : ∇u. We systematically use
+Einstein’s summation convention on repeated indices.
+— For a vector field u and a scalar field P, we recall the notation σ(u, P) = 2 D(u) − P Id
+for the Cauchy stress tensor.
+— We denote by C ≥ 1 any constant that only depends on dimension d, on the constant ϑ
+in Hypothesis 1.1, and on the reference domain U. We use the notation ≲ (resp. ≳)
+for ≤ C×(resp. ≥ 1
+C ×) up to such a multiplicative constant C. We write ≪ (resp. ≫)
+for ≤ C× (resp. ≥ C×) up to a sufficiently large multiplicative constant C. We add
+subscripts to C, ≲, ≳ in order to indicate dependence on other parameters.
+— We write M0 ⊂ Rd×d for the subset of trace-free matrices, Msym
+0
+for the subset of
+symmetric trace-free matrices, and Mskew for the subset of skew-symmetric matrices.
+— The ball centered at x of radius r in Rd is denoted by Br(x), and we set B(x) = B1(x),
+Br = Br(0) and B = B1(0).
+— We use the standard notation ⟨a⟩ := (1 + |a|2)1/2.
+2. Corrector problems
+We start by recalling the relevant correctors for the passive suspension problem as in-
+troduced in [2, 4, 8, 10], and then we define new correctors for the active problem.
+2.1. Passive corrector problems. Correctors for passive suspensions were first defined
+in [8] and are key to the definition of the associated effective viscosity ¯B, cf. (1.3).
+Lemma 2.1 (Passive correctors [8]). Let Hypothesis 1.1 hold. For all E ∈ Msym
+0
+, there
+exist a unique random field ψE ∈ L2(Ω; H1
+loc(Rd)d) and a unique pressure field ΣE ∈
+L2(Ω; L2
+loc(Rd \ I)) such that:
+— almost surely, realizations of ψE and ΣE satisfy
+
+
+
+
+
+
+
+
+
+
+
+−△ψE + ∇ΣE = 0,
+in Rd \ I,
+div(ψE) = 0,
+in Rd \ I,
+D(ψE + Ex) = 0,
+in I,
+´
+∂In σ(ψE + Ex, ΣE)ν = 0,
+∀n,
+´
+∂In Θ(x − εxn) · σ(ψE + Ex, ΣE)ν = 0,
+∀n, ∀Θ ∈ Mskew,
+(2.1)
+
+16
+A. BERNOU, M. DUERINCKX, AND A. GLORIA
+— the corrector gradient ∇ψE and the pressure ΣE1Rd\I are stationary, with
+E
+�
+∇ψE
+�
+= 0,
+E
+�
+ΣE1Rd\I
+�
+= 0,
+E
+�
+|∇ψE|2�
++ E
+�
+Σ2
+E1Rd\I
+�
+≲ λ|E|2,
+and with the anchoring condition
+´
+B ψE = 0.
+In addition, the following properties hold.
+(i) Ergodic theorem: almost surely,
+(∇ψE)( ·
+ε)
+⇀
+E [∇ψE] = 0,
+(ΣE1Rd\I)( ·
+ε)
+⇀
+E
+�
+ΣE1Rd\I
+�
+= 0,
+weakly in L2
+loc(Rd) as ε ↓ 0.
+(ii) Sublinearity: almost surely, for all q <
+2d
+d−2,
+εψE( ·
+ε) → 0
+strongly in Lq
+loc(Rd)d as ε ↓ 0.
+♦
+As in [2, 10], in view of quantitative estimates, we further need to define an associated
+extended flux J and a flux corrector Υ. More precisely, J is a solenoidal extension of the
+natural flux σ(ψE +Ex, ΣE) outside the particles, and Υ is the associated vector potential
+in the Coulomb gauge.
+Lemma 2.2 (Passive flux correctors [2, 10]). Let Hypothesis 1.1 hold. For all E ∈ Msym
+0
+,
+there exists a stationary random 2-tensor field JE = (JE;ij)1≤i,j≤d with finite second mo-
+ment such that, almost surely,
+JE1Rd\I
+=
+σ(ψE + Ex, ΣE)1Rd\I,
+div(JE)
+=
+0,
+and for all n,
+∥JE∥L2(In) ≲ ∥σ(ψE + Ex, ΣE)∥L2((In+B)\In).
+(2.2)
+Moreover, there exists a unique random 3-tensor field ΥE = (ΥE;ijk)1≤i,j,k≤d such that:
+— for all i, j, k, almost surely, realizations of ΥE;ijk belong to H1
+loc(Rd) and satisfy
+− ∆ΥE;ijk = ∂jJE;ik − ∂kJE;ij;
+(2.3)
+— the random field ∇ΥE is stationary, has vanishing expectation, has finite second mo-
+ment, and satisfies the anchoring condition
+ffl
+B ΥE = 0.
+In addition, the following properties hold.
+(i) Skew-symmetry: almost surely, ΥE;ijk = −ΥE;ikj for all i, j, k.
+(ii) Vector potential: almost surely, for all i,
+div(ΥE;i) = JE;i − E[JE;i],
+where we have set ΥE;i = (ΥE;ijk)1≤j,k≤d and JE;i = (JE;ij)1≤j≤d.
+(iii) Ergodic theorem: almost surely,
+(∇ΥE)( ·
+ε) ⇀ E [∇ΥE] = 0weakly in L2
+loc(Rd) as ε ↓ 0.
+(iv) Sublinearity: almost surely, for all q <
+2d
+d−2,
+εΥE( ·
+ε) ⇀ 0
+strongly in Lq
+loc(Rd)d as ε ↓ 0.
+
+HOMOGENIZATION FOR ACTIVE SUSPENSIONS
+17
+(v) Effective constants: the expectation of JE takes the form
+E [JE] = 2 ¯BpasE + (¯b : E) Id,
+in terms of the effective constants ¯Bpas and ¯b defined in (1.21) and (1.22).
+♦
+2.2. Active corrector problems. We turn to the definition of suitable correctors for the
+active suspension problem. These new correctors characterize the contribution of swimming
+forces of the particles in a uniform fluid velocity gradient E ∈ Msym
+0
+.
+Lemma 2.3. Let Hypotheses 1.1, 1.3, and 1.5 hold. For all E ∈ Msym
+0
+, there exist a unique
+random field φE ∈ L2(Ω; H1
+loc(Rd)d) and a unique pressure field ΠE ∈ L2(Ω; L2
+loc(Rd \ I))
+such that:
+— almost surely, realizations of φE and ΠE satisfy
+
+
+
+
+
+
+
+
+
+
+
+−∆φE + ∇ΠE = �
+n fn(E),
+in Rd \ I,
+div(φE) = 0,
+in Rd \ I,
+D(φE) = 0,
+in I,
+´
+∂In σ(φE, ΠE)ν + ¯fn(E) = 0,
+∀n,
+´
+∂In Θ(x − xn) · σ(φE, ΠE)ν + Θ : ˜fn(E) = 0,
+∀n, ∀Θ ∈ Mskew,
+(2.4)
+— the corrector gradient ∇φE and the pressure ΠE1Rd\I are stationary, with
+E
+�
+∇φE
+�
+= 0,
+E
+�
+ΠE1Rd\I
+�
+= 0,
+E
+�
+|∇φE|2�
++ E
+�
+Π2
+E1Rd\I
+�
+≲ λ⟨E⟩2,
+(2.5)
+and with the anchoring condition
+´
+B φE = 0.
+In addition, the following properties hold.
+(i) Ergodic theorem: almost surely,
+(∇φE)( ·
+ε)
+⇀
+E [∇φE] = 0,
+(ΠE1Rd\I)( ·
+ε)
+⇀
+E
+�
+ΠE1Rd\I
+�
+= 0,
+weakly in L2
+loc(Rd) as ε ↓ 0.
+(ii) Sublinearity: almost surely, for all q <
+2d
+d−2,
+εφE( ·
+ε) → 0
+strongly in Lq
+loc(Rd) as ε ↓ 0.
+♦
+Proof. The argument is similar to that in [8, Proposition 2.1] for Lemma 2.1 above, and
+we only briefly show the needed adaptations: we describe the structure of equation (2.4),
+following the first step of the proof of [8, Proposition 2.1], while the rest of the proof
+in [8] is then easily repeated and is skipped here for brevity.
+More precisely, we only
+show the following: if φE is a solution of (2.4) with ∇φE, ΠE stationary with finite second
+moments, then it satisfies for all stationary fields v ∈ L2(Ω; H1
+loc(Rd)d) with div(v) = 0
+and D(v)|I = 0,
+E [∇v : ∇φE] = −E
+� �
+n
+1In
+|In|
+ˆ
+In+B
+v · fn(E)
+�
+,
+(2.6)
+where, by the Cauchy–Schwarz and the Poincaré inequalities, together with the hardcore
+assumption and the property
+´
+In+B fn(E) = 0 (cf. (1.9)), the right-hand side is bounded
+by
+����E
+� �
+n
+1In
+|In|
+ˆ
+In+B
+v · fn(E)
+����� ≲ E
+�
+|∇v|2� 1
+2
+�
+λ E
+� ˆ
+I◦+B
+|f◦(E)|2
+�� 1
+2
+.
+
+18
+A. BERNOU, M. DUERINCKX, AND A. GLORIA
+To prove this claim, we start by noting that the hardcore assumption allows to construct
+almost surely for all R > 0 a cut-off function ηR such that
+ηR|BR = 1,
+ηR|Rd\BR+5 = 0,
+|∇ηR| ≲ 1,
+and such that ηR is constant in In + B for all n. As φE is divergence-free, testing equa-
+tion (2.4) with ηRv and integrating by parts, we find
+2
+ˆ
+Rd\I
+D(ηRv) : D(φE) −
+ˆ
+Rd\I
+∇ηR · vΠE
+=
+�
+n
+ˆ
+(In+B)\In
+ηRv · fn(E) −
+�
+n
+ˆ
+∂In
+ηRv · σ(φE, ΠE)ν.
+(2.7)
+Since D(φE)|I = 0 and ∇ηR|I = 0, the left-hand side of (2.7) writes
+2
+ˆ
+Rd\I
+D(ηRv) : D(φE) −
+ˆ
+Rd\I
+∇ηR · vΠE =
+ˆ
+Rd ∇(ηRv) : ∇φE −
+ˆ
+Rd ∇ηR · vΠE.
+As ηR is constant in In + B and D(v) = 0 in In, we can rewrite the last right-hand side
+term as
+ˆ
+∂In
+ηRv · σ(φE, ΠE)ν = ηR(xn)
+��  
+In
+v
+�
+·
+ˆ
+∂In
+σ(φE, ΠE)ν
++
+�  
+In
+(∇v)skew�
+:
+ˆ
+∂In
+σ(φE, ΠE)ν ⊗ (x − xn)
+�
+,
+and thus, in view of the boundary conditions for (φE, ΠE) in (2.4), using the notation (1.8),
+ˆ
+∂In
+ηRv · σ(φE, ΠE)ν
+=
+−ηR(xn)
+��  
+In
+v
+�
+· ¯fn(E) +
+�  
+In
+∇v
+�
+: ˜fn(E)
+�
+=
+−
+ˆ
+In
+ηRv · fn(E).
+Inserting this into (2.7), we get
+ˆ
+Rd ∇(ηRv) : ∇φE −
+ˆ
+Rd ∇ηR · vΠE =
+�
+n
+ˆ
+In+B
+ηRv · fn(E).
+(2.8)
+Expanding the gradient in the left-hand side, passing to the limit R ↑ ∞, and using the
+stationarity of ∇φE, ΠE and of v, the claim (2.6) follows. From there, we may then refer
+to the proof in [8, Proposition 2.1].
+□
+Next, as in Lemma 2.2, we further need to define an associated extended flux and a flux
+corrector for the active suspension problem. The difficulty, however, is that even in the fluid
+domain Rd \ I the flux σ(φE, ΠE) is not divergence-free. It thus needs to be first suitably
+compensated and we are led to defining the following auxiliary corrector γE. The proof of
+the upcoming lemma (which is a simplified version of Lemma 2.3) is straightforward and
+skipped for brevity.
+Lemma 2.4. Let Hypotheses 1.1, 1.3, and 1.5 hold. For all E ∈ Msym
+0
+, there exists a
+unique random field γE ∈ L2(Ω; H1
+loc(Rd)d) such that:
+
+HOMOGENIZATION FOR ACTIVE SUSPENSIONS
+19
+— almost surely, the realizations of γE satisfy
+− △γE =
+�
+n
+fn(E);
+(2.9)
+— the gradient field ∇γE is stationary with
+E
+�
+∇γE
+�
+= 0,
+E
+�
+|∇γE|2�
+≲ λ⟨E⟩2,
+and with the anchoring condition
+´
+B γE = 0.
+In addition, the following properties hold.
+(i) Ergodic theorem: almost surely,
+(∇γE)( ·
+ε)
+⇀
+E [∇γE] = 0
+weakly in L2
+loc(Rd)d as ε ↓ 0.
+(ii) Sublinearity: almost surely, for all q <
+2d
+d−2,
+εγE( ·
+ε) → 0
+strongly in Lq
+loc(Rd)d as ε ↓ 0.
+♦
+Using the above-defined γE to compensate the divergence of the flux σ(φE, ΠE) in the
+fluid domain, and extending it similarly as in Lemma 2.2 inside the particles, we are now in
+the position to define a flux KE, a divergence-free compensated flux LE, and the associated
+flux corrector θE.
+Lemma 2.5. Under Hypotheses 1.1, 1.3, and 1.5, there exists a stationary random sym-
+metric 2-tensor field KE = (KE;ij)1≤i,j≤d with finite second moment
+E
+�
+|KE|2�
+≲ λ⟨E⟩2,
+(2.10)
+such that, almost surely,
+KE1Rd\I
+=
+σ(φE, ΠE)1Rd\I,
+div(KE)
+=
+−
+�
+n
+fn(E).
+(2.11)
+Moreover, the expectation of KE takes the form
+E [KE] = 2 ¯C(E) + ¯c(E) Id,
+(2.12)
+in terms of the effective tensors ¯C(E) and ¯c(E) defined in (1.24). Next, for the divergence-
+free compensated flux
+LE := KE − E [KE] − ∇γE,
+(2.13)
+there exists a unique random 3-tensor field θE = (θE;ijk)1≤i,j,k≤d such that:
+— for all i, j, k, almost surely, realizations of θE;ijk belong to H1
+loc(Rd) and satisfy
+− △θE;ijk = ∂jLE;ik − ∂kLE;ij;
+(2.14)
+— the random field ∇θE is stationary, has vanishing expectation, has finite second moment,
+and satisfies the anchoring condition
+ffl
+B θE = 0.
+In addition, the following properties hold.
+(i) Skew-symmetry: almost surely, θE;ijk = −θE;ikj for all i, j, k.
+(ii) Vector potential: almost surely, for all i,
+div(θE;i) = LE;i,
+where we have set θE;i = (θE;ijk)1≤j,k≤d and LE;i = (LE;ij)1≤j≤d.
+
+20
+A. BERNOU, M. DUERINCKX, AND A. GLORIA
+(iii) Ergodic theorem: almost surely,
+(∇θE)( ·
+ε)
+⇀
+E [∇θE] = 0
+weakly in L2
+loc(Rd)d as ε ↓ 0.
+(iv) Sublinearity: almost surely, for all q <
+2d
+d−2,
+εθE( ·
+ε) → 0
+strongly in Lq
+loc(Rd)d as ε ↓ 0.
+♦
+Proof. We split the proof into three main steps.
+Step 1. Construction and properties of the extended flux KE.
+For all g ∈ C∞
+c (Rd), equation (2.4) yields by integration by parts,
+ˆ
+Rd\I
+∇g : σ(φE, ΠE)
+=
+�
+n
+ˆ
+(In+B)\In
+g · fn(E) −
+�
+n
+ˆ
+∂In
+g · σ(φE, ΠE)ν,
+and thus, using boundary conditions for (φE, ΠE) in (2.4) and recalling the notation (1.8),
+ˆ
+Rd\I
+∇g : σ(φE, ΠE) +
+�
+n
+ˆ
+∂In
+�
+g −
+�  
+In
+g
+�
+−
+�  
+In
+(∇g)skew�
+(x − xn)
+�
+· σ(φE, ΠE)ν
+=
+�
+n
+ˆ
+In+B
+g · fn(E).
+(2.15)
+For all n, we may then consider the following Neumann problem,
+
+
+
+−△φn
+E + ∇Πn
+E = fn(E),
+in In,
+div(φn
+E) = 0,
+in In,
+σ(φn
+E, Πn
+E)ν = σ(φE, ΠE)ν,
+on ∂In.
+(2.16)
+Note that this only defines φn
+E up to a rigid motion, which is fixed by choosing φn
+E with
+ffl
+In φn
+E = 0 and
+ffl
+In ∇φn
+E ∈ Msym
+0
+. Assuming that this Neumann problem (2.16) is well-
+posed, and setting
+˜qE
+:=
+D(φE) +
+�
+n
+D(φn
+E)1In,
+˜ΠE
+:=
+ΠE1Rd\I +
+�
+n
+Πn
+E1In,
+(2.17)
+we easily deduce that the extended flux
+KE := 2˜qE − ˜ΠE Id
+(2.18)
+satisfies the desired relations (2.11) (recall that D(φE)|I = 0). It remains to check the
+well-posedness of this Neumann problem and to establish the bound (2.10). Note that this
+well-posedness, together with the uniqueness in our construction of the pressures {Πn
+E}n
+below, ensures that ˜qE and ˜ΠE are stationary.
+We split the proof into three further
+substeps.
+Substep 1.1. Well-posedness of the Neumann problem (2.16) for φn
+E.
+The weak formulation of (2.16) reads as follows: for all divergence-free fields v ∈ H1(In)d,
+2
+ˆ
+In
+D(v) : D(φn
+E) = LE(v),
+(2.19)
+
+HOMOGENIZATION FOR ACTIVE SUSPENSIONS
+21
+where LE stands for the linear form
+LE(v) :=
+ˆ
+In
+v · fn(E) +
+ˆ
+∂In
+v · σ(φE, ΠE)ν.
+Using the boundary conditions for (φE, ΠE) in (2.4) and recalling the notation (1.8), the
+latter can be reformulated as
+LE(v) =
+ˆ
+∂In
+�
+v −
+�  
+In
+v
+�
+−
+�  
+In
+(∇v)skew�
+(x − xn)
+�
+· σ(φE, ΠE)ν.
+(2.20)
+We shall show that (2.19) is well-posed in the following Hilbert subspace of H1(In)d,
+T :=
+�
+v ∈ H1(In)d : div(v) = 0,
+ 
+In
+v = 0, and
+ 
+In
+∇v ∈ Msym
+0
+�
+.
+First note that Korn’s inequality yields for all v ∈ T ,
+∥∇v∥L2(In) ≲ ∥D(v)∥L2(In),
+which entails that the bilinear form (v, ψ) �→ 2
+´
+In D(v) : D(ψ) is continuous and coercive
+on T × T . By the Lax–Milgram theorem, in order to prove the well-posedness of (2.19),
+it remains to show that the linear form LE is continuous on T as well.
+In order to deal with the Neumann condition, we consider an extension map
+Tn :
+�
+v ∈ H1(In)d : div(v) = 0
+�
+→
+�
+v ∈ H1
+0(In + B)d : div(v) = 0
+�
+,
+such that Tnv|In = v|In and
+∥∇Tnv∥L2(In+B) ≲ ∥∇v∥L2(In).
+(2.21)
+Smuggling in Tn in (2.20), integrating by parts, and inserting equation (2.4), the linear
+form LE can be rewritten as
+LE(v)
+=
+−
+ˆ
+(In+B)\In
+div
+�
+σ(φE, ΠE) Tn
+�
+v −
+�  
+In
+v
+�
+−
+�  
+In
+(∇v)skew�
+(x − xn)
+��
+=
+ˆ
+(In+B)\In
+Tn
+�
+v −
+�  
+In
+v
+�
+−
+�  
+In
+(∇v)skew�
+(x − xn)
+�
+· fn(E)
+−2
+ˆ
+(In+B)\In
+D(φE) : D
+�
+Tn
+�
+v −
+�  
+In
+v
+�
+−
+�  
+In
+(∇v)skew�
+(x − xn)
+��
+.
+Hence, in view of (2.21) and of Korn’s inequality,
+|LE(v)| ≲ ∥D(v)∥L2(In)
+�
+∥D(φE)∥L2(In+B) + ∥fn(E)∥L2(In+B)
+�
+,
+which proves the continuity of LE on T .
+By the Lax–Milgram theorem, we deduce that there exists a unique solution φn
+E ∈ T
+of (2.19), and that it satisfies
+∥D(φn
+E)∥L2(In) ≲ ∥D(φE)∥L2(In+B) + ∥fn(E)∥L2(In+B).
+By Korn’s inequality, this further yields
+∥∇φn
+E∥L2(In) ≲ ∥D(φE)∥L2(In+B) + ∥fn(E)∥L2(In+B).
+(2.22)
+
+22
+A. BERNOU, M. DUERINCKX, AND A. GLORIA
+Substep 1.2. Construction of the pressure.
+As (2.17) reads ˜qE = D(φE) + D(φn
+E)1In in In + B, combining equation (2.4) for φE and
+equation (2.19) for φn
+E, we find for all v ∈ C∞
+c (In + B)d with div(v) = 0,
+2
+ˆ
+Rd D(v) : ˜qE =
+ˆ
+Rd v · fn(E).
+Appealing e.g. to [26, Proposition 12.10], we deduce that there exists an associated pressure
+field Πn
+E ∈ L2
+loc(In + B), which is unique up to an additive constant, such that for all test
+functions v ∈ C∞
+c (In + B)d,
+ˆ
+Rd D(v) : (2˜qE − Πn
+E Id) =
+ˆ
+Rd v · fn(E).
+(2.23)
+Since for all v ∈ C∞
+c ((In + B) \ In)d we get
+ˆ
+Rd D(v) : σ(φE, Πn
+E) =
+ˆ
+Rd D(v) : (2˜qE − Πn
+E Id) =
+ˆ
+Rd v · fn(E),
+and comparing with equation (2.4), we deduce that Πn
+E can be chosen uniquely to coincide
+with ΠE on (In + B) \ In.
+It remains to estimate the above-constructed pressure. Using that Πn
+E coincides with ΠE
+in (In + B) \ In, we can split
+ˆ
+In+B
+Πn
+E
+=
+ˆ
+(In+B)\In
+ΠE +
+ˆ
+In
+Πn
+E
+=
+ˆ
+(In+B)\In
+ΠE +
+ˆ
+In
+�
+Πn
+E −
+ 
+In+B
+Πn
+E
+�
++ |In|
+ 
+In+B
+ΠE,
+to the effect that
+(|In + B| − |In|)
+���
+ 
+In+B
+Πn
+E
+��� ≤
+���
+ˆ
+(In+B)\In
+ΠE
+��� +
+���
+ˆ
+In
+�
+Πn
+E −
+ 
+In+B
+Πn
+E
+����.
+Hence,
+∥Πn
+E∥L2(In)
+≲
+���Πn
+E −
+ 
+In+B
+Πn
+E
+���
+L2(In+B) +
+���
+ 
+In+B
+Πn
+E
+���
+≲
+���Πn
+E −
+ 
+In+B
+Πn
+E
+���
+L2(In+B) +
+���
+ 
+(In+B)\In
+ΠE
+���.
+Starting from (2.23), a standard argument based on the Bogovskii operator yields
+���Πn
+E −
+ 
+In+B
+Πn
+E
+���
+L2(In+B) ≲ ∥˜qE∥L2(In+B),
+so that the above becomes
+∥Πn
+E∥L2(In) ≲ ∥˜qE∥L2(In+B) + ∥ΠE∥L2((In+B)\In).
+Combining this with (2.22), we get
+∥∇φn
+E∥L2(In) + ∥Πn
+E∥L2(In) ≲ ∥σ(φE, ΠE)∥L2((In+B)\In) + ∥fn(E)∥L2(In+B).
+(2.24)
+
+HOMOGENIZATION FOR ACTIVE SUSPENSIONS
+23
+Substep 1.3. Proof of (2.10).
+By definition (2.18), the bound (2.24) yields for all n,
+∥KE∥L2(In)
+≲
+∥σ(φn
+E, Πn
+E)∥L2(In)
+≲
+∥σ(φE, ΠE)∥L2((In+B)\In) + ∥fn(E)∥L2(In+B).
+(2.25)
+Hence, for all R > 0,
+∥KE∥2
+L2(BR) ≲ ∥σ(φE, ΠE)1Rd\I∥2
+L2(BR+5) +
+�
+n:In∩BR̸=∅
+ˆ
+In+B
+|fn(E)|2,
+and thus, by stationarity, letting R ↑ ∞,
+∥KE∥2
+L2(Ω) ≲ ∥σ(φE, ΠE)1Rd\I∥2
+L2(Ω) + λ E
+� ˆ
+I◦+B
+|f◦(E)|2
+�
+.
+Combined with (2.5), this yields (2.10).
+Step 2. Formula for E [KE] and definition of ¯C(E) and ¯c(E).
+We split the proof into two further substeps, separately proving formula (2.12) for E [KE]
+and establishing the alternative formulas (1.26) for ¯C(E) and ¯c(E).
+Substep 2.1. Proof of (2.12).
+The hardcore assumption allows to construct almost surely for all R > 0 a cut-off func-
+tion ηR such that
+ηR|BR = 1,
+ηR|Rd\BR+5 = 0,
+|∇ηR| ≲ 1,
+and such that ηR is constant in In + B for all n. By definition of KE and (φE, ΠE), we
+have
+E
+�
+KE1Rd\I
+�
+= E
+�
+σ(φE, ΠE)1Rd\I
+�
+= 2E
+�
+D(φE)
+�
+− E
+�
+ΠE1Rd\I
+�
+Id = 0,
+and the ergodic theorem then yields almost surely,
+E [KE] = E
+�
+KE1I
+�
+= lim
+R↑∞ |BR|−1
+ˆ
+I
+ηR KE.
+(2.26)
+By definition of KE and the choice of ηR, integration by parts and the equation (2.16)
+for (φn
+E, Σn
+E) yield
+ˆ
+I
+ηR KE
+=
+�
+n
+ηR(xn)
+ˆ
+In
+σ(φn
+E, Πn
+E)
+=
+�
+n
+ηR(xn)
+ˆ
+∂In
+σ(φn
+E, Πn
+E)ν ⊗ (x − xn)
++
+�
+n
+ηR(xn)
+ˆ
+In
+fn(E) ⊗ (x − xn).
+By the boundary conditions for (φn
+E, Πn
+E) and recalling the notation (1.8), we deduce
+ˆ
+I
+ηR KE
+=
+ˆ
+Rd ηR
+�
+n
+1In
+|In|
+�
+˜fn(E) +
+ˆ
+∂In
+σ(φE, ΠE)ν ⊗ (x − xn)
+�
+.
+Letting R ↑ ∞, the ergodic theorem then entails
+E [KE] = E
+� �
+n
+1In
+|In|
+�
+˜fn(E) +
+ˆ
+∂In
+σ(φE, ΠE)ν ⊗ (· − xn)
+��
+.
+
+24
+A. BERNOU, M. DUERINCKX, AND A. GLORIA
+Since KE is symmetric, taking the symmetric part of this identity yields (2.12) in combi-
+nation with the skew-symmetry of ˜fn(E) and the definition (1.24) of ¯C(E) and ¯c(E).
+Step 2.2. Proof of (1.26).
+We start from (1.24), projected in some direction E′ ∈ Msym
+0
+,
+2E′ : ¯C(E) = E
+� �
+n
+1In
+|In|
+ˆ
+∂In
+E′(x − xn) · σ(φE, ΠE)ν
+�
+,
+which we reformulate using the ergodic theorem as
+2E′ : ¯C(E) = lim
+R↑∞ |BR|−1 �
+n
+ˆ
+∂In
+ηRE′(x − xn) · σ(φE, ΠE)ν,
+(2.27)
+with ηR as above. We turn to a suitable reformulation of the right-hand side. Adding and
+subtracting the passive corrector ψE′, we can write
+�
+n
+ˆ
+∂In
+ηRE′(x − xn) · σ(φE, ΠE)ν
+=
+�
+n
+ˆ
+∂In
+ηR(ψE′ + E′(x − xn)) · σ(φE, ΠE)ν −
+�
+n
+ˆ
+∂In
+ηRψE′ · σ(φE, ΠE)ν.
+(2.28)
+For the first right-hand side term, we use that ψE′ + E′(x − xn) is a rigid motion in In,
+cf. (2.1), we appeal to boundary conditions for (φE, ΠE) in (2.4), and we recall the nota-
+tion (1.8), which leads us to
+�
+n
+ˆ
+∂In
+ηR(ψE′ + E′(x − xn)) · σ(φE, ΠE)ν = −
+�
+n
+ˆ
+In
+ηR(ψE′ + E′(x − xn)) · fn(E).
+In order to reformulate the second right-hand side term of (2.28), we appeal to the weak
+formulation of equation (2.4) for φE: testing this equation with ηRψE′ yields, as in (2.7),
+ˆ
+Rd ∇(ηRψE′) : ∇φE −
+ˆ
+Rd ∇ηR · ψE′ΠE =
+�
+n
+ˆ
+(In+B)\In
+ηRψE′ · fn(E)
+−
+�
+n
+ˆ
+∂In
+ηRψE′ · σ(φE, ΠE)ν.
+Hence, (2.28) turns into
+�
+n
+ˆ
+∂In
+ηRE′(x − xn) · σ(φE, ΠE)ν = −
+�
+n
+ˆ
+In
+ηR(ψE′ + E′(x − xn)) · fn(E)
+−
+ˆ
+Rd ∇ηR · ψE′ΠE +
+ˆ
+Rd ∇(ηRψE′) : ∇φE −
+�
+n
+ˆ
+(In+B)\In
+ηRψE′ · fn(E),
+and thus, after reorganizing the terms, using that the skew-symmetry of ˜fn(E) in (1.8)
+yields
+´
+In E′(x − xn) · fn(E) = 0,
+�
+n
+ˆ
+∂In
+ηRE′(x − xn) · σ(φE, ΠE)ν =
+ˆ
+Rd ∇(ηRψE′) : ∇φE
+−
+ˆ
+Rd ∇ηR · ψE′ΠE −
+�
+n
+ˆ
+In+B
+ηRψE′ · fn(E).
+
+HOMOGENIZATION FOR ACTIVE SUSPENSIONS
+25
+Alternatively, expanding the first right-hand side term,
+�
+n
+ˆ
+∂In
+ηRE′(x − xn) · σ(φE, ΠE)ν =
+ˆ
+Rd ∇ψE′ : ∇(ηRφE) −
+ˆ
+Rd ∇ηR · ψE′ΠE
++
+ˆ
+Rd(ψE′ ⊗ ∇ηR) : ∇φE −
+ˆ
+Rd(φE ⊗ ∇ηR) : ∇ψE′ −
+�
+n
+ˆ
+In+B
+ηRψE′ · fn(E).
+Now testing equation (2.1) for ψE′ with ηRφE, and using that ηRφE is rigid in I, we find
+that the first right-hand side term only yields a term involving ΣE
+�
+n
+ˆ
+∂In
+ηRE′(x − xn) · σ(φE, ΠE)ν = −
+ˆ
+Rd ∇ηR · (ψE′ΠE + φEΣE′)
++
+ˆ
+Rd(ψE′ ⊗ ∇ηR) : ∇φE −
+ˆ
+Rd(φE ⊗ ∇ηR) : ∇ψE′ −
+�
+n
+ˆ
+In+B
+ηRψE′ · fn(E).
+Using the sublinearity of ψE′ and φE and the stationarity of ∇ψE′, ∇φE, ΠE1Rd\I, cf. Lem-
+mas 2.1 and 2.3, we can now pass to the limit R ↑ ∞ in this identity, and we obtain
+lim
+R↑∞ |BR|−1 �
+n
+ˆ
+∂In
+ηRE′(x − xn) · σ(φE, ΠE)ν = −E
+� �
+n
+1In
+|In|
+ˆ
+In+B
+ψE′ · fn(E)
+�
+.
+Inserting this into (2.27), the conclusion (1.26) follows. Finally, the formula for ¯c(E) simply
+follows by taking the trace in (1.24).
+Step 3. Construction of θ.
+This construction is standard: as LE is stationary, it follows from stationary calculus,
+e.g. [26, Chapter 7], that there is a unique solution θE = (θE;ijk)ijk ∈ L2(Ω; H1
+loc(Rd)) of
+−△θE;ijk = ∂jLE;ik − ∂kLE;ij,
+(2.29)
+such that ∇θE is stationary, has vanishing expectation, has finite second moment,
+∥∇θE∥L2(Ω) ≲ ∥LE∥L2(Ω) ≲ ∥KE∥L2(Ω) + ∥∇γE∥L2(Ω),
+and satisfies the anchoring condition
+ffl
+B θE = 0. By uniqueness, the skew-symmetry of θE
+follows from the skew-symmetry of the right-hand side of (2.29) with respect to indices j, k.
+The fact that θE is a vector potential for LE follows from this defining equation as e.g.
+in [20, Section 3.1]. Finally, the sublinearity of θE is a standard property for random fields
+with stationary gradient and vanishing expectation; see e.g. [26, Chapter 7].
+□
+In view of the internal contribution to the effective viscosity, cf. (1.25), we also define
+the following stationary field,
+FE := −
+�
+n
+1In
+|In|
+ˆ
+In+B
+fn(E) ⊗ (x − xn),
+(2.30)
+which is such that
+¯F E = E [FE] .
+(2.31)
+When considering two-scale expansions, since correctors φE, ΠE, γE, θE, fluxes KE, LE,
+and FE are nonlinear with respect to E, we shall need to consider derivatives of these
+objects with respect to E. We introduce a general notation for linearized quantities.
+
+26
+A. BERNOU, M. DUERINCKX, AND A. GLORIA
+Definition 2.6. Given E, E′, E′′ ∈ Msym
+0
+, we use the following notation for directional
+derivatives of swimming forces fn(E) in directions E′, E′′,
+∂E′fn(E)
+:=
+lim
+h→0
+1
+h
+�
+fn(E + hE′) − fn(E)
+�
+,
+∂E′,E′′fn(E)
+:=
+lim
+h→0
+1
+h
+�
+∂E′fn(E + hE′′) − ∂E′fn(E)
+�
+.
+— First linearized correctors: We define (dφE,E′, dΠE,E′) as in Lemma 2.3 with the swim-
+ming forces fn(E) replaced by ∂E′fn(E) (and similarly for dFE,E′).
+We then de-
+fine dγE,E′ as in Lemma 2.4 and dKE,E′, dLE,E′, dθE,E′ as in Lemma 2.5 with fn(E)
+replaced by ∂E′fn(E) and with (φE, ΠE) replaced by (dφE,E′, dΠE,E′).
+— Second linearized correctors: We define (d2φE,E′,E′′, d2ΠE,E′,E′′) as in Lemma 2.3 with
+swimming forces fn(E) replaced by ∂E′,E′′fn(E) (and similarly for d2FE,E′,E′′). We
+then define d2γE,E′,E′′ as in Lemma 2.4 and d2KE,E′,E′′, d2LE,E′,E′′, d2θE,E′,E′′ as in
+Lemma 2.5 with fn(E) replaced by ∂E′,E′′fn(E) and with (φE, ΠE) replaced by (d2φE,E′,E′′,
+d2ΠE,E′,E′′).
+♦
+Finally, we state that the effective maps ¯C, ¯c, ¯F are smooth, with derivatives given in
+terms of linearized correctors. The proof, based on Hypothesis 1.3, is straightforward and
+left to the reader.
+Lemma 2.7. The effective maps ¯C, ¯c, ¯F are smooth and
+| ¯C(E)|, |¯c(E)|, | ¯F (E)| ≲ λ⟨E⟩,
+|∂k ¯C(E)|, |∂k¯c(E)|, |∂k ¯F (E)| ≲ λ.
+(2.32)
+Moreover, we have for all E, F ∈ Msym
+0
+,
+2∂F ¯C(E) + ∂F ¯c(E) Id = E
+� �
+n
+1In
+|In|
+ˆ
+∂In
+σ(dφE,F, dΠE,F )ν ⊗s (x − xn)
+�
+.
+In particular, in view of (2.12), this yields ∂F E [KE] = E [dKE,F].
+♦
+3. Proof of the homogenization result
+This section is devoted to the proof of Theorem 1.7. We quickly establish the well-
+posedness result of Proposition 1.6 before turning to the analysis of the limit ε ↓ 0. For
+any map V and domain D, we henceforth use the short-hand notation V∤D :=
+ffl
+D V .
+3.1. Well-posedness of hydrodynamic model. This section is devoted to the proof of
+Proposition 1.6. We start by recalling the following standard computation (see e.g. [8]),
+which we already partly encountered when proving existence of correctors.
+Lemma 3.1 (e.g. [8]). If vector fields u, h and a scalar field P satisfy the following relations,
+
+
+
+−△u + ∇P = h,
+in Rd \ I,
+div(u) = 0,
+in Rd \ I,
+D(u) = 0,
+in I,
+then the following holds in Rd
+− △u + ∇(P1Rd\I) = h1Rd\I −
+�
+n
+δ∂Inσ(u, P)ν.
+(3.1)
+The same is true if Rd and I are replaced by U and Iε(U), respectively.
+♦
+
+HOMOGENIZATION FOR ACTIVE SUSPENSIONS
+27
+We turn to the proof of Proposition 1.6 and proceed by a fixed-point argument. Let ε > 0
+be fixed.
+Given v ∈ W 1,1(U)d, define Tε(v) ∈ H1
+0(U)d as the unique solution of the
+following linear problem, with associated pressure Pε(v) ∈ L2(U \ Iε(U)),
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+−△Tε(v) + ∇Pε(v)
+= h + κ
+ε
+�
+n∈Nε(U) fn,ε(χδ ∗ D(v)∤εIn),
+in U \ Iε(U),
+div(Tε(v)) = 0,
+in U \ Iε(U),
+D(Tε(v)) = 0,
+in Iε(U),
+´
+ε∂In σ(Tε(v), Pε(v))ν + κ
+ε ¯fn,ε(χδ ∗ D(v)∤εIn) = 0,
+∀n ∈ Nε(U),
+´
+ε∂In Θ(x − εxn) · σ(Tε(v), Pε(v))ν
++κ Θ : ˜fn(χδ ∗ D(v)∤εIn) = 0,
+∀Θ ∈ Mskew, ∀n ∈ Nε(U).
+(3.2)
+We split the proof into three steps. We start by proving the result under the stronger
+smallness condition κℓ−d/2 ≪ 1, before relaxing it to (1.18).
+Step 1. Suboptimal contraction estimate: for all v, w ∈ H1
+0(U)d we have
+ˆ
+U
+|∇(Tε(v) − Tε(w))|2 ≲χ (κℓ− d
+2 )2
+ˆ
+U
+|∇(v − w)|2.
+(3.3)
+This proves that Tε is a contraction on H1
+0(U)d provided that κℓ− d
+2 ≪ 1 is small enough.
+Under this condition, we deduce the well-posedness of the hydrodynamic model (1.7)
+in H1
+0(U)d.
+We turn to the proof of (3.3). For abbreviation, we set Tε(v, w) := Tε(v) − Tε(w) and
+Pε(v, w) = Pε(v)−Pε(w). Testing the equations for Tε(v) and Tε(w) with Tε(v, w) in form
+of (3.1), we find
+ˆ
+U
+|∇Tε(v, w)|2 = −
+�
+n∈Nε(U)
+ˆ
+ε∂In
+Tε(v, w) · σ(Tε(v, w), Pε(v, w))ν
++ κ
+ε
+�
+n∈Nε(U)
+ˆ
+ε(In+B)\εIn
+Tε(v, w) ·
+�
+fn,ε(χδ ∗ D(v)∤εIn) − fn,ε(χδ ∗ D(w)∤εIn)
+�
+,
+and thus, using the rigidity of Tε(v, w) in εIn and using boundary conditions, recalling the
+notation (1.8),
+ˆ
+U
+|∇Tε(v, w)|2 = κ
+ε
+�
+n∈Nε(U)
+ˆ
+ε(In+B)
+Tε(v, w) ·
+�
+fn,ε(χδ ∗ D(v)∤εIn) − fn,ε(χδ ∗ D(w)∤εIn)
+�
+.
+By the neutrality condition (1.9), appealing to Poincaré’s inequality, using the hardcore
+assumption, and recalling that fn,ε is Lipschitz, we get
+ˆ
+U
+|∇Tε(v, w)|2 ≲ κ
+� ˆ
+U
+|∇Tε(v, w)|2� 1
+2
+�
+�
+n∈Nε(U)
+ˆ
+εIn
+|χδ ∗ D(v − w)|2
+� 1
+2
+.
+By Jensen’s inequality, with
+´
+Rd χδ = 1, the second factor is bounded by
+�
+n∈Nε(U)
+ˆ
+εIn
+|χδ ∗ D(v − w)|2 ≲
+ˆ
+U
+�
+�
+n∈Nε(U)
+ˆ
+εIn
+χδ(x − y) dx
+�
+|D(v − w)(y)|2 dy.
+
+28
+A. BERNOU, M. DUERINCKX, AND A. GLORIA
+The expression into brackets can be estimated as follows,
+sup
+y∈Rd
+�
+n∈Nε(U)
+ˆ
+εIn
+χδ(x − y) dx
+≤
+sup
+y∈Rd
+�
+n
+ˆ
+ε
+δ In
+χ(x − y) dx
+≲
+( ε
+δ)d sup
+y∈Rd
+�
+n
+sup
+y+ ε
+δ In
+χ
+≤
+( ε
+δ)d sup
+y∈Rd
+�
+n
+sup
+ε
+δ Bℓ(y+xn)
+χ
+≲
+ℓ−d sup
+y∈Rd
+�
+n
+ˆ
+ε
+δ Bℓ(y+xn)
+�
+sup
+B2ℓ ε
+δ (z)
+χ
+�
+dz,
+and thus, by the hardcore assumption, provided εℓ ≤ δ,
+sup
+y∈Rd
+�
+n∈Nε(U)
+ˆ
+εIn
+χδ(x − y) dx ≲ ℓ−d
+ˆ
+Rd
+�
+sup
+B2(z)
+χ
+�
+dz.
+Inserting this into the above, we get
+�
+n∈Nε(U)
+ˆ
+εIn
+|χδ ∗ D(v − w)|2 ≲χ ℓ−d
+ˆ
+U
+|∇(v − w)|2,
+(3.4)
+and thus,
+ˆ
+U
+|∇Tε(v, w)|2 ≲χ κℓ− d
+2
+� ˆ
+U
+|∇Tε(v, w)|2� 1
+2 � ˆ
+U
+|∇(v − w)|2� 1
+2 ,
+that is, (3.3).
+Step 2. Improved contraction estimate: given 1 < p ≤ 2 and given ℓ ≫p 1 large enough,
+we have for all v, w ∈ W 1,p
+0 (U)d,
+∥∇(Tε(v) − Tε(w))∥Lp(U) ≲p,χ κℓ− d
+p ∥∇(v − w)∥Lp(U).
+This proves that Tε is a contraction on W 1,p
+0 (U)d provided that κℓ− d
+p ≪ 1 is small enough,
+which then implies the well-posedness of the hydrodynamic model (1.7) in W 1,p
+0 (U)d.
+We appeal to the dilute deterministic Lp regularity theory developed by Höfer in [25].
+Given 1 < p ≤ 2, provided that ℓ ≫p 1 is large enough (depending on p and dimension d),
+it allows us to deduce almost surely,
+∥∇(Tε(v) − Tε(w))∥Lp(U) ≲p κ
+���
+�
+n∈Nε(U)
+�
+fn,ε(χδ ∗ D(v)∤εIn) − fn,ε(χδ ∗ D(w)∤εIn)
+����
+Lp(U),
+and thus, using properties of {fn,ε}n,
+∥∇(Tε(v) − Tε(w))∥Lp(U)
+≲p
+κ
+�
+�
+n∈Nε(U)
+ˆ
+ε(In+B)
+��fn,ε(χδ ∗ D(v)∤εIn) − fn,ε(χδ ∗ D(w)∤εIn)
+��p
+� 1
+p
+≲
+κ
+�
+�
+n∈Nε(U)
+ˆ
+εIn
+|χδ ∗ D(v − w)|p
+� 1
+p
+.
+
+HOMOGENIZATION FOR ACTIVE SUSPENSIONS
+29
+Repeating the argument in (3.4), this proves the claim.
+Step 3. Conclusion.
+Testing equation (3.2) with its solution Tε(v) itself, using boundary conditions and recalling
+the notation (1.8), we find
+ˆ
+U
+|∇Tε(v)|2 =
+ˆ
+U\Iε(U)
+h · Tε(v) + κ
+ε
+�
+n∈Nε(U)
+ˆ
+ε(In+B)
+Tε(v) · fn,ε(χδ ∗ D(v)∤εIn),
+and thus, by the neutrality condition (1.9), appealing to Poincaré’s inequality and recalling
+that fn,ε is Lipschitz, arguing exactly as in Step 1,
+ˆ
+U
+|∇Tε(v)|2 ≲ κ2ℓ−d +
+ˆ
+U\Iε(U)
+|h|2 + κ2
+�
+n∈Nε(U)
+ˆ
+εIn
+|χδ ∗ D(v)|2.
+Now using the hardcore condition and Young’s inequality, we deduce for all 1 < p ≤ 2,
+ˆ
+U
+|∇Tε(v)|2 ≲ κ2ℓ−d +
+ˆ
+U\Iε(U)
+|h|2 + κ2∥χδ∥2
+L1 ∩ L2(U)∥∇v∥2
+Lp(U).
+This proves that Tε embeds W 1,p
+0 (U)d into H1
+0(U)d, so that the solution uε = Tε(uε) in
+W 1,p
+0 (U)d constructed in Step 2 actually belongs to H1
+0(U)d, and the a priori estimate (1.19)
+follows by iteration.
+□
+3.2. Qualitative homogenization at fixed δ. This section is devoted to the proof of
+Theorem 1.7.
+Before turning to the proof, we argue for the well-posedness of the ho-
+mogenized equation (1.20).
+Using | ¯C(E)|, | ¯F (E)| ≲ λ⟨E⟩, cf. (2.32), a perturbative
+argument as in the proof of Proposition 1.6 yields the well-posedness of (1.20) with
+(¯uδ, ¯Pδ) ∈ H1
+0(U)d × L2(U)/R provided that κλ ≪ 1 is small enough.
+As λ ≲ ℓ−d,
+this holds in particular under the smallness condition (1.18).
+Interestingly, our proof of qualitative homogenization relies on a semi-quantitative two-
+scale analysis and we do not believe that there exists a simpler and purely qualitative proof.
+In terms of a suitable limiting profile ¯uε (mildly depending on ε and to be identified at the
+end of the proof), we consider the following two-scale expansions for the solution (uε, Pε)
+of the hydrodynamic model (1.7),
+uε
+❀
+¯uε + εψE( ·
+ε)∂E ¯uε + εκφχδ∗D(uε)( ·
+ε),
+(3.5)
+Pε1Rd\εI
+❀
+¯Pε + ¯b : D(¯uε) + κ¯c(χδ ∗ D(uε))
++(ΣE1Rd\I)( ·
+ε)∂E ¯uε + (κΠχδ∗D(uε)(x)1Rd\I)(x
+ε ),
+where uε is implicitely extended by 0 on Rd \ U to ensure that χδ ∗ D(uε) is well-defined,
+and where we implicitly sum over E in an orthonormal basis of Msym
+0
+. We start with two
+comments on the form of these two-scale expansions:
+— The most unusual feature is that correctors φ, Π are evaluated at a background fluid
+deformation χδ ∗ D(uε) depending on the microscopic solution uε. This non-standard
+choice is taken as an intermediate step and happens to be providential in our proof,
+where the limit of this background deformation χδ ∗ D(uε) ∼ χδ ∗ D(¯uε) can only
+be identified at the very end.
+To our knowledge, the necessity of such a two-step
+homogenization argument is new to the literature and gives the present problem an
+interest of its own.
+
+30
+A. BERNOU, M. DUERINCKX, AND A. GLORIA
+— The choice of the non-oscillating part ¯Pε + ¯b : D(¯uε) + κ¯c(χδ ∗ D(uε)) in the two-
+scale expansion of the pressure is dictated by the proof and is similar to the case of
+passive suspensions [8]: the pressure for (1.7) does not converge to the pressure of the
+homogenized problem in its naïve form.
+As χδ ∗ D(uε) is smooth, the maps (x, y) �→ (φχδ∗D(uε)(x))(y), (Πχδ∗D(uε)(x)1Rd\I)(y) are
+Carathéodory functions, which ensures that x �→ φχδ∗D(uε)(x)(x
+ε), (Πχδ∗D(uε)(x)1Rd\I)(x
+ε )
+in (3.5) are measurable. When differentiating such composed functions, some care is needed
+in the notation. Given a smooth field V : Rd → Msym
+0
+, we denote by φV the function of
+two variables (x, y) �→ φV (x)(y), and we use the short-hand notation φV (x
+ε) := φV (x)(x
+ε).
+We use the following notation for the derivative of φV with V frozen,
+(∇φ|V )(x
+ε) := ∇yφV (x)|y= x
+ε ,
+so that the total derivative is then given by
+∇(εφV )(x
+ε) = (∇φ|V )(x
+ε) + εdφV (x),E(x
+ε) ⊗ ∇(E : V (x)),
+(3.6)
+where the linearized corrector dφ is given in Definition 2.6 and where we implicitly sum
+over E in an orthonormal basis of Msym
+0
+. In addition, if V is a smooth random field, we
+use the following notation for the expectation of φV with V frozen: for any random field a,
+E[a(x)φ|V (x
+ε)] := E
+�
+a(x)φE(x
+ε)
+�
+|E=V (x).
+The same notation is used for other correctors and fluxes.
+As usual, correctors are not capturing the relevant behavior close to the boundary of the
+domain U: in particular, the two-scale expansion in (3.5) does not vanish at the boundary.
+To circumvent this, we proceed by truncating correctors in a neighborhood of the boundary.
+We set for abbreviation
+∂rU := {x ∈ U : dist(x, ∂U) < r},
+and, given rε ≥ 4ε (which we shall optimize later on), we choose a smooth cut-off function
+ηε ∈ C∞
+c (U; [0, 1]) such that
+ηε|U\∂2rεU = 1,
+ηε|∂rεU = 0,
+|∇ηε| ≲
+1
+rε ,
+and such that ηε is constant inside each of the fattened particles {ε(In + 1
+2B)}n. Note that
+by definition the set Iε(U) coincides with εI on the support of ηε,
+ηε1Iε(U) = ηε1εI.
+(3.7)
+In these terms, truncating the two-scale expansions (3.5), we are led to considering the
+following truncated two-scale expansion errors,
+wε := uε − u2s
+ε ,
+Qε := Pε1U\Iε(U) − P 2s
+ε ,
+where the two-scale expansion (u2s
+ε , P 2s
+ε ) of (uε, Pε) is given by
+u2s
+ε (x)
+:=
+¯uε(x) + εηε(x)ψE(x
+ε)∂E ¯uε(x) + εκηε(x)φχδ∗D(uε)(x
+ε),
+(3.8)
+P 2s
+ε (x)
+:=
+−P ∗
+ε (x) + ¯Pε(x) + ηε(x)¯b : D(¯uε)(x) + κηε(x)¯c(χδ ∗ D(uε)(x))
++ηε(x)(ΣE1Rd\I)(x
+ε)(∂E ¯uε)(x) + κηε(x)(Πχδ∗D(uε)1Rd\I)(x
+ε),
+where we have further added a locally constant pressure field
+P ∗
+ε := P ′
+ε1U\Iε(U) +
+�
+n∈Nε(U)
+P ′′
+ε,n1εIn,
+(3.9)
+
+HOMOGENIZATION FOR ACTIVE SUSPENSIONS
+31
+in terms of some constants P ′
+ε and {P ′′
+ε,n}n to be suitably chosen later on, cf. (3.44), and
+where the limiting profile (¯uε, ¯Pε) ∈ H1
+0(U)d × L2(U)/R is chosen as the unique solution of
+− div(2 ¯Bpas D(¯uε)) + ∇ ¯Pε = (1 − λ)h + div(2κ ¯Bact(χδ ∗ D(uε))).
+(3.10)
+Again, this equation is viewed as a convenient intermediate step towards the relevant
+homogenized equation (1.20): as in the two-scale expansion (3.8), the background fluid
+deformation χδ ∗ D(uε) is expressed here in terms of the microscopic solution uε itself and
+its limit will only be identified at the very end of the proof. We split the proof into three
+main steps.
+Step 1. Equation for the two-scale expansion error (wε, Qε): in the weak sense in U,
+− △wε + ∇Qε = −div
+�
+(JE1I)( ·
+ε)ηε∂E ¯uε + κ(Kχδ∗D(uε)1I)( ·
+ε) ηε
+�
+−
+�
+n∈Nε(U)
+�
+δε∂Inσ(uε, Pε + P ′
+ε − P ′′
+ε,n)ν + κ
+ε fn,ε
+�
+χδ ∗ D(uε)∤εIn
+�
+1εIn
+�
+− κ
+ε
+�
+n∈Nε(U)
+�
+ηεfn,ε(χδ ∗ D(uε)) − fn,ε
+�
+χδ ∗ D(uε)∤εIn
+��
+− κdFχδ∗D(uε),E( ·
+ε)ηε∇(χδ ∗ ∂Euε)
++ (1 − ηε)(λ − 1Iε(U))h − κ ¯F (χδ ∗ D(uε))∇ηε
+− div
+�
+(1 − ηε)
+�
+2( ¯Bpas − Id) D(¯uε) + 2κ ¯Bact(χδ ∗ D(uε))
+��
++ εdiv
+��
+2ψE ⊗s − Id ⊗ψE − ΥE
+�
+( ·
+ε)∇(ηε∂E¯uε) − ηεh ⊗ (△−1∇1I)( ·
+ε)
++ κ
+�
+2φχδ∗D(uε) ⊗s − Id ⊗φχδ∗D(uε) − θχδ∗D(uε) + γχδ∗D(uε) ⊗ Id
+�
+( ·
+ε)∇ηε
++ κ
+�
+2dφχδ∗D(uε),E ⊗s − Id ⊗dφχδ∗D(uε),E − dθχδ∗D(uε),E
++ dγχδ∗D(uε),E ⊗ Id
+�
+( ·
+ε) ηε∇(χδ ∗ ∂Euε)
+�
++ κε∇i
+�
+(△−1∇idF|χδ∗D(uε),E)( ·
+ε)ηε∇(χδ ∗ ∂Euε)
+�
++ ε(△−1∇j1I)( ·
+ε)∇j(ηεh) − κεγχδ∗D(uε)( ·
+ε)△ηε − κεdγχδ∗D(uε),E( ·
+ε) : ηε△(χδ ∗ ∂Euε)
+− κε(△−1∇idF|χδ∗D(uε),E)( ·
+ε)∇i(ηε∇(χδ ∗ ∂Euε))
++ 2κε
+�
+dθχδ∗D(uε),E − dγχδ∗D(uε),E ⊗ Id
+�
+( ·
+ε) : ∇(χδ ∗ ∂Euε) ⊗ ∇ηε
++ κε
+�
+d2θχδ∗D(uε),E,E′ − d2γχδ∗D(uε),E,E′ ⊗ Id −△−1∇d2F|χδ∗D(uε),E,E′)
+�
+( ·
+ε)
+: ηε∇(χδ ∗ ∂E′uε) ⊗ ∇(χδ ∗ ∂Euε).
+(3.11)
+We split the proof into six substeps.
+Substep 1.1. Reformulation of the equation for uε:
+− △uε + ∇(Pε1U\Iε(U) + P ∗
+ε ) = h1U\Iε(U) + κ
+ε
+�
+n∈Nε(U)
+fn,ε
+�
+χδ ∗ D(uε)∤εIn
+�
+−
+�
+n∈Nε(U)
+�
+δε∂Inσ(uε, Pε + P ′
+ε − P ′′
+ε,n)ν + κ
+εfn,ε
+�
+χδ ∗ D(uε)∤εIn
+�
+1εIn
+�
+.
+(3.12)
+Starting from equation (1.7) in form of (3.1), we find
+− △uε + ∇(Pε1U\Iε(U))
+
+32
+A. BERNOU, M. DUERINCKX, AND A. GLORIA
+= h1U\Iε(U) + κ
+ε
+�
+n∈Nε(U)
+fn,ε
+�
+χδ ∗ D(uε)∤εIn
+�
+1ε(In+B)\εIn −
+�
+n∈Nε(U)
+δε∂Inσ(uε, Pε)ν.
+Adding and subtracting the contribution of swimming forces on the particles, this becomes
+− △uε + ∇(Pε1U\Iε(U)) = h1U\Iε(U) + κ
+ε
+�
+n∈Nε(U)
+fn,ε
+�
+χδ ∗ D(uε)∤εIn
+�
+−
+�
+n∈Nε(U)
+�
+δε∂Inσ(uε, Pε)ν + κ
+εfn,ε
+�
+χδ ∗ D(uε)∤εIn
+�
+1εIn
+�
+.
+Adding ∇P ∗
+ε to both sides, cf. (3.9), the claim (3.12) follows.
+Substep 1.2. Equation for the two-scale expansion for the two-scale expansion error u2s
+ε :
+− △u2s
+ε + ∇(P 2s
+ε + P ∗
+ε ) = ∇
+�
+¯Pε + ηε¯b : D(¯uε) + κηε¯c(χδ ∗ D(uε))
+�
+− T 1
+ε − κT 2
+ε − div
+�
+2(1 − ηε) D(¯uε)
+�
+− εdiv
+�
+2ψE( ·
+ε) ⊗s ∇(ηε∂E¯uε) + 2κφχδ∗D(uε)( ·
+ε) ⊗s ∇ηε
++ 2κdφχδ∗D(uε),Eα( ·
+ε) ⊗s ηε∇(χδ ∗ ∂Eαuε)
+�
++ ε∇
+�
+ψE( ·
+ε) · ∇(ηε∂E ¯uε) + κφχδ∗D(uε)( ·
+ε) · ∇ηε
++ κdφχδ∗D(uε),Eα( ·
+ε) · ηε∇(χδ ∗ ∂Eαuε)
+�
+,
+(3.13)
+where
+T 1
+ε
+:=
+div
+��
+2(D(ψE) + E) − ΣE1Rd\I Id
+�
+( ·
+ε) ηε∂E ¯uε
+�
+,
+T 2
+ε
+:=
+div
+��
+2 D(φ|χδ∗D(uε)) − Πχδ∗D(uε)1Rd\I Id
+�
+( ·
+ε) ηε
+�
+are two terms that we shall further reformulate in the upcoming substeps.
+For the two-scale expansion error (u2s
+ε , P 2s
+ε ) defined in (3.8), we have
+− △u2s
+ε + ∇(P 2s
+ε + P ∗
+ε ) = −△¯uε + ∇
+�
+¯Pε + ηε¯b : D(¯uε) + κηε¯c(χδ ∗ D(uε))
+�
+− △
+�
+εψE( ·
+ε)ηε∂E ¯uε + εκφχδ∗D(uε)( ·
+ε)ηε
+�
++ ∇
+�
+(ΣE1Rd\I)( ·
+ε)ηε∂E ¯uε + κ(Πχδ∗D(uε)1Rd\I)( ·
+ε)ηε
+�
+.
+(3.14)
+It remains to reformulate the penultimate right-hand side term. By the identity
+△h = div(2 D(h)) − ∇div(h),
+we can write
+△
+�
+εψE( ·
+ε)ηε∂E ¯uε
+�
+=
+div
+�
+2 D(εψE( ·
+ε)ηε∂E ¯uε)
+�
+− ∇div
+�
+εψE( ·
+ε)ηε∂E ¯uε
+�
+,
+△
+�
+εφχδ∗D(uε)( ·
+ε)ηε
+�
+=
+div
+�
+2 D(εφχδ∗D(uε)( ·
+ε)ηε)
+�
+− ∇div
+�
+εφχδ∗D(uε)( ·
+ε)ηε
+�
+.
+First, as div(ψE) = 0, we find
+D(εψE( ·
+ε)ηε∂E¯uε)
+=
+D(ψE)( ·
+ε)ηε∂E ¯uε + εψE( ·
+ε) ⊗s ∇(ηε∂E ¯uε),
+div(εψE( ·
+ε)ηε∂E¯uε)
+=
+εψE( ·
+ε) · ∇(ηε∂E ¯uε).
+(3.15)
+
+HOMOGENIZATION FOR ACTIVE SUSPENSIONS
+33
+Second, as div(φE) = 0, further recalling (3.6), we find
+D(εφχδ∗D(uε)( ·
+ε)ηε)
+=
+D(φ|χδ∗D(uε))( ·
+ε)ηε + εdφχδ∗D(uε),E( ·
+ε) ⊗s ηε∇(χδ ∗ ∂Euε)
++εφχδ∗D(uε)( ·
+ε) ⊗s ∇ηε,
+div(εφχδ∗D(uε)( ·
+ε)ηε)
+=
+εφχδ∗D(uε)( ·
+ε) · ∇ηε
++εdφχδ∗D(uε),E( ·
+ε) · ηε∇(χδ ∗ ∂Euε),
+(3.16)
+where we recall that we implicitly sum over E in an orthonormal basis of Msym
+0
+, and where
+the linearized corrector dφ is given in Definition 2.6.
+In these terms, the penultimate
+right-hand side term in (3.14) takes the form
+△
+�
+εψE( ·
+ε)ηε∂E ¯uε + εκφχδ∗D(uε)( ·
+ε)ηε
+�
+=
+div
+�
+2 D(ψE)( ·
+ε)ηε∂E ¯uε + 2κ D(φ|χδ∗D(uε))( ·
+ε)ηε + 2εψE( ·
+ε) ⊗s ∇(ηε∂E ¯uε)
++ 2εκφχδ∗D(uε)( ·
+ε) ⊗s ∇ηε + 2εκdφχδ∗D(uε),Eα( ·
+ε) ⊗s ηε∇(χδ ∗ ∂Eαuε)
+�
+− ∇
+�
+εψE( ·
+ε) · ∇(ηε∂E ¯uε) + εκφχδ∗D(uε)( ·
+ε) · ∇ηε
++εκdφχδ∗D(uε),Eα( ·
+ε) · ηε∇(χδ ∗ ∂Eαuε)
+�
+.
+Inserting this identity into (3.14), using that div(¯uε) = 0, and decomposing
+△¯uε = div(2 D(¯uε)) = div
+�
+2(1 − ηε) D(¯uε)
+�
++ div(2Eηε∂E¯uε),
+the claim (3.13) follows.
+Substep 1.3. Proof of
+T 1
+ε = div
+��
+2(D(ψE) + E) − ΣE1Rd\I Id
+�
+( ·
+ε) ηε∂E ¯uε
+�
+= div
+�
+2ηε ¯Bpas D(¯uε)
+�
++ ∇
+�
+ηε¯b : D(¯uε)
+�
+− εdiv
+�
+ΥE( ·
+ε)∇(ηε∂E ¯uε)
+�
+− div
+�
+(JE1I)( ·
+ε)ηε∂E ¯uε
+�
+.
+(3.17)
+In terms of the extended flux JE, cf. Lemma 2.2, we have
+div
+��
+2(D(ψE) + E) − ΣE1Rd\I Id
+�
+( ·
+ε) ηε∂E ¯uε
+�
+= div
+�
+(JE1Rd\I)( ·
+ε) ηε∂E¯uε
+�
+.
+Recalling that div(JE) = 0, we can decompose
+div
+��
+2(D(ψE) + E) − ΣE1Rd\I Id
+�
+( ·
+ε) ηε∂E¯uε
+�
+= JE( ·
+ε)∇(ηε∂E¯uε) − div
+�
+(JE1I)( ·
+ε) ηε∂E ¯uε
+�
+.
+As Lemma 2.2 further yields JE − E [JE] = div(ΥE) and E [JE] = 2 ¯BpasE + (¯b : E) Id,
+where the flux corrector ΥE is skew-symmetric in its last two indices, the claim (3.17)
+follows. For completeness, we recall the standard argument based on skew-symmetry of Υ
+that leads to this identity: for any smooth scalar field ζ, we have
+(JE − E [JE])∇ζ
+=
+div(ΥE)∇ζ
+=
+ei(∇kΥE;ijk)∇jζ
+=
+ei∇k
+�
+ΥE;ijk
+� �� �
+=−ΥE;ikj
+∇jζ
+�
+− eiΥE;ijk∇2
+jkζ
+�
+��
+�
+=0
+
+34
+A. BERNOU, M. DUERINCKX, AND A. GLORIA
+=
+−div(ΥE∇ζ).
+(3.18)
+Substep 1.4. Proof of
+T 2
+ε = div
+��
+2 D(φ|χδ∗D(uε)) − Πχδ∗D(uε)1Rd\I Id
+�
+( ·
+ε) ηε
+�
+= div(2ηε ¯C(χδ ∗ D(uε))) + ∇(ηε¯c(χδ ∗ D(uε)))
+− 1
+εηε
+�
+n∈Nε(U)
+fn,ε(χδ ∗ D(uε)) − div
+�
+(Kχδ∗D(uε)1I)( ·
+ε) ηε
+�
+− εdiv
+��
+θχδ∗D(uε) − γχδ∗D(uε) ⊗ Id
+�
+( ·
+ε)∇ηε
+�
+− εdiv
+��
+dθχδ∗D(uε),E − dγχδ∗D(uε),E ⊗ Id
+�
+( ·
+ε) ηε∇(χδ ∗ ∂Euε)
+�
+− εγχδ∗D(uε)( ·
+ε)△ηε − εdγχδ∗D(uε),E( ·
+ε) : ηε△(χδ ∗ ∂Euε)
++ 2ε
+�
+dθχδ∗D(uε),E − dγχδ∗D(uε),E ⊗ Id
+�
+( ·
+ε) : ∇(χδ ∗ ∂Euε) ⊗ ∇ηε
++ ε
+�
+d2θχδ∗D(uε),E,E′ − d2γχδ∗D(uε),E,E′ ⊗ Id
+�
+( ·
+ε)
+: ηε∇(χδ ∗ ∂E′uε) ⊗ ∇(χδ ∗ ∂Euε).
+(3.19)
+In terms of the extended flux KE, cf. Lemma 2.5, we have
+div
+��
+2 D(φ|χδ∗D(uε)) − Πχδ∗D(uε)1Rd\I Id
+�
+( ·
+ε) ηε
+�
+= div
+�
+(Kχδ∗D(uε)1Rd\I)( ·
+ε) ηε
+�
+.
+As Lemma 2.5 further yields E [KE] = 2 ¯C(E) + ¯c(E) Id and div(KE) = − �
+n fn(E), and
+appealing to (3.6) and (3.7), we find
+div
+��
+2 D(φ|χδ∗D(uε)) − Πχδ∗D(uε)1Rd\I Id
+�
+( ·
+ε) ηε
+�
+= div(2ηε ¯C(χδ ∗ D(uε))) + ∇(ηε¯c(χδ ∗ D(uε))) − 1
+εηε
+�
+n∈Nε(U)
+fn,ε(χδ ∗ D(uε))
+− div
+�
+(Kχδ∗D(uε)1I)( ·
+ε) ηε
+�
++
+�
+Kχδ∗D(uε)( ·
+ε) − E
+�
+K|χδ∗D(uε)
+��
+∇ηε
++
+�
+dKχδ∗D(uε),E( ·
+ε) − E
+�
+dK|χδ∗D(uε),E( ·
+ε)
+��
+ηε∇(χδ ∗ ∂Euε).
+(3.20)
+It remains to reformulate the last two right-hand side terms. As Lemma 2.5 yields
+KE − E [KE] = div(θE) + ∇γE,
+we get
+�
+Kχδ∗D(uε)( ·
+ε) − E
+�
+K|χδ∗D(uε)
+��
+∇ηε =
+�
+div(θ|χδ∗D(uε)) + ∇γ|χδ∗D(uε)
+�
+( ·
+ε)∇ηε,
+and thus, by Leibniz’ rule, using (3.6) and the skew-symmetry of θ (whence the minus sign
+of the first right-hand side term, cf. (3.18)),
+�
+Kχδ∗D(uε)( ·
+ε) − E
+�
+K|χδ∗D(uε)
+��
+∇ηε
+= −εdiv
+��
+θχδ∗D(uε) − γχδ∗D(uε) ⊗ Id
+�
+( ·
+ε)∇ηε
+�
+− εγχδ∗D(uε)( ·
+ε)△ηε
++ ε
+�
+dθχδ∗D(uε),E − dγχδ∗D(uε),E ⊗ Id
+�
+( ·
+ε) : ∇(χδ ∗ ∂Euε) ⊗ ∇ηε.
+Similarly, with the notation of Definition 2.6, we have
+�
+dKχδ∗D(uε),E( ·
+ε) − E
+�
+dK|χδ∗D(uε),E
+��
+ηε∇(χδ ∗ ∂Euε)
+
+HOMOGENIZATION FOR ACTIVE SUSPENSIONS
+35
+=
+�
+div(dθ|χδ∗D(uε),E) + ∇dγ|χδ∗D(uε),E
+�
+( ·
+ε) ηε∇(χδ ∗ ∂Euε),
+and thus, by Leibniz’ rule, using (3.6) and the skew-symmetry of dθ,
+�
+dKχδ∗D(uε),E( ·
+ε) − E
+�
+dK|χδ∗D(uε),E
+��
+ηε∇(χδ ∗ ∂Euε)
+= −εdiv
+��
+dθχδ∗D(uε),E − dγχδ∗D(uε),E ⊗ Id
+�
+( ·
+ε) ηε∇(χδ ∗ ∂Euε)
+�
++ ε
+�
+dθχδ∗D(uε),E − dγχδ∗D(uε),E ⊗ Id
+�
+( ·
+ε) : ∇(ηε∇(χδ ∗ ∂Euε))
++ ε
+�
+d2θχδ∗D(uε),E,E′ − d2γχδ∗D(uε),E,E′ ⊗ Id
+�
+( ·
+ε) : ηε∇(χδ ∗ ∂E′uε) ⊗ ∇(χδ ∗ ∂Euε).
+Inserting this into (3.20), the claim (3.19) follows.
+Substep 1.5. In order to reconstruct ¯Bact(χδ ∗ D(uε)), we shall need the following identity:
+div(ηε ¯F (χδ ∗ D(uε))) = dFχδ∗D(uε),E( ·
+ε)ηε∇(χδ ∗ ∂Euε) + ¯F (χδ ∗ D(uε))∇ηε
+− ε∇i
+�
+(△−1∇idF|χδ∗D(uε),E)( ·
+ε)ηε∇(χδ ∗ ∂Euε)
+�
++ ε(△−1∇idF|χδ∗D(uε),E)( ·
+ε)∇i(ηε∇(χδ ∗ ∂Euε))
++ ε(△−1∇id2F|χδ∗D(uε),E,E′)( ·
+ε)ηε∇(χδ ∗ ∂Euε)∇i(χδ ∗ ∂E′(uε)).
+(3.21)
+As E [FE] = ¯F (E), cf. (2.31), and using the notation in Definition 2.6, we can decompose
+dFE,E′
+=
+∂E′ ¯F (E) +
+�
+dFE,E′ − E
+�
+dFE,E′��
+=
+∂E′ ¯F (E) + ∇i△−1∇idFE,E′.
+Using this together with Leibniz’ rule, we find
+dFχδ∗D(uε),E( ·
+ε)ηε∇(χδ ∗ ∂Euε) = ∂E ¯F (χδ ∗ D(uε))ηε∇(χδ ∗ ∂Euε)
++ ε∇i
+�
+(△−1∇idF|χδ∗D(uε),E)( ·
+ε)
+�
+ηε∇(χδ ∗ ∂Euε)
+− ε(△−1∇id2F|χδ∗D(uε),E,E′)( ·
+ε)ηε∇(χδ ∗ ∂Euε)∇i(χδ ∗ ∂E′(uε)).
+Further reformulating the first right-hand side term, noting that
+∂E ¯F (χδ ∗ D(uε))∇(χδ ∗ ∂Euε) = div( ¯F (χδ ∗ D(uε))),
+the claim (3.21) follows.
+Substep 1.6. Proof of (3.11).
+Subtracting (3.12) and (3.13), inserting identities (3.17), (3.19), and (3.21), using equa-
+tion (3.10) for ¯uε, and decomposing
+h1U\Iε(U) − (1 − λ)h
+=
+(λ − 1Iε(U))(1 − ηε)h + (λ − 1εI)ηεh
+=
+(λ − 1Iε(U))(1 − ηε)h
+−ε∇j
+�
+(△−1∇j1I)( ·
+ε)ηεh
+�
++ ε(△−1∇j1I)( ·
+ε)∇j(ηεh),
+the claim (3.11) follows after straightforward simplifications.
+In the rest of the proof, for notational convenience, we do not make explicit the dependence
+of estimates wrt κ and ℓ.
+Step 2. Energy estimate for (3.11): for all K ≥ 1,
+
+36
+A. BERNOU, M. DUERINCKX, AND A. GLORIA
+ˆ
+U
+|∇wε|2 ≲
+1
+K
+ˆ
+U
+(Qε)2 +
+�
+1 + ∥(h, ∇¯uε)∥2
+W 1,∞(U) + ∥χδ ∗ D(uε)∥6
+W 2,∞(U)
+�
+×
+� ˆ
+∂3rεU
+|(1, ∇ψ, Σ1Rd\I, ∇φ, Π1Rd\I)( ·
+ε)|2
++ ε2K
+ˆ
+U
+�
+1 + r−2
+ε 1∂3rεU
+�
+|(1, ψ, ∇ψ, Υ, φ, θ, γ, dφ, ∇dφ, dθ, dγ, d2θ, d2γ,
+△−1∇1I, △−1∇dF, △−1∇d2F)( ·
+ε)|2
+�
+,
+(3.22)
+where we use the following notation for correctors,
+|ψ| := sup
+E
+|E|−1|ψE|,
+(3.23)
+and similarly for |Υ|. For the active corrector φ, which depends nonlinearly on the direc-
+tion E, as well as for θ, γ, we rather set
+|φ| :=
+sup
+|E|≤Cδ(h)
+⟨E⟩−1|φE|,
+(3.24)
+where the deterministic constant Cδ(h) > 0 is chosen such that almost surely
+∥χδ ∗ D(uε)∥L∞(U) ≤ ∥χδ∥L2(Rd)∥∇uε∥L2(U) ≤ Cδ(h).
+Such a constant can be chosen in view of (1.19). For the linearized correctors dφ and d2φ,
+as well as for dθ, dγ, d2θ, d2γ, △−1∇dF, △−1∇d2F, we similarly set
+|dφ|
+:=
+sup
+|E|≤Cδ(h)
+sup
+E′ |E′|−1|dφE,E′|,
+|d2φ|
+:=
+sup
+|E|≤Cδ(h)
+sup
+E′,E′′ |E′|−1|E′′|−1|d2φE,E′,E′′|.
+By Poincaré’s inequality in form of
+sup
+ε(In+B)
+���∇¯uε −
+ 
+ε(In+B)
+∇¯uε
+���
+≲
+ε∥∇2¯uε∥L∞(U),
+sup
+ε(In+B)
+���χδ ∗ D(uε) −
+ 
+ε(In+B)
+χδ ∗ D(uε)
+���
+≲
+ε∥χδ ∗ D(uε)∥W 1,∞(U),
+the claim (3.22) follows from
+ˆ
+U
+|∇wε|2 ≲
+1
+K
+ˆ
+U
+(Qε)2
++
+ˆ
+∂3rεU
+|(1, ∇ψ, Σ1Rd\I, ∇φ, Π1Rd\I)( ·
+ε)|2�
+1 + |h|2 + |∇¯uε|2 + [∇¯uε]2
+4ε + |χδ ∗ D(uε)|2�
++ ε2r−2
+ε K
+ˆ
+∂3rεU
+|(ψ, Υ, φ, θ, γ, dφ)( ·
+ε)|2�
+1 + |∇¯uε|2 + |χδ ∗ D(uε)|2�
++ ε2K
+ˆ
+U
+|(1, ψ, Υ, φ, dφ, dθ, dγ, d2θ, d2γ, △−1∇1I, △−1∇dF, △−1∇d2F)( ·
+ε)|2
+×
+�
+|⟨∇⟩h|2 + |∇2¯uε|2 + |∇⟨∇⟩χδ ∗ D(uε)|2 + |∇χδ ∗ D(uε)|4
++ |∇(χδ ∗ ∂Euε)|2�
+1 + [χδ ∗ D(uε)]4
+4ε
+��
+
+HOMOGENIZATION FOR ACTIVE SUSPENSIONS
+37
++ K
+�
+n
+ˆ
+ε(In+B)
+|(ψ, ∇ψ)( ·
+ε)|2���∇¯uε −
+ 
+ε(In+B)
+∇¯uε
+���
+2
++ K
+�
+n
+ˆ
+ε(In+B)
+|(dφ, ∇dφ)( ·
+ε)|2���χδ ∗ D(uε) −
+ 
+ε(In+B)
+χδ ∗ D(uε)
+���
+2
+,
+(3.25)
+where we use the short-hand notation [g]4ε(x) := (
+ffl
+B4ε(x) |g|2)
+1
+2. We split the proof of (3.25)
+into seven substeps.
+Substep 2.1. Preliminary.
+In order to obtain (3.25), we may wish to test equation (3.11) with wε itself. However,
+wε is not rigid inside particles, which prevents us from taking advantage of the boundary
+conditions. To circumvent this issue, we make use of the following truncation maps T ε
+0 , T ε
+1 :
+for all g ∈ C∞
+b (Rd)d,
+T ε
+0 [g]
+:=
+(1 − ρε)g +
+�
+n
+ρε
+n
+�  
+ε(In+B)
+g
+�
+,
+T ε
+1 [g]
+:=
+(1 − ρε)g +
+�
+n
+ρε
+n
+��  
+ε(In+B)
+g
+�
++ (· − εxn)j
+�  
+ε(In+B)
+∂jg
+��
+,
+where for all n we have chosen a cut-off function ρε
+n ∈ C∞
+c (Rd; [0, 1]) with
+ρε
+n|ε(In+ 1
+4 B) = 1,
+ρε
+n|Rd\ε(In+ 1
+2 B) = 0,
+|∇ρε
+n| ≲ 1
+ε,
+and where we have set for abbreviation ρε := �
+n ρε
+n. In these terms, we shall test (3.11)
+with the following modification of the two-scale expansion error wε, cf. (3.8),
+˜wε := uε − T ε
+1 [¯uε] − εηεψE( ·
+ε)T ε
+0 [∂E ¯uε] − εκηεφT ε
+0 [χδ∗D(uε)]( ·
+ε).
+Testing equation (3.11) with ˜wε, we obtain
+Iε
+0 = Iε
+1 + Iε
+2 + Iε
+3 + Iε
+4 + Iε
+5,
+(3.26)
+in terms of
+Iε
+0
+:=
+ˆ
+U
+∇ ˜wε : ∇wε −
+ˆ
+U
+Qε div( ˜wε),
+Iε
+1
+:=
+ˆ
+U
+ηε∇ ˜wε :
+�
+(JE1I)( ·
+ε)∂E ¯uε + κ(Kχδ∗D(uε)1I)( ·
+ε)
+�
+Iε
+2
+:=
+−
+�
+n∈Nε(U)
+� ˆ
+ε∂In
+˜wε · σ(uε, Pε + P ′
+ε − P ′′
+ε,n)ν + κ
+ε
+ˆ
+εIn
+˜wε · fn,ε
+�
+χδ ∗ D(uε)∤εIn
+��
+Iε
+3
+:=
+− κ
+ε
+�
+n∈Nε(U)
+ˆ
+ε(In+B)
+ηε ˜wε ·
+�
+fn,ε(χδ ∗ D(uε)) − fn,ε
+�
+χδ ∗ D(uε)∤εIn
+��
+−κ
+ˆ
+U
+ηε ˜wε ⊗ ∇(χδ ∗ ∂Euε) : dFχδ∗D(uε),E( ·
+ε),
+Iε
+4
+:=
+ˆ
+U
+(1 − ηε)(λ − 1Iε(U)) ˜wε · h + κ
+ε
+�
+n∈Nε(U)
+ˆ
+ε(In+B)
+(1 − ηε) ˜wε · fn,ε
+�
+χδ ∗ D(uε)∤εIn
+�
++2
+ˆ
+U
+(1 − ηε) D( ˜wε) :
+�
+( ¯Bpas − Id) D(¯uε) + κ ¯Bact(χδ ∗ D(uε))
+�
+
+38
+A. BERNOU, M. DUERINCKX, AND A. GLORIA
+−κ
+ˆ
+U
+˜wε ⊗ ∇ηε : ¯F (χδ ∗ D(uε))
+and
+Iε
+5
+:=
+−ε
+ˆ
+U
+∇ ˜wε :
+��
+2ψE ⊗s − Id ⊗ψE − ΥE
+�
+( ·
+ε)∇(ηε∂E ¯uε) − ηεh ⊗ (△−1∇1I)( ·
+ε)
++κ
+�
+2φχδ∗D(uε) ⊗s − Id ⊗φχδ∗D(uε) − θχδ∗D(uε) + γχδ∗D(uε) ⊗ Id
+�
+( ·
+ε)∇ηε
++κ
+�
+2dφχδ∗D(uε),E ⊗s − Id ⊗dφχδ∗D(uε),E − dθχδ∗D(uε),E
++dγχδ∗D(uε),E ⊗ Id
+�
+( ·
+ε) ηε∇(χδ ∗ ∂Euε)
+�
+−κε
+ˆ
+U
+ηε∇i ˜wε ⊗ ∇(χδ ∗ ∂Euε) : (△−1∇idF|χδ∗D(uε))( ·
+ε)
++ε
+ˆ
+U
+˜wε ·
+�
+(△−1∇j1I)( ·
+ε)∇j(ηεh) − κγχδ∗D(uε)( ·
+ε)△ηε
+−κdγχδ∗D(uε),E( ·
+ε) : ηε△(χδ ∗ ∂Euε)
+−κ(△−1∇idF|χδ∗D(uε),E)( ·
+ε)∇i(ηε∇(χδ ∗ ∂Euε))
++2κ
+�
+dθχδ∗D(uε),E − dγχδ∗D(uε),E ⊗ Id
+�
+( ·
+ε) : ∇(χδ ∗ ∂Euε) ⊗ ∇ηε
++κ
+�
+d2θχδ∗D(uε),E,E′ − d2γχδ∗D(uε),E,E′ ⊗ Id −△−1∇id2F|χδ∗D(uε),E,E′
+�
+( ·
+ε)
+: ηε∇(χδ ∗ ∂E′uε) ⊗ ∇(χδ ∗ ∂Euε)
+�
+.
+We analyze the different terms separately in the upcoming six substeps.
+Substep 2.2. Proof that for all K ≥ 1,
+Iε
+0 ≥ (1 −
+1
+2K )
+ˆ
+U
+|∇wε|2 − 1
+K
+ˆ
+U
+(Qε)2 − K
+ˆ
+U
+|∇( ˜wε − wε)|2
+− ε2r−2
+ε CK
+ˆ
+∂2rεU
+|(ψ, φ, dφ)( ·
+ε)|2�
+1 + |∇¯uε|2 + |χδ ∗ D(uε)|2�
+− ε2CK
+ˆ
+U
+|(ψ, φ, dφ)( ·
+ε)|2�
+|∇2¯uε|2 + |∇χδ ∗ D(uε)|2�
+.
+(3.27)
+Adding and subtracting wε to ˜wε, we find from Young’s inequality, for all K ≥ 1,
+Iε
+0 ≥ (1 −
+1
+2K )
+ˆ
+U
+|∇wε|2 − 1
+K
+ˆ
+U
+(Qε)2 − K
+ˆ
+U
+|∇( ˜wε − wε)|2 − K
+ˆ
+U
+|div(wε)|2.
+Since div(uε) = div(¯uε) = 0, we get from (3.15) and (3.16),
+div(wε) = −εψE( ·
+ε) · ∇(ηε∂E ¯uε) − εκφχδ∗D(uε)( ·
+ε) · ∇ηε
+− εκdφχδ∗D(uε),E( ·
+ε) · ηε∇(χδ ∗ ∂Euε),
+(3.28)
+and the claim (3.27) follows using the properties of ηε.
+Substep 2.3. Proof that
+Iε
+1 ≲
+� ˆ
+∂2rεU
+�
+1 + |∇¯uε|2 + |χδ ∗ D(uε)|2�� 1
+2
+×
+� ˆ
+∂3rεU
+|(1, ∇ψ, Σ1Rd\I, ∇φ, Π1Rd\I)( ·
+ε)|2[∇¯uε]2
+4ε
+� 1
+2 .
+(3.29)
+
+HOMOGENIZATION FOR ACTIVE SUSPENSIONS
+39
+First note that for all n we have in εIn, since D(ψE)|In = −E,
+D( ˜wε)|εIn
+=
+−
+�
+D(T ε
+1 [¯uε]) + ηε D(ψE)( ·
+ε)T ε
+0 [∂E ¯uε]
+����
+εIn
+=
+−(1 − ηε)|εIn
+ 
+ε(In+B)
+D(¯uε).
+(3.30)
+As by construction J and K are symmetric matrix fields, we may replace ∇ ˜wε by D( ˜wε)
+in the definition of Iε
+1, and we thus find
+Iε
+1 = −
+�
+n∈Nε(U)
+(ηε(1 − ηε))(εxn)
+ 
+ε(In+B)
+D(¯uε) :
+ˆ
+εIn
+�
+JE( ·
+ε)∂E ¯uε + κ(Kχδ∗D(uε))( ·
+ε)
+�
+.
+By the hardcore assumption, using the properties of ηε, and using (2.2), (2.25), and the
+Lipschitz continuity of E �→ fn(E) (cf. Hypothesis 1.3), the claim (3.29) follows.
+Substep 2.4. Proof that
+Iε
+2 ≲
+� ˆ
+U
+|∇wε|2� 1
+2� ˆ
+∂3rεU
+|∇¯uε|2� 1
+2
++
+� ˆ
+∂3rεU
+|∇¯uε|2� 1
+2� ˆ
+∂3rεU
+|(1, ∇ψ, ∇φ)( ·
+ε)|2�
+1 + |∇¯uε|2 + |χδ ∗ D(uε)|2�� 1
+2
++ ε
+� ˆ
+∂3rεU
+|∇¯uε|2� 1
+2�
+r−2
+ε
+ˆ
+∂3rεU
+|(ψ, φ)( ·
+ε)|2�
+1 + |∇¯uε|2 + |χδ ∗ D(uε)|2�� 1
+2
++ ε
+� ˆ
+∂3rεU
+|∇¯uε|2� 1
+2 � ˆ
+∂3rεU
+|(ψ, dφ)( ·
+ε)|2�
+|∇2¯uε|2 + |∇χδ ∗ D(uε)|2�� 1
+2 .
+(3.31)
+Appealing to the boundary conditions in (1.7), recalling the notation (1.8), and using
+´
+ε∂In ν = 0 and
+´
+ε∂In ν ⊗ (x − εxn) = |εIn| Id, we have
+Iε
+2 = −
+�
+n∈Nε(U)
+�  
+εIn
+D( ˜wε)
+�
+:
+ˆ
+ε∂In
+σ(uε, Pε + P ′
+ε − P ′′
+ε,n)ν ⊗ (x − εxn),
+and thus, using (3.30) again, and noting that the identities
+´
+ε∂In ν ⊗ (· − εxn) = |εIn| Id
+and div(¯uε) = 0 allow to remove any constant from the pressure field Pε in this expression,
+Iε
+2 =
+�
+n∈Nε(U)
+(1 − ηε)(εxn)
+�  
+ε(In+B)
+D(¯uε)
+�
+:
+ˆ
+ε∂In
+σ(uε, Pε − P ′′′
+ε,n)ν ⊗ (x − εxn), (3.32)
+where we have chosen P ′′′
+ε,n :=
+ffl
+ε(In+B)\εIn Pε. In order to estimate the right-hand side, we
+shall turn surface integrals into volume integrals, proceeding as for the construction of KE
+in Lemma 2.5. More precisely, for all n, we consider the following Neumann problem,
+
+
+
+−△un
+ε + ∇P n
+ε = fn,ε
+�
+χδ ∗ D(uε)∤εIn
+�
+,
+in εIn,
+div(un
+ε ) = 0,
+in εIn,
+σ(un
+ε , P n
+ε )ν = σ(uε, Pε − P ′′′
+ε,n)ν,
+on ε∂In.
+As for (2.16), we can show that there is a unique solution un
+ε ∈ H1
+0(εIn)d with
+ffl
+εIn un
+ε = 0
+and
+ffl
+εIn ∇un
+ε ∈ Msym
+0
+, and a unique pressure P n
+ε ∈ L2(εIn)/R, such that
+∥(∇un
+ε , P n
+ε )∥L2(εIn) ≲ ∥D(uε)∥L2(ε(In+B)) +
+��fn,ε
+�
+χδ ∗ D(uε)∤εIn
+���
+L2(ε(In+B)).
+(3.33)
+
+40
+A. BERNOU, M. DUERINCKX, AND A. GLORIA
+In these terms, we can reformulate the surface integral in (3.32) as follows,
+ˆ
+ε∂In
+σ(uε, Pε)ν ⊗ (x − εxn) =
+ˆ
+εIn
+∇j
+�
+σ(un
+ε , P n
+ε )ej ⊗ (x − εxn)
+�
+= −
+ˆ
+εIn
+fn,ε
+�
+χδ ∗ D(uε)∤εIn
+�
+⊗ (x − εxn) +
+ˆ
+εIn
+σ(un
+ε , P n
+ε ),
+and thus, in view of (3.33),
+����
+ˆ
+ε∂In
+σ(uε, Pε)ν ⊗ (x − εxn)
+����
+≲ |εIn|
+1
+2
+�
+|εIn|
+1
+2 + ∥D(uε)∥L2(ε(In+B)) + ∥χδ ∗ D(uε)∥L2(εIn)
+�
+.
+Inserting this into (3.32) and using the properties of ηε, we are led to
+Iε
+2 ≲
+� ˆ
+∂3rεU
+|∇¯uε|2� 1
+2� ˆ
+∂3rεU
+�
+1 + |∇uε|2 + |χδ ∗ D(uε)|2�� 1
+2 .
+Inserting then the two-scale expansion to replace the norm of ∇uε in the right-hand side,
+uε = wε + ¯uε + εψE( ·
+ε)ηε∂E ¯uε + εκφχδ∗D(uε)( ·
+ε)ηε,
+the claim (3.31) follows.
+Substep 2.5. Proof that
+|Iε
+3| ≲ ε
+� ˆ
+U
+|∇ ˜wε|2� 1
+2� ˆ
+U
+|∇(χδ ∗ D(uε))|4 + |∇2(χδ ∗ D(uε))|2
++ |∇(χδ ∗ D(uε))|2�
+1 + [χδ ∗ D(uε)]4
+4ε
+�� 1
+2 .
+(3.34)
+We start by decomposing
+Iε
+3 = − κ
+ε
+�
+n∈Nε(U)
+ˆ
+ε(In+B)
+�
+ηε ˜wε −
+ 
+εIn
+ηε ˜wε
+�
+·
+�
+fn,ε(χδ ∗ D(uε)) − fn,ε
+�
+χδ ∗ D(uε)∤εIn
+��
+− κ
+ε
+�
+n∈Nε(U)
+�  
+εIn
+ηε ˜wε
+�
+·
+ˆ
+ε(In+B)
+�
+fn,ε(χδ ∗ D(uε)) − fn,ε
+�
+χδ ∗ D(uε)∤εIn
+��
+− κ
+ˆ
+U
+ηε ˜wε ⊗ ∇(χδ ∗ ∂Euε) : dFχδ∗D(uε),E( ·
+ε).
+(3.35)
+We now estimate the first right-hand side term. Using the Lipschitz regularity of fn,ε,
+applying Poincaré’s inequality, and using the hardcore assumption, we get
+����
+1
+ε
+�
+n∈Nε(U)
+ˆ
+ε(In+B)
+�
+ηε ˜wε −
+ 
+εIn
+ηε ˜wε
+�
+·
+�
+fn,ε(χδ ∗ D(uε)) − fn,ε
+�
+χδ ∗ D(uε)∤εIn
+������
+≲ ε
+� ˆ
+U
+|∇(ηε ˜wε)|2� 1
+2 � ˆ
+U
+|∇(χδ ∗ D(uε))|2� 1
+2 .
+
+HOMOGENIZATION FOR ACTIVE SUSPENSIONS
+41
+We further appeal to the following version of Poincaré’s inequality for ˜wε ∈ H1
+0(U)d in ∂2rεU
+(this will be used several times in the proof): by the properties of ηε,
+ˆ
+U
+|∇ηε|2 ˜w2
+ε ≲ r−2
+ε
+ˆ
+∂2rεU
+˜w2
+ε ≲
+ˆ
+U
+|∇ ˜wε|2.
+(3.36)
+The above then becomes
+����
+1
+ε
+�
+n∈Nε(U)
+ˆ
+ε(In+B)
+�
+ηε ˜wε −
+ 
+εIn
+ηε ˜wε
+�
+·
+�
+fn,ε(χδ ∗ D(uε)) − fn,ε
+�
+χδ ∗ D(uε)∤εIn
+������
+≲ ε
+� ˆ
+U
+|∇ ˜wε|2� 1
+2 � ˆ
+U
+|∇(χδ ∗ D(uε))|2� 1
+2 .
+(3.37)
+We turn to the analysis of the last two terms in (3.35). By a Taylor expansion (of the form
+|u(a + b) − u(a) − cu′(a)| ≲ b2 sup |u′′| + |b − c||u′(a)|), using the C2 regularity of fn,ε, we
+can estimate
+����
+ˆ
+ε(In+B)
+�
+fn,ε(χδ ∗ D(uε)) − fn,ε
+�
+χδ ∗ D(uε)∤εIn
+��
+−
+�  
+εIn
+∇i(χδ ∗ ∂Euε)
+� ˆ
+ε(In+B)
+(x − εxn)i ∂Efn,ε(χδ ∗ D(uε)∤εIn)
+����
+≲ ε2
+ˆ
+ε(In+B)
+|∇(χδ ∗ D(uε))|2 + ε2
+ˆ
+ε(In+B)
+|∇2(χδ ∗ D(uε))|.
+Summing over n and recognizing the definition of dF, cf. (2.30),
+dFE,E′ := −
+�
+n
+1In
+|In|
+ˆ
+In+B
+∂E′fn(E) ⊗ (x − xn),
+we are led to
+����
+1
+ε
+�
+n∈Nε(U)
+�  
+εIn
+ηε ˜wε
+�
+·
+ˆ
+ε(In+B)
+�
+fn,ε(χδ ∗ D(uε)) − fn,ε
+�
+χδ ∗ D(uε)∤εIn
+��
++
+ˆ
+U
+ηε ˜wε ⊗ ∇(χδ ∗ ∂Euε) : dFχδ∗D(uε),E( ·
+ε)
+����
+≲ ε
+� ˆ
+U
+|ηε ˜wε|2� 1
+2 � ˆ
+U
+|∇(χδ ∗ D(uε))|4 + |∇2(χδ ∗ D(uε))|2� 1
+2
++ ε
+� ˆ
+U
+|∇(ηε ˜wε)|2� 1
+2 � ˆ
+U
+|∇(χδ ∗ D(uε))|2�
+1 + [χδ ∗ D(uε)]4
+4ε
+�� 1
+2 .
+Further appealing to Poincaré’s inequality and to (3.36), this becomes
+����
+1
+ε
+�
+n∈Nε(U)
+�  
+εIn
+ηε ˜wε
+�
+·
+ˆ
+ε(In+B)
+�
+fn,ε(χδ ∗ D(uε)) − fn,ε
+�
+χδ ∗ D(uε)∤εIn
+��
++
+ˆ
+U
+ηε ˜wε ⊗ ∇(χδ ∗ ∂Euε) : dFχδ∗D(uε),E( ·
+ε)
+����
+≲ ε
+� ˆ
+U
+|∇ ˜wε|2� 1
+2� ˆ
+U
+|∇(χδ ∗ D(uε))|4 + |∇2(χδ ∗ D(uε))|2
+
+42
+A. BERNOU, M. DUERINCKX, AND A. GLORIA
++ |∇(χδ ∗ D(uε))|2�
+1 + [χδ ∗ D(uε)]4
+4ε
+�� 1
+2 .
+Combining this with (3.35) and (3.37), the claim (3.34) follows.
+Substep 2.6. Proof that
+Iε
+4
+≲
+� ˆ
+U
+|∇ ˜wε|2� 1
+2� ˆ
+∂3rεU
+�
+1 + |h|2 + |∇¯uε|2 + |χδ ∗ D(uε)|2�� 1
+2 ,
+(3.38)
+Iε
+5
+≲
+ε
+� ˆ
+U
+|∇ ˜wε|2� 1
+2 �
+r−2
+ε
+ˆ
+∂2rεU
+|(ψ, Υ, φ, θ, γ)|2�
+1 + |∇¯uε|2 + |χδ ∗ D(uε)|2�� 1
+2
++ ε
+� ˆ
+U
+|∇ ˜wε|2� 1
+2 � ˆ
+U
+|(ψ, Υ, dφ, dθ, dγ, △−1∇1I, △−1∇dF)|2
+×
+�
+|⟨∇⟩h|2 + |∇2¯uε|2 + |∇⟨∇⟩χδ ∗ D(uε)|2�� 1
+2
++ ε
+� ˆ
+U
+|∇ ˜wε|2� 1
+2 � ˆ
+U
+|(d2θ, d2γ, △−1∇d2F)( ·
+ε)|2|∇χδ ∗ D(uε)|4� 1
+2 .
+(3.39)
+Using properties of ηε, the neutrality condition (1.9), and Poincaré’s inequality, a direct
+estimate yields
+Iε
+4 ≲
+� ˆ
+∂3rεU
+�
+1 + |h|2 + |∇¯uε|2 + |χδ ∗ D(uε)|2�� 1
+2� ˆ
+U
+|∇ ˜wε|2 +
+ˆ
+U
+|∇ηε|2| ˜wε|2� 1
+2 ,
+and the claimed estimate (3.38) follows by applying (3.36) again. The bound (3.39) on Iε
+5
+is obtained by similar straightforward computations.
+Substep 2.7. Proof of (3.25).
+Starting from (3.26), combining estimates (3.27), (3.29), (3.31), (3.34), (3.38), and (3.39),
+adding and subtracting wε to ˜wε in the right-hand side, and applying Young’s inequality,
+we get after straightforward simplifications, for all K ≥ 1,
+ˆ
+U
+|∇wε|2 ≲
+1
+K
+ˆ
+U
+(Qε)2 + K
+ˆ
+U
+|∇( ˜wε − wε)|2
++
+ˆ
+∂3rεU
+|(1, ∇ψ, Σ1Rd\I, ∇φ, Π1Rd\I)( ·
+ε)|2�
+1 + |h|2 + |∇¯uε|2 + [∇¯uε]2
+4ε + |χδ ∗ D(uε)|2�
++ ε2r−2
+ε K
+ˆ
+∂3rεU
+|(ψ, Υ, φ, θ, γ, dφ)( ·
+ε)|2�
+1 + |∇¯uε|2 + |χδ ∗ D(uε)|2�
++ ε2K
+ˆ
+U
+|(1, ψ, Υ, φ, dφ, dθ, dγ, d2θ, d2γ, △−1∇1I, △−1∇dF, △−1∇d2F)( ·
+ε)|2
+×
+�
+|⟨∇⟩h|2 + |∇2¯uε|2 + |∇⟨∇⟩χδ ∗ D(uε)|2 + |∇χδ ∗ D(uε)|4
++ |∇(χδ ∗ D(uε))|2�
+1 + [χδ ∗ D(uε)]4
+4ε
+��
+.
+(3.40)
+It remains to evaluate the norm of ∇( ˜wε − wε). By definition of ˜wε, we have
+∇( ˜wε − wε) = ∇(¯uε − T ε
+1 [¯uε]) + ∇
+�
+εψE( ·
+ε)ηε(∂E ¯uε − T ε
+0 [∂E ¯uε])
+�
++ ∇
+�
+εκφχδ∗D(uε)( ·
+ε)ηε − εκφT ε
+0 [χδ∗D(uε)]( ·
+ε)ηε
+�
+,
+
+HOMOGENIZATION FOR ACTIVE SUSPENSIONS
+43
+and thus, inserting the definition of the truncation operators T ε
+0 , T ε
+1 , and noting that ηε is
+constant in the support of the cut-off functions {ρn
+ε }n by definition,
+∇( ˜wε − wε) =
+�
+n
+ρε
+n
+�
+∇¯uε −
+ 
+ε(In+B)
+∇¯uε
+�
++
+�
+n
+∇ψE( ·
+ε)ηερε
+n
+�
+∂E ¯uε −
+ 
+ε(In+B)
+∂E ¯uε
+�
++ κ
+�
+n
+�
+∇φ|χδ∗D(uε)( ·
+ε) − ∇φχδ∗D(uε)∤ε(In+B)( ·
+ε)
+�
+ηερε
+n
++
+�
+n
+εψE( ·
+ε)ηερε
+n∇∂E ¯uε +
+�
+n
+εκdφχδ∗D(uε),E( ·
+ε) ⊗ ηερε
+n∇(χδ ∗ ∂Euε)
++
+�
+n
+�
+¯uε −
+�  
+ε(In+B)
+¯uε
+�
+−
+�  
+ε(In+B)
+∇¯uε
+�
+(x − εxn)
+�
+⊗ ∇ρε
+n
++
+�
+n
+εψE( ·
+ε) ⊗ ηε∇ρε
+n
+�
+∂E¯uε −
+ 
+ε(In+B)
+∂E ¯uε
+�
++
+�
+n
+εκ
+�
+φχδ∗D(uε)( ·
+ε) − φχδ∗D(uε)∤ε(In+B)( ·
+ε)
+�
+⊗ ηε∇ρε
+n.
+Using that
+ˆ
+ε(In+B)
+��φχδ∗D(uε)( ·
+ε) − φχδ∗D(uε)∤ε(In+B)( ·
+ε)
+��2
+≲ ε2
+ˆ
+ε(In+B)
+|dφ( ·
+ε)|2���χδ ∗ D(uε) −
+ 
+ε(In+B)
+χδ ∗ D(uε)
+���
+2
+,
+and
+ˆ
+ε(In+B)
+��∇φ|χδ∗D(uε)( ·
+ε) − ∇φχδ∗D(uε)∤ε(In+B)( ·
+ε)
+��2
+≲ ε2
+ˆ
+ε(In+B)
+|(∇dφ)( ·
+ε)|2���χδ ∗ D(uε) −
+ 
+ε(In+B)
+χδ ∗ D(uε)
+���
+2
+,
+a direct computation then leads us to
+ˆ
+U
+|∇( ˜wε − wε)|2 ≲ ε2
+ˆ
+U
+|(1, ψ)( ·
+ε)|2|∇2¯uε|2 + ε2
+ˆ
+U
+|dφ( ·
+ε)|2|∇χδ ∗ D(uε)|2
++
+�
+n
+ˆ
+ε(In+B)
+|(ψ, ∇ψ)( ·
+ε)|2���∇¯uε −
+ 
+ε(In+B)
+∇¯uε
+���
+2
++
+�
+n
+ˆ
+ε(In+B)
+|(dφ, ∇dφ)( ·
+ε)|2���χδ ∗ D(uε) −
+ 
+ε(In+B)
+χδ ∗ D(uε)
+���
+2
+.
+(3.41)
+Inserting this into (3.40), the conclusion (3.25) follows.
+Step 3. Pressure estimate for (3.11):
+ˆ
+U
+(Qε)2 ≲
+ˆ
+U
+|∇wε|2 + rε
+�
+1 + ∥(h, ∇¯uε, χδ ∗ D(uε))∥2
+L∞(U)
+�
++ ε2�
+1 + ∥(h, ∇¯uε, ¯Pε)∥2
+W 1,∞(U) + ∥χδ ∗ D(uε)∥6
+W 2,∞(U)
+�
+×
+ˆ
+U
+�
+1 + r−2
+ε 1∂3rεU
+����
+1, ψ, Υ, Σ1Rd\I, φ, θ, γ, dφ, dθ, dΠ1Rd\I, dγ, d2θ, d2γ,
+
+44
+A. BERNOU, M. DUERINCKX, AND A. GLORIA
+△−1∇1I, △−1∇dF, △−1∇d2F
+�
+( ·
+ε)
+��2,
+(3.42)
+which follows from
+ˆ
+U
+(Qε)2 ≲
+ˆ
+U
+|∇wε|2 +
+ˆ
+∂3rεU
+�
+1 + |h|2 + |∇¯uε|2 + |χδ ∗ D(uε)|2�
++ ε2r−2
+ε
+ˆ
+∂3rεU
+|(ψ, Υ, φ, θ, γ, dθ, dγ)( ·
+ε)|2�
+1 + |∇¯uε|2 + |χδ ∗ D(uε)|2�
++ ε2
+ˆ
+U
+|(1, ψ, Υ, dφ, dθ, dγ, d2θ, d2γ, △−1∇1I, △−1∇dF, △−1∇d2F)( ·
+ε)|2
+×
+�
+|⟨∇⟩h|2 + |∇2¯uε|2 + |∇ ¯Pε|2 + |∇⟨∇⟩χδ ∗ D(uε)|2 + |∇χδ ∗ D(uε)|4
++ |∇(χδ ∗ D(uε))|2�
+1 + [χδ ∗ D(uε)]4
+4ε
+��
++
+�
+n
+ˆ
+ε(In+B)
+|(Σ1Rd\I)( ·
+ε)|2���∇¯uε −
+ 
+ε(In+B)
+∇¯uε
+���
+2
++
+�
+n
+ˆ
+ε(In+B)
+|(dΠ1Rd\I)( ·
+ε)|2���χδ ∗ D(uε) −
+ 
+ε(In+B)
+χδ ∗ D(uε)
+���
+2
+,
+(3.43)
+after taking uniform norms of h, ¯uε, χδ ∗ D(uε) and using Poincaré’s inequality in the last
+two summands.
+Similarly as in Step 2, we shall appeal to a truncated version of Qε,
+˜Qε := Pε1Rd\εI + P ∗
+ε − T ε
+0 [ ¯Pε] − ηε¯b : T ε
+0 [D(¯uε)] − κηε¯c(T ε
+0 [χδ ∗ D(uε)])
+− (ΣE1Rd\I)( ·
+ε)ηεT ε
+0 [∂E ¯uε] − κ(ΠT ε
+0 [χδ∗D(uε)]1Rd\I)( ·
+ε)ηε,
+where we recall that P ∗
+ε stands for some locally constant pressure field, cf. (3.9),
+P ∗
+ε := P ′
+ε1U\Iε(U) +
+�
+n∈Nε(U)
+P ′′
+ε,n1εIn,
+and where we now choose the constants P ′
+ε and {P ′′
+ε,n}n in such a way that
+˜Qε|Iε(U) = 0
+and
+ˆ
+U
+˜Qε = 0.
+(3.44)
+Using the Bogovskii operator as in [4], we can construct a vector field Sε ∈ H1
+0(U)d such
+that Sε|εIn is a constant for all n ∈ Nε(U) and such that
+div(Sε) = − ˜Qε
+in U,
+ˆ
+U
+|∇Sε|2 ≲
+ˆ
+U
+| ˜Qε|2.
+(3.45)
+Testing equation (3.11) with Sε, using the property that Sε|εIn is a constant for all n ∈
+Nε(U), and using the boundary conditions in (1.10), we find
+ˆ
+U
+˜QεQε =
+−
+ˆ
+U
+∇Sε : ∇wε − κ
+ˆ
+U
+ηεSε ⊗ ∇(χδ ∗ ∂Euε) : dFχδ∗D(uε),E( ·
+ε)
+− κ
+ε
+�
+n∈Nε(U)
+ˆ
+ε(In+B)
+ηεSε ·
+�
+fn,ε(χδ ∗ D(uε)) − fn,ε
+�
+χδ ∗ D(uε)∤εIn
+��
+
+HOMOGENIZATION FOR ACTIVE SUSPENSIONS
+45
++
+ˆ
+U
+(1 − ηε)(λ − 1Iε(U))Sε · h − κ
+ˆ
+U
+Sε ⊗ ∇ηε : ¯F (χδ ∗ D(uε))
++ κ
+ε
+�
+n∈Nε(U)
+ˆ
+ε(In+B)
+(1 − ηε)Sε · fn,ε
+�
+χδ ∗ D(uε)∤εIn
+�
++
+ˆ
+U
+(1 − ηε)∇Sε :
+�
+2( ¯Bpas − Id) D(¯uε) + 2κ ¯C(χδ ∗ D(uε)) + κ ¯F (χδ ∗ D(uε))
+�
+−ε
+ˆ
+U
+∇Sε :
+��
+2ψE ⊗s − Id ⊗ψE − ΥE
+�
+( ·
+ε)∇(ηε∂E ¯uε) − ηεh ⊗ (△−1∇1I)( ·
+ε)
++κ
+�
+2φχδ∗D(uε) ⊗s − Id ⊗φχδ∗D(uε) − θχδ∗D(uε) + γχδ∗D(uε) ⊗ Id
+�
+( ·
+ε)∇ηε
++κ
+�
+2dφχδ∗D(uε),E ⊗s − Id ⊗dφχδ∗D(uε),E − dθχδ∗D(uε),E
++dγχδ∗D(uε),E ⊗ Id
+�
+( ·
+ε)ηε∇(χδ ∗ ∂Euε)
+�
+−ε
+ˆ
+U
+ηε∇iSε ⊗ ∇(χδ ∗ ∂Euε) : (△−1∇idF|χδ∗D(uε),E)( ·
+ε)
++ε
+ˆ
+U
+Sε ·
+�
+(△−1∇j1I)( ·
+ε)∇j(ηεh) − κγχδ∗D(uε)( ·
+ε)△ηε
+−κdγχδ∗D(uε),E( ·
+ε) : ηε△(χδ ∗ ∂Euε)
+−κ(△−1∇idF|χδ∗D(uε),E)( ·
+ε)∇i(ηε∇(χδ ∗ ∂Euε))
++2κ
+�
+dθχδ∗D(uε),E − dγχδ∗D(uε),E ⊗ Id
+�
+( ·
+ε) : ∇(χδ ∗ ∂Euε) ⊗ ∇ηε
++κ
+�
+d2θχδ∗D(uε),E,E′ − d2γχδ∗D(uε),E,E′ ⊗ Id −△−1∇d2F|χδ∗D(uε),E,E′
+�
+( ·
+ε)
+: ηε∇(χδ ∗ ∂E′uε) ⊗ ∇(χδ ∗ ∂Euε)
+�
+.
+Adding and subtracting Qε to ˜Qε in the left-hand side, and proceeding as in Step 2 to
+estimate the different contributions, we deduce
+ˆ
+U
+(Qε)2 ≲
+ˆ
+U
+( ˜Qε − Qε)2 +
+� ˆ
+U
+|∇Sε|2� 1
+2 � ˆ
+U
+|∇wε|2� 1
+2
++
+� ˆ
+U
+|∇Sε|2� 1
+2 � ˆ
+∂3rεU
+�
+1 + |h|2 + |∇¯uε|2 + |χδ ∗ D(uε)|2�� 1
+2
++ε
+� ˆ
+U
+|∇Sε|2� 1
+2 �
+r−2
+ε
+ˆ
+∂3rεU
+|(ψ, Υ, φ, θ, γ, dθ, dγ)|2�
+1 + |∇¯uε|2 + |χδ ∗ D(uε)|2�� 1
+2
++ε
+� ˆ
+U
+|∇Sε|2� 1
+2
+� ˆ
+U
+|(1, ψ, Υ, dφ, dθ, dγ, d2θ, d2γ, △−1∇1I, △−1∇dF, △−1∇d2F)|2
+×
+�
+|⟨∇⟩h|2 + |∇2¯uε|2 + |∇⟨∇⟩χδ ∗ D(uε)|2 + |∇χδ ∗ D(uε)|4
++|∇(χδ ∗ D(uε))|2�
+1 + [χδ ∗ D(uε)]4
+4ε
+��� 1
+2
+.
+Hence, by (3.45) and Young’s inequality,
+ˆ
+U
+(Qε)2 ≲
+ˆ
+U
+( ˜Qε − Qε)2 +
+ˆ
+U
+|∇wε|2 +
+ˆ
+∂3rεU
+�
+1 + |h|2 + |∇¯uε|2 + |χδ ∗ D(uε)|2�
+
+46
+A. BERNOU, M. DUERINCKX, AND A. GLORIA
++ ε2r−2
+ε
+ˆ
+∂3rεU
+|(ψ, Υ, φ, θ, γ, dθ, dγ)|2�
+1 + |∇¯uε|2 + |χδ ∗ D(uε)|2�
++ ε2
+ˆ
+U
+|(1, ψ, Υ, dφ, dθ, dγ, d2θ, d2γ, △−1∇1I, △−1∇dF, △−1∇d2F)|2
+×
+�
+|⟨∇⟩h|2 + |∇2¯uε|2 + |∇⟨∇⟩χδ ∗ D(uε)|2 + |∇χδ ∗ D(uε)|4
++ |∇(χδ ∗ D(uε))|2�
+1 + [χδ ∗ D(uε)]4
+4ε
+��
+.
+(3.46)
+It remains to estimate the norm of ˜Qε − Qε in the right-hand side. By definition of ˜Qε, we
+have
+˜Qε − Qε = ¯Pε − T ε
+0 [ ¯Pε] + ηε¯b :
+�
+D(¯uε) − T ε
+0 [D(¯uε)]
+�
++ κηε
+�
+¯c(χδ ∗ D(uε)) − ¯c(T ε
+0 [χδ ∗ D uε)])
+�
++ (ΣE1Rd\I)( ·
+ε)ηε(∂E ¯uε − T ε
+0 [∂E ¯uε])
++ κ
+�
+Πχδ∗D(uε)1Rd\I − ΠT ε
+0 [χδ∗D(uε)]1Rd\I
+�
+( ·
+ε) ηε,
+and thus, using (2.32) and proceeding as for (3.41), we get
+ˆ
+U
+( ˜Qε − Qε)2 ≲ ε2
+ˆ
+U
+�
+|∇2¯uε|2 + |∇ ¯Pε|2 + |∇χδ ∗ D(uε)|2�
++
+�
+n
+ˆ
+ε(In+B)
+|(Σ1Rd\I)( ·
+ε)|2���∇¯uε −
+ 
+ε(In+B)
+∇¯uε
+���
+2
++
+�
+n
+ˆ
+ε(In+B)
+|(dΠ1Rd\I)( ·
+ε)|2���χδ ∗ D(uε) −
+ 
+ε(In+B)
+χδ ∗ D(uε)
+���
+2
+.
+(3.47)
+Inserting this into (3.46), the conclusion (3.43) follows.
+Step 4. Conclusion.
+Choosing K ≥ 1 large enough to absorb part of the pressure into the left-hand side, the
+combination of (3.22) and (3.42) yields
+ˆ
+U
+|∇wε|2 +
+ˆ
+U
+(Qε)2 ≲
+�
+1 + ∥(h, ∇¯uε, ¯Pε)∥2
+W 1,∞(U) + ∥χδ ∗ D(uε)∥6
+W 2,∞(U)
+�
+×
+� ˆ
+∂3rεU
+|(1, ∇ψ, Σ1Rd\I, ∇φ, Π1Rd\I)( ·
+ε)|2
++ ε2
+ˆ
+U
+�
+1 + r−2
+ε 1∂3rεU
+�
+|(1, ψ, ∇ψ, Υ, Σ1Rd\I, φ, θ, γ, dφ, ∇dφ, dθ, dΠ1Rd\I, dγ,
+d2θ, d2γ, △−1∇1I, △−1∇dF, △−1∇d2F)( ·
+ε)|2
+�
+.
+(3.48)
+By Proposition 1.6, we have ∥∇uε∥L2(U) ≲ 1 + ∥h∥L2(U), and thus for all s > 0,
+∥χδ ∗ D(uε)∥W 1+s,∞(U) ≲ ∥χδ∥H1+s(Rd)(1 + ∥h∥L2(U)).
+By the regularity theory for the Stokes equation, using (2.32), we then deduce that the
+solution (¯uε, ¯Pε) of (3.10) satisfies for all s > 0,
+∥(∇¯uε, ¯Pε)∥W 1,∞(U)
+≲
+∥h∥W s,∞(U) + ∥ ¯Bact(χδ ∗ D(uε))∥W 1+s,∞(U)
+≲
+∥h∥W s,∞(U) + ∥χδ∥H1+s(Rd)(1 + ∥h∥L2(U)).
+(3.49)
+
+HOMOGENIZATION FOR ACTIVE SUSPENSIONS
+47
+Inserting these estimates into (3.48), we get
+ˆ
+U
+|∇wε|2 +
+ˆ
+U
+(Qε)2
+≲χδ
+�
+1 + ∥h∥6
+W 1,∞(U)
+�� ˆ
+∂3rεU
+���
+1, ∇ψ, Σ1Rd\I, ∇φ, Π1Rd\I
+�
+( ·
+ε)
+��2
++ ε2
+ˆ
+U
+�
+1 + r−2
+ε 1∂3rεU
+�
+|(1, ψ, ∇ψ, Υ, Σ1Rd\I, φ, θ, γ, dφ, ∇dφ, dθ, dΠ1Rd\I, dγ,
+d2θ, d2γ, △−1∇1I, △−1∇dF, △−1∇d2F)( ·
+ε)|2
+�
+.
+(3.50)
+By the ergodic theorem and by the sublinearity of correctors, cf. Lemmas 2.1, 2.2, 2.3, 2.4,
+and 2.5, we have for any fixed r > 0, almost surely,
+lim
+ε↓0
+ˆ
+∂3rU
+���
+1, ∇ψ, Σ1Rd\I, ∇φ, Π1Rd\I
+�
+( ·
+ε)
+��2 ≲ Cr,
+(3.51)
+and
+lim
+ε↓0 ε2
+ˆ
+U
+�
+1 + r−21∂3rU
+�
+|(1, ψ, ∇ψ, Υ, Σ1Rd\I, φ, θ, γ, dφ, ∇dφ, dθ, dΠ1Rd\I, dγ,
+d2θ, d2γ, △−1∇1I, △−1∇dF, △−1∇d2F)( ·
+ε)|2 = 0.
+(3.52)
+Some care is however needed to prove these convergence results as we take suprema in
+the notation (3.23)–(3.24): while the linear dependence of ∇ψE on E makes the suprema
+|∇ψ| = supE |E|−1|∇ψE| trivial, the same is not true for ∇φE. In that case, we use the
+Sobolev embedding in form of
+|(∇φ)( ·
+ε)|2 ≤
+sup
+|E|≤Cδ(h)
+|∇φE( ·
+ε)|2 ≲
+k
+�
+l=0
+ˆ
+|E|≤Cδ(h)
+|(∇dlφE)( ·
+ε)|2 dE,
+(3.53)
+for some k > 1
+2 dim Msym
+0
+, and thus
+ˆ
+∂3rU
+|(∇φ)( ·
+ε)|2 ≲
+k
+�
+l=0
+ˆ
+|E|≤Cδ(h)
+� ˆ
+∂3rU
+|(∇dlφE)( ·
+ε)|2�
+dE,
+to which the ergodic theorem for (linearized) correctors ∇dlφ in Lemma 2.3 can now be
+applied, leading to the claim (3.51). Similarly, writing
+ε2
+ˆ
+U
+�
+1 + r−21∂3rU
+�
+|φ( ·
+ε)|2
+≤
+ε2
+ˆ
+U
+�
+1 + r−21∂3rU
+�
+sup
+|E|≤Cδ(h)
+|φE( ·
+ε)|2
+≲
+k
+�
+l=0
+ˆ
+|E|≤Cδ(h)
+�
+ε2
+ˆ
+U
+�
+1 + r−21∂3rU
+�
+|dlφE( ·
+ε)|2
+�
+dE,
+the claim (3.52) indeed follows from the sublinearity of the (linearized) correctors dlφ in
+Lemma 2.3.
+Next, inserting (3.51)–(3.52) into (3.50) and appealing to a diagonalization argument, we
+conclude that there exists a (random) sequence rε ↓ 0 such that for this choice we have,
+
+48
+A. BERNOU, M. DUERINCKX, AND A. GLORIA
+almost surely,
+lim
+ε↓0
+� ˆ
+U
+|∇wε|2 +
+ˆ
+U
+(Qε)2�
+= 0,
+that is, wε → 0 in H1
+0(U) and Qε → 0 in L2(U). On the other hand, note that a priori
+estimates (1.19) and (3.49) entail that up to an extraction we have uε ⇀ u0 and ¯uε ⇀ ¯u0
+in H1
+0(U), for some u0, ¯u0 ∈ H1
+0(U)d. Passing to the weak limit in equation (3.10) along
+this subsequence, we find that ¯u0 satisfies
+− div(2 ¯Bpas D(¯u0)) + ∇ ¯P0 = (1 − λ)h + div(2κ ¯Bact(χδ ∗ D(u0))).
+(3.54)
+Now, by definition of the two-scale expansion error wε, cf. (3.8), together with the sub-
+linearity of correctors, cf. Lemmas 2.1 and 2.3, the convergence wε → 0 in H1
+0(U) implies
+uε − ¯uε → 0 in L2(U), and thus u0 = ¯u0. From (3.54), we deduce that ¯u := u0 = ¯u0
+actually satisfies the homogenized equation (1.20). In view of the well-posedness for the
+latter, we conclude uε ⇀ ¯u in H1
+0(U) independently of extractions.
+We turn to the convergence of the pressure field. Recall that we have shown Qε → 0
+in L2(U). The a priori estimate (3.49) ensures ¯Pε ⇀ ¯P in L2(U), where ¯P is the unique pres-
+sure field in L2(U)/R for the homogenized equation (3.54). By definition of Qε, cf. (3.8),
+together with the ergodic theorem for corrector pressures, cf. Lemmas 2.1 and 2.3, and
+with the choice (3.44) of P ′
+ε, the convergence of the pressure follows.
+□
+3.3. Quantitative homogenization and limit δ ↓ 0. This section is devoted to the
+proof of Theorem 1.8. As the above proof is semi-quantitative, one can infer convergence
+rates provided that quantitative mixing assumptions such as Hypothesis 1.2 are further
+made on the statistical ensemble of inclusions. Quantitative rates then allow in particular
+to let the parameter δ tend to 0 in a nontrivial regime. We split the proof into three steps.
+Step 1. Convergence of the homogenized equation (1.20) as δ ↓ 0.
+Writing equation (1.20) as
+−div(2 ¯Bpas D(¯uδ)) + ∇ ¯Pδ = (1 − λ)h + div(2κ ¯Bact(χδ ∗ D(¯uδ))),
+and appealing to the regularity theory for the Stokes equation, the unique solution (¯uδ, ¯Pδ) ∈
+H1
+0(U)d × L2(U)/R satisfies for all 0 < η < 1,
+∥(∇¯uδ, ¯Pδ)∥W 1−η,∞(U) ≲η ∥h∥L∞(U) + κ∥ ¯Bact(χδ ∗ D(¯uδ))∥W 1−η,∞(U).
+Using (2.32), we deduce
+∥(∇¯uδ, ¯Pδ)∥W 1−η,∞(U)
+≲η
+∥h∥L∞(U) + κλ∥χδ ∗ D(¯uδ)∥W 1−η,∞(U)
+≲η
+∥h∥L∞(U) + κλ∥∇¯uδ∥W 1−η,∞(U).
+The smallness condition (1.18) yields κλ ≲ κℓ−d ≪ 1, and we thus infer
+∥(∇¯uδ, ¯Pδ)∥W 1−η,∞(U) ≲η ∥h∥L∞(U).
+Up to an extraction, this implies (∇¯uδ, ¯Pδ) → (∇¯u0, ¯P0) in L∞(U), for some limit (¯u0, ¯P0) ∈
+H1
+0(U)d × L2(U)/R. Passing to the limit in equation (1.20), we find that (¯u0, ¯P0) satisfies
+equation (1.28). Provided that κλ ≪ 1 is small enough, which is ensured by the smallness
+condition (1.18), the well-posedness of (1.28) follows from the same argument as for (1.20).
+We conclude that (¯u0, ¯P0) = (¯u, ¯P) is the unique solution of (1.28) and that ¯uδ → ¯u
+in W 1,∞(U) and ¯Pδ → ¯P in L∞(U).
+
+HOMOGENIZATION FOR ACTIVE SUSPENSIONS
+49
+Combining this with Theorem 1.7, by a diagonalization argument, we deduce that there is
+a (random) sequence δ◦
+ε ↓ 0 such that, for any sequence 0 < δε ≤ δ◦
+ε, the solution (uε, Pε)
+of (1.10) with δ = δε satisfies, as ε ↓ 0,
+uε
+⇀
+¯u,
+in H1
+0(U)d,
+Pε1U\Iε(U)
+⇀
+(1 − λ) ¯P + (1 − λ)¯b : D(¯u)
++(1 − λ)κ
+�
+¯c(D(¯u)) −
+ffl
+U ¯c(D(¯u))
+�
+,
+in L2(U).
+(3.55)
+To improve on such a diagonal result, we need to prove a quantitative version of Theo-
+rem 1.7 and capture the precise dependence on δ. This is the purpose of the next two
+steps.
+Step 2. Corrector estimates.
+As we proved in [10] for passive correctors, under a quantitative mixing assumption such
+as Hypothesis 1.2, we have for all s < ∞,
+E
+�
+|(∇ψ, Σ1Rd\I, ∇Υ)(x)|s� 1
+s ≲s 1,
+E [|(ψ, Υ)(x)|s]
+1
+s ≲s µd(x) :=
+�
+log
+1
+2(2 + |x|)
+:
+d = 2,
+1
+:
+d > 2.
+(3.56)
+which optimally quantifies the sublinearity of passive correctors. The method in [10] applies
+mutadis mutandis to active correctors, and yields the following: for all E, E′, E′′ ∈ Msym
+0
+and s < ∞, we get
+E
+�
+|(∇φE, ΠE1Rd\I, ∇θE, ∇dφE, ∇dθE,E′, ∇d2φE,E′,E′′, ∇d2θE,E′,E′′)(x)|s� 1
+s ≲s,E,E′,E′′ 1,
+E
+�
+|(φE, γE, θE, dφE,E′, dγE,E′, dθE,E′,
+d2φE,E′,E′′, d2γE,E′,E′′, d2θE,E′,E′′)(x)|s� 1
+s ≲s,E,E′,E′′ µd(x).
+(3.57)
+Yet, for our purposes, we further need corresponding estimates on suprema such as
+|∇φ| =
+sup
+|E|≤Cδ(h)
+⟨E⟩−1|∇φE|,
+which is not trivial due to the nonlinear dependence on E. As we aim at capturing the best
+dependence on δ, we cannot appeal to brutal Sobolev estimates as in (3.53). Instead, we
+shall take advantage of the above moment estimates (3.57) together with Hypothesis 1.4.
+More precisely, we decompose
+|∇φ| ≤ sup
+E
+⟨E⟩−1|∇φ∞
+E | + sup
+E
+⟨E⟩−1|∇(φE − φ∞
+E )|,
+(3.58)
+where we compare φE to the random field φ∞
+E that is defined via the same corrector prob-
+lem (2.4) & (2.5) with the swimming forces {fn(E)}n replaced by their large-E approxi-
+mations {f ∞(E)ξn}n, cf. Hypothesis 1.4. On the one hand, as f ∞(E) is a deterministic
+function of E, the supremum of ∇φ∞
+E over E becomes trivial and moment estimates can
+be established in the following form, for all s < ∞,
+E
+��
+supE ⟨E⟩−1|∇φ∞
+E |
+�s� 1
+s ≲s 1.
+
+50
+A. BERNOU, M. DUERINCKX, AND A. GLORIA
+On the other hand, by the Sobolev embedding, we can bound for all s < ∞, provided
+s > dim Msym
+0
+,
+sup
+E
+⟨E⟩−1|∇(φE − φ∞
+E )| ≲
+� ˆ
+Msym
+0
+⟨E⟩−s�
+|∇(φE − φ∞
+E )| + |∇(dφE − dφ∞
+E )|
+�s dE
+� 1
+s
+and thus, using the proof of moment estimates (3.57) together with Hypothesis 1.4 in form
+of
+E
+��
+|∇(φE − φ∞
+E )| + |∇(dφE − dφ∞
+E )|
+�s� 1
+s ≲s ⟨E⟩1−γ,
+we deduce for all s < ∞ with s > (1 ∨ 1
+γ ) dim Msym
+0
+,
+E
+��
+supE ⟨E⟩−1|∇(φE − φ∞
+E )|
+�s� 1
+s ≲
+� ˆ
+Msym
+0
+⟨E⟩−γs dE
+� 1
+s
+≲s 1.
+Combining these bounds with (3.58), we deduce for all s < ∞,
+E [|∇φ|s]
+1
+s ≲s 1,
+uniformly with respect to δ > 0. This string of arguments allows us to post-process (3.57)
+into
+E
+�
+|(∇φ, Π1Rd\I, ∇θ, ∇dφ, ∇dθ, ∇d2φ, ∇d2θ)(x)|s� 1
+s
+≲s
+1,
+E
+�
+|(φ, γ, θ, dφ, dγ, dθ, d2φ, d2γ, d2θ)(x)|s� 1
+s
+≲s
+µd(x).
+Step 3. Conclusion.
+In order to capture the best dependence on δ, we need a version of (3.48) where the
+dependence on ¯uε, χδ ∗ D(uε) does not deteriorate in terms of uniform norms.
+Rather
+combining (3.25) and (3.43), and choosing K ≥ 1 large enough, we get
+ˆ
+U
+|∇wε|2 +
+ˆ
+U
+(Qε)2
+≲
+ˆ
+∂3rεU
+|(1, ∇ψ, Σ1Rd\I, ∇φ, Π1Rd\I)( ·
+ε)|2�
+1 + |h|2 + |∇¯uε|2 + [∇¯uε]2
+4ε + |χδ ∗ D(uε)|2�
++ ε2r−2
+ε
+ˆ
+∂3rεU
+|(ψ, Υ, φ, θ, γ, dφ, dθ, dγ)( ·
+ε )|2�
+1 + |∇¯uε|2 + |χδ ∗ D(uε)|2�
++ ε2
+ˆ
+U
+|(1, ψ, Υ, φ, dφ, dθ, dγ, d2θ, d2γ, △−1∇1I, △−1∇dF, △−1∇d2F)( ·
+ε)|2
+×
+�
+|⟨∇⟩h|2 + |∇2¯uε|2 + |∇ ¯Pε|2 + |∇⟨∇⟩χδ ∗ D(uε)|2 + |∇χδ ∗ D(uε)|4
++ |∇(χδ ∗ D(uε))|2�
+1 + [χδ ∗ D(uε)]4
+4ε
+��
++
+�
+n
+ˆ
+ε(In+B)
+|(ψ, ∇ψ, Σ1Rd\I)( ·
+ε)|2���∇¯uε −
+ 
+ε(In+B)
+∇¯uε
+���
+2
++
+�
+n
+ˆ
+ε(In+B)
+|(dφ, ∇dφ, dΠ1Rd\I)( ·
+ε)|2���χδ ∗ D(uε) −
+ 
+ε(In+B)
+χδ ∗ D(uε)
+���
+2
+.
+
+HOMOGENIZATION FOR ACTIVE SUSPENSIONS
+51
+By Proposition 1.6, we have ∥∇uε∥L2(U) ≲ 1 + ∥h∥L2(U) and thus, for all k ≥ 0 and s ≥ 2,
+∥χδ ∗ D(uε)∥W k,s(U)
+≲
+δ−k−d( 1
+2− 1
+s)�
+1 + ∥h∥L2(U)
+�
+,
+∥χδ ∗ D(uε)∥W k,s(∂rεU)
+≲
+δ−k− d
+2 �
+δd ∧ rε
+� 1
+s �
+1 + ∥h∥L2(U)
+�
+.
+By the regularity theory for the Stokes equation, using (2.32), we then deduce that the
+solution (¯uε, ¯Pε) of (3.10) satisfies for all s ≥ 2 and η > 0,
+∥(∇¯uε, ¯Pε)∥W 1,s(U)
+≲
+δ−1−d( 1
+2 − 1
+s)�
+1 + ∥h∥L2∨s(U)
+�
+,
+∥(∇¯uε, ¯Pε)∥Ls(∂3rεU)
+≲
+δ− d
+2 �
+δd ∧ (rεδ−η)
+� 1
+s �
+1 + ∥h∥L∞(U)
+�
+.
+Inserting these estimates into the above, we get for all s ≥ 1,
+ˆ
+U
+|∇wε|2 +
+ˆ
+U
+(Qε)2
+≲
+�
+rε + (rεδ−d−η) ∧ (rεδ−d)
+1
+s
+��  
+∂3rεU
+|(1, ∇ψ, Σ1Rd\I, ∇φ, Π1Rd\I)( ·
+ε)|2s� 1
+s
++ ε2r−2
+ε
+�
+rε + (rεδ−d−η) ∧ (rεδ−d)
+1
+s
+��  
+∂3rεU
+|(ψ, Υ, φ, θ, γ, dφ, dθ, dγ)( ·
+ε )|2s� 1
+s
++ ε2δ−2−d(2+ 1
+s )� ˆ
+U
+|(1, ψ, Υ, ∇ψ, Σ1Rd\I, φ, dφ, ∇dφ, dθ, dΠ1Rd\I, dγ,
+d2θ, d2γ, △−1∇1I, △−1∇dF, △−1∇d2F)( ·
+ε)|2s� 1
+s ,
+where the multiplicative constant depends on the W 1,∞(U) norm of h. Hence, taking the
+expectation and using the corrector estimates of Step 2, we get for all s < ∞,
+E
+�� ˆ
+U
+|∇wε|2�s� 1
+s
++ E
+�� ˆ
+U
+(Qε)2�s� 1
+s
+≲h
+�
+1 + ε2µd(1
+ε)2r−2
+ε
+��
+rε + (rεδ−d−η) ∧ (rεδ−d)
+1
+s
+�
++ ε2µd(1
+ε)2δ−2−d(2+ 1
+s)
+Choosing rε := εµd(1
+ε), this becomes
+E
+�� ˆ
+U
+|∇wε|2�s� 1
+s
++ E
+�� ˆ
+U
+(Qε)2�s� 1
+s
+≲h εµd(1
+ε) + εµd(1
+ε)δ−d−η + (εµd(1
+ε))2δ−2−d(2+ 1
+s).
+As ε, δ ↓ 0 in the regime (1.27), we thus get wε → 0 in Ls(Ω; H1(U)) and Qε → 0 in
+Ls(Ω; L2(U)). Arguing as for Theorem 1.7, and further using the result of Step 1, the
+conclusion follows.
+□
+4. Dilute expansion of the effective viscosity
+This section is devoted to the proof of Theorem 1.9, that is, the first-order dilute ex-
+pansion of the effective viscosity ¯Btot. We recall that Einstein’s formula for the passive
+contribution was already established in [5] (see also [16, 15, 18, 17]), in form of
+�� ¯Bpas − Id −λ1 ¯B(1)
+pas
+�� ≲ λ2|log λ1|,
+
+52
+A. BERNOU, M. DUERINCKX, AND A. GLORIA
+and it remains to prove
+�� ¯Bact(E) − λ1 ¯B(1)
+act(E)
+�� ≲ ⟨E⟩
+�
+λ2|log λ1| + (λ1λ2 + λ1λ3|log λ1|)
+1
+2
+�
+.
+(4.1)
+We split the proof into four steps.
+Step 1. Periodic approximation.
+Define a periodized version PL of the point process P = {xn}n on the cube QL := [− L
+2 , L
+2 )d,
+PL := {xn : n ∈ NL},
+NL := {n : xn ∈ QL−4},
+and consider the corresponding random set
+IL :=
+�
+n∈NL
+In,
+In := xn + I◦
+n.
+For notational convenience, we choose an enumeration PL := {xn,L}n and we set In,L :=
+xn,L + I◦
+n,L. By definition, under Hypothesis 1.1, for all L, the periodized random set
+IL + LZd satisfies the same regularity and hardcore conditions as in Hypothesis 1.1.
+Moreover, we emphasize the stabilization property PL|QL−4 = P|QL−4. Next, we define
+ψE;L ∈ L2(Ω; H1
+per(QL)d) as the unique almost sure solution of the periodic version of (2.1),
+
+
+
+
+
+
+
+
+
+
+
+−△ψE;L + ∇ΣE;L = 0,
+in QL \ IL,
+div(ψE;L) = 0,
+in QL \ IL,
+D(ψE;L + Ex) = 0,
+in IL,
+´
+∂In,L σ(ψE;L + Ex, ΣE;L)ν = 0,
+∀n,
+´
+∂In,L Θ(x − εxn,L) · σ(ψE;L + Ex, ΣE;L)ν = 0,
+∀n, ∀Θ ∈ Mskew.
+(4.2)
+It is easily checked that the map ¯Bact defined in (1.26) can be reformulated as
+E′ : 2 ¯Bact(E) = lim
+L↑∞ E′ : 2 ¯Bact;L(E),
+E′ : 2 ¯Bact;L(E) := −E
+�
+L−d �
+n∈QL
+ˆ
+In,L+B
+(ψE′;L + E′(x − xn,L)) · fn,L(E)
+�
+.
+(4.3)
+We start by decomposing
+E′ : 2 ¯Bact;L(E) = E′ : ¯B(1)
+act;L(E) + E′ : R1;L(E) + E′ : R2;L(E),
+(4.4)
+where we have set
+E′ : ¯B(1)
+act;L(E) := −L−dE
+� �
+n
+ˆ
+In,L+B
+(ψn
+E′;L + E′(x − xn,L)) · fn,L(E)
+�
+,
+and where the remainders R1;L(E), R2;L(E) are given by
+E′ : R1;L(E)
+:=
+−L−dE
+� �
+n̸=m
+ˆ
+In,L+B
+ψm
+E′;L · fn,L(E)
+�
+,
+E′ : R2;L(E)
+:=
+−L−dE
+� �
+n
+ˆ
+In,L+B
+�
+ψE′;L −
+�
+m
+ψm
+E′;L
+�
+· fn,L(E)
+�
+,
+
+HOMOGENIZATION FOR ACTIVE SUSPENSIONS
+53
+in terms of the solution ψn
+E′;L of the single-particle periodized problem
+
+
+
+
+
+
+
+
+
+
+
+
+
+−△ψn
+E;L + ∇Σn
+E;L = 0,
+in QL \ In,L,
+div(ψn
+E;L) = 0,
+in QL \ In,L,
+D(ψn
+E;L + Ex) = 0,
+in In,L,
+ffl
+∂In,L σ(ψn
+E;L + Ex, Σn
+E;L)ν = 0,
+∀n,
+ffl
+∂In,L Θ(x − εxn,L) · σ(ψn
+E;L + Ex, Σn
+E;L)ν = 0,
+∀n, ∀Θ ∈ Mskew.
+(4.5)
+Step 2. Proof that
+|R1;L| ≲ λ2
+�
+|log λ1| + log L
+L
+�
+.
+(4.6)
+As by assumption the point process is independent of particles’ shapes and swimming
+forces, we can write, in terms of the two-point intensity g2, cf. (1.29),
+E′ : R1;L(E) = −L−d
+¨
+QL−4×QL−4
+E
+� ˆ
+QL
+ψ◦
+E′;L(· + x − y) · ˜f ◦(E)
+�
+g2(x, y) dxdy,
+where ˜f ◦ is an iid copy of f ◦, hence independent of ψ◦
+E′;L. Noting that the periodicity
+of ψ◦
+E′;L yields
+ˆ
+QL
+� ˆ
+QL
+ψ◦
+E′;L(· + x − y) · ˜f ◦(E)
+�
+dx = 0,
+(4.7)
+we can replace the two-point density g2 by the correlation function h2 = g2 − λ2
+1, to the
+effect of
+E′ : R1;L(E) = −L−d
+¨
+QL×QL−4
+E
+� ˆ
+QL
+ψ◦
+E′;L(· + x − y) · ˜f ◦(E)
+�
+h2(x, y) dxdy
++ L−d
+¨
+(QL\QL−4)×QL−4
+E
+� ˆ
+QL
+ψ◦
+E′;L(· + x − y) · ˜f ◦(E)
+�
+g2(x, y) dxdy.
+The neutrality condition (1.9) entails
+���
+ˆ
+QL
+ψ◦
+E′;L(· + x − y) · ˜f ◦(E)
+��� ≲ ⟨E⟩
+� ˆ
+2B
+|∇ψ◦
+E′;L(· + x − y)|2� 1
+2,
+and thus, using standard decay estimates, see e.g. [5, Lemma 4.2],
+���
+ˆ
+QL
+ψ◦
+E′;L(· + x − y) · ˜f ◦(E)
+��� ≲ ⟨E⟩⟨x − y⟩−d.
+The above then becomes
+|R1;L| ≲ L−d
+¨
+QL×QL
+⟨x − y⟩−d |h2(x, y)| dxdy
++ L−d
+¨
+(QL\QL−4)×QL
+⟨x − y⟩−d g2(x, y) dxdy.
+The definition of two-point intensity (1.30) and the decay of correlations (1.31) yield
+|h2(x′, y′)| ≲ (λ2
+1 + g2(x, y)) ∧ ⟨x − y⟩−γ,
+g2(x, y) = g2(x, y)1|x−y|≥2ℓ,
+ 
+Bℓ(x)×Bℓ(y)
+g2(x′, y′) dx′dy′ ≲ λ2,
+
+54
+A. BERNOU, M. DUERINCKX, AND A. GLORIA
+and the claim (4.6) easily follows.
+Step 3. Proof that
+|R2;L| ≲ (λ1)
+1
+2(λ2 + λ3|log λ1|)
+1
+2 .
+(4.8)
+We start by decomposing
+E′ : R2;L(E) = −L−dE
+� �
+n
+ˆ
+(In,L+B)\In,L
+�
+ψE′;L −
+�
+m
+ψm
+E′;L
+�
+· fn,L(E)
+�
+(4.9)
++ L−dE
+� �
+n
+ˆ
+∂In,L
+�
+ψE′;L −
+�
+m
+ψm
+E′;L
+�
+· σ(φE;L, ΠE;L)ν
+�
+− L−dE
+� �
+n
+ˆ
+In,L+B
+�
+ψE′;L −
+�
+m
+ψm
+E′;L
+�
+·
+�
+fn,L(E)1In,L + δ∂In,Lσ(φE;L, ΠE;L)ν
+��
+.
+The first two terms can be recovered in the weak formulation of the equation for φE;L
+(periodized version of (2.4) as in (4.2)), when tested with ψE′;L − �
+m ψm
+E′;L,
+− L−dE
+� �
+n
+ˆ
+(In,L+B)\In,L
+�
+ψE′;L −
+�
+m
+ψm
+E′;L
+�
+· fn,L(E)
+�
++ L−dE
+� �
+n
+ˆ
+∂In,L
+�
+ψE′;L −
+�
+m
+ψm
+E′;L
+�
+· σ(φE;L, ΠE;L)ν
+�
+= −
+ˆ
+QL
+∇φE;L : ∇
+�
+ψE′;L −
+�
+m
+ψm
+E′;L
+�
+,
+and thus, similarly testing the equation for ψE′;L − �
+m ψm
+E′;L with φE;L, using boundary
+conditions and the rigidity condition for φE;L in IL,
+− L−dE
+� �
+n
+ˆ
+(In,L+B)\In,L
+�
+ψE′;L −
+�
+m
+ψm
+E′;L
+�
+· fn,L(E)
+�
++ L−dE
+� �
+n
+ˆ
+∂In,L
+�
+ψE′;L −
+�
+m
+ψm
+E′;L
+�
+· σ(φE;L, ΠE;L)ν
+�
+= 0.
+Next, using boundary conditions for φE;L, noting that ψE′;L − ψn
+E′;L is rigid in In,L, the
+last term in (4.9) can be rewritten as
+− L−dE
+� �
+n
+ˆ
+In,L+B
+�
+ψE′;L −
+�
+m
+ψm
+E′;L
+�
+·
+�
+fn,L(E)1In,L + δ∂In,Lσ(φE;L, ΠE;L)ν
+��
+= L−dE
+� �
+n̸=m
+ˆ
+In,L+B
+ψm
+E′;L ·
+�
+fn,L(E)1In,L + δ∂In,Lσ(φE;L, ΠE;L)ν
+��
+.
+Inserting these identities into (4.9), we get
+E′ : R2;L(E) = L−dE
+� �
+n̸=m
+ˆ
+In,L+B
+ψm
+E′;L ·
+�
+fn,L(E)1In,L + δ∂In,Lσ(φE;L, ΠE;L)ν
+��
+.
+Using boundary conditions for φE;L to replace ψm
+E′;L by ψm
+E′;L −
+ffl
+In,L ψm
+E′;L, using the
+Poincaré inequality, and a trace estimate, this can be estimated as
+
+HOMOGENIZATION FOR ACTIVE SUSPENSIONS
+55
+|E′ : R2;L(E)| ≲ E
+�
+L−d �
+n
+ˆ
+In,L+B
+���
+�
+m:m̸=n
+∇ψm
+E′;L
+���
+2� 1
+2
+× E
+�
+L−d �
+n
+ˆ
+In,L+B
+|fn,L(E)|2 + L−d �
+n
+ˆ
+In,L+B
+|∇φE;L|2
+� 1
+2
+.
+Taking advantage of explicit renormalizations as in [5, Section 4.4], the first factor can
+easily be estimated by
+E
+�
+L−d �
+n
+ˆ
+In,L+B
+���
+�
+m:m̸=n
+∇ψm
+E′;L
+���
+2�
+≲ (λ2 + λ3|log λ1|)|E′|2.
+Further noting that
+E
+�
+L−d �
+n
+ˆ
+In,L+B
+|fn,L(E)|2
+�
+≲ λ1
+ˆ
+2B
+|f◦(E)|2 ≲ λ1⟨E⟩2,
+(4.10)
+and using the hardcore condition, we deduce
+|R2;L(E)| ≲ (λ2 + λ3|log λ1|)
+1
+2
+�
+λ1⟨E⟩2 + E
+�  
+QL
+|∇φE;L|2
+�� 1
+2
+.
+(4.11)
+It remains to estimate the last integral: starting from the energy identity for φE;L,
+ˆ
+QL
+|∇φE;L|2 =
+�
+n
+ˆ
+In,L+B
+φE;L · fn,L(E),
+and using boundary conditions and the Poincaré inequality to estimate the right-hand side,
+we find
+ˆ
+QL
+|∇φE;L|2 ≲
+� �
+n
+ˆ
+In,L+B
+|fn,L(E)|2
+� 1
+2� �
+n
+ˆ
+In,L+B
+|∇φE;L|2
+� 1
+2
+.
+Hence, using the hardcore condition, absorbing the last factor, taking the expectation, and
+combining with (4.10), we obtain
+E
+�  
+QL
+|∇φE;L|2
+�
+≲ E
+�
+L−d �
+n
+ˆ
+In,L+B
+|fn,L(E)|2
+�
+≲ λ1⟨E⟩2.
+Inserting this into (4.11), the claim (4.8) follows.
+Step 4. Conclusion.
+In view of Steps 1, 2 and 3, it remains to examine the limit of the main term ¯B(1)
+act;L in (4.4).
+As by assumption the point process {xn} is independent of particles’ shapes and swimming
+forces, we can write
+E′ : ¯B(1)
+act;L(E) = −λ1L−d|QL−4|E
+� ˆ
+2B
+(ψ◦
+E′;L + E′x) · f ◦(E)
+�
+,
+and thus, as L ↑ ∞,
+lim
+L↑∞ E′ : ¯B(1)
+act;L(E) = −λ1E
+� ˆ
+2B
+(ψ◦
+E′ + E′x) · f ◦(E)
+�
+,
+
+56
+A. BERNOU, M. DUERINCKX, AND A. GLORIA
+in terms of the solution ψ◦
+E′ of the whole-space single-particle problem (1.33). In case of
+spherical particles, I◦ = B, the latter is explicitly solvable, cf. [21, Section 2.1.3],
+ψ◦
+E′(x) =
+�
+−E′x
+:
+|x| ≤ 1,
+− d+2
+2
+(x·E′x)x
+|x|d+2
+�
+1 −
+1
+|x|2
+�
+−
+E′x
+|x|d+2
+:
+|x| > 1,
+and the conclusion follows.
+□
+Acknowledgements
+Mitia Duerinckx acknowledges financial support from F.R.S.-FNRS, and Armand Bernou
+and Antoine Gloria from the European Research Council (ERC) under the European
+Union’s Horizon 2020 research and innovation programme (Grant Agreement n◦ 864066).
+References
+[1] L. H. Cisneros, R. Cortez, C. Dombrowski, R. E. Goldstein, and J. O. Kessler. Fluid dynamics of
+self-propelled microorganisms, from individuals to concentrated populations. Experiments in Fluids,
+43(5):737–753, 2007.
+[2] M. Duerinckx. Effective viscosity of random suspensions without uniform separation. Ann. Inst. H.
+Poincaré C Anal. Non Linéaire, 39(5):1009–1052, 2022.
+[3] M. Duerinckx. Semi-dilute mean-field description of particle suspensions. In preparation, 2023.
+[4] M. Duerinckx and A. Gloria. Continuum percolation in stochastic homogenization and the effective
+viscosity problem. Preprint, arXiv:2108.09654.
+[5] M. Duerinckx and A. Gloria. On Einstein’s effective viscosity formula. Preprint, arXiv:2008.03837.
+[6] M. Duerinckx and A. Gloria. Multiscale functional inequalities in probability: concentration proper-
+ties. ALEA Lat. Am. J. Probab. Math. Stat., 17(1):133–157, 2020.
+[7] M. Duerinckx and A. Gloria. Multiscale functional inequalities in probability: constructive approach.
+Ann. H. Lebesgue, 3:825–872, 2020.
+[8] M. Duerinckx and A. Gloria. Corrector equations in fluid mechanics: effective viscosity of colloidal
+suspensions. Arch. Ration. Mech. Anal., 239(2):1025–1060, 2021.
+[9] M. Duerinckx and A. Gloria. Effective viscosity of semi-dilute suspensions. Séminaire Laurent
+Schwartz, EDP et applications, 2021-2022. Exposé n◦III.
+[10] M. Duerinckx and A. Gloria. Quantitative homogenization theory for random suspensions in steady
+stokes flow. J. Ec. Polytech. - Math., 9:1183–1244, 2022.
+[11] M. Duerinckx and A. Gloria. Sedimentation of random suspensions and the effect of hyperuniformity.
+Annals of PDE, 8(2), 2022.
+[12] A. Einstein. Eine Neue Bestimmung der Moleküldimensionen. Ann. Phys., 19:289–306, 1906.
+[13] G. A. Francfort, A. Gloria, and O. Lopez-Pamies. Enhancement of elasto-dielectrics by homogenization
+of active charges. J. Math. Pures Appl. (9), 156:392–419, 2021.
+[14] J. Frenkel. Kinetic Theory of Liquids. Clarendon Press, Oxford, 1946.
+[15] D. Gérard-Varet. Derivation of the Batchelor-Green formula for random suspensions. J. Math. Pures
+Appl. (9), 152:211–250, 2021.
+[16] D. Gérard-Varet and M. Hillairet. Analysis of the viscosity of dilute suspensions beyond Einstein’s
+formula. Arch. Ration. Mech. Anal., 238(3):1349–1411, 2020.
+[17] D. Gérard-Varet and R. M. Höfer. Mild assumptions for the derivation of Einstein’s effective viscosity
+formula. Comm. Partial Differential Equations, 46(4):611–629, 2021.
+[18] D. Gérard-Varet and A. Mecherbet. On the correction to Einstein’s formula for the effective viscosity.
+Ann. Inst. H. Poincaré C Anal. Non Linéaire, 39(1):87–119, 2022.
+[19] D. Girodroux-Lavigne. Derivation of an effective rheology for dilute suspensions of micro-swimmers.
+Preprint, arXiv:2204.04967.
+[20] A. Gloria, S. Neukamm, and F. Otto. A regularity theory for random elliptic operators. Milan J.
+Math., 88(1):99–170, 2020.
+[21] E. Guazzelli and J. Morris. A Physical Introduction to Suspension Dynamics. Cambridge University
+Press, 2011.
+
+HOMOGENIZATION FOR ACTIVE SUSPENSIONS
+57
+[22] B. M. Haines, I. S. Aronson, L. Berlyand, and D. A. Karpeev. Effective viscosity of dilute bacterial
+suspensions: a two-dimensional model. Phys. Biol., 5(4):046003, 2008.
+[23] B. M. Haines, A. Sokolov, I. S. Aranson, L. Berlyand, and D. A. Karpeev. Three-dimensional model
+for the effective viscosity of bacterial suspensions. Phys. Rev. E, 80(4):041922, 2009.
+[24] Y. Hatwalne, S. Ramaswamy, M. Rao, and R. A. Simha. Rheology of active-particle suspensions. Phys.
+Rev. Lett., 92(11):118101, 2004.
+[25] R. M. Höfer. Convergence of the method of reflections for particle suspensions in Stokes flows. J.
+Differ. Equ., 297:81–109, 2021.
+[26] V. V. Jikov, S. M. Kozlov, and O. A. Ole˘ınik. Homogenization of differential operators and integral
+functionals. Springer-Verlag, Berlin, 1994.
+[27] H. M. López, J. Gachelin, C. Douarche, H. Auradou, and É. Clément. Turning bacteria suspensions
+into superfluids. Phys. Rev. Lett., 115:028301, 2015.
+[28] N. H. Mendelson, A. Bourque, K. Wilkening, K. R. Anderson, and J. C. Watkins. Organized cell
+swimming motions in bacillus subtilis colonies: Patterns of short-lived whirls and jets. Journal of
+Bacteriology, 181(2):600–609, 1999.
+[29] W. F. Paxton, K. C. Kistler, C. C. Olmeda, A. Sen, S. K. St. Angelo, Y. Cao, T. E. Mallouk, P. E.
+Lammert, and V. H. Crespi. Catalytic nanomotors: Autonomous movement of striped nanorods. J.
+Am. Chem. Soc., 126(41):13424–13431, 2004.
+[30] M. Potomkin, M. Tournus, L. V. Berlyand, and I. S. Aranson. Flagella bending affects macroscopic
+properties of bacterial suspensions. Journal of The Royal Society Interface, 14(130):20161031, 2017.
+[31] S. Rafaï, L. Jibuti, and P. Peyla. Effective viscosity of microswimmer suspensions. Phys. Rev. Lett.,
+104(9):098102, 2010.
+[32] S. D. Ryan, B. M. Haines, L. Berlyand, F. Ziebert, and I. S. Aranson. Viscosity of bacterial suspensions:
+Hydrodynamic interactions and self-induced noise. Phys. Rev. E, 83:050904, 2011.
+[33] D. Saintillan. Rheology of active fluids. Annu. Rev. Fluid Mech., 50(1):563–592, 2018.
+[34] D. Saintillan and M. J. Shelley. Theory of active suspensions. In Complex Fluids in Biological Systems,
+pages 319–355. Springer New York, 2014.
+[35] A. Sokolov and I. S. Aranson. Reduction of viscosity in suspension of swimming bacteria. Phys. Rev.
+Lett., 103(3):148101, 2009.
+[36] A. Sokolov, R. E. Goldstein, F. I. Feldchtein, and I. S. Aranson. Enhanced mixing and spatial insta-
+bility in concentrated bacterial suspensions. Phys. Rev. E, 80(3):031903, 2009.
+[37] O. Yasa, P. Erkoc, Y. Alapan, and M. Sitti. Microalga-powered microswimmers toward active cargo
+delivery. Adv. Mater., 30(45):1804130, 2018.
+(Armand Bernou) Sorbonne Université, CNRS, Université de Paris, Laboratoire Jacques-
+Louis Lions, 75005 Paris, France
+Email address: armand.bernou@sorbonne-universite.fr
+(Mitia Duerinckx) Université Libre de Bruxelles, Département de Mathématique, 1050 Brus-
+sels, Belgium
+Email address: mitia.duerinckx@ulb.be
+(Antoine Gloria) Sorbonne Université, CNRS, Université de Paris, Laboratoire Jacques-
+Louis Lions, 75005 Paris, France & Institut Universitaire de France & Université Libre de
+Bruxelles, Département de Mathématique, 1050 Brussels, Belgium
+Email address: antoine.gloria@sorbonne-universite.fr
+
diff --git a/YdAyT4oBgHgl3EQfWffL/content/tmp_files/load_file.txt b/YdAyT4oBgHgl3EQfWffL/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..2b7ab177d313be7e75313e19e4da50ea3152eb9b
--- /dev/null
+++ b/YdAyT4oBgHgl3EQfWffL/content/tmp_files/load_file.txt
@@ -0,0 +1,2076 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf,len=2075
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='00166v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='AP]  31 Dec 2022  HOMOGENIZATION OF ACTIVE SUSPENSIONS  AND REDUCTION OF EFFECTIVE VISCOSITY  ARMAND BERNOU, MITIA DUERINCKX, AND ANTOINE GLORIA  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' We consider a suspension of active rigid particles (swimmers) in a steady  Stokes flow, using for simplicity a steady-state model where particles are distributed ac-  cording to a stationary ergodic random process, and we study its homogenization in the  macroscopic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' A key point in the model is that swimmers are allowed to adapt their  propulsion to the surrounding fluid deformation: swimming forces are not prescribed  a priori, but are rather obtained through the retroaction of the fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Qualitative ho-  mogenization of this nonlinear model requires an unusual proof that crucially relies on  a semi-quantitative two-scale analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Thanks to the introduction of new correctors  that accurately capture spatial oscillations created by swimming forces, we identify the  contribution of the activity to the effective viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' In agreement with the physics liter-  ature, an analysis in the dilute regime shows that the activity of the particles can either  increase or decrease the effective viscosity (depending on the swimming mechanism), in  a way that strongly differs from the well-known effect of passive suspensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Contents  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Introduction and main results  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  General overview  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Reminder on passive suspensions  3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Hydrodynamic model for active suspensions  5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Heuristics and relation to the physics literature  8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Main results: well-posedness, homogenization, and dilute regime  10  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Viscosity reduction  14  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Corrector problems  15  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Passive corrector problems  15  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Active corrector problems  17  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Proof of the homogenization result  26  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Well-posedness of hydrodynamic model  26  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Qualitative homogenization at fixed δ  29  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Quantitative homogenization and limit δ ↓ 0  48  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Dilute expansion of the effective viscosity  51  Acknowledgements  56  References  56  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Introduction and main results  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' General overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' This work is devoted to the large-scale rheology of suspensions  of active particles in viscous fluids, where active particles are devices that can propel  1  2  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' BERNOU, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' DUERINCKX, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' GLORIA  themselves in the fluid (in a direction that can depend on the surrounding fluid flow itself).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Our main purpose is to establish rigorously, in a simple (yet somewhat realistic) physical  model, that the presence of active particles in a fluid can strongly reduce its viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Important examples include suspensions of bacteria [36], micro-algae [37], nanomo-  tors [29], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Compared to passive systems, active suspensions exhibit a particularly  rich phenomenology, with the experimental observation of pattern formations [1] and un-  steady whirls and jets [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Due to this complexity, the response to an external forcing can  defy intuition, with rheological measurements displaying in some settings a transition to a  superfluid-like behavior [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Our contribution is threefold: we formulate a suitable steady-state model for active  suspensions based on the physics literature, we prove homogenization for this model and  identify the effective rheology on large scales, and we finally analyze the latter in the dilute  regime — then recovering standard explicit results of the physics literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  As reviewed in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='2 below, Einstein’s celebrated viscosity formula [12] shows that  the presence of passive rigid particles can only increase the plain fluid viscosity on large  scales as those particles hinder the fluid flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' The case of active suspensions is radically  different: the activity of micro-swimmers may overcome viscous dissipation and reduce  the effective viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' As first pointed out in [24], this is related to the emergence of a  state of collective orientation of the particles under a given external force: the preferred  orientation happens to determine the increase or decrease in viscosity and was predicted to  depend on the type of swimming mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' In 2009, Sokolov and Aranson [35] observed  experimentally a striking reduction of the effective viscosity of a suspension of Bacillus  subtilis in a dilute regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  In contrast, for motile microalgae such as Chlamydomonas  reinhardtii, a drastic increase in the viscosity was observed experimentally in [31] — drastic  even compared to suspensions with the same amount of dead cells!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' This confirmed that  the swimming mechanism plays a key role in the modification of the viscosity on large  scales and subsequent studies have classified these into two categories:  — “pusher” particles, such as Bacillus subtilis, move into the fluid via an extensile motion,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' using a flagella, and tend to decrease the effective viscosity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  — “puller” particles, such as Chlamydomonas reinhardtii, are contractile swimmers, and  their activity can increase the effective viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Those experimental results have attracted much attention in the physics community over  the past decade and several models have been introduced to explain the effect of the  swimming mechanism on the effective viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' We refer in particular to the heuristic  analysis in [22, 23, 32], where the preferred orientation of the particles is formally computed  depending on the external shear flow, and is shown to affect the effective viscosity in the  dilute regime, in agreement with experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' In these works, the swimming mechanism  is usually modeled for simplicity by point dipole forces at particle locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  We refer  to [30, 33, 34] and references therein for the physical background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  From a modeling point of view, a key question in the study of suspensions is how to  describe the positions and orientations of the suspended particles and how they are coupled  to the fluid flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' There are two natural ways to proceed:  (I) Dynamical model: Particles follow the fluid flow and propel themselves, so their  positions and orientations evolve in time accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' The geometry of the array  of particles can then adapt dynamically to external forces, leading to a nonlinear  response (non-Newtonian effects).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  HOMOGENIZATION FOR ACTIVE SUSPENSIONS  3  (II) Nonlinear steady-state model: The statistics of the positions are assumed to be  given, as well as the statistics of the orientations, but the latter are allowed to depend  on the surrounding fluid flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' The problem is then reduced to understanding the  retroaction of the fluid on the self-propulsion of the particles as their orientations  adapt to the surrounding fluid deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  From the rigorous mathematical perspective, the literature is particularly scarce, with the  exception of the recent work [19] by Girodroux-Lavigne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' In this work, the author considers  a dilute steady-state model, consisting of a steady Stokes fluid with a dilute suspension  of active particles for which swimming forces are fully prescribed in advance (hence, are  part of the data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' This reduces the analysis to a combination of homogenization (due to  the presence of rigid particles, via a dilute analysis) with averaging of the active part (in  form of a source term).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' This can be viewed as a very first step towards both situations (I)  and (II): it remains to be combined with the retroaction of the fluid, which can be modeled  either by the particle dynamics (I), or by the dependence of particle orientations on the  surrounding fluid in a nonlinear steady-state model (II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Rigorous results in a realistic  dynamical setting (I) seem out of reach at the moment beyond dilute regimes [3], and  the present contribution is rather devoted to the steady-state setting (II) in a general  non-dilute regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  The sequel of this introduction is organized as follows: In Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='2, we recall the  steady-state model for a steady Stokes fluid with a suspension of passive rigid particles,  and we state the associated homogenization result previously obtained by the last two  authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' In Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='3, we introduce a corresponding model for active suspensions in a  steady Stokes flow, as inspired by the review articles [33, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' In Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='4, we relate this  model to the physics literature and show how it can be used to recover standard results  of the physics literature on a heuristic level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='5 is dedicated to the main results  of this paper: the rigorous homogenization of this nonlinear model, and the analysis of  the effective rheology in the dilute regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Last, in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='6, based on these results,  we investigate the contribution of active particles to the effective viscosity, and rigorously  establish that a significant reduction can take place in the case of pushers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Reminder on passive suspensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Given an underlying probability space (Ω, P)  (with expectation E [·]), let {xn}n be a random point process on the ambient space Rd,  consider an associated collection of random shapes {I◦  n}n, where each I◦  n is a connected  random open subset of the unit ball B centered at the origin (in the sense that  ´  I◦n y dy = 0),  and then define the corresponding inclusions  In := xn + I◦  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Note that random shapes are not required to be independent of the point process {xn}n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  We then consider the random set  I :=  �  n  In,  which we assume to satisfy the following for some ϑ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Hypothesis 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='1 (Particle suspension).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (a) Stationarity and ergodicity: The random set I is stationary and ergodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (b) Uniform C2 regularity: The random shapes {I◦  n}n satisfy interior and exterior ball  conditions with radius ϑ almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  4  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' BERNOU, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' DUERINCKX, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' GLORIA  (c) Uniform hardcore condition: For some ℓ ≥ ϑ, there holds (In + ℓB) ∩ (Im + ℓB) = ∅  almost surely for all n ̸= m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  We let ℓ be the largest such value, that is, half the  interparticle distance  ℓ := 1  2 inf  n̸=m dist(In, Im).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='1)  ♦  Now consider a tank, described as a bounded Lipschitz domain U ⊂ Rd, and assume  that it is filled with a steady Stokes fluid with a suspension of particles of size ε, described  as the ε-rescaling of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' More precisely, we only consider particles that are included in U  and remove those close to the boundary: define Nε(U) as the set of indices n such that  ε(In + ℓB) ⊂ U, and set  Iε(U) :=  �  n∈Nε(U)  εIn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  We write uε for the fluid velocity, Pε for the corresponding pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' We assume Dirichlet  conditions uε = 0 on ∂U, and we extend the fluid velocity inside particles with the rigidity  constraint  D(uε) := 1  2(∇uε + (∇uε)T ) = 0,  in Iε(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Recall the definition of the Cauchy stress tensor  σ(uε, Pε) := 2 D(uε) − Pε Id,  where Id denotes the identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Given an internal force h, the fluid velocity uε ∈  H1  0(U)d and the associated pressure Pε ∈ L2(U \\ Iε(U)) are then given as the solutions of  the Stokes system  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  −△uε + ∇Pε = h  in U \\ Iε(U),  div(uε) = 0,  in U \\ Iε(U),  D(uε) = 0,  in Iε(U),  ´  ε∂In σ(uε, Pε)ν = 0,  ∀n ∈ Nε(U),  ´  ε∂In Θ(x − εxn) · σ(uε, Pε)ν = 0,  ∀n ∈ Nε(U), Θ ∈ Mskew,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='2)  where ν stands for the outward normal to ∂In, where Mskew is the set of skew-symmetric  matrices, and where we assume the additional anchoring condition  ˆ  U\\Iε(U)  Pε = 0,  which we shall abbreviate as choosing Pε ∈ L2(U \\ Iε(U))/R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  The homogenization of  the Stokes system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='2) was the object of [2, 4, 8], where the last two authors proved  that (uε, Pε1U\\Iε(U)) converges almost surely weakly in H1  0(U) × L2(U)/R to the unique  solution (¯u, ¯P) of the homogenized Stokes system  �  −div(2 ¯Bpas D(¯u)) + ∇ ¯P = (1 − λ)f  in U,  div(¯u) = 0,  in U,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='3)  where λ stands for the particle volume fraction  λ := E [1I] ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='4)  and where the effective viscosity ¯Bpas is a symmetric linear map on the set Msym  0  of  symmetric trace-free matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' We recall that the latter satisfies E : ¯BpasE > |E|2 for all  E ∈ Msym  0  as soon as λ > 0, meaning that the presence of (passive) rigid particles always  increases the effective viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' We refer to [9] for a review of the topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  HOMOGENIZATION FOR ACTIVE SUSPENSIONS  5  In view of the quantitative homogenization results that we shall need later, we occa-  sionally make quantitative ergodicity assumptions in form of the validity of the following  multiscale variance inequality introduced by the last two authors in [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' This assumption  holds for instance for hardcore Poisson point process with exponentially decaying π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Hypothesis 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='2 (Quantitative mixing assumption).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' There exists a non-increasing weight  function π : R+ → R+ with superalgebraic decay (that is,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' π(t) ≤ Cp⟨t⟩−p for all p < ∞)  such that the random set I satisfies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' for all σ(I)-measurable random variables Y (I),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Var [Y (I)] ≤ E  � ˆ ∞  0  ˆ  Rd  �  ∂osc  I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='Bt(x)Y (I)  �2  dx ⟨t⟩−dπ(t) dt  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  where the “oscillation” ∂osc of the random variable Y (I) is defined by  ∂osc  I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='Bt(x)Y (I) := sup ess  �  Y (I′) : I′ ∩ (Rd \\ Bt(x)) = I ∩ (Rd \\ Bt(x))  �  − inf ess  �  Y (I′) : I′ ∩ (Rd \\ Bt(x)) = I ∩ (Rd \\ Bt(x))  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  ♦  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Hydrodynamic model for active suspensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' As opposed to passive particles,  active particles propel themselves by applying a force on the surrounding fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' In a steady-  state perspective, we assume that we are given a random ensemble of particle positions and  swimming directions, and we aim to evaluate the associated large-scale rheology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Swim-  ming directions should not be taken as uniformly distributed, but should depend on the  surrounding fluid deformation, which leads to a nontrivial interaction between the fluid  flow and particles’ swimming forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' More precisely, our model is based on the following  assumption: if the fluid is locally deformed, then the distribution of orientations depends  on some local average of the symmetrized velocity gradient of the surrounding fluid around  each particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Although this steady-state perspective is certainly simplistic, our model does  not prescribe the retroaction of the fluid on the particles a priori, but leaves it as part of  the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' We start by modeling the swimming mechanism for a single particle, before  combining it with (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='2) into a model for the whole active suspension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Single-particle swimming mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Let us place ourselves at the scale of an iso-  lated particle I, and denote by u the fluid velocity outside I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Given a nonnegative smooth  kernel χ with unit mass  ´  Rd χ = 1, the locally averaged fluid deformation felt by the particle  is taken of the form  EI(u) :=  I  χ ∗ D(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='5)  The precise choice of this operator does not matter in our analysis, provided that it is a  compact operator applied to a restriction of D(u) around I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Given a value EI(u) = E of  this averaged fluid deformation, the particle adapts its random swimming direction: we  denote by ¯f(E) ∈ Rd the resulting propulsion force and by ˜f(E) ∈ Mskew the resulting  torque on the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' By the action-reaction principle, this force and torque must result  from an action of the particle on the surrounding fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' The detail of this action depends  on the details of the swimming mechanism (flagella, cilia, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' The force field exerted by  the particle on the fluid is denoted by f(E) = f(·, E), depending on the deformation E,  and is taken to be supported in the immediate neighborhood (I + B) \\ I of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  6  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' BERNOU, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' DUERINCKX, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' GLORIA  By the action-reaction principle for forces and torques, we must have  ¯f(E) +  ˆ  (I+B)\\I  f(E)  =  0,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='6)  Θ : ˜f(E) +  ˆ  (I+B)\\I  Θx · f(E)  =  0,  ∀Θ ∈ Mskew,  since the barycenter of particle I is  ´  I y dy = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' The relation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='6) reads as a local neutrality  condition that actually entails that swimming forces act as dipoles in the fluid equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  We emphasize that this is fundamentally different from the sedimentation problem studied  in [11], for which the force on particles originates from gravity and is not compensated  locally by opposite forces in the surrounding fluid — the backflow is then uniform and  leads to more important large-scale effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  We further make the following assumptions on the regularity of the swimming force with  respect to the fluid deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Hypothesis 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='3 (Swimming mechanism).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' The random force field defines almost surely a  smooth map Msym  0  → L∞((I + B) \\ I)d : E �→ f(E) such that, for all k ≥ 1,  ∥f(E)∥L∞((I+B)\\I) ≤ C⟨E⟩,  ∥∂kf(E)∥L∞((I+B)\\I) ≤ Ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  ♦  In view of quantitative homogenization results, we shall occasionally need to further as-  sume that for large strain rates E the swimming direction becomes a deterministic function  of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' This technical assumption is physically reasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Hypothesis 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='4 (Swimming in large strain rate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' There exist a deterministic direction field  f ∞ : Msym  0  → Rd, a random strength field S ∈ L∞((I + B) \\ I), and an exponent γ > 0,  such that for all E ∈ Msym  0  and k ≥ 1 we have almost surely  ∥f(E) − Sf ∞(E)∥L∞((I+B)\\I) ≤ C⟨E⟩1−γ,  ∥∂kf(E) − S∂kf ∞(E)∥L∞((I+B)\\I) ≤ Ck⟨E⟩−γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  ♦  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Resulting system for many particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Before including the above single-particle swim-  ming mechanism into the passive suspension model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='2), we start by making an assumption  on the joint law of particles’ swimming forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Hypothesis 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='5 (Joint swimming forces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Let {fn(E)}n be a sequence of random maps,  such that fn satisfies Hypothesis 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='3 with I = In for all n, and such that I and �  n fn(E)  are jointly stationary for all E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  ♦  In order to include these swimming forces into the model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='2) for a suspension of  small particles {εIn}n, they need to be properly rescaled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' The natural scaling happens to  be O(1  ε), which is indeed the only scaling giving rise to a nontrivial and finite contribution  in the macroscopic limit ε ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' We add a coupling parameter κ, which stands for the  activity strength and will need to be chosen small enough to perform the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' In  this ε-rescaling, the kernel χ defining the local averaged fluid deformations felt by the  particles (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='5) should naturally be replaced by χε := ε−dχ( ·  ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  This however leads to  important difficulties due to the highly oscillatory local behavior of the fluid flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Instead,  we need to replace it by χδ := δ−dχ( ·  δ), for some intermediate averaging scale ε ≪ δ ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  HOMOGENIZATION FOR ACTIVE SUSPENSIONS  7  The resulting hydrodynamic model takes on the following guise,  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  −△uε + ∇Pε  = h + κ  ε  �  n∈Nε(U) fn,ε(  ffl  εIn χδ ∗ D(uε)),  in U \\ Iε(U),  div(uε) = 0,  in U \\ Iε(U),  uε = 0,  on ∂U,  D(uε) = 0,  in Iε(U),  ´  ε∂In σ(uε, Pε)ν + κ  ε ¯fn(  ffl  εIn χδ ∗ D(uε)) = 0,  ∀n ∈ Nε(U),  ´  ε∂In Θ(· − εxn) · σ(uε, Pε)ν  + κ  ε Θ : ˜fn(  ffl  εIn χδ ∗ D(uε)) = 0,  ∀n ∈ Nε(U), Θ ∈ Mskew,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='7)  where we have set  fn,ε(E) := fn,ε(·, E) := ε−dfn  � ·  ε, E  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  As above, the pressure is anchored via  ´  U\\Iε(U) Pε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  In what follows, it will be convenient to use an equivalent formulation of swimming  forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' While each force field fn is supported in the particle neighborhood (In + B) \\ In in  the fluid domain, we may naturally extend fn inside the particle domain In to match its  propulsion force and torque.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' More precisely, we can uniquely define fn in In as an affine  function such that  ¯fn(E) =  ˆ  In  fn(E),  ˜fn(E) =  ˆ  In  fn(E) ⊗ (x − xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='8)  In terms of these extensions, the neutrality condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='6) takes the simpler form  ˆ  In+B  fn(E)  =  0,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='9)  ˆ  In+B  Θ(x − xn) · fn(E)  =  0,  ∀Θ ∈ Mskew,  and the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='7) then becomes  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  −△uε + ∇Pε  = h + κ  ε  �  n∈Nε(U) fn,ε(  ffl  εIn χδ ∗ D(uε)),  in U \\ Iε(U),  div(uε) = 0,  in U \\ Iε(U),  uε = 0,  on ∂U,  D(uε) = 0,  in Iε(U),  ´  ε∂In σ(uε, Pε)ν + κ  ε  ´  εIn fn,ε(  ffl  εIn χδ ∗ D(uε)) = 0,  ∀n ∈ Nε(U),  ´  ε∂In Θ(x − εxn) · σ(uε, Pε)ν  + κ  ε  ´  εIn Θ(x − εxn) · fn,ε(  ffl  εIn χδ ∗ D(uε)) = 0,  ∀n ∈ Nε(U), Θ ∈ Mskew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='10)  With this reformulation, we may readily check that the solution uε can be viewed as the  orthogonal projection on {u ∈ H1  0(U)d : D(u)|Iε(U) = 0, div(u) = 0} of the solution of  \uf8f1  \uf8f2  \uf8f3  −△vε + ∇Rε = h + κ  ε  �  n∈Nε(U) fn,ε(  ffl  εIn χδ ∗ D(uε)),  in U,  div(vε) = 0,  in U,  vε = 0,  on ∂U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='11)  This observation is not used in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  8  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' BERNOU, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' DUERINCKX, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' GLORIA  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Heuristics and relation to the physics literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' In the physics literature, one  usually considers a forcing term in form of an imposed strain rate E ∈ Msym  0  at infinity —  in which case it is natural to replace (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='5) by E itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' The velocity field of the suspension  on microscopic scale is then given by uE + Ex, where uE is a suitable solution of the  following infinite-volume problem,  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  −△uE + ∇PE = κ �  n fn(E),  in Rd \\ I,  div(uE) = 0,  in Rd \\ I,  D(uE + Ex) = 0,  in I,  ´  ∂In σ(uE + Ex, PE)ν + κ  ´  In fn(E) = 0,  ∀n,  ´  ∂In Θ(x − xn) · σ(uE + Ex, PE)ν  +κ  ´  In Θ(x − xn) · fn(E) = 0,  ∀n, Θ ∈ Mskew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='12)  In these terms, the effective viscosity ¯Btot(E) of the suspension in direction E is obtained  as the associated ensemble-averaged stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Splitting the contributions of the stress in the  fluid domain and in the particles, and taking into account swimming forces, we get for  all E′ ∈ Msym  0  ,  E′ : 2 ¯Btot(E) = E′ : E  �  σ(uE + Ex, PE)1Rd\\I  �  + E′ : E  �  σE1I  �  − κE  � �  n  E′(x − xn) · fn(E)  �  ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='13)  where σE stands for the stress inside the particles, which we shall define in the proof of  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='5 below via the extension problem  �  −div(σE) = κfn(E),  in In,  σEν = σ(uE + Ex, PE)ν,  on ∂In.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='14)  Noting that rigidity constraints D(uE + Ex) = 0 in I yield  σ(uE + Ex, PE)1Rd\\I = 2 D(uE + Ex) − PE Id 1Rd\\I,  and recalling that E [D(uE)] = 0 (as the average of a gradient), the first contribution  in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='13) takes the form  E′ : E  �  σ(uE + Ex, PE)1Rd\\I  �  = 2E′ : E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Next, using (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='14), integrating by parts, and using (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='8), and the skew-symmetry of ˜fn(E),  one can reformulate the second contribution in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='13) as the ensemble-averaged stresslet  on the particles  E  �  σE1I  �  =  E  � �  n  1In  |In|  ˆ  In  σE  �  =  E  � �  n  1In  |In|  ˆ  ∂In  σ(uE + Ex, PE)ν ⊗s (x − xn)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  The effective viscosity (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='13) thus takes the form  E′ : 2 ¯Btot(E) = 2E′ : E  + E  � �  n  1In  |In|  ˆ  ∂In  E′(x − xn) · σ(uE + Ex, PE)ν  �  − κE  � �  n  E′(x − xn) · fn(E)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='15)  HOMOGENIZATION FOR ACTIVE SUSPENSIONS  9  For convenience, we shall distinguish the passive from the active contributions in this  expression, and we decompose the solution uE as uE = ψE + κφE in terms of the so-called  passive and active correctors ψE and φE, defined as suitable solutions of  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  −△ψE + ∇ΣE = 0,  in Rd \\ I,  div(ψE) = 0,  in Rd \\ I,  D(ψE) + E = 0,  in I,  ´  ∂In σ(Ex + ψE, ΣE)ν = 0,  ∀n,  ´  ∂In Θ(x − xn) · σ(Ex + ψE, ΣE)ν = 0,  ∀Θ ∈ Mskew, ∀n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  and  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  −△φE + ∇ΠE = �  n fn(E),  in Rd \\ I,  div(φE) = 0,  in Rd \\ I,  D(φE) = 0,  in I,  ´  ∂In σ(φE, ΠE)ν +  ´  In fn(E) = 0,  ∀n,  ´  ∂In Θ(x − xn) · σ(φE, ΠE)ν +  ´  In Θ(x − xn) · fn(E) = 0,  ∀Θ ∈ Mskew, ∀n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Precise definitions of these correctors are postponed to Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  In these terms, the  effective viscosity takes the form  ¯Btot(E) = ¯BpasE + κ ¯Bact(E),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='16)  where the passive and active contributions are given by  E′ : 2 ¯BpasE  :=  2E′ : E + E  � �  n  1In  |In|  ˆ  ∂In  E′(x − xn) · σ(ψE + Ex, ΣE)ν  �  ,  E′ : 2 ¯Bact(E)  :=  E  � �  n  1In  |In|  ˆ  ∂In  E′(x − xn) · σ(φE, ΠE)ν  �  −E  � �  n  E′(x − xn) · fn(E)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Using equations for correctors, these expressions are equivalently given by  E : 2 ¯BpasE  =  E  �  2|D(ψE) + E|2�  ,  E′ : 2 ¯Bact(E)  =  −E  � �  n  (ψE′ + E′(x − xn)) · fn(E)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='17)  As one could have expected, the active contribution E′ : ¯Bact(E) coincides with the aver-  aged swimming force along the passive corrector in direction E′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Note in particular that the  active corrector φE does not appear in that formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' The first main contribution of the  present article is to properly justify these effective viscosity formulas using homogenization  theory, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  While these general formulas are difficult to analyze in practice without resorting to  numerical simulations, it is a classical problem in the physics community to derive simpler  approximate formulas in the dilute regime, which are easier to interpret and provide a  useful grasp at the physical behavior of suspensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' This is made possible by replacing  correctors by explicit solutions of single-particle problems: we refer to Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='9 and  Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='6 below for justification of such dilute approximations based on the methods  introduced by the last two authors in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  10  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' BERNOU, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' DUERINCKX, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' GLORIA  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Main results: well-posedness, homogenization, and dilute regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' We turn  to the statement of our main results and we start with the well-posedness of the hy-  drodynamic model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' It requires either the coupling constant κ to be small enough  or the interparticle distance ℓ to be large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Note that condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='18) below is  nearly almost optimal in general: the same condition with η = 0 is required to ensure  the perturbative well-posedness of the homogenized equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='20), see first paragraph of  Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='6 (Well-posedness of the hydrodynamic model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Let Hypotheses 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='3,  and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='5 hold, and assume εℓ ≤ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Provided for some η > 0 we have that  κℓη−d ≪ 1  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='18)  is small enough (only depending on η, the dimension d, the domain U, and the unscaled  kernel χ), the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='7) is well-posed almost surely for any h ∈ L2(U)d: there exists  a unique almost sure weak solution (uε, Pε) ∈ L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' H1  0(U)d × L2(U \\ Iε(U))/R) and it  satisfies almost surely  ˆ  U  |∇uε|2 +  ˆ  U\\Iε(U)  (Pε)2 ≲η (1 + κ2)  �  κ2ℓ−d +  ˆ  U\\Iε(U)  |h|2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='19)  ♦  We now state the homogenization result for this model in the macroscopic limit ε ↓ 0,  in the simplified situation when the mesoscopic averaging scale δ is fixed (see the proof for  the associated corrector result).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='7 (Homogenization at fixed δ > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Let Hypotheses 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='3, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='5 hold, as  well as the smallness condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='18) to ensure well-posedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' For any h ∈ W 1,∞(U)d,  as ε ↓ 0 with δ > 0 fixed, the almost sure weak solution (uε, Pε) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='7) satisfies almost  surely  uε  ⇀  ¯uδ,  in H1  0(U),  Pε1U\\Iε(U)  ⇀  (1 − λ) ¯Pδ + (1 − λ)¯b : D(¯uδ)  +(1 − λ)κ  �  ¯c(χδ ∗ D(¯uδ)) −  ffl  U ¯c(χδ ∗ D(¯uδ))  �  ,  in L2(U),  where (¯uδ, ¯Pδ) ∈ H1  0(U)d × L2(U)/R is the unique solution of the well-posed macroscopic  system  \uf8f1  \uf8f2  \uf8f3  −div(2 ¯Bpas D(¯uδ)) − div(2κ ¯Bact(χδ ∗ D(¯uδ))) + ∇ ¯Pδ = (1 − λ)h,  in U,  div(¯uδ) = 0,  in U,  ¯uδ = 0,  on ∂U,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='20)  where λ := E [1I] is the particle volume fraction, and where the effective tensors ¯Bpas,  ¯Bact, ¯b, ¯c are defined as follows, in terms of the correctors (ψ, Σ) and (φ, Π) given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='1)  and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='4) below,  The passive effective viscosity ¯Bpas is a positive definite symmetric linear map on the  space of symmetric trace-free matrices Msym  0  : together with the associated symmetric  trace-free matrix ¯b ∈ Msym  0  , it is defined for all E ∈ Msym  0  by  2( ¯Bpas − Id)E + (¯b : E) Id := E  � �  n  1In  |In|  ˆ  ∂In  σ(ψE + Ex, ΣE)ν ⊗s (x − xn)  �  ,  HOMOGENIZATION FOR ACTIVE SUSPENSIONS  11  or equivalently,  E : 2 ¯BpasE  :=  E  �  2|D(ψE) + E|2�  ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='21)  ¯b : E  :=  1  d E  � �  n  1In  |In|  ˆ  ∂In  (x − xn) · σ(ψE + Ex, ΣE)ν  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='22)  The active effective viscosity ¯Bact is given by  ¯Bact := ¯C + 1  2 ¯F ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='23)  where the map ¯C : Msym  0  → Msym  0  , together with the associated map ¯c : Msym  0  → R, is  defined for all E ∈ Msym  0  by  2 ¯C(E) + ¯c(E) Id := E  � �  n  1In  |In|  ˆ  ∂In  σ(φE, ΠE)ν ⊗s (x − xn)  �  ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='24)  and where the map ¯F : Msym  0  → Msym  0  is defined for all E, E′ ∈ Msym  0  by  E′ : ¯F (E) := −E  � �  n  1In  |In|  ˆ  In+B  E′(x − xn) · fn(E)  �  ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='25)  or equivalently for all E, E′ ∈ Msym  0  ,  E′ : 2 ¯Bact(E)  =  −E  � �  n  1In  |In|  ˆ  In+B  (ψE′ + E′(x − xn)) · fn(E)  �  ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='26)  ¯c(E)  =  1  dE  � �  n  1In  |In|  � ˆ  ∂In  (x − xn) · σ(φE, ΠE)ν  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  ♦  Next, we combine the above (nonlocal) homogenization limit with the (local) limit δ ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  In order to get beyond a purely diagonal regime, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='55) below, we need to appeal to the  quantitative homogenization techniques developed in [10] by the last two authors in the  context of the Stokes equation with rigid inclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' This requires a quantitative mixing  assumption such as Hypothesis 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='2 (which holds in particular for hardcore Poisson point  processes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Although from the modeling viewpoint the choice δ ∼ ε could be more natural,  it is not accessible to our analysis, and we are restricted to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='27) below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='8 (Homogenization as δ ↓ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Let Hypotheses 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='3, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='5 hold, as well  as the smallness condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='18) to ensure well-posedness, and further let the quantitative  mixing Hypothesis 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='2 and the technical Hypothesis 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='4 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Then, for any h ∈ W 1,∞(U)d,  as ε, δ ↓ 0, in the regime  δ−sε → 0,  for some s > d + 1,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='27)  the almost sure weak solution (uε, Pε) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='7) satisfies almost surely  uε  ⇀  ¯u,  in H1  0(U)d,  Pε1U\\Iε(U)  ⇀  (1 − λ) ¯P + (1 − λ)¯b : D(¯u)  +(1 − λ)κ  �  ¯c(D(¯u)) −  ffl  U ¯c(D(¯u))  �  ,  in L2(U),  where (¯u, ¯P) ∈ H1  0(U)d × L2(U)/R is the unique solution of  \uf8f1  \uf8f2  \uf8f3  −div(2 ¯Btot(D(¯u))) + ∇ ¯P = (1 − λ)h,  in U,  div(¯u) = 0,  in U,  ¯u = 0,  on ∂U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='28)  12  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' BERNOU, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' DUERINCKX, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' GLORIA  in terms of the total effective viscosity  ¯Btot(E) := ¯BpasE + κ ¯Bact(E),  where we recall that λ, ¯Bpas, ¯Bact, ¯b, ¯c are defined in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='7 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  ♦  The above shows that the effective stress-strain constitutive relation E �→ 2 ¯BpasE is  replaced by the nonlinear (non-Newtonian) relation E �→ 2 ¯BpasE + 2 ¯Bact(E) due to the  effect of particle activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Our last result concerns the analysis of the latter in the dilute regime, and we establish  the active counterpart to Einstein’s effective viscosity formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Before stating the result,  we need to recall some notation from [5]: Denote by Qr(x) = x + [− r  2, r  2)d the cube of  sidelength r centered at x, and set Q(x) = Q1(x), Qr = Qr(0), and Q = Q1(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' The  intensity of the point process {xn}n is  λ1 := E  �  ♯{n : xn ∈ Q}  �  ,  and we further define the two- and three-point intensities as  λ2  :=  ℓ−2d sup  x∈Rd E  �  ♯  �  (n, m) : n ̸= m, xn ∈ Qℓ, xm ∈ Qℓ(x)  ��  ,  λ3  :=  ℓ−3d  sup  x1,x2∈Rd E  �  ♯  �  (n0, n1, n2) : n0, n1, n2 pairwise distinct,  xn0 ∈ Qℓ, xn1 ∈ Qℓ(x1), xn2 ∈ Qℓ(x2)  ��  ,  where we recall that ℓ stands for (half) the interparticle distance, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Note that by  definition the intensity can be compared to the particle volume fraction λ, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='4), and  we have for k = 2, 3,  λ1 ≃ λ ≲ ℓ−d  and  λk ≲ ℓ−kd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  We further recall that the two- and three-point densities g2, g3 of the point process are  defined by the following relations, for all ζ2 ∈ C∞  c ((Rd)2) and ζ3 ∈ C∞  c ((Rd)3),  ¨  (Rd)2 ζ2(x, y) g2(x, y) dxdy  =  E  � �  n̸=m  ζ2(xn, xm)  �  ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='29)  ¨  (Rd)3 ζ3(x, y, z) g3(x, y, z) dxdydz  =  E  �  �  n0,n1,n2  distinct  ζ3(xn0, xn1, xn2)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  In these terms, the above definitions of the two- and three-point intensities are equivalently  written as  λ2  =  sup  x∈Rd  Qℓ  Qℓ(x)  g2(y, z) dydz,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='30)  λ3  =  sup  x1,x2∈Rd  Qℓ  Qℓ(x1)  Qℓ(x2)  g3(x, y, z) dxdydz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  With this notation at hand, we may now state the following result on the first-order dilute  expansion of the effective viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' The expansion of the passive contribution ¯Bpas was  already established in [5], and we extend it here to the active setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='9 (Dilute expansion of the effective viscosity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Let Hypotheses 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='3, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='5  hold, and further assume  HOMOGENIZATION FOR ACTIVE SUSPENSIONS  13  (a) Independence condition: The random shapes and swimming forces {I◦  n, fn}n are iid  copies of a given random open subset I◦ and of a random map f ◦, independently of the  point process {xn}n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (b) Decay of correlations: The point process {xn}n is strongly mixing, and the two- and  three-point correlation functions  h2(x, y)  :=  g2(x, y) − λ2  1,  h3(x, y, z)  :=  g3(x, y, z) − λ3  1 − λ1(h2(x, y) + h2(x, z) + h2(y, z)),  have algebraic decay: there exists γ > 0 such that for all x, y, z,  |h2(x, y)| ≲ ⟨x − y⟩−γ,  |h3(x, y, z)| ≲ ⟨x − y⟩−γ ∧ ⟨x − z⟩−γ ∧ ⟨y − z⟩−γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='31)  Then, we have  ��� ¯Btot(E) −  �  E + λ1 ¯B(1)  pasE + κλ1 ¯B(1)  act(E)  ����  ≲ ⟨E⟩  �  λ2|log λ1| + (λ1λ2 + λ1λ3|log λ1|)  1  2  �  ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='32)  where we have set  E′ : 2 ¯B(1)  pasE  :=  E  � ˆ  ∂I◦ E′x · σ(ψ◦  E + Ex, Σ◦  E)ν  �  ,  E′ : 2 ¯B(1)  act(E)  :=  −E  � ˆ  2B  (ψ◦  E′ + E′x) · f ◦(E)  �  ,  in terms of the solution (ψ◦  E, Σ◦  E) of the single-particle problem  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  −△ψ◦  E + ∇Σ◦  E = 0,  in Rd \\ I◦,  div(ψ◦  E) = 0,  in Rd \\ I◦,  D(ψ◦  E + Ex) = 0,  in I◦,  ´  ∂I◦ σ(ψ◦  E + Ex, Σ◦  E)ν = 0,  ´  ∂I◦ Θx · σ(ψ◦  E + Ex, Σ◦  E)ν = 0,  ∀Θ ∈ Mskew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='33)  In particular, in case of spherical particles I◦ = B, these expressions take the explicit forms  E′ : 2 ¯B(1)  pasE := (d + 2)|B|E′ : E,  E′ : 2 ¯B(1)  act(E) := −  ˆ  Rd\\B  �  1 −  1  |x|d+2  �  E′x · E [f ◦(E)]  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='34)  + d+2  2  ˆ  Rd\\B  �  1 −  1  |x|2  � (x·E′x)x  |x|d+2 · E [f ◦(E)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  ♦  Note that the error bound in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='32) is ≪ λ1 provided that λ2|log λ1| ≪ λ1, and is in  particular bounded by ℓ−3d/2 ≪ ℓ−d in the regime ℓ ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Alternatively, arguing similarly as for the reformulation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='26) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='24)–(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='25), the dilute  active effective viscosity can be written as  E′ : 2 ¯B(1)  act(E) = E′ : E  � �  n  1In  |In|  � ˆ  ∂In  σ(φn  E, Πn  E)ν⊗s(x−xn)−  ˆ  In+B  fn(E)⊗s(x−xn)  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  14  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' BERNOU, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' DUERINCKX, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' GLORIA  In particular, if fn(E) is rotationally symmetric around the direction ¯fn(E), we find by  symmetry,  2 ¯B(1)  act(E) = λ1E  �  α(E)  � ¯fn(E)⊗ ¯fn(E)  | ¯fn(E)|2  − 1  d Id  ��  ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='35)  for some random prefactor α(E) ∈ R, the sign of which is actually of critical interest and  depends on the shape of the swimming mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Viscosity reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' The main motivation of this work is to introduce a nontrivial  (and hopefully somewhat realistic) model for active suspensions and rigorously establish  a reduction of the viscosity of the plain fluid due to the activity of the particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' In the  dilute regime, the above formulas provide a rigorous contribution to this celebrated topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Although the question can be reduced to understanding the sign of α(E) in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='35), we take  a shorter path here based on (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' To make computations explicit, we restrict ourselves  to the following simplified model for the swimming mechanism, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' [19, 22],  f ◦(E) := ¯f(E)(|B|−11B − δx(E)),  where the force exerted by the particle on the surrounding fluid is reduced to a Dirac force  at a surrounding point x(E) ∈ Rd\\B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Consider a shear deformation E = s  2(e1⊗e2+e2⊗e1)  for some s ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' In these terms, the formula (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='34) reads  E : ¯B(1)  act(E) = s  2  �  1 −  1  |x(E)|d+2  ��  x(E)1 ¯f(E)2 + x(E)2 ¯f(E)1  �  − s d+2  2  �  1 −  1  |x(E)|2  �x(E)1x(E)2  |x(E)|d+2 x(E) · ¯f(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  For a spherical particle with its swimming device viewed as a rigid elongated particle, the  motion in shear flow has been well-studied on the formal level: in the limit of a strong  angular diffusion, a standard heuristic computation shows that the preferred orientation  of the particle is e :=  1  √  2(e1 + e2) or its opposite;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' [14, Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' This leads us  to choosing  ¯f(E) = ±| ¯f(E)|e  and  x(E) = ±γ|x(E)|e,  where γ = 1 in case when the Dirac swimming force is ahead of the particle (“puller”  particle) and γ = −1 in case when it is behind the particle (“pusher” particle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Hence, we  get  E : ¯B(1)  act(E) = γ s  2|x(E)|| ¯f(E)|  �  1 − d+2  2  1  |x(E)|d + d  2  1  |x(E)|d+2  �  ,  which is negative (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' positive) in case of a pusher (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' puller).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' This is in full agreement  with well-known experiments and predictions of [35, 36, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  To conclude on the extent of the possible viscosity reduction, we come back to Theo-  rem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='9 in case ℓ ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' As stated, the error bound in the result is then of order O(ℓ−3d/2),  ¯Btot(E) =  �  1 + d+2  2 |B|λ1  �  E + κλ1 ¯B(1)  act(E) + O(ℓ−3d/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='36)  If ¯B(1)  act(E) < 0, we infer that the total effective viscosity is smaller than the viscosity of  the plain fluid,  E : ¯Btot(E) < |E|2,  provided that the activity is strong enough κ ≫ 1 in such a way that the active contribution  exceeds the passive one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Moreover, as λ1 = O(ℓ−d), we see that the viscosity reduction  could become of order 1 if κ = O(ℓd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  This drastic reduction of viscosity is however  HOMOGENIZATION FOR ACTIVE SUSPENSIONS  15  prohibited by the (only nearly optimal) condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='18), which is used to ensure the well-  posedness of the microscopic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' In some sense, the above analysis can be compared  to the enhancement of elastostriction by active charges analyzed in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Outline of the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' The article is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Section 2 is dedicated to  the introduction of correctors, the key quantities in homogenization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' The proofs of the  homogenization results of Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='7 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='8 are the object of Section 3, while the  dilute analysis and the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='9 are postponed to Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  — For vector fields u, u′, and matrix fields T, T ′, we set (∇u)ij = ∇jui, div(T)i = ∇jTij,  T : T ′ = TijT ′  ij, (u ⊗ u′)ij = uiu′  j, (T s)ij = 1  2(Tij + Tji), D(u) = (∇u)s, (u ⊗s u′) =  (u ⊗ u′)s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' For a 3-tensor field S, the matrix div(S) is defined by div(S)ij = ∇kSijk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  For a matrix E and a vector field u, we write ∂Eu = E : ∇u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' We systematically use  Einstein’s summation convention on repeated indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  — For a vector field u and a scalar field P, we recall the notation σ(u, P) = 2 D(u) − P Id  for the Cauchy stress tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  — We denote by C ≥ 1 any constant that only depends on dimension d, on the constant ϑ  in Hypothesis 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='1, and on the reference domain U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' We use the notation ≲ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' ≳)  for ≤ C×(resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' ≥ 1  C ×) up to such a multiplicative constant C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' We write ≪ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' ≫)  for ≤ C× (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' ≥ C×) up to a sufficiently large multiplicative constant C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' We add  subscripts to C, ≲, ≳ in order to indicate dependence on other parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  — We write M0 ⊂ Rd×d for the subset of trace-free matrices, Msym  0  for the subset of  symmetric trace-free matrices, and Mskew for the subset of skew-symmetric matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  — The ball centered at x of radius r in Rd is denoted by Br(x), and we set B(x) = B1(x),  Br = Br(0) and B = B1(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  — We use the standard notation ⟨a⟩ := (1 + |a|2)1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Corrector problems  We start by recalling the relevant correctors for the passive suspension problem as in-  troduced in [2, 4, 8, 10], and then we define new correctors for the active problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Passive corrector problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Correctors for passive suspensions were first defined  in [8] and are key to the definition of the associated effective viscosity ¯B, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='1 (Passive correctors [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Let Hypothesis 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='1 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' For all E ∈ Msym  0  , there  exist a unique random field ψE ∈ L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' H1  loc(Rd)d) and a unique pressure field ΣE ∈  L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' L2  loc(Rd \\ I)) such that:  — almost surely, realizations of ψE and ΣE satisfy  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  −△ψE + ∇ΣE = 0,  in Rd \\ I,  div(ψE) = 0,  in Rd \\ I,  D(ψE + Ex) = 0,  in I,  ´  ∂In σ(ψE + Ex, ΣE)ν = 0,  ∀n,  ´  ∂In Θ(x − εxn) · σ(ψE + Ex, ΣE)ν = 0,  ∀n, ∀Θ ∈ Mskew,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='1)  16  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' BERNOU, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' DUERINCKX, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' GLORIA  — the corrector gradient ∇ψE and the pressure ΣE1Rd\\I are stationary, with  E  �  ∇ψE  �  = 0,  E  �  ΣE1Rd\\I  �  = 0,  E  �  |∇ψE|2�  + E  �  Σ2  E1Rd\\I  �  ≲ λ|E|2,  and with the anchoring condition  ´  B ψE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  In addition, the following properties hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (i) Ergodic theorem: almost surely,  (∇ψE)( ·  ε)  ⇀  E [∇ψE] = 0,  (ΣE1Rd\\I)( ·  ε)  ⇀  E  �  ΣE1Rd\\I  �  = 0,  weakly in L2  loc(Rd) as ε ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (ii) Sublinearity: almost surely, for all q <  2d  d−2,  εψE( ·  ε) → 0  strongly in Lq  loc(Rd)d as ε ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  ♦  As in [2, 10], in view of quantitative estimates, we further need to define an associated  extended flux J and a flux corrector Υ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' More precisely, J is a solenoidal extension of the  natural flux σ(ψE +Ex, ΣE) outside the particles, and Υ is the associated vector potential  in the Coulomb gauge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='2 (Passive flux correctors [2, 10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Let Hypothesis 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='1 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' For all E ∈ Msym  0  ,  there exists a stationary random 2-tensor field JE = (JE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='ij)1≤i,j≤d with finite second mo-  ment such that, almost surely,  JE1Rd\\I  =  σ(ψE + Ex, ΣE)1Rd\\I,  div(JE)  =  0,  and for all n,  ∥JE∥L2(In) ≲ ∥σ(ψE + Ex, ΣE)∥L2((In+B)\\In).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='2)  Moreover, there exists a unique random 3-tensor field ΥE = (ΥE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='ijk)1≤i,j,k≤d such that:  — for all i, j, k, almost surely, realizations of ΥE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='ijk belong to H1  loc(Rd) and satisfy  − ∆ΥE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='ijk = ∂jJE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='ik − ∂kJE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='ij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='3)  — the random field ∇ΥE is stationary, has vanishing expectation, has finite second mo-  ment, and satisfies the anchoring condition  ffl  B ΥE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  In addition, the following properties hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (i) Skew-symmetry: almost surely, ΥE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='ijk = −ΥE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='ikj for all i, j, k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (ii) Vector potential: almost surely, for all i,  div(ΥE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='i) = JE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='i − E[JE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='i],  where we have set ΥE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='i = (ΥE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='ijk)1≤j,k≤d and JE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='i = (JE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='ij)1≤j≤d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (iii) Ergodic theorem: almost surely,  (∇ΥE)( ·  ε) ⇀ E [∇ΥE] = 0weakly in L2  loc(Rd) as ε ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (iv) Sublinearity: almost surely, for all q <  2d  d−2,  εΥE( ·  ε) ⇀ 0  strongly in Lq  loc(Rd)d as ε ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  HOMOGENIZATION FOR ACTIVE SUSPENSIONS  17  (v) Effective constants: the expectation of JE takes the form  E [JE] = 2 ¯BpasE + (¯b : E) Id,  in terms of the effective constants ¯Bpas and ¯b defined in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='21) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  ♦  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Active corrector problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' We turn to the definition of suitable correctors for the  active suspension problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' These new correctors characterize the contribution of swimming  forces of the particles in a uniform fluid velocity gradient E ∈ Msym  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Let Hypotheses 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='3, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='5 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' For all E ∈ Msym  0  , there exist a unique  random field φE ∈ L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' H1  loc(Rd)d) and a unique pressure field ΠE ∈ L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' L2  loc(Rd \\ I))  such that:  — almost surely, realizations of φE and ΠE satisfy  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  −∆φE + ∇ΠE = �  n fn(E),  in Rd \\ I,  div(φE) = 0,  in Rd \\ I,  D(φE) = 0,  in I,  ´  ∂In σ(φE, ΠE)ν + ¯fn(E) = 0,  ∀n,  ´  ∂In Θ(x − xn) · σ(φE, ΠE)ν + Θ : ˜fn(E) = 0,  ∀n, ∀Θ ∈ Mskew,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='4)  — the corrector gradient ∇φE and the pressure ΠE1Rd\\I are stationary, with  E  �  ∇φE  �  = 0,  E  �  ΠE1Rd\\I  �  = 0,  E  �  |∇φE|2�  + E  �  Π2  E1Rd\\I  �  ≲ λ⟨E⟩2,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='5)  and with the anchoring condition  ´  B φE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  In addition, the following properties hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (i) Ergodic theorem: almost surely,  (∇φE)( ·  ε)  ⇀  E [∇φE] = 0,  (ΠE1Rd\\I)( ·  ε)  ⇀  E  �  ΠE1Rd\\I  �  = 0,  weakly in L2  loc(Rd) as ε ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (ii) Sublinearity: almost surely, for all q <  2d  d−2,  εφE( ·  ε) → 0  strongly in Lq  loc(Rd) as ε ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  ♦  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' The argument is similar to that in [8, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='1] for Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='1 above, and  we only briefly show the needed adaptations: we describe the structure of equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='4),  following the first step of the proof of [8, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='1], while the rest of the proof  in [8] is then easily repeated and is skipped here for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  More precisely, we only  show the following: if φE is a solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='4) with ∇φE, ΠE stationary with finite second  moments, then it satisfies for all stationary fields v ∈ L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' H1  loc(Rd)d) with div(v) = 0  and D(v)|I = 0,  E [∇v : ∇φE] = −E  � �  n  1In  |In|  ˆ  In+B  v · fn(E)  �  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='6)  where, by the Cauchy–Schwarz and the Poincaré inequalities, together with the hardcore  assumption and the property  ´  In+B fn(E) = 0 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='9)), the right-hand side is bounded  by  ����E  � �  n  1In  |In|  ˆ  In+B  v · fn(E)  ����� ≲ E  �  |∇v|2� 1  2  �  λ E  � ˆ  I◦+B  |f◦(E)|2  �� 1  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  18  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' BERNOU, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' DUERINCKX, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' GLORIA  To prove this claim, we start by noting that the hardcore assumption allows to construct  almost surely for all R > 0 a cut-off function ηR such that  ηR|BR = 1,  ηR|Rd\\BR+5 = 0,  |∇ηR| ≲ 1,  and such that ηR is constant in In + B for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' As φE is divergence-free, testing equa-  tion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='4) with ηRv and integrating by parts, we find  2  ˆ  Rd\\I  D(ηRv) : D(φE) −  ˆ  Rd\\I  ∇ηR · vΠE  =  �  n  ˆ  (In+B)\\In  ηRv · fn(E) −  �  n  ˆ  ∂In  ηRv · σ(φE, ΠE)ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='7)  Since D(φE)|I = 0 and ∇ηR|I = 0, the left-hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='7) writes  2  ˆ  Rd\\I  D(ηRv) : D(φE) −  ˆ  Rd\\I  ∇ηR · vΠE =  ˆ  Rd ∇(ηRv) : ∇φE −  ˆ  Rd ∇ηR · vΠE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  As ηR is constant in In + B and D(v) = 0 in In, we can rewrite the last right-hand side  term as  ˆ  ∂In  ηRv · σ(φE, ΠE)ν = ηR(xn)  ��  In  v  � ˆ  ∂In  σ(φE, ΠE)ν  +  �  In  (∇v)skew�  :  ˆ  ∂In  σ(φE, ΠE)ν ⊗ (x − xn)  �  ,  and thus, in view of the boundary conditions for (φE, ΠE) in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='4), using the notation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='8),  ˆ  ∂In  ηRv · σ(φE, ΠE)ν  =  −ηR(xn)  ��  In  v  �  ¯fn(E) +  �  In  ∇v  �  : ˜fn(E)  �  =  −  ˆ  In  ηRv · fn(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Inserting this into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='7), we get  ˆ  Rd ∇(ηRv) : ∇φE −  ˆ  Rd ∇ηR · vΠE =  �  n  ˆ  In+B  ηRv · fn(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='8)  Expanding the gradient in the left-hand side, passing to the limit R ↑ ∞, and using the  stationarity of ∇φE, ΠE and of v, the claim (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='6) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' From there, we may then refer  to the proof in [8, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  □  Next, as in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='2, we further need to define an associated extended flux and a flux  corrector for the active suspension problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' The difficulty, however, is that even in the fluid  domain Rd \\ I the flux σ(φE, ΠE) is not divergence-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' It thus needs to be first suitably  compensated and we are led to defining the following auxiliary corrector γE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' The proof of  the upcoming lemma (which is a simplified version of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='3) is straightforward and  skipped for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Let Hypotheses 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='3, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='5 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' For all E ∈ Msym  0  , there exists a  unique random field γE ∈ L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' H1  loc(Rd)d) such that:  HOMOGENIZATION FOR ACTIVE SUSPENSIONS  19  — almost surely, the realizations of γE satisfy  − △γE =  �  n  fn(E);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='9)  — the gradient field ∇γE is stationary with  E  �  ∇γE  �  = 0,  E  �  |∇γE|2�  ≲ λ⟨E⟩2,  and with the anchoring condition  ´  B γE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  In addition, the following properties hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (i) Ergodic theorem: almost surely,  (∇γE)( ·  ε)  ⇀  E [∇γE] = 0  weakly in L2  loc(Rd)d as ε ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (ii) Sublinearity: almost surely, for all q <  2d  d−2,  εγE( ·  ε) → 0  strongly in Lq  loc(Rd)d as ε ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  ♦  Using the above-defined γE to compensate the divergence of the flux σ(φE, ΠE) in the  fluid domain, and extending it similarly as in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='2 inside the particles, we are now in  the position to define a flux KE, a divergence-free compensated flux LE, and the associated  flux corrector θE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Under Hypotheses 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='3, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='5, there exists a stationary random sym-  metric 2-tensor field KE = (KE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='ij)1≤i,j≤d with finite second moment  E  �  |KE|2�  ≲ λ⟨E⟩2,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='10)  such that, almost surely,  KE1Rd\\I  =  σ(φE, ΠE)1Rd\\I,  div(KE)  =  −  �  n  fn(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='11)  Moreover, the expectation of KE takes the form  E [KE] = 2 ¯C(E) + ¯c(E) Id,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='12)  in terms of the effective tensors ¯C(E) and ¯c(E) defined in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Next, for the divergence-  free compensated flux  LE := KE − E [KE] − ∇γE,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='13)  there exists a unique random 3-tensor field θE = (θE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='ijk)1≤i,j,k≤d such that:  — for all i, j, k, almost surely, realizations of θE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='ijk belong to H1  loc(Rd) and satisfy  − △θE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='ijk = ∂jLE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='ik − ∂kLE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='ij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='14)  — the random field ∇θE is stationary, has vanishing expectation, has finite second moment,  and satisfies the anchoring condition  ffl  B θE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  In addition, the following properties hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (i) Skew-symmetry: almost surely, θE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='ijk = −θE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='ikj for all i, j, k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (ii) Vector potential: almost surely, for all i,  div(θE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='i) = LE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='i,  where we have set θE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='i = (θE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='ijk)1≤j,k≤d and LE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='i = (LE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='ij)1≤j≤d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  20  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' BERNOU, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' DUERINCKX, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' GLORIA  (iii) Ergodic theorem: almost surely,  (∇θE)( ·  ε)  ⇀  E [∇θE] = 0  weakly in L2  loc(Rd)d as ε ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (iv) Sublinearity: almost surely, for all q <  2d  d−2,  εθE( ·  ε) → 0  strongly in Lq  loc(Rd)d as ε ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  ♦  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' We split the proof into three main steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Construction and properties of the extended flux KE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  For all g ∈ C∞  c (Rd), equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='4) yields by integration by parts,  ˆ  Rd\\I  ∇g : σ(φE, ΠE)  =  �  n  ˆ  (In+B)\\In  g · fn(E) −  �  n  ˆ  ∂In  g · σ(φE, ΠE)ν,  and thus, using boundary conditions for (φE, ΠE) in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='4) and recalling the notation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='8),  ˆ  Rd\\I  ∇g : σ(φE, ΠE) +  �  n  ˆ  ∂In  �  g −  �  In  g  �  −  �  In  (∇g)skew�  (x − xn)  �  σ(φE, ΠE)ν  =  �  n  ˆ  In+B  g · fn(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='15)  For all n, we may then consider the following Neumann problem,  \uf8f1  \uf8f2  \uf8f3  −△φn  E + ∇Πn  E = fn(E),  in In,  div(φn  E) = 0,  in In,  σ(φn  E, Πn  E)ν = σ(φE, ΠE)ν,  on ∂In.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='16)  Note that this only defines φn  E up to a rigid motion, which is fixed by choosing φn  E with  ffl  In φn  E = 0 and  ffl  In ∇φn  E ∈ Msym  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Assuming that this Neumann problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='16) is well-  posed, and setting  ˜qE  :=  D(φE) +  �  n  D(φn  E)1In,  ˜ΠE  :=  ΠE1Rd\\I +  �  n  Πn  E1In,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='17)  we easily deduce that the extended flux  KE := 2˜qE − ˜ΠE Id  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='18)  satisfies the desired relations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='11) (recall that D(φE)|I = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' It remains to check the  well-posedness of this Neumann problem and to establish the bound (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Note that this  well-posedness, together with the uniqueness in our construction of the pressures {Πn  E}n  below, ensures that ˜qE and ˜ΠE are stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  We split the proof into three further  substeps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Substep 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Well-posedness of the Neumann problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='16) for φn  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  The weak formulation of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='16) reads as follows: for all divergence-free fields v ∈ H1(In)d,  2  ˆ  In  D(v) : D(φn  E) = LE(v),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='19)  HOMOGENIZATION FOR ACTIVE SUSPENSIONS  21  where LE stands for the linear form  LE(v) :=  ˆ  In  v · fn(E) +  ˆ  ∂In  v · σ(φE, ΠE)ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Using the boundary conditions for (φE, ΠE) in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='4) and recalling the notation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='8), the  latter can be reformulated as  LE(v) =  ˆ  ∂In  �  v −  �  In  v  �  −  �  In  (∇v)skew�  (x − xn)  �  σ(φE, ΠE)ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='20)  We shall show that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='19) is well-posed in the following Hilbert subspace of H1(In)d,  T :=  �  v ∈ H1(In)d : div(v) = 0,  In  v = 0, and  In  ∇v ∈ Msym  0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  First note that Korn’s inequality yields for all v ∈ T ,  ∥∇v∥L2(In) ≲ ∥D(v)∥L2(In),  which entails that the bilinear form (v, ψ) �→ 2  ´  In D(v) : D(ψ) is continuous and coercive  on T × T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' By the Lax–Milgram theorem, in order to prove the well-posedness of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='19),  it remains to show that the linear form LE is continuous on T as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  In order to deal with the Neumann condition, we consider an extension map  Tn :  �  v ∈ H1(In)d : div(v) = 0  �  →  �  v ∈ H1  0(In + B)d : div(v) = 0  �  ,  such that Tnv|In = v|In and  ∥∇Tnv∥L2(In+B) ≲ ∥∇v∥L2(In).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='21)  Smuggling in Tn in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='20), integrating by parts, and inserting equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='4), the linear  form LE can be rewritten as  LE(v)  =  −  ˆ  (In+B)\\In  div  �  σ(φE, ΠE) Tn  �  v −  �  In  v  �  −  �  In  (∇v)skew�  (x − xn)  ��  =  ˆ  (In+B)\\In  Tn  �  v −  �  In  v  �  −  �  In  (∇v)skew�  (x − xn)  �  fn(E)  −2  ˆ  (In+B)\\In  D(φE) : D  �  Tn  �  v −  �  In  v  �  −  �  In  (∇v)skew�  (x − xn)  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Hence, in view of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='21) and of Korn’s inequality,  |LE(v)| ≲ ∥D(v)∥L2(In)  �  ∥D(φE)∥L2(In+B) + ∥fn(E)∥L2(In+B)  �  ,  which proves the continuity of LE on T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  By the Lax–Milgram theorem, we deduce that there exists a unique solution φn  E ∈ T  of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='19), and that it satisfies  ∥D(φn  E)∥L2(In) ≲ ∥D(φE)∥L2(In+B) + ∥fn(E)∥L2(In+B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  By Korn’s inequality, this further yields  ∥∇φn  E∥L2(In) ≲ ∥D(φE)∥L2(In+B) + ∥fn(E)∥L2(In+B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='22)  22  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' BERNOU, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' DUERINCKX, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' GLORIA  Substep 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Construction of the pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  As (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='17) reads ˜qE = D(φE) + D(φn  E)1In in In + B, combining equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='4) for φE and  equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='19) for φn  E, we find for all v ∈ C∞  c (In + B)d with div(v) = 0,  2  ˆ  Rd D(v) : ˜qE =  ˆ  Rd v · fn(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Appealing e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' to [26, Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='10], we deduce that there exists an associated pressure  field Πn  E ∈ L2  loc(In + B), which is unique up to an additive constant, such that for all test  functions v ∈ C∞  c (In + B)d,  ˆ  Rd D(v) : (2˜qE − Πn  E Id) =  ˆ  Rd v · fn(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='23)  Since for all v ∈ C∞  c ((In + B) \\ In)d we get  ˆ  Rd D(v) : σ(φE, Πn  E) =  ˆ  Rd D(v) : (2˜qE − Πn  E Id) =  ˆ  Rd v · fn(E),  and comparing with equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='4), we deduce that Πn  E can be chosen uniquely to coincide  with ΠE on (In + B) \\ In.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  It remains to estimate the above-constructed pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Using that Πn  E coincides with ΠE  in (In + B) \\ In, we can split  ˆ  In+B  Πn  E  =  ˆ  (In+B)\\In  ΠE +  ˆ  In  Πn  E  =  ˆ  (In+B)\\In  ΠE +  ˆ  In  �  Πn  E −  In+B  Πn  E  �  + |In|  In+B  ΠE,  to the effect that  (|In + B| − |In|)  ���  In+B  Πn  E  ��� ≤  ���  ˆ  (In+B)\\In  ΠE  ��� +  ���  ˆ  In  �  Πn  E −  In+B  Πn  E  ����.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Hence,  ∥Πn  E∥L2(In)  ≲  ���Πn  E −  In+B  Πn  E  ���  L2(In+B) +  ���  In+B  Πn  E  ���  ≲  ���Πn  E −  In+B  Πn  E  ���  L2(In+B) +  ���  (In+B)\\In  ΠE  ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Starting from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='23), a standard argument based on the Bogovskii operator yields  ���Πn  E −  In+B  Πn  E  ���  L2(In+B) ≲ ∥˜qE∥L2(In+B),  so that the above becomes  ∥Πn  E∥L2(In) ≲ ∥˜qE∥L2(In+B) + ∥ΠE∥L2((In+B)\\In).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Combining this with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='22), we get  ∥∇φn  E∥L2(In) + ∥Πn  E∥L2(In) ≲ ∥σ(φE, ΠE)∥L2((In+B)\\In) + ∥fn(E)∥L2(In+B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='24)  HOMOGENIZATION FOR ACTIVE SUSPENSIONS  23  Substep 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Proof of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  By definition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='18), the bound (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='24) yields for all n,  ∥KE∥L2(In)  ≲  ∥σ(φn  E, Πn  E)∥L2(In)  ≲  ∥σ(φE, ΠE)∥L2((In+B)\\In) + ∥fn(E)∥L2(In+B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='25)  Hence, for all R > 0,  ∥KE∥2  L2(BR) ≲ ∥σ(φE, ΠE)1Rd\\I∥2  L2(BR+5) +  �  n:In∩BR̸=∅  ˆ  In+B  |fn(E)|2,  and thus, by stationarity, letting R ↑ ∞,  ∥KE∥2  L2(Ω) ≲ ∥σ(φE, ΠE)1Rd\\I∥2  L2(Ω) + λ E  � ˆ  I◦+B  |f◦(E)|2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Combined with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='5), this yields (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Formula for E [KE] and definition of ¯C(E) and ¯c(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  We split the proof into two further substeps, separately proving formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='12) for E [KE]  and establishing the alternative formulas (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='26) for ¯C(E) and ¯c(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Substep 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Proof of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  The hardcore assumption allows to construct almost surely for all R > 0 a cut-off func-  tion ηR such that  ηR|BR = 1,  ηR|Rd\\BR+5 = 0,  |∇ηR| ≲ 1,  and such that ηR is constant in In + B for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' By definition of KE and (φE, ΠE), we  have  E  �  KE1Rd\\I  �  = E  �  σ(φE, ΠE)1Rd\\I  �  = 2E  �  D(φE)  �  − E  �  ΠE1Rd\\I  �  Id = 0,  and the ergodic theorem then yields almost surely,  E [KE] = E  �  KE1I  �  = lim  R↑∞ |BR|−1  ˆ  I  ηR KE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='26)  By definition of KE and the choice of ηR, integration by parts and the equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='16)  for (φn  E, Σn  E) yield  ˆ  I  ηR KE  =  �  n  ηR(xn)  ˆ  In  σ(φn  E, Πn  E)  =  �  n  ηR(xn)  ˆ  ∂In  σ(φn  E, Πn  E)ν ⊗ (x − xn)  +  �  n  ηR(xn)  ˆ  In  fn(E) ⊗ (x − xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  By the boundary conditions for (φn  E, Πn  E) and recalling the notation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='8), we deduce  ˆ  I  ηR KE  =  ˆ  Rd ηR  �  n  1In  |In|  �  ˜fn(E) +  ˆ  ∂In  σ(φE, ΠE)ν ⊗ (x − xn)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Letting R ↑ ∞, the ergodic theorem then entails  E [KE] = E  � �  n  1In  |In|  �  ˜fn(E) +  ˆ  ∂In  σ(φE, ΠE)ν ⊗ (· − xn)  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  24  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' BERNOU, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' DUERINCKX, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' GLORIA  Since KE is symmetric, taking the symmetric part of this identity yields (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='12) in combi-  nation with the skew-symmetry of ˜fn(E) and the definition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='24) of ¯C(E) and ¯c(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Proof of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  We start from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='24), projected in some direction E′ ∈ Msym  0  ,  2E′ : ¯C(E) = E  � �  n  1In  |In|  ˆ  ∂In  E′(x − xn) · σ(φE, ΠE)ν  �  ,  which we reformulate using the ergodic theorem as  2E′ : ¯C(E) = lim  R↑∞ |BR|−1 �  n  ˆ  ∂In  ηRE′(x − xn) · σ(φE, ΠE)ν,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='27)  with ηR as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' We turn to a suitable reformulation of the right-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Adding and  subtracting the passive corrector ψE′, we can write  �  n  ˆ  ∂In  ηRE′(x − xn) · σ(φE, ΠE)ν  =  �  n  ˆ  ∂In  ηR(ψE′ + E′(x − xn)) · σ(φE, ΠE)ν −  �  n  ˆ  ∂In  ηRψE′ · σ(φE, ΠE)ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='28)  For the first right-hand side term, we use that ψE′ + E′(x − xn) is a rigid motion in In,  cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='1), we appeal to boundary conditions for (φE, ΠE) in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='4), and we recall the nota-  tion (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='8), which leads us to  �  n  ˆ  ∂In  ηR(ψE′ + E′(x − xn)) · σ(φE, ΠE)ν = −  �  n  ˆ  In  ηR(ψE′ + E′(x − xn)) · fn(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  In order to reformulate the second right-hand side term of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='28), we appeal to the weak  formulation of equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='4) for φE: testing this equation with ηRψE′ yields, as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='7),  ˆ  Rd ∇(ηRψE′) : ∇φE −  ˆ  Rd ∇ηR · ψE′ΠE =  �  n  ˆ  (In+B)\\In  ηRψE′ · fn(E)  −  �  n  ˆ  ∂In  ηRψE′ · σ(φE, ΠE)ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Hence, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='28) turns into  �  n  ˆ  ∂In  ηRE′(x − xn) · σ(φE, ΠE)ν = −  �  n  ˆ  In  ηR(ψE′ + E′(x − xn)) · fn(E)  −  ˆ  Rd ∇ηR · ψE′ΠE +  ˆ  Rd ∇(ηRψE′) : ∇φE −  �  n  ˆ  (In+B)\\In  ηRψE′ · fn(E),  and thus, after reorganizing the terms, using that the skew-symmetry of ˜fn(E) in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='8)  yields  ´  In E′(x − xn) · fn(E) = 0,  �  n  ˆ  ∂In  ηRE′(x − xn) · σ(φE, ΠE)ν =  ˆ  Rd ∇(ηRψE′) : ∇φE  −  ˆ  Rd ∇ηR · ψE′ΠE −  �  n  ˆ  In+B  ηRψE′ · fn(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  HOMOGENIZATION FOR ACTIVE SUSPENSIONS  25  Alternatively, expanding the first right-hand side term,  �  n  ˆ  ∂In  ηRE′(x − xn) · σ(φE, ΠE)ν =  ˆ  Rd ∇ψE′ : ∇(ηRφE) −  ˆ  Rd ∇ηR · ψE′ΠE  +  ˆ  Rd(ψE′ ⊗ ∇ηR) : ∇φE −  ˆ  Rd(φE ⊗ ∇ηR) : ∇ψE′ −  �  n  ˆ  In+B  ηRψE′ · fn(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Now testing equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='1) for ψE′ with ηRφE, and using that ηRφE is rigid in I, we find  that the first right-hand side term only yields a term involving ΣE  �  n  ˆ  ∂In  ηRE′(x − xn) · σ(φE, ΠE)ν = −  ˆ  Rd ∇ηR · (ψE′ΠE + φEΣE′)  +  ˆ  Rd(ψE′ ⊗ ∇ηR) : ∇φE −  ˆ  Rd(φE ⊗ ∇ηR) : ∇ψE′ −  �  n  ˆ  In+B  ηRψE′ · fn(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Using the sublinearity of ψE′ and φE and the stationarity of ∇ψE′, ∇φE, ΠE1Rd\\I, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Lem-  mas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='3, we can now pass to the limit R ↑ ∞ in this identity, and we obtain  lim  R↑∞ |BR|−1 �  n  ˆ  ∂In  ηRE′(x − xn) · σ(φE, ΠE)ν = −E  � �  n  1In  |In|  ˆ  In+B  ψE′ · fn(E)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Inserting this into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='27), the conclusion (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='26) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Finally, the formula for ¯c(E) simply  follows by taking the trace in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Construction of θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  This construction is standard: as LE is stationary, it follows from stationary calculus,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' [26, Chapter 7], that there is a unique solution θE = (θE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='ijk)ijk ∈ L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' H1  loc(Rd)) of  −△θE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='ijk = ∂jLE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='ik − ∂kLE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='ij,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='29)  such that ∇θE is stationary, has vanishing expectation, has finite second moment,  ∥∇θE∥L2(Ω) ≲ ∥LE∥L2(Ω) ≲ ∥KE∥L2(Ω) + ∥∇γE∥L2(Ω),  and satisfies the anchoring condition  ffl  B θE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' By uniqueness, the skew-symmetry of θE  follows from the skew-symmetry of the right-hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='29) with respect to indices j, k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  The fact that θE is a vector potential for LE follows from this defining equation as e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  in [20, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Finally, the sublinearity of θE is a standard property for random fields  with stationary gradient and vanishing expectation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' [26, Chapter 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  □  In view of the internal contribution to the effective viscosity, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='25), we also define  the following stationary field,  FE := −  �  n  1In  |In|  ˆ  In+B  fn(E) ⊗ (x − xn),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='30)  which is such that  ¯F E = E [FE] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='31)  When considering two-scale expansions, since correctors φE, ΠE, γE, θE, fluxes KE, LE,  and FE are nonlinear with respect to E, we shall need to consider derivatives of these  objects with respect to E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' We introduce a general notation for linearized quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  26  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' BERNOU, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' DUERINCKX, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' GLORIA  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Given E, E′, E′′ ∈ Msym  0  , we use the following notation for directional  derivatives of swimming forces fn(E) in directions E′, E′′,  ∂E′fn(E)  :=  lim  h→0  1  h  �  fn(E + hE′) − fn(E)  �  ,  ∂E′,E′′fn(E)  :=  lim  h→0  1  h  �  ∂E′fn(E + hE′′) − ∂E′fn(E)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  — First linearized correctors: We define (dφE,E′, dΠE,E′) as in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='3 with the swim-  ming forces fn(E) replaced by ∂E′fn(E) (and similarly for dFE,E′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  We then de-  fine dγE,E′ as in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='4 and dKE,E′, dLE,E′, dθE,E′ as in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='5 with fn(E)  replaced by ∂E′fn(E) and with (φE, ΠE) replaced by (dφE,E′, dΠE,E′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  — Second linearized correctors: We define (d2φE,E′,E′′, d2ΠE,E′,E′′) as in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='3 with  swimming forces fn(E) replaced by ∂E′,E′′fn(E) (and similarly for d2FE,E′,E′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' We  then define d2γE,E′,E′′ as in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='4 and d2KE,E′,E′′, d2LE,E′,E′′, d2θE,E′,E′′ as in  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='5 with fn(E) replaced by ∂E′,E′′fn(E) and with (φE, ΠE) replaced by (d2φE,E′,E′′,  d2ΠE,E′,E′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  ♦  Finally, we state that the effective maps ¯C, ¯c, ¯F are smooth, with derivatives given in  terms of linearized correctors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' The proof, based on Hypothesis 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='3, is straightforward and  left to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' The effective maps ¯C, ¯c, ¯F are smooth and  | ¯C(E)|, |¯c(E)|, | ¯F (E)| ≲ λ⟨E⟩,  |∂k ¯C(E)|, |∂k¯c(E)|, |∂k ¯F (E)| ≲ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='32)  Moreover, we have for all E, F ∈ Msym  0  ,  2∂F ¯C(E) + ∂F ¯c(E) Id = E  � �  n  1In  |In|  ˆ  ∂In  σ(dφE,F, dΠE,F )ν ⊗s (x − xn)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  In particular, in view of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='12), this yields ∂F E [KE] = E [dKE,F].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  ♦  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Proof of the homogenization result  This section is devoted to the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' We quickly establish the well-  posedness result of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='6 before turning to the analysis of the limit ε ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' For  any map V and domain D, we henceforth use the short-hand notation V∤D :=  ffl  D V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Well-posedness of hydrodynamic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' This section is devoted to the proof of  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' We start by recalling the following standard computation (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' [8]),  which we already partly encountered when proving existence of correctors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='1 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' If vector fields u, h and a scalar field P satisfy the following relations,  \uf8f1  \uf8f2  \uf8f3  −△u + ∇P = h,  in Rd \\ I,  div(u) = 0,  in Rd \\ I,  D(u) = 0,  in I,  then the following holds in Rd  − △u + ∇(P1Rd\\I) = h1Rd\\I −  �  n  δ∂Inσ(u, P)ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='1)  The same is true if Rd and I are replaced by U and Iε(U), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  ♦  HOMOGENIZATION FOR ACTIVE SUSPENSIONS  27  We turn to the proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='6 and proceed by a fixed-point argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Let ε > 0  be fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Given v ∈ W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='1(U)d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' define Tε(v) ∈ H1  0(U)d as the unique solution of the  following linear problem,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' with associated pressure Pε(v) ∈ L2(U \\ Iε(U)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  −△Tε(v) + ∇Pε(v)  = h + κ  ε  �  n∈Nε(U) fn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='ε(χδ ∗ D(v)∤εIn),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  in U \\ Iε(U),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  div(Tε(v)) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  in U \\ Iε(U),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  D(Tε(v)) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  in Iε(U),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  ´  ε∂In σ(Tε(v),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Pε(v))ν + κ  ε ¯fn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='ε(χδ ∗ D(v)∤εIn) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  ∀n ∈ Nε(U),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  ´  ε∂In Θ(x − εxn) · σ(Tε(v),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Pε(v))ν  +κ Θ : ˜fn(χδ ∗ D(v)∤εIn) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  ∀Θ ∈ Mskew,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' ∀n ∈ Nε(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='2)  We split the proof into three steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' We start by proving the result under the stronger  smallness condition κℓ−d/2 ≪ 1, before relaxing it to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Suboptimal contraction estimate: for all v, w ∈ H1  0(U)d we have  ˆ  U  |∇(Tε(v) − Tε(w))|2 ≲χ (κℓ− d  2 )2  ˆ  U  |∇(v − w)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='3)  This proves that Tε is a contraction on H1  0(U)d provided that κℓ− d  2 ≪ 1 is small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Under this condition, we deduce the well-posedness of the hydrodynamic model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='7)  in H1  0(U)d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  We turn to the proof of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' For abbreviation, we set Tε(v, w) := Tε(v) − Tε(w) and  Pε(v, w) = Pε(v)−Pε(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Testing the equations for Tε(v) and Tε(w) with Tε(v, w) in form  of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='1), we find  ˆ  U  |∇Tε(v, w)|2 = −  �  n∈Nε(U)  ˆ  ε∂In  Tε(v, w) · σ(Tε(v, w), Pε(v, w))ν  + κ  ε  �  n∈Nε(U)  ˆ  ε(In+B)\\εIn  Tε(v, w) ·  �  fn,ε(χδ ∗ D(v)∤εIn) − fn,ε(χδ ∗ D(w)∤εIn)  �  ,  and thus, using the rigidity of Tε(v, w) in εIn and using boundary conditions, recalling the  notation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='8),  ˆ  U  |∇Tε(v, w)|2 = κ  ε  �  n∈Nε(U)  ˆ  ε(In+B)  Tε(v, w) ·  �  fn,ε(χδ ∗ D(v)∤εIn) − fn,ε(χδ ∗ D(w)∤εIn)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  By the neutrality condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='9), appealing to Poincaré’s inequality, using the hardcore  assumption, and recalling that fn,ε is Lipschitz, we get  ˆ  U  |∇Tε(v, w)|2 ≲ κ  � ˆ  U  |∇Tε(v, w)|2� 1  2  �  �  n∈Nε(U)  ˆ  εIn  |χδ ∗ D(v − w)|2  � 1  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  By Jensen’s inequality, with  ´  Rd χδ = 1, the second factor is bounded by  �  n∈Nε(U)  ˆ  εIn  |χδ ∗ D(v − w)|2 ≲  ˆ  U  �  �  n∈Nε(U)  ˆ  εIn  χδ(x − y) dx  �  |D(v − w)(y)|2 dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  28  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' BERNOU, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' DUERINCKX, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' GLORIA  The expression into brackets can be estimated as follows,  sup  y∈Rd  �  n∈Nε(U)  ˆ  εIn  χδ(x − y) dx  ≤  sup  y∈Rd  �  n  ˆ  ε  δ In  χ(x − y) dx  ≲  ( ε  δ)d sup  y∈Rd  �  n  sup  y+ ε  δ In  χ  ≤  ( ε  δ)d sup  y∈Rd  �  n  sup  ε  δ Bℓ(y+xn)  χ  ≲  ℓ−d sup  y∈Rd  �  n  ˆ  ε  δ Bℓ(y+xn)  �  sup  B2ℓ ε  δ (z)  χ  �  dz,  and thus, by the hardcore assumption, provided εℓ ≤ δ,  sup  y∈Rd  �  n∈Nε(U)  ˆ  εIn  χδ(x − y) dx ≲ ℓ−d  ˆ  Rd  �  sup  B2(z)  χ  �  dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Inserting this into the above, we get  �  n∈Nε(U)  ˆ  εIn  |χδ ∗ D(v − w)|2 ≲χ ℓ−d  ˆ  U  |∇(v − w)|2,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='4)  and thus,  ˆ  U  |∇Tε(v, w)|2 ≲χ κℓ− d  2  � ˆ  U  |∇Tε(v, w)|2� 1  2 � ˆ  U  |∇(v − w)|2� 1  2 ,  that is, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Improved contraction estimate: given 1 < p ≤ 2 and given ℓ ≫p 1 large enough,  we have for all v, w ∈ W 1,p  0 (U)d,  ∥∇(Tε(v) − Tε(w))∥Lp(U) ≲p,χ κℓ− d  p ∥∇(v − w)∥Lp(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  This proves that Tε is a contraction on W 1,p  0 (U)d provided that κℓ− d  p ≪ 1 is small enough,  which then implies the well-posedness of the hydrodynamic model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='7) in W 1,p  0 (U)d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  We appeal to the dilute deterministic Lp regularity theory developed by Höfer in [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Given 1 < p ≤ 2, provided that ℓ ≫p 1 is large enough (depending on p and dimension d),  it allows us to deduce almost surely,  ∥∇(Tε(v) − Tε(w))∥Lp(U) ≲p κ  ���  �  n∈Nε(U)  �  fn,ε(χδ ∗ D(v)∤εIn) − fn,ε(χδ ∗ D(w)∤εIn)  ����  Lp(U),  and thus, using properties of {fn,ε}n,  ∥∇(Tε(v) − Tε(w))∥Lp(U)  ≲p  κ  �  �  n∈Nε(U)  ˆ  ε(In+B)  ��fn,ε(χδ ∗ D(v)∤εIn) − fn,ε(χδ ∗ D(w)∤εIn)  ��p  � 1  p  ≲  κ  �  �  n∈Nε(U)  ˆ  εIn  |χδ ∗ D(v − w)|p  � 1  p  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  HOMOGENIZATION FOR ACTIVE SUSPENSIONS  29  Repeating the argument in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='4), this proves the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Testing equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='2) with its solution Tε(v) itself, using boundary conditions and recalling  the notation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='8), we find  ˆ  U  |∇Tε(v)|2 =  ˆ  U\\Iε(U)  h · Tε(v) + κ  ε  �  n∈Nε(U)  ˆ  ε(In+B)  Tε(v) · fn,ε(χδ ∗ D(v)∤εIn),  and thus, by the neutrality condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='9), appealing to Poincaré’s inequality and recalling  that fn,ε is Lipschitz, arguing exactly as in Step 1,  ˆ  U  |∇Tε(v)|2 ≲ κ2ℓ−d +  ˆ  U\\Iε(U)  |h|2 + κ2  �  n∈Nε(U)  ˆ  εIn  |χδ ∗ D(v)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Now using the hardcore condition and Young’s inequality, we deduce for all 1 < p ≤ 2,  ˆ  U  |∇Tε(v)|2 ≲ κ2ℓ−d +  ˆ  U\\Iε(U)  |h|2 + κ2∥χδ∥2  L1 ∩ L2(U)∥∇v∥2  Lp(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  This proves that Tε embeds W 1,p  0 (U)d into H1  0(U)d, so that the solution uε = Tε(uε) in  W 1,p  0 (U)d constructed in Step 2 actually belongs to H1  0(U)d, and the a priori estimate (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='19)  follows by iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Qualitative homogenization at fixed δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' This section is devoted to the proof of  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Before turning to the proof, we argue for the well-posedness of the ho-  mogenized equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Using | ¯C(E)|, | ¯F (E)| ≲ λ⟨E⟩, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='32), a perturbative  argument as in the proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='6 yields the well-posedness of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='20) with  (¯uδ, ¯Pδ) ∈ H1  0(U)d × L2(U)/R provided that κλ ≪ 1 is small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  As λ ≲ ℓ−d,  this holds in particular under the smallness condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Interestingly, our proof of qualitative homogenization relies on a semi-quantitative two-  scale analysis and we do not believe that there exists a simpler and purely qualitative proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  In terms of a suitable limiting profile ¯uε (mildly depending on ε and to be identified at the  end of the proof), we consider the following two-scale expansions for the solution (uε, Pε)  of the hydrodynamic model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='7),  uε  ❀  ¯uε + εψE( ·  ε)∂E ¯uε + εκφχδ∗D(uε)( ·  ε),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='5)  Pε1Rd\\εI  ❀  ¯Pε + ¯b : D(¯uε) + κ¯c(χδ ∗ D(uε))  +(ΣE1Rd\\I)( ·  ε)∂E ¯uε + (κΠχδ∗D(uε)(x)1Rd\\I)(x  ε ),  where uε is implicitely extended by 0 on Rd \\ U to ensure that χδ ∗ D(uε) is well-defined,  and where we implicitly sum over E in an orthonormal basis of Msym  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' We start with two  comments on the form of these two-scale expansions:  — The most unusual feature is that correctors φ, Π are evaluated at a background fluid  deformation χδ ∗ D(uε) depending on the microscopic solution uε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' This non-standard  choice is taken as an intermediate step and happens to be providential in our proof,  where the limit of this background deformation χδ ∗ D(uε) ∼ χδ ∗ D(¯uε) can only  be identified at the very end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  To our knowledge, the necessity of such a two-step  homogenization argument is new to the literature and gives the present problem an  interest of its own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  30  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' BERNOU, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' DUERINCKX, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' GLORIA  — The choice of the non-oscillating part ¯Pε + ¯b : D(¯uε) + κ¯c(χδ ∗ D(uε)) in the two-  scale expansion of the pressure is dictated by the proof and is similar to the case of  passive suspensions [8]: the pressure for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='7) does not converge to the pressure of the  homogenized problem in its naïve form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  As χδ ∗ D(uε) is smooth, the maps (x, y) �→ (φχδ∗D(uε)(x))(y), (Πχδ∗D(uε)(x)1Rd\\I)(y) are  Carathéodory functions, which ensures that x �→ φχδ∗D(uε)(x)(x  ε), (Πχδ∗D(uε)(x)1Rd\\I)(x  ε )  in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='5) are measurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' When differentiating such composed functions, some care is needed  in the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Given a smooth field V : Rd → Msym  0  , we denote by φV the function of  two variables (x, y) �→ φV (x)(y), and we use the short-hand notation φV (x  ε) := φV (x)(x  ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  We use the following notation for the derivative of φV with V frozen,  (∇φ|V )(x  ε) := ∇yφV (x)|y= x  ε ,  so that the total derivative is then given by  ∇(εφV )(x  ε) = (∇φ|V )(x  ε) + εdφV (x),E(x  ε) ⊗ ∇(E : V (x)),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='6)  where the linearized corrector dφ is given in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='6 and where we implicitly sum  over E in an orthonormal basis of Msym  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' In addition, if V is a smooth random field, we  use the following notation for the expectation of φV with V frozen: for any random field a,  E[a(x)φ|V (x  ε)] := E  �  a(x)φE(x  ε)  �  |E=V (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  The same notation is used for other correctors and fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  As usual, correctors are not capturing the relevant behavior close to the boundary of the  domain U: in particular, the two-scale expansion in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='5) does not vanish at the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  To circumvent this, we proceed by truncating correctors in a neighborhood of the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  We set for abbreviation  ∂rU := {x ∈ U : dist(x, ∂U) < r},  and, given rε ≥ 4ε (which we shall optimize later on), we choose a smooth cut-off function  ηε ∈ C∞  c (U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' [0, 1]) such that  ηε|U\\∂2rεU = 1,  ηε|∂rεU = 0,  |∇ηε| ≲  1  rε ,  and such that ηε is constant inside each of the fattened particles {ε(In + 1  2B)}n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Note that  by definition the set Iε(U) coincides with εI on the support of ηε,  ηε1Iε(U) = ηε1εI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='7)  In these terms, truncating the two-scale expansions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='5), we are led to considering the  following truncated two-scale expansion errors,  wε := uε − u2s  ε ,  Qε := Pε1U\\Iε(U) − P 2s  ε ,  where the two-scale expansion (u2s  ε , P 2s  ε ) of (uε, Pε) is given by  u2s  ε (x)  :=  ¯uε(x) + εηε(x)ψE(x  ε)∂E ¯uε(x) + εκηε(x)φχδ∗D(uε)(x  ε),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='8)  P 2s  ε (x)  :=  −P ∗  ε (x) + ¯Pε(x) + ηε(x)¯b : D(¯uε)(x) + κηε(x)¯c(χδ ∗ D(uε)(x))  +ηε(x)(ΣE1Rd\\I)(x  ε)(∂E ¯uε)(x) + κηε(x)(Πχδ∗D(uε)1Rd\\I)(x  ε),  where we have further added a locally constant pressure field  P ∗  ε := P ′  ε1U\\Iε(U) +  �  n∈Nε(U)  P ′′  ε,n1εIn,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='9)  HOMOGENIZATION FOR ACTIVE SUSPENSIONS  31  in terms of some constants P ′  ε and {P ′′  ε,n}n to be suitably chosen later on, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='44), and  where the limiting profile (¯uε, ¯Pε) ∈ H1  0(U)d × L2(U)/R is chosen as the unique solution of  − div(2 ¯Bpas D(¯uε)) + ∇ ¯Pε = (1 − λ)h + div(2κ ¯Bact(χδ ∗ D(uε))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='10)  Again, this equation is viewed as a convenient intermediate step towards the relevant  homogenized equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='20): as in the two-scale expansion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='8), the background fluid  deformation χδ ∗ D(uε) is expressed here in terms of the microscopic solution uε itself and  its limit will only be identified at the very end of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' We split the proof into three  main steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Equation for the two-scale expansion error (wε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Qε): in the weak sense in U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  − △wε + ∇Qε = −div  �  (JE1I)( ·  ε)ηε∂E ¯uε + κ(Kχδ∗D(uε)1I)( ·  ε) ηε  �  −  �  n∈Nε(U)  �  δε∂Inσ(uε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Pε + P ′  ε − P ′′  ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='n)ν + κ  ε fn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='ε  �  χδ ∗ D(uε)∤εIn  �  1εIn  �  − κ  ε  �  n∈Nε(U)  �  ηεfn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='ε(χδ ∗ D(uε)) − fn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='ε  �  χδ ∗ D(uε)∤εIn  ��  − κdFχδ∗D(uε),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='E( ·  ε)ηε∇(χδ ∗ ∂Euε)  + (1 − ηε)(λ − 1Iε(U))h − κ ¯F (χδ ∗ D(uε))∇ηε  − div  �  (1 − ηε)  �  2( ¯Bpas − Id) D(¯uε) + 2κ ¯Bact(χδ ∗ D(uε))  ��  + εdiv  ��  2ψE ⊗s − Id ⊗ψE − ΥE  �  ( ·  ε)∇(ηε∂E¯uε) − ηεh ⊗ (△−1∇1I)( ·  ε)  + κ  �  2φχδ∗D(uε) ⊗s − Id ⊗φχδ∗D(uε) − θχδ∗D(uε) + γχδ∗D(uε) ⊗ Id  �  ( ·  ε)∇ηε  + κ  �  2dφχδ∗D(uε),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='E ⊗s − Id ⊗dφχδ∗D(uε),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='E − dθχδ∗D(uε),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='E  + dγχδ∗D(uε),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='E ⊗ Id  �  ( ·  ε) ηε∇(χδ ∗ ∂Euε)  �  + κε∇i  �  (△−1∇idF|χδ∗D(uε),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='E)( ·  ε)ηε∇(χδ ∗ ∂Euε)  �  + ε(△−1∇j1I)( ·  ε)∇j(ηεh) − κεγχδ∗D(uε)( ·  ε)△ηε − κεdγχδ∗D(uε),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='E( ·  ε) : ηε△(χδ ∗ ∂Euε)  − κε(△−1∇idF|χδ∗D(uε),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='E)( ·  ε)∇i(ηε∇(χδ ∗ ∂Euε))  + 2κε  �  dθχδ∗D(uε),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='E − dγχδ∗D(uε),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='E ⊗ Id  �  ( ·  ε) : ∇(χδ ∗ ∂Euε) ⊗ ∇ηε  + κε  �  d2θχδ∗D(uε),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='E′ − d2γχδ∗D(uε),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='E′ ⊗ Id −△−1∇d2F|χδ∗D(uε),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='E′)  �  ( ·  ε)  : ηε∇(χδ ∗ ∂E′uε) ⊗ ∇(χδ ∗ ∂Euε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='11)  We split the proof into six substeps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Substep 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Reformulation of the equation for uε:  − △uε + ∇(Pε1U\\Iε(U) + P ∗  ε ) = h1U\\Iε(U) + κ  ε  �  n∈Nε(U)  fn,ε  �  χδ ∗ D(uε)∤εIn  �  −  �  n∈Nε(U)  �  δε∂Inσ(uε, Pε + P ′  ε − P ′′  ε,n)ν + κ  εfn,ε  �  χδ ∗ D(uε)∤εIn  �  1εIn  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='12)  Starting from equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='7) in form of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='1), we find  − △uε + ∇(Pε1U\\Iε(U))  32  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' BERNOU, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' DUERINCKX, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' GLORIA  = h1U\\Iε(U) + κ  ε  �  n∈Nε(U)  fn,ε  �  χδ ∗ D(uε)∤εIn  �  1ε(In+B)\\εIn −  �  n∈Nε(U)  δε∂Inσ(uε, Pε)ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Adding and subtracting the contribution of swimming forces on the particles, this becomes  − △uε + ∇(Pε1U\\Iε(U)) = h1U\\Iε(U) + κ  ε  �  n∈Nε(U)  fn,ε  �  χδ ∗ D(uε)∤εIn  �  −  �  n∈Nε(U)  �  δε∂Inσ(uε, Pε)ν + κ  εfn,ε  �  χδ ∗ D(uε)∤εIn  �  1εIn  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Adding ∇P ∗  ε to both sides, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='9), the claim (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='12) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Substep 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Equation for the two-scale expansion for the two-scale expansion error u2s  ε :  − △u2s  ε + ∇(P 2s  ε + P ∗  ε ) = ∇  �  ¯Pε + ηε¯b : D(¯uε) + κηε¯c(χδ ∗ D(uε))  �  − T 1  ε − κT 2  ε − div  �  2(1 − ηε) D(¯uε)  �  − εdiv  �  2ψE( ·  ε) ⊗s ∇(ηε∂E¯uε) + 2κφχδ∗D(uε)( ·  ε) ⊗s ∇ηε  + 2κdφχδ∗D(uε),Eα( ·  ε) ⊗s ηε∇(χδ ∗ ∂Eαuε)  �  + ε∇  �  ψE( ·  ε) · ∇(ηε∂E ¯uε) + κφχδ∗D(uε)( ·  ε) · ∇ηε  + κdφχδ∗D(uε),Eα( ·  ε) · ηε∇(χδ ∗ ∂Eαuε)  �  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='13)  where  T 1  ε  :=  div  ��  2(D(ψE) + E) − ΣE1Rd\\I Id  �  ( ·  ε) ηε∂E ¯uε  �  ,  T 2  ε  :=  div  ��  2 D(φ|χδ∗D(uε)) − Πχδ∗D(uε)1Rd\\I Id  �  ( ·  ε) ηε  �  are two terms that we shall further reformulate in the upcoming substeps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  For the two-scale expansion error (u2s  ε , P 2s  ε ) defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='8), we have  − △u2s  ε + ∇(P 2s  ε + P ∗  ε ) = −△¯uε + ∇  �  ¯Pε + ηε¯b : D(¯uε) + κηε¯c(χδ ∗ D(uε))  �  − △  �  εψE( ·  ε)ηε∂E ¯uε + εκφχδ∗D(uε)( ·  ε)ηε  �  + ∇  �  (ΣE1Rd\\I)( ·  ε)ηε∂E ¯uε + κ(Πχδ∗D(uε)1Rd\\I)( ·  ε)ηε  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='14)  It remains to reformulate the penultimate right-hand side term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' By the identity  △h = div(2 D(h)) − ∇div(h),  we can write  △  �  εψE( ·  ε)ηε∂E ¯uε  �  =  div  �  2 D(εψE( ·  ε)ηε∂E ¯uε)  �  − ∇div  �  εψE( ·  ε)ηε∂E ¯uε  �  ,  △  �  εφχδ∗D(uε)( ·  ε)ηε  �  =  div  �  2 D(εφχδ∗D(uε)( ·  ε)ηε)  �  − ∇div  �  εφχδ∗D(uε)( ·  ε)ηε  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  First, as div(ψE) = 0, we find  D(εψE( ·  ε)ηε∂E¯uε)  =  D(ψE)( ·  ε)ηε∂E ¯uε + εψE( ·  ε) ⊗s ∇(ηε∂E ¯uε),  div(εψE( ·  ε)ηε∂E¯uε)  =  εψE( ·  ε) · ∇(ηε∂E ¯uε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='15)  HOMOGENIZATION FOR ACTIVE SUSPENSIONS  33  Second, as div(φE) = 0, further recalling (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='6), we find  D(εφχδ∗D(uε)( ·  ε)ηε)  =  D(φ|χδ∗D(uε))( ·  ε)ηε + εdφχδ∗D(uε),E( ·  ε) ⊗s ηε∇(χδ ∗ ∂Euε)  +εφχδ∗D(uε)( ·  ε) ⊗s ∇ηε,  div(εφχδ∗D(uε)( ·  ε)ηε)  =  εφχδ∗D(uε)( ·  ε) · ∇ηε  +εdφχδ∗D(uε),E( ·  ε) · ηε∇(χδ ∗ ∂Euε),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='16)  where we recall that we implicitly sum over E in an orthonormal basis of Msym  0  , and where  the linearized corrector dφ is given in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  In these terms, the penultimate  right-hand side term in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='14) takes the form  △  �  εψE( ·  ε)ηε∂E ¯uε + εκφχδ∗D(uε)( ·  ε)ηε  �  =  div  �  2 D(ψE)( ·  ε)ηε∂E ¯uε + 2κ D(φ|χδ∗D(uε))( ·  ε)ηε + 2εψE( ·  ε) ⊗s ∇(ηε∂E ¯uε)  + 2εκφχδ∗D(uε)( ·  ε) ⊗s ∇ηε + 2εκdφχδ∗D(uε),Eα( ·  ε) ⊗s ηε∇(χδ ∗ ∂Eαuε)  �  − ∇  �  εψE( ·  ε) · ∇(ηε∂E ¯uε) + εκφχδ∗D(uε)( ·  ε) · ∇ηε  +εκdφχδ∗D(uε),Eα( ·  ε) · ηε∇(χδ ∗ ∂Eαuε)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Inserting this identity into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='14), using that div(¯uε) = 0, and decomposing  △¯uε = div(2 D(¯uε)) = div  �  2(1 − ηε) D(¯uε)  �  + div(2Eηε∂E¯uε),  the claim (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='13) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Substep 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Proof of  T 1  ε = div  ��  2(D(ψE) + E) − ΣE1Rd\\I Id  �  ( ·  ε) ηε∂E ¯uε  �  = div  �  2ηε ¯Bpas D(¯uε)  �  + ∇  �  ηε¯b : D(¯uε)  �  − εdiv  �  ΥE( ·  ε)∇(ηε∂E ¯uε)  �  − div  �  (JE1I)( ·  ε)ηε∂E ¯uε  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='17)  In terms of the extended flux JE, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='2, we have  div  ��  2(D(ψE) + E) − ΣE1Rd\\I Id  �  ( ·  ε) ηε∂E ¯uε  �  = div  �  (JE1Rd\\I)( ·  ε) ηε∂E¯uε  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Recalling that div(JE) = 0, we can decompose  div  ��  2(D(ψE) + E) − ΣE1Rd\\I Id  �  ( ·  ε) ηε∂E¯uε  �  = JE( ·  ε)∇(ηε∂E¯uε) − div  �  (JE1I)( ·  ε) ηε∂E ¯uε  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  As Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='2 further yields JE − E [JE] = div(ΥE) and E [JE] = 2 ¯BpasE + (¯b : E) Id,  where the flux corrector ΥE is skew-symmetric in its last two indices, the claim (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='17)  follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' For completeness, we recall the standard argument based on skew-symmetry of Υ  that leads to this identity: for any smooth scalar field ζ, we have  (JE − E [JE])∇ζ  =  div(ΥE)∇ζ  =  ei(∇kΥE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='ijk)∇jζ  =  ei∇k  �  ΥE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='ijk  � �� �  =−ΥE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='ikj  ∇jζ  �  − eiΥE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='ijk∇2  jkζ  �  ��  �  =0  34  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' BERNOU, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' DUERINCKX, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' GLORIA  =  −div(ΥE∇ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='18)  Substep 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Proof of  T 2  ε = div  ��  2 D(φ|χδ∗D(uε)) − Πχδ∗D(uε)1Rd\\I Id  �  ( ·  ε) ηε  �  = div(2ηε ¯C(χδ ∗ D(uε))) + ∇(ηε¯c(χδ ∗ D(uε)))  − 1  εηε  �  n∈Nε(U)  fn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='ε(χδ ∗ D(uε)) − div  �  (Kχδ∗D(uε)1I)( ·  ε) ηε  �  − εdiv  ��  θχδ∗D(uε) − γχδ∗D(uε) ⊗ Id  �  ( ·  ε)∇ηε  �  − εdiv  ��  dθχδ∗D(uε),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='E − dγχδ∗D(uε),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='E ⊗ Id  �  ( ·  ε) ηε∇(χδ ∗ ∂Euε)  �  − εγχδ∗D(uε)( ·  ε)△ηε − εdγχδ∗D(uε),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='E( ·  ε) : ηε△(χδ ∗ ∂Euε)  + 2ε  �  dθχδ∗D(uε),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='E − dγχδ∗D(uε),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='E ⊗ Id  �  ( ·  ε) : ∇(χδ ∗ ∂Euε) ⊗ ∇ηε  + ε  �  d2θχδ∗D(uε),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='E′ − d2γχδ∗D(uε),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='E′ ⊗ Id  �  ( ·  ε)  : ηε∇(χδ ∗ ∂E′uε) ⊗ ∇(χδ ∗ ∂Euε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='19)  In terms of the extended flux KE, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='5, we have  div  ��  2 D(φ|χδ∗D(uε)) − Πχδ∗D(uε)1Rd\\I Id  �  ( ·  ε) ηε  �  = div  �  (Kχδ∗D(uε)1Rd\\I)( ·  ε) ηε  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  As Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='5 further yields E [KE] = 2 ¯C(E) + ¯c(E) Id and div(KE) = − �  n fn(E), and  appealing to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='6) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='7), we find  div  ��  2 D(φ|χδ∗D(uε)) − Πχδ∗D(uε)1Rd\\I Id  �  ( ·  ε) ηε  �  = div(2ηε ¯C(χδ ∗ D(uε))) + ∇(ηε¯c(χδ ∗ D(uε))) − 1  εηε  �  n∈Nε(U)  fn,ε(χδ ∗ D(uε))  − div  �  (Kχδ∗D(uε)1I)( ·  ε) ηε  �  +  �  Kχδ∗D(uε)( ·  ε) − E  �  K|χδ∗D(uε)  ��  ∇ηε  +  �  dKχδ∗D(uε),E( ·  ε) − E  �  dK|χδ∗D(uε),E( ·  ε)  ��  ηε∇(χδ ∗ ∂Euε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='20)  It remains to reformulate the last two right-hand side terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' As Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='5 yields  KE − E [KE] = div(θE) + ∇γE,  we get  �  Kχδ∗D(uε)( ·  ε) − E  �  K|χδ∗D(uε)  ��  ∇ηε =  �  div(θ|χδ∗D(uε)) + ∇γ|χδ∗D(uε)  �  ( ·  ε)∇ηε,  and thus, by Leibniz’ rule, using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='6) and the skew-symmetry of θ (whence the minus sign  of the first right-hand side term, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='18)),  �  Kχδ∗D(uε)( ·  ε) − E  �  K|χδ∗D(uε)  ��  ∇ηε  = −εdiv  ��  θχδ∗D(uε) − γχδ∗D(uε) ⊗ Id  �  ( ·  ε)∇ηε  �  − εγχδ∗D(uε)( ·  ε)△ηε  + ε  �  dθχδ∗D(uε),E − dγχδ∗D(uε),E ⊗ Id  �  ( ·  ε) : ∇(χδ ∗ ∂Euε) ⊗ ∇ηε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Similarly, with the notation of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='6, we have  �  dKχδ∗D(uε),E( ·  ε) − E  �  dK|χδ∗D(uε),E  ��  ηε∇(χδ ∗ ∂Euε)  HOMOGENIZATION FOR ACTIVE SUSPENSIONS  35  =  �  div(dθ|χδ∗D(uε),E) + ∇dγ|χδ∗D(uε),E  �  ( ·  ε) ηε∇(χδ ∗ ∂Euε),  and thus, by Leibniz’ rule, using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='6) and the skew-symmetry of dθ,  �  dKχδ∗D(uε),E( ·  ε) − E  �  dK|χδ∗D(uε),E  ��  ηε∇(χδ ∗ ∂Euε)  = −εdiv  ��  dθχδ∗D(uε),E − dγχδ∗D(uε),E ⊗ Id  �  ( ·  ε) ηε∇(χδ ∗ ∂Euε)  �  + ε  �  dθχδ∗D(uε),E − dγχδ∗D(uε),E ⊗ Id  �  ( ·  ε) : ∇(ηε∇(χδ ∗ ∂Euε))  + ε  �  d2θχδ∗D(uε),E,E′ − d2γχδ∗D(uε),E,E′ ⊗ Id  �  ( ·  ε) : ηε∇(χδ ∗ ∂E′uε) ⊗ ∇(χδ ∗ ∂Euε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Inserting this into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='20), the claim (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='19) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Substep 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' In order to reconstruct ¯Bact(χδ ∗ D(uε)), we shall need the following identity:  div(ηε ¯F (χδ ∗ D(uε))) = dFχδ∗D(uε),E( ·  ε)ηε∇(χδ ∗ ∂Euε) + ¯F (χδ ∗ D(uε))∇ηε  − ε∇i  �  (△−1∇idF|χδ∗D(uε),E)( ·  ε)ηε∇(χδ ∗ ∂Euε)  �  + ε(△−1∇idF|χδ∗D(uε),E)( ·  ε)∇i(ηε∇(χδ ∗ ∂Euε))  + ε(△−1∇id2F|χδ∗D(uε),E,E′)( ·  ε)ηε∇(χδ ∗ ∂Euε)∇i(χδ ∗ ∂E′(uε)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='21)  As E [FE] = ¯F (E), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='31), and using the notation in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='6, we can decompose  dFE,E′  =  ∂E′ ¯F (E) +  �  dFE,E′ − E  �  dFE,E′��  =  ∂E′ ¯F (E) + ∇i△−1∇idFE,E′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Using this together with Leibniz’ rule, we find  dFχδ∗D(uε),E( ·  ε)ηε∇(χδ ∗ ∂Euε) = ∂E ¯F (χδ ∗ D(uε))ηε∇(χδ ∗ ∂Euε)  + ε∇i  �  (△−1∇idF|χδ∗D(uε),E)( ·  ε)  �  ηε∇(χδ ∗ ∂Euε)  − ε(△−1∇id2F|χδ∗D(uε),E,E′)( ·  ε)ηε∇(χδ ∗ ∂Euε)∇i(χδ ∗ ∂E′(uε)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Further reformulating the first right-hand side term, noting that  ∂E ¯F (χδ ∗ D(uε))∇(χδ ∗ ∂Euε) = div( ¯F (χδ ∗ D(uε))),  the claim (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='21) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Substep 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Proof of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Subtracting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='12) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='13), inserting identities (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='17), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='19), and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='21), using equa-  tion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='10) for ¯uε, and decomposing  h1U\\Iε(U) − (1 − λ)h  =  (λ − 1Iε(U))(1 − ηε)h + (λ − 1εI)ηεh  =  (λ − 1Iε(U))(1 − ηε)h  −ε∇j  �  (△−1∇j1I)( ·  ε)ηεh  �  + ε(△−1∇j1I)( ·  ε)∇j(ηεh),  the claim (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='11) follows after straightforward simplifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  In the rest of the proof, for notational convenience, we do not make explicit the dependence  of estimates wrt κ and ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Energy estimate for (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='11): for all K ≥ 1,  36  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' BERNOU, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' DUERINCKX, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' GLORIA  ˆ  U  |∇wε|2 ≲  1  K  ˆ  U  (Qε)2 +  �  1 + ∥(h, ∇¯uε)∥2  W 1,∞(U) + ∥χδ ∗ D(uε)∥6  W 2,∞(U)  �  ×  � ˆ  ∂3rεU  |(1, ∇ψ, Σ1Rd\\I, ∇φ, Π1Rd\\I)( ·  ε)|2  + ε2K  ˆ  U  �  1 + r−2  ε 1∂3rεU  �  |(1, ψ, ∇ψ, Υ, φ, θ, γ, dφ, ∇dφ, dθ, dγ, d2θ, d2γ,  △−1∇1I, △−1∇dF, △−1∇d2F)( ·  ε)|2  �  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='22)  where we use the following notation for correctors,  |ψ| := sup  E  |E|−1|ψE|,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='23)  and similarly for |Υ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' For the active corrector φ, which depends nonlinearly on the direc-  tion E, as well as for θ, γ, we rather set  |φ| :=  sup  |E|≤Cδ(h)  ⟨E⟩−1|φE|,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='24)  where the deterministic constant Cδ(h) > 0 is chosen such that almost surely  ∥χδ ∗ D(uε)∥L∞(U) ≤ ∥χδ∥L2(Rd)∥∇uε∥L2(U) ≤ Cδ(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Such a constant can be chosen in view of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' For the linearized correctors dφ and d2φ,  as well as for dθ, dγ, d2θ, d2γ, △−1∇dF, △−1∇d2F, we similarly set  |dφ|  :=  sup  |E|≤Cδ(h)  sup  E′ |E′|−1|dφE,E′|,  |d2φ|  :=  sup  |E|≤Cδ(h)  sup  E′,E′′ |E′|−1|E′′|−1|d2φE,E′,E′′|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  By Poincaré’s inequality in form of  sup  ε(In+B)  ���∇¯uε −  ε(In+B)  ∇¯uε  ���  ≲  ε∥∇2¯uε∥L∞(U),  sup  ε(In+B)  ���χδ ∗ D(uε) −  ε(In+B)  χδ ∗ D(uε)  ���  ≲  ε∥χδ ∗ D(uε)∥W 1,∞(U),  the claim (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='22) follows from  ˆ  U  |∇wε|2 ≲  1  K  ˆ  U  (Qε)2  +  ˆ  ∂3rεU  |(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' ∇ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Σ1Rd\\I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' ∇φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Π1Rd\\I)( ·  ε)|2�  1 + |h|2 + |∇¯uε|2 + [∇¯uε]2  4ε + |χδ ∗ D(uε)|2�  + ε2r−2  ε K  ˆ  ∂3rεU  |(ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Υ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' dφ)( ·  ε)|2�  1 + |∇¯uε|2 + |χδ ∗ D(uε)|2�  + ε2K  ˆ  U  |(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Υ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' dφ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' dθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' dγ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' d2θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' d2γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' △−1∇1I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' △−1∇dF,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' △−1∇d2F)( ·  ε)|2  ×  �  |⟨∇⟩h|2 + |∇2¯uε|2 + |∇⟨∇⟩χδ ∗ D(uε)|2 + |∇χδ ∗ D(uε)|4  + |∇(χδ ∗ ∂Euε)|2�  1 + [χδ ∗ D(uε)]4  4ε  ��  HOMOGENIZATION FOR ACTIVE SUSPENSIONS  37  + K  �  n  ˆ  ε(In+B)  |(ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' ∇ψ)( ·  ε)|2���∇¯uε −  ε(In+B)  ∇¯uε  ���  2  + K  �  n  ˆ  ε(In+B)  |(dφ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' ∇dφ)( ·  ε)|2���χδ ∗ D(uε) −  ε(In+B)  χδ ∗ D(uε)  ���  2  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='25)  where we use the short-hand notation [g]4ε(x) := (  ffl  B4ε(x) |g|2)  1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' We split the proof of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='25)  into seven substeps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Substep 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Preliminary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  In order to obtain (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='25), we may wish to test equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='11) with wε itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' However,  wε is not rigid inside particles, which prevents us from taking advantage of the boundary  conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' To circumvent this issue, we make use of the following truncation maps T ε  0 , T ε  1 :  for all g ∈ C∞  b (Rd)d,  T ε  0 [g]  :=  (1 − ρε)g +  �  n  ρε  n  �  ε(In+B)  g  �  ,  T ε  1 [g]  :=  (1 − ρε)g +  �  n  ρε  n  ��  ε(In+B)  g  �  + (· − εxn)j  �  ε(In+B)  ∂jg  ��  ,  where for all n we have chosen a cut-off function ρε  n ∈ C∞  c (Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' [0, 1]) with  ρε  n|ε(In+ 1  4 B) = 1,  ρε  n|Rd\\ε(In+ 1  2 B) = 0,  |∇ρε  n| ≲ 1  ε,  and where we have set for abbreviation ρε := �  n ρε  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' In these terms, we shall test (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='11)  with the following modification of the two-scale expansion error wε, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='8),  ˜wε := uε − T ε  1 [¯uε] − εηεψE( ·  ε)T ε  0 [∂E ¯uε] − εκηεφT ε  0 [χδ∗D(uε)]( ·  ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Testing equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='11) with ˜wε, we obtain  Iε  0 = Iε  1 + Iε  2 + Iε  3 + Iε  4 + Iε  5,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='26)  in terms of  Iε  0  :=  ˆ  U  ∇ ˜wε : ∇wε −  ˆ  U  Qε div( ˜wε),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Iε  1  :=  ˆ  U  ηε∇ ˜wε :  �  (JE1I)( ·  ε)∂E ¯uε + κ(Kχδ∗D(uε)1I)( ·  ε)  �  Iε  2  :=  −  �  n∈Nε(U)  � ˆ  ε∂In  ˜wε · σ(uε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Pε + P ′  ε − P ′′  ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='n)ν + κ  ε  ˆ  εIn  ˜wε · fn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='ε  �  χδ ∗ D(uε)∤εIn  ��  Iε  3  :=  − κ  ε  �  n∈Nε(U)  ˆ  ε(In+B)  ηε ˜wε ·  �  fn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='ε(χδ ∗ D(uε)) − fn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='ε  �  χδ ∗ D(uε)∤εIn  ��  −κ  ˆ  U  ηε ˜wε ⊗ ∇(χδ ∗ ∂Euε) : dFχδ∗D(uε),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='E( ·  ε),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Iε  4  :=  ˆ  U  (1 − ηε)(λ − 1Iε(U)) ˜wε · h + κ  ε  �  n∈Nε(U)  ˆ  ε(In+B)  (1 − ηε) ˜wε · fn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='ε  �  χδ ∗ D(uε)∤εIn  �  +2  ˆ  U  (1 − ηε) D( ˜wε) :  �  ( ¯Bpas − Id) D(¯uε) + κ ¯Bact(χδ ∗ D(uε))  �  38  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' BERNOU, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' DUERINCKX, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' GLORIA  −κ  ˆ  U  ˜wε ⊗ ∇ηε : ¯F (χδ ∗ D(uε))  and  Iε  5  :=  −ε  ˆ  U  ∇ ˜wε :  ��  2ψE ⊗s − Id ⊗ψE − ΥE  �  ( ·  ε)∇(ηε∂E ¯uε) − ηεh ⊗ (△−1∇1I)( ·  ε)  +κ  �  2φχδ∗D(uε) ⊗s − Id ⊗φχδ∗D(uε) − θχδ∗D(uε) + γχδ∗D(uε) ⊗ Id  �  ( ·  ε)∇ηε  +κ  �  2dφχδ∗D(uε),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='E ⊗s − Id ⊗dφχδ∗D(uε),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='E − dθχδ∗D(uε),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='E  +dγχδ∗D(uε),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='E ⊗ Id  �  ( ·  ε) ηε∇(χδ ∗ ∂Euε)  �  −κε  ˆ  U  ηε∇i ˜wε ⊗ ∇(χδ ∗ ∂Euε) : (△−1∇idF|χδ∗D(uε))( ·  ε)  +ε  ˆ  U  ˜wε ·  �  (△−1∇j1I)( ·  ε)∇j(ηεh) − κγχδ∗D(uε)( ·  ε)△ηε  −κdγχδ∗D(uε),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='E( ·  ε) : ηε△(χδ ∗ ∂Euε)  −κ(△−1∇idF|χδ∗D(uε),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='E)( ·  ε)∇i(ηε∇(χδ ∗ ∂Euε))  +2κ  �  dθχδ∗D(uε),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='E − dγχδ∗D(uε),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='E ⊗ Id  �  ( ·  ε) : ∇(χδ ∗ ∂Euε) ⊗ ∇ηε  +κ  �  d2θχδ∗D(uε),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='E′ − d2γχδ∗D(uε),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='E′ ⊗ Id −△−1∇id2F|χδ∗D(uε),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='E′  �  ( ·  ε)  : ηε∇(χδ ∗ ∂E′uε) ⊗ ∇(χδ ∗ ∂Euε)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  We analyze the different terms separately in the upcoming six substeps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Substep 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Proof that for all K ≥ 1,  Iε  0 ≥ (1 −  1  2K )  ˆ  U  |∇wε|2 − 1  K  ˆ  U  (Qε)2 − K  ˆ  U  |∇( ˜wε − wε)|2  − ε2r−2  ε CK  ˆ  ∂2rεU  |(ψ, φ, dφ)( ·  ε)|2�  1 + |∇¯uε|2 + |χδ ∗ D(uε)|2�  − ε2CK  ˆ  U  |(ψ, φ, dφ)( ·  ε)|2�  |∇2¯uε|2 + |∇χδ ∗ D(uε)|2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='27)  Adding and subtracting wε to ˜wε, we find from Young’s inequality, for all K ≥ 1,  Iε  0 ≥ (1 −  1  2K )  ˆ  U  |∇wε|2 − 1  K  ˆ  U  (Qε)2 − K  ˆ  U  |∇( ˜wε − wε)|2 − K  ˆ  U  |div(wε)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Since div(uε) = div(¯uε) = 0, we get from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='15) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='16),  div(wε) = −εψE( ·  ε) · ∇(ηε∂E ¯uε) − εκφχδ∗D(uε)( ·  ε) · ∇ηε  − εκdφχδ∗D(uε),E( ·  ε) · ηε∇(χδ ∗ ∂Euε),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='28)  and the claim (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='27) follows using the properties of ηε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Substep 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Proof that  Iε  1 ≲  � ˆ  ∂2rεU  �  1 + |∇¯uε|2 + |χδ ∗ D(uε)|2�� 1  2  ×  � ˆ  ∂3rεU  |(1, ∇ψ, Σ1Rd\\I, ∇φ, Π1Rd\\I)( ·  ε)|2[∇¯uε]2  4ε  � 1  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='29)  HOMOGENIZATION FOR ACTIVE SUSPENSIONS  39  First note that for all n we have in εIn, since D(ψE)|In = −E,  D( ˜wε)|εIn  =  −  �  D(T ε  1 [¯uε]) + ηε D(ψE)( ·  ε)T ε  0 [∂E ¯uε]  ����  εIn  =  −(1 − ηε)|εIn  ε(In+B)  D(¯uε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='30)  As by construction J and K are symmetric matrix fields, we may replace ∇ ˜wε by D( ˜wε)  in the definition of Iε  1, and we thus find  Iε  1 = −  �  n∈Nε(U)  (ηε(1 − ηε))(εxn)  ε(In+B)  D(¯uε) :  ˆ  εIn  �  JE( ·  ε)∂E ¯uε + κ(Kχδ∗D(uε))( ·  ε)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  By the hardcore assumption, using the properties of ηε, and using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='2), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='25), and the  Lipschitz continuity of E �→ fn(E) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Hypothesis 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='3), the claim (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='29) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Substep 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Proof that  Iε  2 ≲  � ˆ  U  |∇wε|2� 1  2� ˆ  ∂3rεU  |∇¯uε|2� 1  2  +  � ˆ  ∂3rεU  |∇¯uε|2� 1  2� ˆ  ∂3rεU  |(1, ∇ψ, ∇φ)( ·  ε)|2�  1 + |∇¯uε|2 + |χδ ∗ D(uε)|2�� 1  2  + ε  � ˆ  ∂3rεU  |∇¯uε|2� 1  2�  r−2  ε  ˆ  ∂3rεU  |(ψ, φ)( ·  ε)|2�  1 + |∇¯uε|2 + |χδ ∗ D(uε)|2�� 1  2  + ε  � ˆ  ∂3rεU  |∇¯uε|2� 1  2 � ˆ  ∂3rεU  |(ψ, dφ)( ·  ε)|2�  |∇2¯uε|2 + |∇χδ ∗ D(uε)|2�� 1  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='31)  Appealing to the boundary conditions in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='7), recalling the notation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='8), and using  ´  ε∂In ν = 0 and  ´  ε∂In ν ⊗ (x − εxn) = |εIn| Id, we have  Iε  2 = −  �  n∈Nε(U)  �  εIn  D( ˜wε)  �  :  ˆ  ε∂In  σ(uε, Pε + P ′  ε − P ′′  ε,n)ν ⊗ (x − εxn),  and thus, using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='30) again, and noting that the identities  ´  ε∂In ν ⊗ (· − εxn) = |εIn| Id  and div(¯uε) = 0 allow to remove any constant from the pressure field Pε in this expression,  Iε  2 =  �  n∈Nε(U)  (1 − ηε)(εxn)  �  ε(In+B)  D(¯uε)  �  :  ˆ  ε∂In  σ(uε, Pε − P ′′′  ε,n)ν ⊗ (x − εxn), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='32)  where we have chosen P ′′′  ε,n :=  ffl  ε(In+B)\\εIn Pε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' In order to estimate the right-hand side, we  shall turn surface integrals into volume integrals, proceeding as for the construction of KE  in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' More precisely, for all n, we consider the following Neumann problem,  \uf8f1  \uf8f2  \uf8f3  −△un  ε + ∇P n  ε = fn,ε  �  χδ ∗ D(uε)∤εIn  �  ,  in εIn,  div(un  ε ) = 0,  in εIn,  σ(un  ε , P n  ε )ν = σ(uε, Pε − P ′′′  ε,n)ν,  on ε∂In.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  As for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='16), we can show that there is a unique solution un  ε ∈ H1  0(εIn)d with  ffl  εIn un  ε = 0  and  ffl  εIn ∇un  ε ∈ Msym  0  , and a unique pressure P n  ε ∈ L2(εIn)/R, such that  ∥(∇un  ε , P n  ε )∥L2(εIn) ≲ ∥D(uε)∥L2(ε(In+B)) +  ��fn,ε  �  χδ ∗ D(uε)∤εIn  ���  L2(ε(In+B)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='33)  40  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' BERNOU, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' DUERINCKX, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' GLORIA  In these terms, we can reformulate the surface integral in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='32) as follows,  ˆ  ε∂In  σ(uε, Pε)ν ⊗ (x − εxn) =  ˆ  εIn  ∇j  �  σ(un  ε , P n  ε )ej ⊗ (x − εxn)  �  = −  ˆ  εIn  fn,ε  �  χδ ∗ D(uε)∤εIn  �  ⊗ (x − εxn) +  ˆ  εIn  σ(un  ε , P n  ε ),  and thus, in view of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='33),  ����  ˆ  ε∂In  σ(uε, Pε)ν ⊗ (x − εxn)  ����  ≲ |εIn|  1  2  �  |εIn|  1  2 + ∥D(uε)∥L2(ε(In+B)) + ∥χδ ∗ D(uε)∥L2(εIn)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Inserting this into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='32) and using the properties of ηε, we are led to  Iε  2 ≲  � ˆ  ∂3rεU  |∇¯uε|2� 1  2� ˆ  ∂3rεU  �  1 + |∇uε|2 + |χδ ∗ D(uε)|2�� 1  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Inserting then the two-scale expansion to replace the norm of ∇uε in the right-hand side,  uε = wε + ¯uε + εψE( ·  ε)ηε∂E ¯uε + εκφχδ∗D(uε)( ·  ε)ηε,  the claim (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='31) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Substep 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Proof that  |Iε  3| ≲ ε  � ˆ  U  |∇ ˜wε|2� 1  2� ˆ  U  |∇(χδ ∗ D(uε))|4 + |∇2(χδ ∗ D(uε))|2  + |∇(χδ ∗ D(uε))|2�  1 + [χδ ∗ D(uε)]4  4ε  �� 1  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='34)  We start by decomposing  Iε  3 = − κ  ε  �  n∈Nε(U)  ˆ  ε(In+B)  �  ηε ˜wε −  εIn  ηε ˜wε  � �  fn,ε(χδ ∗ D(uε)) − fn,ε  �  χδ ∗ D(uε)∤εIn  ��  − κ  ε  �  n∈Nε(U)  �  εIn  ηε ˜wε  � ˆ  ε(In+B)  �  fn,ε(χδ ∗ D(uε)) − fn,ε  �  χδ ∗ D(uε)∤εIn  ��  − κ  ˆ  U  ηε ˜wε ⊗ ∇(χδ ∗ ∂Euε) : dFχδ∗D(uε),E( ·  ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='35)  We now estimate the first right-hand side term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Using the Lipschitz regularity of fn,ε,  applying Poincaré’s inequality, and using the hardcore assumption, we get  ����  1  ε  �  n∈Nε(U)  ˆ  ε(In+B)  �  ηε ˜wε −  εIn  ηε ˜wε  � �  fn,ε(χδ ∗ D(uε)) − fn,ε  �  χδ ∗ D(uε)∤εIn  ������  ≲ ε  � ˆ  U  |∇(ηε ˜wε)|2� 1  2 � ˆ  U  |∇(χδ ∗ D(uε))|2� 1  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  HOMOGENIZATION FOR ACTIVE SUSPENSIONS  41  We further appeal to the following version of Poincaré’s inequality for ˜wε ∈ H1  0(U)d in ∂2rεU  (this will be used several times in the proof): by the properties of ηε,  ˆ  U  |∇ηε|2 ˜w2  ε ≲ r−2  ε  ˆ  ∂2rεU  ˜w2  ε ≲  ˆ  U  |∇ ˜wε|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='36)  The above then becomes  ����  1  ε  �  n∈Nε(U)  ˆ  ε(In+B)  �  ηε ˜wε −  εIn  ηε ˜wε  � �  fn,ε(χδ ∗ D(uε)) − fn,ε  �  χδ ∗ D(uε)∤εIn  ������  ≲ ε  � ˆ  U  |∇ ˜wε|2� 1  2 � ˆ  U  |∇(χδ ∗ D(uε))|2� 1  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='37)  We turn to the analysis of the last two terms in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' By a Taylor expansion (of the form  |u(a + b) − u(a) − cu′(a)| ≲ b2 sup |u′′| + |b − c||u′(a)|), using the C2 regularity of fn,ε, we  can estimate  ����  ˆ  ε(In+B)  �  fn,ε(χδ ∗ D(uε)) − fn,ε  �  χδ ∗ D(uε)∤εIn  ��  −  �  εIn  ∇i(χδ ∗ ∂Euε)  � ˆ  ε(In+B)  (x − εxn)i ∂Efn,ε(χδ ∗ D(uε)∤εIn)  ����  ≲ ε2  ˆ  ε(In+B)  |∇(χδ ∗ D(uε))|2 + ε2  ˆ  ε(In+B)  |∇2(χδ ∗ D(uε))|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Summing over n and recognizing the definition of dF, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='30),  dFE,E′ := −  �  n  1In  |In|  ˆ  In+B  ∂E′fn(E) ⊗ (x − xn),  we are led to  ����  1  ε  �  n∈Nε(U)  �  εIn  ηε ˜wε  � ˆ  ε(In+B)  �  fn,ε(χδ ∗ D(uε)) − fn,ε  �  χδ ∗ D(uε)∤εIn  ��  +  ˆ  U  ηε ˜wε ⊗ ∇(χδ ∗ ∂Euε) : dFχδ∗D(uε),E( ·  ε)  ����  ≲ ε  � ˆ  U  |ηε ˜wε|2� 1  2 � ˆ  U  |∇(χδ ∗ D(uε))|4 + |∇2(χδ ∗ D(uε))|2� 1  2  + ε  � ˆ  U  |∇(ηε ˜wε)|2� 1  2 � ˆ  U  |∇(χδ ∗ D(uε))|2�  1 + [χδ ∗ D(uε)]4  4ε  �� 1  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Further appealing to Poincaré’s inequality and to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='36), this becomes  ����  1  ε  �  n∈Nε(U)  �  εIn  ηε ˜wε  � ˆ  ε(In+B)  �  fn,ε(χδ ∗ D(uε)) − fn,ε  �  χδ ∗ D(uε)∤εIn  ��  +  ˆ  U  ηε ˜wε ⊗ ∇(χδ ∗ ∂Euε) : dFχδ∗D(uε),E( ·  ε)  ����  ≲ ε  � ˆ  U  |∇ ˜wε|2� 1  2� ˆ  U  |∇(χδ ∗ D(uε))|4 + |∇2(χδ ∗ D(uε))|2  42  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' BERNOU, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' DUERINCKX, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' GLORIA  + |∇(χδ ∗ D(uε))|2�  1 + [χδ ∗ D(uε)]4  4ε  �� 1  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Combining this with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='35) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='37), the claim (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='34) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Substep 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Proof that  Iε  4  ≲  � ˆ  U  |∇ ˜wε|2� 1  2� ˆ  ∂3rεU  �  1 + |h|2 + |∇¯uε|2 + |χδ ∗ D(uε)|2�� 1  2 ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='38)  Iε  5  ≲  ε  � ˆ  U  |∇ ˜wε|2� 1  2 �  r−2  ε  ˆ  ∂2rεU  |(ψ, Υ, φ, θ, γ)|2�  1 + |∇¯uε|2 + |χδ ∗ D(uε)|2�� 1  2  + ε  � ˆ  U  |∇ ˜wε|2� 1  2 � ˆ  U  |(ψ, Υ, dφ, dθ, dγ, △−1∇1I, △−1∇dF)|2  ×  �  |⟨∇⟩h|2 + |∇2¯uε|2 + |∇⟨∇⟩χδ ∗ D(uε)|2�� 1  2  + ε  � ˆ  U  |∇ ˜wε|2� 1  2 � ˆ  U  |(d2θ, d2γ, △−1∇d2F)( ·  ε)|2|∇χδ ∗ D(uε)|4� 1  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='39)  Using properties of ηε, the neutrality condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='9), and Poincaré’s inequality, a direct  estimate yields  Iε  4 ≲  � ˆ  ∂3rεU  �  1 + |h|2 + |∇¯uε|2 + |χδ ∗ D(uε)|2�� 1  2� ˆ  U  |∇ ˜wε|2 +  ˆ  U  |∇ηε|2| ˜wε|2� 1  2 ,  and the claimed estimate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='38) follows by applying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='36) again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' The bound (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='39) on Iε  5  is obtained by similar straightforward computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Substep 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Proof of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Starting from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='26), combining estimates (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='27), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='29), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='31), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='34), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='38), and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='39),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  adding and subtracting wε to ˜wε in the right-hand side,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' and applying Young’s inequality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  we get after straightforward simplifications,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' for all K ≥ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  ˆ  U  |∇wε|2 ≲  1  K  ˆ  U  (Qε)2 + K  ˆ  U  |∇( ˜wε − wε)|2  +  ˆ  ∂3rεU  |(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' ∇ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Σ1Rd\\I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' ∇φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Π1Rd\\I)( ·  ε)|2�  1 + |h|2 + |∇¯uε|2 + [∇¯uε]2  4ε + |χδ ∗ D(uε)|2�  + ε2r−2  ε K  ˆ  ∂3rεU  |(ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Υ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' dφ)( ·  ε)|2�  1 + |∇¯uε|2 + |χδ ∗ D(uε)|2�  + ε2K  ˆ  U  |(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Υ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' dφ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' dθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' dγ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' d2θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' d2γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' △−1∇1I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' △−1∇dF,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' △−1∇d2F)( ·  ε)|2  ×  �  |⟨∇⟩h|2 + |∇2¯uε|2 + |∇⟨∇⟩χδ ∗ D(uε)|2 + |∇χδ ∗ D(uε)|4  + |∇(χδ ∗ D(uε))|2�  1 + [χδ ∗ D(uε)]4  4ε  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='40)  It remains to evaluate the norm of ∇( ˜wε − wε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' By definition of ˜wε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' we have  ∇( ˜wε − wε) = ∇(¯uε − T ε  1 [¯uε]) + ∇  �  εψE( ·  ε)ηε(∂E ¯uε − T ε  0 [∂E ¯uε])  �  + ∇  �  εκφχδ∗D(uε)( ·  ε)ηε − εκφT ε  0 [χδ∗D(uε)]( ·  ε)ηε  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  HOMOGENIZATION FOR ACTIVE SUSPENSIONS  43  and thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' inserting the definition of the truncation operators T ε  0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' T ε  1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' and noting that ηε is  constant in the support of the cut-off functions {ρn  ε }n by definition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  ∇( ˜wε − wε) =  �  n  ρε  n  �  ∇¯uε −  ε(In+B)  ∇¯uε  �  +  �  n  ∇ψE( ·  ε)ηερε  n  �  ∂E ¯uε −  ε(In+B)  ∂E ¯uε  �  + κ  �  n  �  ∇φ|χδ∗D(uε)( ·  ε) − ∇φχδ∗D(uε)∤ε(In+B)( ·  ε)  �  ηερε  n  +  �  n  εψE( ·  ε)ηερε  n∇∂E ¯uε +  �  n  εκdφχδ∗D(uε),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='E( ·  ε) ⊗ ηερε  n∇(χδ ∗ ∂Euε)  +  �  n  �  ¯uε −  �  ε(In+B)  ¯uε  �  −  �  ε(In+B)  ∇¯uε  �  (x − εxn)  �  ⊗ ∇ρε  n  +  �  n  εψE( ·  ε) ⊗ ηε∇ρε  n  �  ∂E¯uε −  ε(In+B)  ∂E ¯uε  �  +  �  n  εκ  �  φχδ∗D(uε)( ·  ε) − φχδ∗D(uε)∤ε(In+B)( ·  ε)  �  ⊗ ηε∇ρε  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Using that  ˆ  ε(In+B)  ��φχδ∗D(uε)( ·  ε) − φχδ∗D(uε)∤ε(In+B)( ·  ε)  ��2  ≲ ε2  ˆ  ε(In+B)  |dφ( ·  ε)|2���χδ ∗ D(uε) −  ε(In+B)  χδ ∗ D(uε)  ���  2  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  and  ˆ  ε(In+B)  ��∇φ|χδ∗D(uε)( ·  ε) − ∇φχδ∗D(uε)∤ε(In+B)( ·  ε)  ��2  ≲ ε2  ˆ  ε(In+B)  |(∇dφ)( ·  ε)|2���χδ ∗ D(uε) −  ε(In+B)  χδ ∗ D(uε)  ���  2  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  a direct computation then leads us to  ˆ  U  |∇( ˜wε − wε)|2 ≲ ε2  ˆ  U  |(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' ψ)( ·  ε)|2|∇2¯uε|2 + ε2  ˆ  U  |dφ( ·  ε)|2|∇χδ ∗ D(uε)|2  +  �  n  ˆ  ε(In+B)  |(ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' ∇ψ)( ·  ε)|2���∇¯uε −  ε(In+B)  ∇¯uε  ���  2  +  �  n  ˆ  ε(In+B)  |(dφ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' ∇dφ)( ·  ε)|2���χδ ∗ D(uε) −  ε(In+B)  χδ ∗ D(uε)  ���  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='41)  Inserting this into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='40), the conclusion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='25) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Pressure estimate for (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='11):  ˆ  U  (Qε)2 ≲  ˆ  U  |∇wε|2 + rε  �  1 + ∥(h, ∇¯uε, χδ ∗ D(uε))∥2  L∞(U)  �  + ε2�  1 + ∥(h, ∇¯uε, ¯Pε)∥2  W 1,∞(U) + ∥χδ ∗ D(uε)∥6  W 2,∞(U)  �  ×  ˆ  U  �  1 + r−2  ε 1∂3rεU  ����  1, ψ, Υ, Σ1Rd\\I, φ, θ, γ, dφ, dθ, dΠ1Rd\\I, dγ, d2θ, d2γ,  44  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' BERNOU, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' DUERINCKX, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' GLORIA  △−1∇1I, △−1∇dF, △−1∇d2F  �  ( ·  ε)  ��2,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='42)  which follows from  ˆ  U  (Qε)2 ≲  ˆ  U  |∇wε|2 +  ˆ  ∂3rεU  �  1 + |h|2 + |∇¯uε|2 + |χδ ∗ D(uε)|2�  + ε2r−2  ε  ˆ  ∂3rεU  |(ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Υ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' dθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' dγ)( ·  ε)|2�  1 + |∇¯uε|2 + |χδ ∗ D(uε)|2�  + ε2  ˆ  U  |(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Υ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' dφ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' dθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' dγ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' d2θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' d2γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' △−1∇1I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' △−1∇dF,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' △−1∇d2F)( ·  ε)|2  ×  �  |⟨∇⟩h|2 + |∇2¯uε|2 + |∇ ¯Pε|2 + |∇⟨∇⟩χδ ∗ D(uε)|2 + |∇χδ ∗ D(uε)|4  + |∇(χδ ∗ D(uε))|2�  1 + [χδ ∗ D(uε)]4  4ε  ��  +  �  n  ˆ  ε(In+B)  |(Σ1Rd\\I)( ·  ε)|2���∇¯uε −  ε(In+B)  ∇¯uε  ���  2  +  �  n  ˆ  ε(In+B)  |(dΠ1Rd\\I)( ·  ε)|2���χδ ∗ D(uε) −  ε(In+B)  χδ ∗ D(uε)  ���  2  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='43)  after taking uniform norms of h, ¯uε, χδ ∗ D(uε) and using Poincaré’s inequality in the last  two summands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Similarly as in Step 2, we shall appeal to a truncated version of Qε,  ˜Qε := Pε1Rd\\εI + P ∗  ε − T ε  0 [ ¯Pε] − ηε¯b : T ε  0 [D(¯uε)] − κηε¯c(T ε  0 [χδ ∗ D(uε)])  − (ΣE1Rd\\I)( ·  ε)ηεT ε  0 [∂E ¯uε] − κ(ΠT ε  0 [χδ∗D(uε)]1Rd\\I)( ·  ε)ηε,  where we recall that P ∗  ε stands for some locally constant pressure field, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='9),  P ∗  ε := P ′  ε1U\\Iε(U) +  �  n∈Nε(U)  P ′′  ε,n1εIn,  and where we now choose the constants P ′  ε and {P ′′  ε,n}n in such a way that  ˜Qε|Iε(U) = 0  and  ˆ  U  ˜Qε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='44)  Using the Bogovskii operator as in [4], we can construct a vector field Sε ∈ H1  0(U)d such  that Sε|εIn is a constant for all n ∈ Nε(U) and such that  div(Sε) = − ˜Qε  in U,  ˆ  U  |∇Sε|2 ≲  ˆ  U  | ˜Qε|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='45)  Testing equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='11) with Sε, using the property that Sε|εIn is a constant for all n ∈  Nε(U), and using the boundary conditions in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='10),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' we find  ˆ  U  ˜QεQε =  −  ˆ  U  ∇Sε : ∇wε − κ  ˆ  U  ηεSε ⊗ ∇(χδ ∗ ∂Euε) : dFχδ∗D(uε),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='E( ·  ε)  − κ  ε  �  n∈Nε(U)  ˆ  ε(In+B)  ηεSε ·  �  fn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='ε(χδ ∗ D(uε)) − fn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='ε  �  χδ ∗ D(uε)∤εIn  ��  HOMOGENIZATION FOR ACTIVE SUSPENSIONS  45  +  ˆ  U  (1 − ηε)(λ − 1Iε(U))Sε · h − κ  ˆ  U  Sε ⊗ ∇ηε : ¯F (χδ ∗ D(uε))  + κ  ε  �  n∈Nε(U)  ˆ  ε(In+B)  (1 − ηε)Sε · fn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='ε  �  χδ ∗ D(uε)∤εIn  �  +  ˆ  U  (1 − ηε)∇Sε :  �  2( ¯Bpas − Id) D(¯uε) + 2κ ¯C(χδ ∗ D(uε)) + κ ¯F (χδ ∗ D(uε))  �  −ε  ˆ  U  ∇Sε :  ��  2ψE ⊗s − Id ⊗ψE − ΥE  �  ( ·  ε)∇(ηε∂E ¯uε) − ηεh ⊗ (△−1∇1I)( ·  ε)  +κ  �  2φχδ∗D(uε) ⊗s − Id ⊗φχδ∗D(uε) − θχδ∗D(uε) + γχδ∗D(uε) ⊗ Id  �  ( ·  ε)∇ηε  +κ  �  2dφχδ∗D(uε),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='E ⊗s − Id ⊗dφχδ∗D(uε),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='E − dθχδ∗D(uε),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='E  +dγχδ∗D(uε),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='E ⊗ Id  �  ( ·  ε)ηε∇(χδ ∗ ∂Euε)  �  −ε  ˆ  U  ηε∇iSε ⊗ ∇(χδ ∗ ∂Euε) : (△−1∇idF|χδ∗D(uε),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='E)( ·  ε)  +ε  ˆ  U  Sε ·  �  (△−1∇j1I)( ·  ε)∇j(ηεh) − κγχδ∗D(uε)( ·  ε)△ηε  −κdγχδ∗D(uε),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='E( ·  ε) : ηε△(χδ ∗ ∂Euε)  −κ(△−1∇idF|χδ∗D(uε),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='E)( ·  ε)∇i(ηε∇(χδ ∗ ∂Euε))  +2κ  �  dθχδ∗D(uε),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='E − dγχδ∗D(uε),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='E ⊗ Id  �  ( ·  ε) : ∇(χδ ∗ ∂Euε) ⊗ ∇ηε  +κ  �  d2θχδ∗D(uε),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='E′ − d2γχδ∗D(uε),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='E′ ⊗ Id −△−1∇d2F|χδ∗D(uε),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='E′  �  ( ·  ε)  : ηε∇(χδ ∗ ∂E′uε) ⊗ ∇(χδ ∗ ∂Euε)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Adding and subtracting Qε to ˜Qε in the left-hand side,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' and proceeding as in Step 2 to  estimate the different contributions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' we deduce  ˆ  U  (Qε)2 ≲  ˆ  U  ( ˜Qε − Qε)2 +  � ˆ  U  |∇Sε|2� 1  2 � ˆ  U  |∇wε|2� 1  2  +  � ˆ  U  |∇Sε|2� 1  2 � ˆ  ∂3rεU  �  1 + |h|2 + |∇¯uε|2 + |χδ ∗ D(uε)|2�� 1  2  +ε  � ˆ  U  |∇Sε|2� 1  2 �  r−2  ε  ˆ  ∂3rεU  |(ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Υ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' dθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' dγ)|2�  1 + |∇¯uε|2 + |χδ ∗ D(uε)|2�� 1  2  +ε  � ˆ  U  |∇Sε|2� 1  2  � ˆ  U  |(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Υ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' dφ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' dθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' dγ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' d2θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' d2γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' △−1∇1I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' △−1∇dF,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' △−1∇d2F)|2  ×  �  |⟨∇⟩h|2 + |∇2¯uε|2 + |∇⟨∇⟩χδ ∗ D(uε)|2 + |∇χδ ∗ D(uε)|4  +|∇(χδ ∗ D(uε))|2�  1 + [χδ ∗ D(uε)]4  4ε  ��� 1  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Hence, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='45) and Young’s inequality,  ˆ  U  (Qε)2 ≲  ˆ  U  ( ˜Qε − Qε)2 +  ˆ  U  |∇wε|2 +  ˆ  ∂3rεU  �  1 + |h|2 + |∇¯uε|2 + |χδ ∗ D(uε)|2�  46  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' BERNOU, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' DUERINCKX, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' GLORIA  + ε2r−2  ε  ˆ  ∂3rεU  |(ψ, Υ, φ, θ, γ, dθ, dγ)|2�  1 + |∇¯uε|2 + |χδ ∗ D(uε)|2�  + ε2  ˆ  U  |(1, ψ, Υ, dφ, dθ, dγ, d2θ, d2γ, △−1∇1I, △−1∇dF, △−1∇d2F)|2  ×  �  |⟨∇⟩h|2 + |∇2¯uε|2 + |∇⟨∇⟩χδ ∗ D(uε)|2 + |∇χδ ∗ D(uε)|4  + |∇(χδ ∗ D(uε))|2�  1 + [χδ ∗ D(uε)]4  4ε  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='46)  It remains to estimate the norm of ˜Qε − Qε in the right-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' By definition of ˜Qε, we  have  ˜Qε − Qε = ¯Pε − T ε  0 [ ¯Pε] + ηε¯b :  �  D(¯uε) − T ε  0 [D(¯uε)]  �  + κηε  �  ¯c(χδ ∗ D(uε)) − ¯c(T ε  0 [χδ ∗ D uε)])  �  + (ΣE1Rd\\I)( ·  ε)ηε(∂E ¯uε − T ε  0 [∂E ¯uε])  + κ  �  Πχδ∗D(uε)1Rd\\I − ΠT ε  0 [χδ∗D(uε)]1Rd\\I  �  ( ·  ε) ηε,  and thus, using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='32) and proceeding as for (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='41), we get  ˆ  U  ( ˜Qε − Qε)2 ≲ ε2  ˆ  U  �  |∇2¯uε|2 + |∇ ¯Pε|2 + |∇χδ ∗ D(uε)|2�  +  �  n  ˆ  ε(In+B)  |(Σ1Rd\\I)( ·  ε)|2���∇¯uε −  ε(In+B)  ∇¯uε  ���  2  +  �  n  ˆ  ε(In+B)  |(dΠ1Rd\\I)( ·  ε)|2���χδ ∗ D(uε) −  ε(In+B)  χδ ∗ D(uε)  ���  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='47)  Inserting this into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='46), the conclusion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='43) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Choosing K ≥ 1 large enough to absorb part of the pressure into the left-hand side, the  combination of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='22) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='42) yields  ˆ  U  |∇wε|2 +  ˆ  U  (Qε)2 ≲  �  1 + ∥(h, ∇¯uε, ¯Pε)∥2  W 1,∞(U) + ∥χδ ∗ D(uε)∥6  W 2,∞(U)  �  ×  � ˆ  ∂3rεU  |(1, ∇ψ, Σ1Rd\\I, ∇φ, Π1Rd\\I)( ·  ε)|2  + ε2  ˆ  U  �  1 + r−2  ε 1∂3rεU  �  |(1, ψ, ∇ψ, Υ, Σ1Rd\\I, φ, θ, γ, dφ, ∇dφ, dθ, dΠ1Rd\\I, dγ,  d2θ, d2γ, △−1∇1I, △−1∇dF, △−1∇d2F)( ·  ε)|2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='48)  By Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='6, we have ∥∇uε∥L2(U) ≲ 1 + ∥h∥L2(U), and thus for all s > 0,  ∥χδ ∗ D(uε)∥W 1+s,∞(U) ≲ ∥χδ∥H1+s(Rd)(1 + ∥h∥L2(U)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  By the regularity theory for the Stokes equation, using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='32), we then deduce that the  solution (¯uε, ¯Pε) of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='10) satisfies for all s > 0,  ∥(∇¯uε, ¯Pε)∥W 1,∞(U)  ≲  ∥h∥W s,∞(U) + ∥ ¯Bact(χδ ∗ D(uε))∥W 1+s,∞(U)  ≲  ∥h∥W s,∞(U) + ∥χδ∥H1+s(Rd)(1 + ∥h∥L2(U)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='49)  HOMOGENIZATION FOR ACTIVE SUSPENSIONS  47  Inserting these estimates into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='48), we get  ˆ  U  |∇wε|2 +  ˆ  U  (Qε)2  ≲χδ  �  1 + ∥h∥6  W 1,∞(U)  �� ˆ  ∂3rεU  ���  1, ∇ψ, Σ1Rd\\I, ∇φ, Π1Rd\\I  �  ( ·  ε)  ��2  + ε2  ˆ  U  �  1 + r−2  ε 1∂3rεU  �  |(1, ψ, ∇ψ, Υ, Σ1Rd\\I, φ, θ, γ, dφ, ∇dφ, dθ, dΠ1Rd\\I, dγ,  d2θ, d2γ, △−1∇1I, △−1∇dF, △−1∇d2F)( ·  ε)|2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='50)  By the ergodic theorem and by the sublinearity of correctors, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='2, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='3, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='4,  and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='5, we have for any fixed r > 0, almost surely,  lim  ε↓0  ˆ  ∂3rU  ���  1, ∇ψ, Σ1Rd\\I, ∇φ, Π1Rd\\I  �  ( ·  ε)  ��2 ≲ Cr,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='51)  and  lim  ε↓0 ε2  ˆ  U  �  1 + r−21∂3rU  �  |(1, ψ, ∇ψ, Υ, Σ1Rd\\I, φ, θ, γ, dφ, ∇dφ, dθ, dΠ1Rd\\I, dγ,  d2θ, d2γ, △−1∇1I, △−1∇dF, △−1∇d2F)( ·  ε)|2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='52)  Some care is however needed to prove these convergence results as we take suprema in  the notation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='23)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='24): while the linear dependence of ∇ψE on E makes the suprema  |∇ψ| = supE |E|−1|∇ψE| trivial, the same is not true for ∇φE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' In that case, we use the  Sobolev embedding in form of  |(∇φ)( ·  ε)|2 ≤  sup  |E|≤Cδ(h)  |∇φE( ·  ε)|2 ≲  k  �  l=0  ˆ  |E|≤Cδ(h)  |(∇dlφE)( ·  ε)|2 dE,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='53)  for some k > 1  2 dim Msym  0  , and thus  ˆ  ∂3rU  |(∇φ)( ·  ε)|2 ≲  k  �  l=0  ˆ  |E|≤Cδ(h)  � ˆ  ∂3rU  |(∇dlφE)( ·  ε)|2�  dE,  to which the ergodic theorem for (linearized) correctors ∇dlφ in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='3 can now be  applied, leading to the claim (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='51).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Similarly, writing  ε2  ˆ  U  �  1 + r−21∂3rU  �  |φ( ·  ε)|2  ≤  ε2  ˆ  U  �  1 + r−21∂3rU  �  sup  |E|≤Cδ(h)  |φE( ·  ε)|2  ≲  k  �  l=0  ˆ  |E|≤Cδ(h)  �  ε2  ˆ  U  �  1 + r−21∂3rU  �  |dlφE( ·  ε)|2  �  dE,  the claim (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='52) indeed follows from the sublinearity of the (linearized) correctors dlφ in  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Next, inserting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='51)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='52) into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='50) and appealing to a diagonalization argument, we  conclude that there exists a (random) sequence rε ↓ 0 such that for this choice we have,  48  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' BERNOU, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' DUERINCKX, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' GLORIA  almost surely,  lim  ε↓0  � ˆ  U  |∇wε|2 +  ˆ  U  (Qε)2�  = 0,  that is, wε → 0 in H1  0(U) and Qε → 0 in L2(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' On the other hand, note that a priori  estimates (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='19) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='49) entail that up to an extraction we have uε ⇀ u0 and ¯uε ⇀ ¯u0  in H1  0(U), for some u0, ¯u0 ∈ H1  0(U)d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Passing to the weak limit in equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='10) along  this subsequence, we find that ¯u0 satisfies  − div(2 ¯Bpas D(¯u0)) + ∇ ¯P0 = (1 − λ)h + div(2κ ¯Bact(χδ ∗ D(u0))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='54)  Now, by definition of the two-scale expansion error wε, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='8), together with the sub-  linearity of correctors, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='3, the convergence wε → 0 in H1  0(U) implies  uε − ¯uε → 0 in L2(U), and thus u0 = ¯u0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='54), we deduce that ¯u := u0 = ¯u0  actually satisfies the homogenized equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' In view of the well-posedness for the  latter, we conclude uε ⇀ ¯u in H1  0(U) independently of extractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  We turn to the convergence of the pressure field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Recall that we have shown Qε → 0  in L2(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' The a priori estimate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='49) ensures ¯Pε ⇀ ¯P in L2(U), where ¯P is the unique pres-  sure field in L2(U)/R for the homogenized equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' By definition of Qε, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='8),  together with the ergodic theorem for corrector pressures, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='3, and  with the choice (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='44) of P ′  ε, the convergence of the pressure follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Quantitative homogenization and limit δ ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' This section is devoted to the  proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' As the above proof is semi-quantitative, one can infer convergence  rates provided that quantitative mixing assumptions such as Hypothesis 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='2 are further  made on the statistical ensemble of inclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Quantitative rates then allow in particular  to let the parameter δ tend to 0 in a nontrivial regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' We split the proof into three steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Convergence of the homogenized equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='20) as δ ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Writing equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='20) as  −div(2 ¯Bpas D(¯uδ)) + ∇ ¯Pδ = (1 − λ)h + div(2κ ¯Bact(χδ ∗ D(¯uδ))),  and appealing to the regularity theory for the Stokes equation, the unique solution (¯uδ, ¯Pδ) ∈  H1  0(U)d × L2(U)/R satisfies for all 0 < η < 1,  ∥(∇¯uδ, ¯Pδ)∥W 1−η,∞(U) ≲η ∥h∥L∞(U) + κ∥ ¯Bact(χδ ∗ D(¯uδ))∥W 1−η,∞(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='32), we deduce  ∥(∇¯uδ, ¯Pδ)∥W 1−η,∞(U)  ≲η  ∥h∥L∞(U) + κλ∥χδ ∗ D(¯uδ)∥W 1−η,∞(U)  ≲η  ∥h∥L∞(U) + κλ∥∇¯uδ∥W 1−η,∞(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  The smallness condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='18) yields κλ ≲ κℓ−d ≪ 1, and we thus infer  ∥(∇¯uδ, ¯Pδ)∥W 1−η,∞(U) ≲η ∥h∥L∞(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Up to an extraction, this implies (∇¯uδ, ¯Pδ) → (∇¯u0, ¯P0) in L∞(U), for some limit (¯u0, ¯P0) ∈  H1  0(U)d × L2(U)/R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Passing to the limit in equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='20), we find that (¯u0, ¯P0) satisfies  equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Provided that κλ ≪ 1 is small enough, which is ensured by the smallness  condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='18), the well-posedness of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='28) follows from the same argument as for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  We conclude that (¯u0, ¯P0) = (¯u, ¯P) is the unique solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='28) and that ¯uδ → ¯u  in W 1,∞(U) and ¯Pδ → ¯P in L∞(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  HOMOGENIZATION FOR ACTIVE SUSPENSIONS  49  Combining this with Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='7, by a diagonalization argument, we deduce that there is  a (random) sequence δ◦  ε ↓ 0 such that, for any sequence 0 < δε ≤ δ◦  ε, the solution (uε, Pε)  of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='10) with δ = δε satisfies, as ε ↓ 0,  uε  ⇀  ¯u,  in H1  0(U)d,  Pε1U\\Iε(U)  ⇀  (1 − λ) ¯P + (1 − λ)¯b : D(¯u)  +(1 − λ)κ  �  ¯c(D(¯u)) −  ffl  U ¯c(D(¯u))  �  ,  in L2(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='55)  To improve on such a diagonal result, we need to prove a quantitative version of Theo-  rem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='7 and capture the precise dependence on δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' This is the purpose of the next two  steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Corrector estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  As we proved in [10] for passive correctors, under a quantitative mixing assumption such  as Hypothesis 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='2, we have for all s < ∞,  E  �  |(∇ψ, Σ1Rd\\I, ∇Υ)(x)|s� 1  s ≲s 1,  E [|(ψ, Υ)(x)|s]  1  s ≲s µd(x) :=  �  log  1  2(2 + |x|)  :  d = 2,  1  :  d > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='56)  which optimally quantifies the sublinearity of passive correctors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' The method in [10] applies  mutadis mutandis to active correctors, and yields the following: for all E, E′, E′′ ∈ Msym  0  and s < ∞, we get  E  �  |(∇φE, ΠE1Rd\\I, ∇θE, ∇dφE, ∇dθE,E′, ∇d2φE,E′,E′′, ∇d2θE,E′,E′′)(x)|s� 1  s ≲s,E,E′,E′′ 1,  E  �  |(φE, γE, θE, dφE,E′, dγE,E′, dθE,E′,  d2φE,E′,E′′, d2γE,E′,E′′, d2θE,E′,E′′)(x)|s� 1  s ≲s,E,E′,E′′ µd(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='57)  Yet, for our purposes, we further need corresponding estimates on suprema such as  |∇φ| =  sup  |E|≤Cδ(h)  ⟨E⟩−1|∇φE|,  which is not trivial due to the nonlinear dependence on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' As we aim at capturing the best  dependence on δ, we cannot appeal to brutal Sobolev estimates as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='53).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Instead, we  shall take advantage of the above moment estimates (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='57) together with Hypothesis 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  More precisely, we decompose  |∇φ| ≤ sup  E  ⟨E⟩−1|∇φ∞  E | + sup  E  ⟨E⟩−1|∇(φE − φ∞  E )|,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='58)  where we compare φE to the random field φ∞  E that is defined via the same corrector prob-  lem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='4) & (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='5) with the swimming forces {fn(E)}n replaced by their large-E approxi-  mations {f ∞(E)ξn}n, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Hypothesis 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' On the one hand, as f ∞(E) is a deterministic  function of E, the supremum of ∇φ∞  E over E becomes trivial and moment estimates can  be established in the following form, for all s < ∞,  E  ��  supE ⟨E⟩−1|∇φ∞  E |  �s� 1  s ≲s 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  50  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' BERNOU, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' DUERINCKX, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' GLORIA  On the other hand, by the Sobolev embedding, we can bound for all s < ∞, provided  s > dim Msym  0  ,  sup  E  ⟨E⟩−1|∇(φE − φ∞  E )| ≲  � ˆ  Msym  0  ⟨E⟩−s�  |∇(φE − φ∞  E )| + |∇(dφE − dφ∞  E )|  �s dE  � 1  s  and thus, using the proof of moment estimates (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='57) together with Hypothesis 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='4 in form  of  E  ��  |∇(φE − φ∞  E )| + |∇(dφE − dφ∞  E )|  �s� 1  s ≲s ⟨E⟩1−γ,  we deduce for all s < ∞ with s > (1 ∨ 1  γ ) dim Msym  0  ,  E  ��  supE ⟨E⟩−1|∇(φE − φ∞  E )|  �s� 1  s ≲  � ˆ  Msym  0  ⟨E⟩−γs dE  � 1  s  ≲s 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Combining these bounds with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='58), we deduce for all s < ∞,  E [|∇φ|s]  1  s ≲s 1,  uniformly with respect to δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' This string of arguments allows us to post-process (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='57)  into  E  �  |(∇φ, Π1Rd\\I, ∇θ, ∇dφ, ∇dθ, ∇d2φ, ∇d2θ)(x)|s� 1  s  ≲s  1,  E  �  |(φ, γ, θ, dφ, dγ, dθ, d2φ, d2γ, d2θ)(x)|s� 1  s  ≲s  µd(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  In order to capture the best dependence on δ, we need a version of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='48) where the  dependence on ¯uε, χδ ∗ D(uε) does not deteriorate in terms of uniform norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Rather  combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='25) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='43),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' and choosing K ≥ 1 large enough,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' we get  ˆ  U  |∇wε|2 +  ˆ  U  (Qε)2  ≲  ˆ  ∂3rεU  |(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' ∇ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Σ1Rd\\I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' ∇φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Π1Rd\\I)( ·  ε)|2�  1 + |h|2 + |∇¯uε|2 + [∇¯uε]2  4ε + |χδ ∗ D(uε)|2�  + ε2r−2  ε  ˆ  ∂3rεU  |(ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Υ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' dφ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' dθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' dγ)( ·  ε )|2�  1 + |∇¯uε|2 + |χδ ∗ D(uε)|2�  + ε2  ˆ  U  |(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Υ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' dφ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' dθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' dγ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' d2θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' d2γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' △−1∇1I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' △−1∇dF,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' △−1∇d2F)( ·  ε)|2  ×  �  |⟨∇⟩h|2 + |∇2¯uε|2 + |∇ ¯Pε|2 + |∇⟨∇⟩χδ ∗ D(uε)|2 + |∇χδ ∗ D(uε)|4  + |∇(χδ ∗ D(uε))|2�  1 + [χδ ∗ D(uε)]4  4ε  ��  +  �  n  ˆ  ε(In+B)  |(ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' ∇ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Σ1Rd\\I)( ·  ε)|2���∇¯uε −  ε(In+B)  ∇¯uε  ���  2  +  �  n  ˆ  ε(In+B)  |(dφ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' ∇dφ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' dΠ1Rd\\I)( ·  ε)|2���χδ ∗ D(uε) −  ε(In+B)  χδ ∗ D(uε)  ���  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  HOMOGENIZATION FOR ACTIVE SUSPENSIONS  51  By Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='6, we have ∥∇uε∥L2(U) ≲ 1 + ∥h∥L2(U) and thus, for all k ≥ 0 and s ≥ 2,  ∥χδ ∗ D(uε)∥W k,s(U)  ≲  δ−k−d( 1  2− 1  s)�  1 + ∥h∥L2(U)  �  ,  ∥χδ ∗ D(uε)∥W k,s(∂rεU)  ≲  δ−k− d  2 �  δd ∧ rε  � 1  s �  1 + ∥h∥L2(U)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  By the regularity theory for the Stokes equation, using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='32), we then deduce that the  solution (¯uε, ¯Pε) of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='10) satisfies for all s ≥ 2 and η > 0,  ∥(∇¯uε, ¯Pε)∥W 1,s(U)  ≲  δ−1−d( 1  2 − 1  s)�  1 + ∥h∥L2∨s(U)  �  ,  ∥(∇¯uε, ¯Pε)∥Ls(∂3rεU)  ≲  δ− d  2 �  δd ∧ (rεδ−η)  � 1  s �  1 + ∥h∥L∞(U)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Inserting these estimates into the above, we get for all s ≥ 1,  ˆ  U  |∇wε|2 +  ˆ  U  (Qε)2  ≲  �  rε + (rεδ−d−η) ∧ (rεδ−d)  1  s  ��  ∂3rεU  |(1, ∇ψ, Σ1Rd\\I, ∇φ, Π1Rd\\I)( ·  ε)|2s� 1  s  + ε2r−2  ε  �  rε + (rεδ−d−η) ∧ (rεδ−d)  1  s  ��  ∂3rεU  |(ψ, Υ, φ, θ, γ, dφ, dθ, dγ)( ·  ε )|2s� 1  s  + ε2δ−2−d(2+ 1  s )� ˆ  U  |(1, ψ, Υ, ∇ψ, Σ1Rd\\I, φ, dφ, ∇dφ, dθ, dΠ1Rd\\I, dγ,  d2θ, d2γ, △−1∇1I, △−1∇dF, △−1∇d2F)( ·  ε)|2s� 1  s ,  where the multiplicative constant depends on the W 1,∞(U) norm of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Hence, taking the  expectation and using the corrector estimates of Step 2, we get for all s < ∞,  E  �� ˆ  U  |∇wε|2�s� 1  s  + E  �� ˆ  U  (Qε)2�s� 1  s  ≲h  �  1 + ε2µd(1  ε)2r−2  ε  ��  rε + (rεδ−d−η) ∧ (rεδ−d)  1  s  �  + ε2µd(1  ε)2δ−2−d(2+ 1  s)  Choosing rε := εµd(1  ε), this becomes  E  �� ˆ  U  |∇wε|2�s� 1  s  + E  �� ˆ  U  (Qε)2�s� 1  s  ≲h εµd(1  ε) + εµd(1  ε)δ−d−η + (εµd(1  ε))2δ−2−d(2+ 1  s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  As ε, δ ↓ 0 in the regime (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='27), we thus get wε → 0 in Ls(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' H1(U)) and Qε → 0 in  Ls(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' L2(U)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Arguing as for Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='7, and further using the result of Step 1, the  conclusion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Dilute expansion of the effective viscosity  This section is devoted to the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='9, that is, the first-order dilute ex-  pansion of the effective viscosity ¯Btot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' We recall that Einstein’s formula for the passive  contribution was already established in [5] (see also [16, 15, 18, 17]), in form of  �� ¯Bpas − Id −λ1 ¯B(1)  pas  �� ≲ λ2|log λ1|,  52  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' BERNOU, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' DUERINCKX, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' GLORIA  and it remains to prove  �� ¯Bact(E) − λ1 ¯B(1)  act(E)  �� ≲ ⟨E⟩  �  λ2|log λ1| + (λ1λ2 + λ1λ3|log λ1|)  1  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='1)  We split the proof into four steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Periodic approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Define a periodized version PL of the point process P = {xn}n on the cube QL := [− L  2 , L  2 )d,  PL := {xn : n ∈ NL},  NL := {n : xn ∈ QL−4},  and consider the corresponding random set  IL :=  �  n∈NL  In,  In := xn + I◦  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  For notational convenience, we choose an enumeration PL := {xn,L}n and we set In,L :=  xn,L + I◦  n,L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' By definition, under Hypothesis 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='1, for all L, the periodized random set  IL + LZd satisfies the same regularity and hardcore conditions as in Hypothesis 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Moreover, we emphasize the stabilization property PL|QL−4 = P|QL−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Next, we define  ψE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L ∈ L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' H1  per(QL)d) as the unique almost sure solution of the periodic version of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='1),  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  −△ψE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L + ∇ΣE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L = 0,  in QL \\ IL,  div(ψE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L) = 0,  in QL \\ IL,  D(ψE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L + Ex) = 0,  in IL,  ´  ∂In,L σ(ψE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L + Ex, ΣE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L)ν = 0,  ∀n,  ´  ∂In,L Θ(x − εxn,L) · σ(ψE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L + Ex, ΣE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L)ν = 0,  ∀n, ∀Θ ∈ Mskew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='2)  It is easily checked that the map ¯Bact defined in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='26) can be reformulated as  E′ : 2 ¯Bact(E) = lim  L↑∞ E′ : 2 ¯Bact;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L(E),  E′ : 2 ¯Bact;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L(E) := −E  �  L−d �  n∈QL  ˆ  In,L+B  (ψE′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L + E′(x − xn,L)) · fn,L(E)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='3)  We start by decomposing  E′ : 2 ¯Bact;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L(E) = E′ : ¯B(1)  act;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L(E) + E′ : R1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L(E) + E′ : R2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L(E),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='4)  where we have set  E′ : ¯B(1)  act;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L(E) := −L−dE  � �  n  ˆ  In,L+B  (ψn  E′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L + E′(x − xn,L)) · fn,L(E)  �  ,  and where the remainders R1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L(E), R2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L(E) are given by  E′ : R1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L(E)  :=  −L−dE  � �  n̸=m  ˆ  In,L+B  ψm  E′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L · fn,L(E)  �  ,  E′ : R2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L(E)  :=  −L−dE  � �  n  ˆ  In,L+B  �  ψE′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L −  �  m  ψm  E′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L  �  fn,L(E)  �  ,  HOMOGENIZATION FOR ACTIVE SUSPENSIONS  53  in terms of the solution ψn  E′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L of the single-particle periodized problem  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  −△ψn  E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L + ∇Σn  E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L = 0,  in QL \\ In,L,  div(ψn  E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L) = 0,  in QL \\ In,L,  D(ψn  E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L + Ex) = 0,  in In,L,  ffl  ∂In,L σ(ψn  E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L + Ex, Σn  E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L)ν = 0,  ∀n,  ffl  ∂In,L Θ(x − εxn,L) · σ(ψn  E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L + Ex, Σn  E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L)ν = 0,  ∀n, ∀Θ ∈ Mskew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='5)  Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Proof that  |R1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L| ≲ λ2  �  |log λ1| + log L  L  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='6)  As by assumption the point process is independent of particles’ shapes and swimming  forces, we can write, in terms of the two-point intensity g2, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='29),  E′ : R1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L(E) = −L−d  ¨  QL−4×QL−4  E  � ˆ  QL  ψ◦  E′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L(· + x − y) · ˜f ◦(E)  �  g2(x, y) dxdy,  where ˜f ◦ is an iid copy of f ◦, hence independent of ψ◦  E′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Noting that the periodicity  of ψ◦  E′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L yields  ˆ  QL  � ˆ  QL  ψ◦  E′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L(· + x − y) · ˜f ◦(E)  �  dx = 0,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='7)  we can replace the two-point density g2 by the correlation function h2 = g2 − λ2  1, to the  effect of  E′ : R1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L(E) = −L−d  ¨  QL×QL−4  E  � ˆ  QL  ψ◦  E′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L(· + x − y) · ˜f ◦(E)  �  h2(x, y) dxdy  + L−d  ¨  (QL\\QL−4)×QL−4  E  � ˆ  QL  ψ◦  E′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L(· + x − y) · ˜f ◦(E)  �  g2(x, y) dxdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  The neutrality condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='9) entails  ���  ˆ  QL  ψ◦  E′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L(· + x − y) · ˜f ◦(E)  ��� ≲ ⟨E⟩  � ˆ  2B  |∇ψ◦  E′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L(· + x − y)|2� 1  2,  and thus, using standard decay estimates, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' [5, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='2],  ���  ˆ  QL  ψ◦  E′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L(· + x − y) · ˜f ◦(E)  ��� ≲ ⟨E⟩⟨x − y⟩−d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  The above then becomes  |R1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L| ≲ L−d  ¨  QL×QL  ⟨x − y⟩−d |h2(x, y)| dxdy  + L−d  ¨  (QL\\QL−4)×QL  ⟨x − y⟩−d g2(x, y) dxdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  The definition of two-point intensity (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='30) and the decay of correlations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='31) yield  |h2(x′, y′)| ≲ (λ2  1 + g2(x, y)) ∧ ⟨x − y⟩−γ,  g2(x, y) = g2(x, y)1|x−y|≥2ℓ,  Bℓ(x)×Bℓ(y)  g2(x′, y′) dx′dy′ ≲ λ2,  54  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' BERNOU, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' DUERINCKX, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' GLORIA  and the claim (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='6) easily follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Proof that  |R2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L| ≲ (λ1)  1  2(λ2 + λ3|log λ1|)  1  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='8)  We start by decomposing  E′ : R2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L(E) = −L−dE  � �  n  ˆ  (In,L+B)\\In,L  �  ψE′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L −  �  m  ψm  E′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L  �  fn,L(E)  �  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='9)  + L−dE  � �  n  ˆ  ∂In,L  �  ψE′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L −  �  m  ψm  E′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L  �  σ(φE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L, ΠE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L)ν  �  − L−dE  � �  n  ˆ  In,L+B  �  ψE′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L −  �  m  ψm  E′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L  � �  fn,L(E)1In,L + δ∂In,Lσ(φE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L, ΠE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L)ν  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  The first two terms can be recovered in the weak formulation of the equation for φE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L  (periodized version of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='4) as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='2)), when tested with ψE′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L − �  m ψm  E′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L,  − L−dE  � �  n  ˆ  (In,L+B)\\In,L  �  ψE′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L −  �  m  ψm  E′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L  �  fn,L(E)  �  + L−dE  � �  n  ˆ  ∂In,L  �  ψE′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L −  �  m  ψm  E′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L  �  σ(φE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L, ΠE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L)ν  �  = −  ˆ  QL  ∇φE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L : ∇  �  ψE′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L −  �  m  ψm  E′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L  �  ,  and thus, similarly testing the equation for ψE′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L − �  m ψm  E′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L with φE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L, using boundary  conditions and the rigidity condition for φE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L in IL,  − L−dE  � �  n  ˆ  (In,L+B)\\In,L  �  ψE′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L −  �  m  ψm  E′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L  �  fn,L(E)  �  + L−dE  � �  n  ˆ  ∂In,L  �  ψE′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L −  �  m  ψm  E′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L  �  σ(φE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L, ΠE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L)ν  �  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Next, using boundary conditions for φE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L, noting that ψE′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L − ψn  E′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L is rigid in In,L, the  last term in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='9) can be rewritten as  − L−dE  � �  n  ˆ  In,L+B  �  ψE′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L −  �  m  ψm  E′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L  � �  fn,L(E)1In,L + δ∂In,Lσ(φE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L, ΠE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L)ν  ��  = L−dE  � �  n̸=m  ˆ  In,L+B  ψm  E′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L ·  �  fn,L(E)1In,L + δ∂In,Lσ(φE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L, ΠE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L)ν  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Inserting these identities into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='9), we get  E′ : R2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L(E) = L−dE  � �  n̸=m  ˆ  In,L+B  ψm  E′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L ·  �  fn,L(E)1In,L + δ∂In,Lσ(φE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L, ΠE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L)ν  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Using boundary conditions for φE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L to replace ψm  E′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L by ψm  E′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L −  ffl  In,L ψm  E′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L, using the  Poincaré inequality, and a trace estimate, this can be estimated as  HOMOGENIZATION FOR ACTIVE SUSPENSIONS  55  |E′ : R2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L(E)| ≲ E  �  L−d �  n  ˆ  In,L+B  ���  �  m:m̸=n  ∇ψm  E′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L  ���  2� 1  2  × E  �  L−d �  n  ˆ  In,L+B  |fn,L(E)|2 + L−d �  n  ˆ  In,L+B  |∇φE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L|2  � 1  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Taking advantage of explicit renormalizations as in [5, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='4], the first factor can  easily be estimated by  E  �  L−d �  n  ˆ  In,L+B  ���  �  m:m̸=n  ∇ψm  E′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L  ���  2�  ≲ (λ2 + λ3|log λ1|)|E′|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Further noting that  E  �  L−d �  n  ˆ  In,L+B  |fn,L(E)|2  �  ≲ λ1  ˆ  2B  |f◦(E)|2 ≲ λ1⟨E⟩2,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='10)  and using the hardcore condition, we deduce  |R2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L(E)| ≲ (λ2 + λ3|log λ1|)  1  2  �  λ1⟨E⟩2 + E  �  QL  |∇φE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L|2  �� 1  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='11)  It remains to estimate the last integral: starting from the energy identity for φE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L,  ˆ  QL  |∇φE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L|2 =  �  n  ˆ  In,L+B  φE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L · fn,L(E),  and using boundary conditions and the Poincaré inequality to estimate the right-hand side,  we find  ˆ  QL  |∇φE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L|2 ≲  � �  n  ˆ  In,L+B  |fn,L(E)|2  � 1  2� �  n  ˆ  In,L+B  |∇φE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L|2  � 1  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Hence, using the hardcore condition, absorbing the last factor, taking the expectation, and  combining with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='10), we obtain  E  �  QL  |∇φE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L|2  �  ≲ E  �  L−d �  n  ˆ  In,L+B  |fn,L(E)|2  �  ≲ λ1⟨E⟩2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Inserting this into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='11), the claim (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='8) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  In view of Steps 1, 2 and 3, it remains to examine the limit of the main term ¯B(1)  act;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  As by assumption the point process {xn} is independent of particles’ shapes and swimming  forces, we can write  E′ : ¯B(1)  act;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L(E) = −λ1L−d|QL−4|E  � ˆ  2B  (ψ◦  E′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L + E′x) · f ◦(E)  �  ,  and thus, as L ↑ ∞,  lim  L↑∞ E′ : ¯B(1)  act;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='L(E) = −λ1E  � ˆ  2B  (ψ◦  E′ + E′x) · f ◦(E)  �  ,  56  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' BERNOU, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' DUERINCKX, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' GLORIA  in terms of the solution ψ◦  E′ of the whole-space single-particle problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' In case of  spherical particles, I◦ = B, the latter is explicitly solvable, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' [21, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='3],  ψ◦  E′(x) =  �  −E′x  :  |x| ≤ 1,  − d+2  2  (x·E′x)x  |x|d+2  �  1 −  1  |x|2  �  −  E′x  |x|d+2  :  |x| > 1,  and the conclusion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  □  Acknowledgements  Mitia Duerinckx acknowledges financial support from F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='-FNRS, and Armand Bernou  and Antoine Gloria from the European Research Council (ERC) under the European  Union’s Horizon 2020 research and innovation programme (Grant Agreement n◦ 864066).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  References  [1] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Cisneros, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Cortez, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Dombrowski, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Goldstein, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Kessler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Fluid dynamics of  self-propelled microorganisms, from individuals to concentrated populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Experiments in Fluids,  43(5):737–753, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  [2] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Duerinckx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Effective viscosity of random suspensions without uniform separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Poincaré C Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Non Linéaire, 39(5):1009–1052, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  [3] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Duerinckx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Semi-dilute mean-field description of particle suspensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' In preparation, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  [4] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Duerinckx and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Gloria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Continuum percolation in stochastic homogenization and the effective  viscosity problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Preprint, arXiv:2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='09654.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  [5] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Duerinckx and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Gloria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' On Einstein’s effective viscosity formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Preprint, arXiv:2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='03837.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  [6] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Duerinckx and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Gloria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Multiscale functional inequalities in probability: concentration proper-  ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' ALEA Lat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=', 17(1):133–157, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  [7] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Duerinckx and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Gloria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Multiscale functional inequalities in probability: constructive approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Lebesgue, 3:825–872, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  [8] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Duerinckx and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Gloria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Corrector equations in fluid mechanics: effective viscosity of colloidal  suspensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Arch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Ration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=', 239(2):1025–1060, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  [9] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Duerinckx and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Gloria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Effective viscosity of semi-dilute suspensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Séminaire Laurent  Schwartz, EDP et applications, 2021-2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Exposé n◦III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  [10] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Duerinckx and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Gloria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Quantitative homogenization theory for random suspensions in steady  stokes flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Ec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Polytech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' - Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=', 9:1183–1244, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  [11] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Duerinckx and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Gloria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Sedimentation of random suspensions and the effect of hyperuniformity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Annals of PDE, 8(2), 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  [12] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Einstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Eine Neue Bestimmung der Moleküldimensionen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=', 19:289–306, 1906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  [13] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Francfort, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Gloria, and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Lopez-Pamies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Enhancement of elasto-dielectrics by homogenization  of active charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Pures Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' (9), 156:392–419, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  [14] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Frenkel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Kinetic Theory of Liquids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Clarendon Press, Oxford, 1946.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  [15] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Gérard-Varet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Derivation of the Batchelor-Green formula for random suspensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Pures  Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' (9), 152:211–250, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  [16] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Gérard-Varet and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Hillairet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Analysis of the viscosity of dilute suspensions beyond Einstein’s  formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Arch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Ration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=', 238(3):1349–1411, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  [17] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Gérard-Varet and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Höfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Mild assumptions for the derivation of Einstein’s effective viscosity  formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Partial Differential Equations, 46(4):611–629, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  [18] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Gérard-Varet and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Mecherbet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' On the correction to Einstein’s formula for the effective viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Poincaré C Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Non Linéaire, 39(1):87–119, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  [19] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Girodroux-Lavigne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Derivation of an effective rheology for dilute suspensions of micro-swimmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Preprint, arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='04967.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  [20] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Gloria, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Neukamm, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Otto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' A regularity theory for random elliptic operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Milan J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=', 88(1):99–170, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  [21] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Guazzelli and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Morris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' A Physical Introduction to Suspension Dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Cambridge University  Press, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  HOMOGENIZATION FOR ACTIVE SUSPENSIONS  57  [22] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Haines, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Aronson, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Berlyand, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Karpeev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Effective viscosity of dilute bacterial  suspensions: a two-dimensional model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Biol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=', 5(4):046003, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  [23] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Haines, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Sokolov, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Aranson, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Berlyand, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Karpeev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Three-dimensional model  for the effective viscosity of bacterial suspensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' E, 80(4):041922, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  [24] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Hatwalne, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Ramaswamy, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Rao, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Simha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Rheology of active-particle suspensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=', 92(11):118101, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  [25] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Höfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Convergence of the method of reflections for particle suspensions in Stokes flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Equ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=', 297:81–109, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  [26] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Jikov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Kozlov, and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Ole˘ınik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Homogenization of differential operators and integral  functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Springer-Verlag, Berlin, 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  [27] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' López, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Gachelin, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Douarche, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Auradou, and É.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Clément.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Turning bacteria suspensions  into superfluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=', 115:028301, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  [28] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Mendelson, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Bourque, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Wilkening, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Anderson, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Watkins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Organized cell  swimming motions in bacillus subtilis colonies: Patterns of short-lived whirls and jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Journal of  Bacteriology, 181(2):600–609, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  [29] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Paxton, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Kistler, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Olmeda, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Sen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Angelo, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Cao, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Mallouk, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Lammert, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Crespi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Catalytic nanomotors: Autonomous movement of striped nanorods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=', 126(41):13424–13431, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  [30] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Potomkin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Tournus, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Berlyand, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Aranson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Flagella bending affects macroscopic  properties of bacterial suspensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Journal of The Royal Society Interface, 14(130):20161031, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  [31] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Rafaï, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Jibuti, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Peyla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Effective viscosity of microswimmer suspensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=',  104(9):098102, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  [32] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Ryan, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Haines, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Berlyand, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Ziebert, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Aranson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Viscosity of bacterial suspensions:  Hydrodynamic interactions and self-induced noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' E, 83:050904, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  [33] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Saintillan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Rheology of active fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Annu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=', 50(1):563–592, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  [34] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Saintillan and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Shelley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Theory of active suspensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' In Complex Fluids in Biological Systems,  pages 319–355.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Springer New York, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  [35] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Sokolov and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Aranson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Reduction of viscosity in suspension of swimming bacteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=', 103(3):148101, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  [36] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Sokolov, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Goldstein, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Feldchtein, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Aranson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Enhanced mixing and spatial insta-  bility in concentrated bacterial suspensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' E, 80(3):031903, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  [37] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Yasa, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Erkoc, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Alapan, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Sitti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Microalga-powered microswimmers toward active cargo  delivery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content=', 30(45):1804130, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='  (Armand Bernou) Sorbonne Université, CNRS, Université de Paris, Laboratoire Jacques-  Louis Lions, 75005 Paris, France  Email address: armand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='bernou@sorbonne-universite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='fr  (Mitia Duerinckx) Université Libre de Bruxelles, Département de Mathématique, 1050 Brus-  sels, Belgium  Email address: mitia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='duerinckx@ulb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='be  (Antoine Gloria) Sorbonne Université, CNRS, Université de Paris, Laboratoire Jacques-  Louis Lions, 75005 Paris, France & Institut Universitaire de France & Université Libre de  Bruxelles, Département de Mathématique, 1050 Brussels, Belgium  Email address: antoine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='gloria@sorbonne-universite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
+page_content='fr' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfWffL/content/2301.00166v1.pdf'}
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+arXiv:2301.12059v1  [cond-mat.mtrl-sci]  28 Jan 2023
+Potential energy surface prediction of Alumina polymorphs using graph neural
+network
+Soumya Sanyal,1, ∗ Arun Kumar Sagotra,2, † Narendra Kumar,3 Sharad Rathi,4 Mohana
+Krishna,4 Nagesh Somayajula,5 Duraivelan Palanisamy,6 Ram R. Ratnakar,7 Suchismita
+Sanyal,6 Partha Talukdar,8, ∗ Umesh Waghmare,3 and Janakiraman Balachandran2, †
+1University of Southern California, USA
+2
+3Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore
+4Vellore Institute of Technology, Vellore
+5Centura, Colarado, USA
+6Shell Technology Centre Bangalore, Bangalore
+7Shell Technology Centre Houston, USA
+8Google Research, Bangalore, India
+The process of design and discovery of new
+materials
+can
+be
+significantly
+expedited
+and
+simplified if we can learn effectively from avail-
+able data. Deep learning (DL) approaches have
+recently received a lot of interest for their ability
+to speed up the design of novel materials by pre-
+dicting material properties with precision close
+to experiments and ab-initio calculations.
+The
+application of deep learning to predict materials
+properties measured by experiments are valuable
+yet challenging due to the limited amount of ex-
+perimental data. Most of the existing approaches
+to predict properties from computational data
+have also been directed towards specific material
+properties.
+In this work, we extend this ap-
+proach, by proposing Landscape Crystal Graph
+Convolution Network(LCGCN), an accurate and
+transferable deep learning framework based on
+graph convolutional networks.
+LCGCN directly
+learns the potential energy surface (PES) from
+atomic configurations. This approach can enable
+transferable models that can predict different
+material properties.
+We apply this framework
+to bulk crystals (i.e.
+Al2O3), and test it by
+calculating potential energy surfaces at different
+temperatures
+and
+across
+different
+phases
+of
+crystal.
+I.
+INTRODUCTION
+The development of next generation technologies in
+energy, chemicals, electronics and transportation indus-
+tries are closely tied to the development of new materials
+with the desired properties [1]. For example in case of
+∗ Work done while at Indian Institute of Science Bangalore
+† Corresponding
+Author
+and
+Work
+done
+while
+at
+Shell
+Technology
+Centre
+Bangalore.;
+arunsagotra91@gmail.com and janakiraman.balachandran@gmail.com
+energy, the need of the hour is the development of in-
+expensive and environmentally safe materials that can
+generate and store electrons with very high density and
+efficiency while operating at high power. In case of chemi-
+cals, there is a critical need to develop new materials with
+better catalytic performance [2], as well as those which
+can make chemicals from alternative feedstock such as
+recycled plastics.
+Historically, discovery of materials has been driven by
+empiricism, through the knowledge and intuition of re-
+searchers working on these problems. Over the past few
+decades, they are being augmented by the insights and
+knowledge derived from physics based models, in partic-
+ular, atomistic models and simulations that can predict
+the various properties of interest for a wide range of ma-
+terials [3, 4]. However, these physics based models are
+computationally expensive and as a result, only aid us
+to explore a small subset of the materials search space.
+Further, within this approach, there is no formal way
+to transfer information and knowledge from one system
+to another, and from one application to another. Data
+driven models based on statistics and machine learning
+can help in speeding up the materials discovery process
+by providing a framework where algorithms can learn
+from existing data obtained from models and experi-
+ments on a subset of materials and predict the properties
+and functionalities of new materials. In order to facilitate
+such developments, various research groups from across
+the world have also provided free public access to the
+datasets of material structures [5–9] and properties ob-
+tained predominantly from atomistic simulations.
+These datasets and other datasets obtained from ex-
+periments have been used in the recent literature to de-
+velop data driven models to predict material proper-
+ties [10–13].
+Most of these models are based on hand
+crafted descriptors that are suitable only for certain sub-
+set of materials and for targeted properties of interest.
+However, a transferable model needs to operate on raw
+data to remove these biases introduced through hand
+crafted descriptors. In case of materials and molecules,
+such raw input data are the atoms and their position
+coordinates.
+This basic materials information can be
+
+2
+naturally expressed in the form of graphs. Graph Neu-
+ral Networks (GNNs) are a class of deep learning mod-
+els that have been successfully used to model graph-
+structured data [14–16]. GNNs can learn to effectively
+represent a molecular/crystal graph from raw data of
+atom/bond features and use neural networks to perform
+regression and classification tasks [17–20]. Recently, Xie
+et al. [13] developed crystal graph convolution neural
+network (CGCNN) that was able to take this basic struc-
+tural and chemical information of a crystal as graph to
+perform graph convolutions on these graphs. They were
+able to obtain excellent results to predict properties of a
+wide range of crystals, in their ground-state equilibrium
+structure. Later, Sanyal et al. [21] integrated CGCNN
+with multi-task learning to further improve the predic-
+tion accuracy.
+II.
+OUTLINE OF THE WORK
+Although CGCNN has been used to predict equilib-
+rium properties of materials, many thermodynamic, ki-
+netic and transport properties at non-zero temperatures
+are determined only after incorporating the phononic
+contributions that distort the crystal and its atomic com-
+ponents from the equilibrium position. These distortion
+and the resultant phononic contributions depend upon
+the potential energy surface (PES) associated with it.
+PES is a highly non-linear complex function of the chem-
+ical composition, symmetry and atomic arrangement of
+the crystals. Even a small change in these parameters
+can significantly change the PES. The PES is the input
+to various statistical approaches such as Monte Carlo and
+molecular dynamics simulations to predict the phononic
+degrees of freedom and their influence on various material
+properties.
+Conventionally PES are fitted to explicit function
+forms from ab-initio simulation results. However, over
+the past few years gaussian process based machine learn-
+ing has been applied to develop interatomic potentials
+for various materials ranging from elemental (amorphous
+carbon, graphene, boron) [22–24] to multi-elemental
+(methane) [25]. Although very successful, development of
+Gaussian Approximation Potential (GAP) [26] requires
+careful design of experiments to simulate small systems
+whose inputs are needed to fit the interatomic potentials.
+Further, there are also challenges in transfering the po-
+tential to a new system.
+These shortcomings can potentially be overcome by
+GNN based deep learning approach, in which a model
+learns from the raw materials data.
+These models
+can possibily lead to better transferability across dif-
+ferent materials composition, crystal symmetry, atomic
+arrangements and environments. In this work, we pro-
+vide an initial demonstration of such transferability and
+propose Landscape Crystal Graph Convolution
+Network (LCGCN), an extension of CGCNN architec-
+ture, to predict the PES of alumina (Al2O3), a material
+that serves as the base material for various adsorbent and
+catalytic applications [27–30].
+Specifically, we demon-
+strate the generalizability of this approach by developing
+LCGCN models that not only accurately predict PES,
+but are also transferable across scales, environmental
+conditions (temperature) and different crystallographic
+polymorphs, trigonal phase (space group R3c), cubic
+phase (space group Ia3), Orthorhombic (space groups
+Pbca and Pna21) of alumina.
+III.
+METHODOLOGY
+A.
+Molecular Dynamics
+The data required to train, validate and test the
+CGCNN models were obtained by performing Molecu-
+lar Dynamics (MD) simulations.
+We have performed
+MD simulations under (N, V, T ) conditions with the
+LAMMPS code [31]. In these simulations the temper-
+ature is kept fluctuating around a setpoint value by us-
+ing Nos´e - Hoover thermostats. We employ simulation
+boxes, typically varying from 30 to 480 atoms, and apply
+periodic boundary conditions along the three Cartesian
+directions. Newton’s equations of motion are integrated
+using the customary Verlet’s algorithm with a time-step
+length of 10−3 ps. The typical duration of a MD run is
+of 10 ps. We used interatomic potentials of the Second-
+Moment tight-Binding-QEq (SMTB-Q) form [32].
+In this work, employing the above MD protocol, we
+simulate Al2O3 systems comprising of different sizes, en-
+vironmental temperatures and polymorphs.
+We pick
+10,000 configurations of these systems after thermal equi-
+libriation. Unless specified otherwise, the 10,000 config-
+urations were divided into a ratio of 6:2:2 for training,
+validation and testing of the LCGCN model.
+B.
+Landscape Crystal Graph Convolution
+Network(LCGCN)
+CGCNN [13] is a graph neural network which focuses
+on building a crystal graph from a given crystal struc-
+ture and utilizes graph convolution networks (GCNs) to
+model the crystal graph and predict their properties with
+accuracy comparable to ab initio physics models.
+We
+propose Landscape Crystal Graph Convolution
+Network (LCGCN), an extension of CGCNN, that can
+be used to predict the potential energy surface (PES) of
+crystal structures. Next, we discuss the details of pro-
+posed method LCGCN in detail.
+A
+crystal graph G
+is an undirected
+multigraph
+defined
+by
+nodes
+representing
+atoms
+and
+edges
+representing
+bonds
+in
+a
+crystal.
+Formally,
+let
+G=(A, E, V, U) denote a crystal graph.
+Here A rep-
+resents the set of atoms in the crystal structure,
+E={(i, j)k: kth bond between atom i and j}, is the set
+of undirected edges denoting the bonds and |A|=N is
+
+3
+FIG. 1. Schematic representation of using LCGCN to predict potential energy of distorted alumina crystals. Initially, the
+crystal structure is represented as a graph with nodes and edges representing the atoms and bonds in the crystal respectively.
+Here, vi, vj and uij denote the node, neighbor and edge embeddings respectively. Then, CGCNN is used to learn a crystal
+level embedding vG which is used to predict the potential energy. Please refer to Sec. III B for more details.
+the number of atoms in the crystal graph. vi ∈ V con-
+tains the features of the ith atom which encodes various
+properties of the atom. u(i,j)k ∈ U is the feature vector
+for the kth bond between atoms i and j that captures
+properties of the edges. To capture the influence of local
+neighborhood in the crystal structure, a graph convolu-
+tion formulation similar to CGCNN [13] is used defined
+as follows,
+v(t+1)
+i
+= v(t)
+i
++
+�
+j,k
+σ(z(t)
+(i,j)kW (t)
+c
++b(t)
+c )⊙g(z(t)
+(i,j)kW (t)
+s
++b(t)
+s ),
+where z(t)
+(i,j)k = v(t)
+i
+⊕v(t)
+j ⊕u(i,j)k denotes the concate-
+nation of atom and bond feature vectors of the neighbors
+of ith atom. ⊙ denotes element-wise multiplication and
+σ denotes a sigmoid function.
+The σ(·) factor acts as
+a learned weight matrix to incorporate different interac-
+tion strengths between neighbors. W (t)
+c , W (t)
+s , b(t)
+c
+and
+b(t)
+s
+are the convolution weight matrix, self weight ma-
+trix, convolution bias and self bias of the t-th layer of
+GCN respectively, and g(·) is the activation function for
+introducing non-linear coupling between layers.
+Graph pooling [33] is a technique used to learn graph
+representations given the learnt node and edge features.
+The learnt atom features are pooled to get a vector rep-
+resentation of the crystal (vG). The crystal representa-
+tion vG is then fed to a network of fully-connected layers
+with non-linearities that learns to predict a property of
+the crystal in a supervised manner. The workflow is de-
+picted in Fig. 1. To train the network, we calculate the
+loss using mean squared error between the true values
+obtained using MD simulations and the model predic-
+tions followed by back-propagation [34]. The accuracy of
+the model is quantified through mean absolute error per
+atom (MAE/atom) by comparing the model predictions
+on test configurations against the true values obtained
+using MD simulations.
+Pooling Operator
+Formulation
+Average
+1
+N
+�N
+i=1 vi
+Max
+max(vi),
+∀i ∈ [1, N]
+FC-Max
+max(viWpool + bpool),
+∀i ∈ [1, N]
+TABLE I. Formulation of different graph pooling operators.
+Here, vi denotes the atom representation and max is an ele-
+mentwise max operator. Wpool and bpool are learnable pool-
+ing parameters. Please refer to Sec. IV C for more details.
+IV.
+RESULTS AND DISCUSSION
+A.
+Effect of different pooling operators
+In this section, we explore some of the existing graph
+pooling operators - Average, Max and FC-max.
+In
+Average pooling, the graph embedding is defined as
+the mean of all the node embeddings.
+Similarly, for
+Max pooling, it is defined as the elementwise max of
+all the node embeddings. Hamilton et al. [33] proposed
+FC-max, a learnable pooling operator, which transforms
+node embeddings using a fully-connected neural network
+and then applies Max pooling operator. The formula-
+tion of these pooling operators are defined in Table I.
+To analyze the effect of pooling, we perform the experi-
+ment where we use alumina trigonal polymorph data for
+training and predict performance on other alumina poly-
+morphs. The results are shown in Table II. We observe
+that FC-max performs the best among all of the pooling
+operators. And hence, we use FC-max as the pooling
+operator in all further analysis of our LCGCN architec-
+ture.
+
+X1
+X2
+Predicted
+LCGCN
+FC
+Build Graph
+Value
+Perturb1
+0
+0
+0
+0
+0
+0
+Crystal Structure
+Predicted
+FC
+LCGCN
+Build Grap
+Value
+Final Fully
+Potential Energy
+Perturbed Crystal
+Connected Layer
+Crystal Graphs
+Crystal
+Structures
+Embeddings4
+Pooling Operator
+Trigonal Orthorhombic (Pbca) Orthorhombic (Pna21)
+Average
+0.0009
+0.0349
+0.1796
+Max
+0.0007
+0.4256
+0.0789
+FC-max
+0.0007
+0.0112
+0.047
+TABLE II. MAE/atom of LCGCN model for different pooling operators on test data of 240 atom alumina orthorhombic (Pbca)
+and Orthorhombic (Pna21) polymorph trained on trigonal data. Please refer to Sec. IV A for details.
+CGCNN
+LCGCN
+Phase
+Error/atom
+Relative Error
+Error/atom
+Relative Error
+Trigonal
+-0.0466
+0
+-0.0018
+0
+Orthorhombic (Pbca)
+-0.0208
+0.0258
+0.0001
+0.0019
+Orthorhombic (Pna21)
+-0.0302
+0.0164
+0.0031
+0.0049
+TABLE III. Error/atom (eV/atom) and Relative Error (w.r.t. trigonal alumina phase) of LCGCN and CGCNN model on test
+dataset of different polymorphs using models trained on setups as defined in Sec. IV B. We observe that LCGCN consistently
+outperforms CGCNN for all alumina phase equilibrium predictions.
+B.
+Prediction of energy at equilibrium
+Xie et al. [13] train their CGCNN model to predict
+formation energy per atom (∆Ef) using a large number
+of equilibrium crystal structures obtained from the Ma-
+terials Project [5]. Since (∆Ef) for a large number of
+crystals can span across a wide range (approximately 8.5
+eV/atom), such a model will not be able to accurately
+predict the small energy differences across different poly-
+morphs of the same composition. However, in this work
+we train LCGCN model to predict the PES of various
+polymorphs of alumina through training on distorted in-
+put structures obtained from MD simulations. The re-
+sults are shown in table III. Due to this training data, we
+expect our model to perform better in predicting the en-
+ergy differences of the alumina polymorphs. The errors
+of predictions from the two approaches are shown in ta-
+ble III. We report both ’signed’ error and relative error.
+For relative error, we choose the energy of the trigonal
+phase as the reference value. As expected, LCGCN sig-
+nificantly outperforms the previous CGCNN model sug-
+gesting that LCGCN is well-equipped than CGCNN to
+predict small energy changes typically seen in distortions.
+In the next upcoming subsections, we will discuss the
+PES predictions of alumina across supercell sizes, tem-
+perature and polymorphs.
+C.
+Prediction of potential energy
+The LCGCN model was used to predict the potential
+energy of the trigonal alumina polymorph of different su-
+percell sizes – namely 120 and 480 atoms. The data to
+train, validate and test was obtained from MD simual-
+tions as described in Sec. III A under a temperature ramp
+of 300K-500K. We quantify the error by reporting mean
+absolute error per atom (MAE/atom) (eV). The results
+for the two different supercells are shown in Fig. 2. As
+expected, increasing the amount of training data reduces
+the error. We observe that with 0.6 times the total alu-
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+T
+raining Dataset Size (x10000)
+0.0006
+0.0008
+0.0010
+0.0012
+0.0014
+0.0016
+MAE/atom (eV)
+120 atoms
+480 atoms
+FIG. 2. Plotting test MAE/atom of LCGCN model in pre-
+dicting potential energy of 120 and 480 atoms systems of
+alumina trigonal polymorph with increasing size of training
+dataset. Please refer to Sec. IV C for more details.
+mina data (i.e.
+6000 samples) for training, our model
+is able to achieve an MAE/atom which is close to the
+accuracy of DFT (1 meV/atom).
+D.
+Transferable LCGCN models across scale
+In many scenarios, the property of interest can only
+be obtained from performing large scale atomistic and
+quantum mechanical simulations. However, the scaling
+of such simulations with system size can be non-linear.
+For example in case of DFT, the cpu time of a simula-
+tion increases as O(N 3) where N is the number of atoms
+that limits the system size. Hence, an ML model that
+is transferable from smaller scale systems to larger scale
+systems are of importance in the field. Firstly, we note
+that the cpu time for inference using our model increases
+as O(N) with the number of nodes N. We perform a
+time analysis of the LCGCN model. For this, we use a
+
+5
+Training
+Validation
+Testing
+120 atoms
+480 atoms
+480 atoms
+480 atoms
+MAE/atom
+1000
+2400
+600
+2000
+0.0017
+3000
+2400
+600
+2000
+0.0017
+6000
+2400
+600
+2000
+0.0011
+10000
+2400
+600
+2000
+0.0011
+TABLE IV. MAE/atom of LCGCN model on test data of 480 atoms with increasing training dataset of 120 atoms. Please
+refer to Sec. IV D for details.
+Training
+Validation
+Testing
+120 atoms
+480 atoms
+120 atoms
+480 atoms
+480 atoms
+MAE/atom
+2400
+0
+600
+0
+2000
+0.0023
+2400
+800
+600
+200
+2000
+0.0014
+2400
+1600
+600
+400
+2000
+0.0013
+2400
+2400
+600
+600
+2000
+0.0011
+TABLE V. MAE/atom of LCGCN model on test data of 480 atoms with increasing training data of 480 atom configurations.
+Please refer to Sec. IV D for further details.
+100
+200
+300
+400
+500
+600
+Number of Atoms
+0.005
+0.010
+0.015
+0.020
+0.025
+0.030
+Time (Seconds)
+FIG. 3. Time analysis of the LCGCN code with respect to
+the CPU run, as a function of number of atoms.
+pre-trained LCGCN model to predict potential energy
+for a set of 100 configurations of 120 atom alumina trig-
+onal polymorph. Next, we repeat this for 240, 360, 480
+and 600 atoms supercells. The average cpu time for each
+setup is depicted in Fig. 3. We observe that LCGCN is
+significantly faster than DFT and MD simulations, and
+in contrast to MD and DFT the LCGCN scales linearly
+with the system size and easily parallelized [35, 36].
+Next, we demonstrate how increasing the training data
+of the 120 atoms supercell of alumina trigonal polymorph
+can reduce the test error of predicting the potential en-
+ergy of the 480 atom supercell configurations. For this,
+we consider a setup where there are a fixed 2400 con-
+figurations of 480 atoms for model training.
+Also, we
+have 10000 configurations of 120 atoms that we progres-
+sively add to the training set. As observed from Table
+IV, an increase in the number of 120 atom configura-
+tions in training data leads to a reduction in the test er-
+ror for the 480 atom configuration. After adding all the
+10000 configurations of 120 atoms in training data, the
+error in MAE/atom for 480 atom configuration is 0.0011
+eV/atom. This is shown in Fig. 4a. In a second set of ex-
+periments, the data of 120 atoms is fixed in training set,
+while the number of 480 atoms is increased in training.
+The results are reported in Table V and depicted in Fig.
+4b. We find that an increase in the 480 atom configura-
+tion in training set leads to significant improvements in
+model performance on predicting potential energy for 480
+atoms. From these set of experiments we can conclude
+that although it is not trivial for LCGCN to generalize,
+although with comparitively higher error to predict PES
+of the larger supercell (480 atoms) even when these larger
+supercell configurations are not included in the training
+data (1st row of Table V). However, this error goes down
+significantly even if a small amount of the 480 atoms con-
+figurations are included into the training data (4th row
+of Table V).
+E.
+Transferable LCGCN models across
+environmental conditions
+Envrionmental conditions, in particular temperature,
+play an important role in determining the dynamics of
+the materials and the resultant phase stability and mate-
+rial properties [1, 37]. Higher temperature molecular dy-
+namics has been employed as a tool in different problems
+to sample the configurations rapidly and to predict the
+resultant low energy configuration and their correspond-
+ing properties [38]. In this section, we analyze LCGCN’s
+transferability in predicting potential energies of configu-
+rations of trigonal alumina across different environmental
+conditions. We specifically focus on variation of temper-
+ature and test the effectiveness of the model in three
+temperature values for alumina - 300K, 500K and 800K.
+We pretrain a LCGCN model using all the data (10,000
+configurations) from one temperature and predict poten-
+tial energy at a different temperature. The results are
+shown in Table VI.
+
+6
+2000
+4000
+6000
+8000
+10000
+T
+raining Samples (120 atoms)
+0.0011
+0.0012
+0.0013
+0.0014
+0.0015
+0.0016
+0.0017
+MAE/atom (eV)
+(a)
+0
+500
+1000
+1500
+2000
+2500
+T
+raining Samples (480 atoms)
+0.0012
+0.0014
+0.0016
+0.0018
+0.0020
+0.0022
+MAE/atom (eV)
+(b)
+FIG. 4. (a) Test MAE/atom of LCGCN model in predicting
+potential energy of 480 atoms of alumina trigonal polymorph
+as a function of increasing training dataset of 120 atoms. Re-
+fer Sec. IV D for details. (b) Test MAE/atom of LCGCN
+model in predicting potential energy of 480 atoms of alumina
+trigonal polymorph as a function of increasing number of 480
+atom configurations in training dataset. Refer to Sec. IV D
+for details.
+In α-Al2O3, two types of Al-O bonds (1.87 angstrom
+and 1.98 angstrom) are present at 0 K. But under clas-
+sical MD simulation, these two bonds vary drastically at
+a given temperature T. At low temperature (say 300 K),
+bond-bond distribution function shows two peaks (at 1.85
+and 2.03 angstrom) while at high temperatures (500 K &
+800 K), the peak corresponding to longer (weaker) Al-O
+bond (2.03 angstrom peak in 300 K configuration) gets
+disappeared, see Fig. 5a. It means that weakly bonded
+Al and O got separated much compared to strongly
+bonded Al and O due to thermal stress.
+Next, we observe the effect of mixing the data across
+two different temperatures (resulting in a total of 20,000
+configurations split into training, validation and test set)
+in predicting energies for data at an unseen temperatures.
+The results are shown in Table VII. We see when the
+model is fed with a mixture of two temperatures data for
+training the prediction error reduces significantly for test
+data at unseen temperatures. Specifically, we observe a
+significant decrease in the MAE for prediction at 800K,
+when data for both 300K and 500K are mixed. In order
+to delineate if the improvement in model prediction is due
+to training data volume, or due to data from two different
+Temp.
+300K
+500K
+800K
+300K
+0.0003
+0.0212
+0.0766
+500K
+0.0151
+0.0004
+0.0254
+800K
+0.0372
+0.0192
+0.0007
+TABLE VI. MAE/atom of LCGCN model on test data of 240
+atom alumina trigonal polymorph at three different temper-
+atures. Row temperature indicates that model was trained
+using the data of that particular temperature and the column
+temperature denotes that the trained model was used to pre-
+dict potential energy at that temperature. Please refer to Sec.
+IV E for more details.
+Mixture
+Target
+MAE/atom
+300K & 500K
+800K
+0.02
+300K & 800K
+500K
+0.0048
+500K & 800K
+300K
+0.0035
+300K & 500K(5000 each)
+800K
+0.0169
+TABLE VII. MAE/atom of LCGCN model on test data of
+240 atom alumina trigonal polymorph at a target tempera-
+ture using a mixture of data for two different temperatures in
+training data. Please refer to Sec. IV E for details.
+temperatures, we train a model with 5000 configurations
+of 300K and 500K each and test the model on data from
+800K. The resulting MAE/atom (4th row of Table VII)
+is still significantly lower than predictions at 800K using
+only 300K or 500K data for training. This suggests that
+the model is able to learn better when it is trained with
+heterogeneous data from two different temperatures.
+F.
+Transferable LCGCN models across different
+Polymorphs
+One of the major challenges in materials and chem-
+istry arises from the difficulty in transfering information,
+model and knowledge across different crystal structures.
+A model that can learn properties such as potential en-
+ergy, from equilibrium and distorted configurations of
+some crystal structures and predict for new crystal struc-
+tures with minimal data would be a very useful tool in
+accelerating materials discovery. As a preliminary step
+toward this goal, in this work, we study the transfer-
+ability of LCGCN models across different polymorphs
+of alumina. We consider three phases of alumina - or-
+thorhombic (Pbca and Pna21) and trigonal. For the first
+set of experiments, we predict the potential energy of
+the different polymorphs based on the training data from
+another polymorph (aka zero shot learning). Similar to
+temperature study, here we use all the data pertaining to
+a polymorph (10,000 configurations) to train the model.
+The results are shown in Table VIII.
+To understand the trends in the transferability of
+LCGCN across different polymorphs, we plotted the
+Al − O bond distribution and configurational energies
+for the three alumina polymorphs as depicted in Fig. 5b
+respectively. We observe that, the more the overlap be-
+
+7
+Phase
+Orthorhombic (Pbca) Trigonal Orthorhombic (Pna21)
+Orthorhombic (Pbca)
+0.0007
+0.0265
+0.037
+Trigonal
+0.0112
+0.0007
+0.047
+Orthorhombic (Pna21)
+0.9689
+0.1542
+0.0006
+TABLE VIII. MAE/atom of LCGCN model on test data of 240 atom alumina for three different polymorphs. The row phase
+denotes that the model was trained using the data for that particular polymorph and column phase indicates that the trained
+model was used to predict the potential energy for that polymorph. Refer to Sec. IV F for more details.
+1.6
+1.8
+2.0
+2.2
+2.4
+Bond Length  (Å)
+0
+2
+4
+6
+8
+Count
+(a)
+300 K
+500 K
+800 K
+1.6
+1.8
+2.0
+2.2
+2.4
+Bond Length (Å)
+0
+2
+4
+6
+8
+Count
+(b)
+Orthorhombic (Pbca)
+Orthorhombic (Pna21)
+T
+rigonal
+FIG. 5. (a) Al-O bond distribution of α-Al2O3 at 300 K ,
+500 K and 800 K temperatures.(b)Al-O bond distribution of
+Alumina polymorphs. (c) Density of energy distribution of
+Alumina polymorphs.
+Mixture
+MAE/atom
+Trigonal + Orthorhombic (Pna21)(5000 each)
+0.1804
+Trigonal + Orthorhombic (Pna21)
+0.1312
+TABLE IX. MAE/atom of LCGCN model on test data of
+240 atom alumina othorhombic (Pbca) polymorph using a
+mixture of data for different alumina polymorphs in training
+data. Please refer to Sec. IV F for details.
+tween Al − O bond distribution of different polymorphs,
+the better transferability, and hence lower the error in
+LCGCN’s prediction. For example, the overlap between
+the Orthorhombic (Pbca) and Trigonal polymorphs is
+greater than that between the Orthorhombic (Pbca) and
+Orthorhombic (Pna21) polymorphs.
+As seen in Table
+VIII, our LCGCN model, when trained on Orthorhom-
+bic (Pbca) data, predicts the Trigonal polymorph with
+a lower error compared to when it is trained on Or-
+thorhombic (Pna21). This is likely due to the fact that
+the LCGCN model’s accuracy is highly dependent on
+bond length, and transferability is better when predict-
+ing systems with similar bond lengths.
+In order to understand how the model accuracy im-
+proves when trained with heterogeneous data, we train
+the model with data from multiple polymorphs to pre-
+dict potential energy of orthorhombic (Pbca) polymorph.
+We choose orthorhombic (Pbca) for our analysis because
+the error in PES prediction was the highest as shown in
+Table VIII. The results for this analysis is shown in Ta-
+ble IX. We observe that if we restrict the training data
+to a total of 10,000 configurations (each polymorph has
+5,000 data points), the error in prediction is about 0.1804
+eV/atom suggesting that training on diverse dataset can
+improve the transferability of the model. However, if we
+incorporate all the data from the two polymorphs (20,000
+configurations in total), the model error further reduces
+to 0.1312 eV/atom suggesting that the transferability of
+the model can be improved by increasing both the volume
+and variety of training data.
+V.
+CONCLUSION
+In this work, we proposed LCGCN, a graph neural
+network framework to directly predict high-dimensional
+potential energy surface for alumina (Al2O3), a base ma-
+terial for various adsorbent and catalytic applications,
+from atomic configurations.
+This approach enables to
+bypass calculating the derivatives of the PES which are
+often the computational bottlenecks of existing ab-initio
+methods.
+We
+further
+demonstrated
+the
+transferability
+of
+LCGCN across scales, environmental conditions and dif-
+ferent alumina polymorphs. Finally, We validated that,
+with appropriate conditioning, LCGCN can obtain ac-
+curacies that are close to DFT (1 meV/atom). Therefore,
+we anticipate that LCGCN will significantly accelerate
+the atomistic studies of other materials with its scala-
+bility, versatility, and excellent potential energy surface
+prediction accuracy.
+DATA AVAILABILITY
+The data that support the findings of this study are
+available from the corresponding author (J.B. and A.K.S)
+upon reasonable request.
+
+8
+ACKNOWLEDGMENTS
+The authors would like to acknowledge Shell, JNCASR
+and IISc for their support.
+AUTHOR CONTRIBUTIONS
+J.B, U.V.W and P.T conceived the study and planned
+the research. A.K.S. and N.K. performed the MD sim-
+ulations to generate the dataset. S.S , M. K, S. R and
+A.K.S implemented and optimized the LCGCN model.
+All authors discussed the results and their implications
+and contributed to the writing of the article.
+COMPETING INTERESTS
+The authors declare no competing interests.
+[1] Arun K Sagotra, Dewei Chu, and Claudio Cazorla. In-
+fluence of lattice dynamics on lithium-ion conductivity:
+A first-principles study. Physical Review Materials, 3(3):
+035405, 2019.
+[2] Mohamed Eisa, Dovile Ragauskaite, Sushil Adhikari,
+Federico Bella, and Jonas Baltrusaitis. Role and respon-
+sibility of sustainable chemistry and engineering in pro-
+viding safe and sufficient nitrogen fertilizer supply at tur-
+bulent times. ACS Sustainable Chemistry & Engineering,
+10(28):8997–9001, 2022.
+[3] National Science and Technology Council (US). Materi-
+als genome initiative for global competitiveness. Execu-
+tive Office of the President, National Science and Tech-
+nology Council, 2011.
+[4] Narendra Kumar, Padmini Rajagopalan, Praveen Panka-
+jakshan,
+Arnab
+Bhattacharyya,
+Suchismita
+Sanyal,
+Janakiraman Balachandran, and Umesh V Waghmare.
+Machine learning constrained with dimensional analysis
+and scaling laws: simple, transferable, and interpretable
+models of materials from small datasets. Chemistry of
+Materials, 31(2):314–321, 2018.
+[5] Anubhav Jain, Shyue Ping Ong, Geoffroy Hautier, Wei
+Chen,
+William Davidson Richards,
+Stephen Dacek,
+Shreyas Cholia,
+Dan Gunter,
+David Skinner,
+Ger-
+brand Ceder, and Kristin a. Persson.
+The Materials
+Project: A materials genome approach to accelerating
+materials innovation.
+APL Materials,
+1(1):011002,
+2013.
+ISSN 2166532X.
+doi:10.1063/1.4812323.
+URL
+http://link.aip.org/link/AMPADS/v1/i1/p011002/s1&Agg=doi.
+[6] Stefano Curtarolo, Wahyu Setyawan, Gus LW Hart,
+Michal Jahnatek, Roman V Chepulskii, Richard H Tay-
+lor, Shidong Wang, Junkai Xue, Kesong Yang, Ohad
+Levy, et al.
+Aflow: an automatic framework for high-
+throughput materials discovery. Computational Materi-
+als Science, 58:218–226, 2012.
+[7] Computational
+materials
+database
+and
+repositories-
+nomad. URL https://nomad-coe.eu/.
+[8] Arun
+Mannodi-Kanakkithodi,
+Gregory
+M.
+Treich,
+Tran Doan Huan, Rui Ma, Mattewos Tefferi, Yang
+Cao,
+Gregory A. Sotzing,
+and Rampi Ramprasad.
+Rational Co-Design
+of Polymer
+Dielectrics
+for
+En-
+ergy Storage.
+Adv. Mater. (Weinheim,
+Ger.),
+28
+(30):6277–6291, August 2016.
+ISSN 0935-9648.
+doi:
+10.1002/adma.201600377.
+[9] James E. Saal, Scott Kirklin, Muratahan Aykol, Bryce
+Meredig, and C. Wolverton. Materials Design and Dis-
+covery with High-Throughput Density Functional The-
+ory: The Open Quantum Materials Database (OQMD).
+JOM, 65(11):1501–1509, November 2013.
+ISSN 1047-
+4838, 1543-1851. doi:10.1007/s11837-013-0755-4.
+[10] Ghanshyam
+Pilania,
+Chenchen
+Wang,
+Xun
+Jiang,
+Sanguthevar Rajasekaran, and Ramamurthy Ramprasad.
+Accelerating materials property predictions using ma-
+chine learning. Scientific reports, 3:2810, 2013.
+[11] Logan Ward, Ankit Agrawal, Alok Choudhary, and
+Christopher Wolverton.
+A general-purpose machine
+learning framework for predicting properties of inorganic
+materials. npj Computational Materials, 2:16028, 2016.
+[12] Chiho Kim, Ghanshyam Pilania, and Ramamurthy Ram-
+prasad.
+From organized high-throughput data to phe-
+nomenological theory using machine learning: the exam-
+ple of dielectric breakdown. Chemistry of Materials, 28
+(5):1304–1311, 2016.
+[13] Tian
+Xie
+and
+Jeffrey
+C.
+Grossman.
+Crystal
+graph
+convolutional
+neural
+networks
+for
+an
+ac-
+curate
+and
+interpretable
+prediction
+of
+material
+properties.
+Phys.
+Rev.
+Lett.,
+120:145301,
+Apr
+2018.
+doi:10.1103/PhysRevLett.120.145301.
+URL
+https://link.aps.org/doi/10.1103/PhysRevLett.120.145301.
+[14] Joan Bruna, Wojciech Zaremba, Arthur Szlam, and Yann
+LeCun.
+Spectral networks and locally connected net-
+works on graphs. In International Conference on Learn-
+ing Representations (ICLR), 2014.
+[15] Michael M. Bronstein, Joan Bruna, Yann LeCun, Arthur
+Szlam, and Pierre Vandergheynst. Geometric deep learn-
+ing: Going beyond euclidean data. IEEE Signal Process.
+Mag., 2017.
+[16] Thomas N. Kipf and Max Welling. Semi-supervised clas-
+sification with graph convolutional networks. In Interna-
+tional Conference on Learning Representations (ICLR),
+2017.
+[17] David K Duvenaud, Dougal Maclaurin, Jorge Ipar-
+raguirre,
+Rafael
+Bombarell,
+Timothy
+Hirzel,
+Alan
+Aspuru-Guzik, and Ryan P Adams. Convolutional net-
+works on graphs for learning molecular fingerprints.
+In Advances in Neural Information Processing Systems
+(NIPS) 28, pages 2224–2232. Curran Associates, Inc.,
+2015.
+
+9
+[18] Steven Kearnes, Kevin McCloskey, Marc Berndl, Vijay
+Pande, and Patrick Riley. Molecular graph convolutions:
+moving beyond fingerprints. Journal of Computer-Aided
+Molecular Design (CAMD), 30(8):595–608, 2016.
+[19] Justin Gilmer, Samuel S. Schoenholz, Patrick F. Riley,
+Oriol Vinyals, and George E. Dahl. Neural message pass-
+ing for quantum chemistry. In Proceedings of the 34th
+International Conference on Machine Learning (ICML),
+pages 1263–1272, 2017.
+[20] Han Altae-Tran, Bharath Ramsundar, Aneesh S. Pappu,
+and Vijay Pande. Low data drug discovery with one-shot
+learning. ACS Central Science, 3(4):283–293, 2017.
+[21] Soumya Sanyal, Janakiraman Balachandran, Naganand
+Yadati, Abhishek Kumar, Padmini Rajagopalan, Suchis-
+mita Sanyal, and Partha Talukdar.
+MT-CGCNN: In-
+tegrating crystal graph convolutional neural network
+with multitask learning for material property prediction.
+arXiv preprint arXiv:1811.05660, 2018.
+[22] Volker
+L.
+Deringer
+and
+G´abor
+Cs´anyi.
+Machine
+learning
+based
+interatomic
+potential
+for
+amor-
+phous
+carbon.
+Phys.
+Rev.
+B,
+95:094203,
+Mar
+2017.
+doi:10.1103/PhysRevB.95.094203.
+URL
+https://link.aps.org/doi/10.1103/PhysRevB.95.094203.
+[23] Patrick Rowe, G´abor Cs´anyi, Dario Alf`e, and Ange-
+los Michaelides.
+Development of a machine learning
+potential for graphene.
+Phys. Rev. B, 97:054303,
+Feb 2018.
+doi:10.1103/PhysRevB.97.054303.
+URL
+https://link.aps.org/doi/10.1103/PhysRevB.97.054303.
+[24] Volker
+L.
+Deringer,
+Chris
+J.
+Pickard,
+and
+G´abor
+Cs´anyi. Data-driven learning of total and local energies
+in elemental boron.
+Phys. Rev. Lett.,
+120:156001,
+Apr 2018.
+doi:10.1103/PhysRevLett.120.156001.
+URL
+https://link.aps.org/doi/10.1103/PhysRevLett.120.156001.
+[25] Max
+Veit,
+Sandeep
+Kumar
+Jain,
+Satyanarayana
+Bonakala,
+Indranil Rudra,
+Detlef Hohl,
+and G´abor
+Cs´anyi.
+Equation of state of fluid methane from first
+principles with machine learning potentials.
+Jour-
+nal
+of
+Chemical
+Theory
+and
+Computation,
+15(4):
+2574–2586, 2019.
+doi:10.1021/acs.jctc.8b01242.
+URL
+https://doi.org/10.1021/acs.jctc.8b01242.
+PMID:
+30794393.
+[26] Albert
+P.
+Bart´ok,
+Mike
+C.
+Payne,
+Risi
+Kondor,
+and G´abor Cs´anyi.
+Gaussian approximation poten-
+tials:
+The accuracy of quantum mechanics, without
+the electrons.
+Phys. Rev. Lett.,
+104:136403,
+Apr
+2010.
+doi:10.1103/PhysRevLett.104.136403.
+URL
+https://link.aps.org/doi/10.1103/PhysRevLett.104.136403.
+[27] J´ozsef Hlavay and Kl´ara Poly´ak. Determination of sur-
+face properties of iron hydroxide-coated alumina adsor-
+bent prepared for removal of arsenic from drinking water.
+Journal of Colloid and Interface Science, 284(1):71–77,
+2005.
+[28] Vinod K Gupta, Shilpi Agarwal, and Tawfik A Saleh.
+Synthesis and characterization of alumina-coated carbon
+nanotubes and their application for lead removal. Journal
+of hazardous materials, 185(1):17–23, 2011.
+[29] Herman Pines and Werner O Haag. Alumina: catalyst
+and support. i. alumina, its intrinsic acidity and catalytic
+activity1. Journal of the American Chemical Society, 82
+(10):2471–2483, 1960.
+[30] Monica Trueba and Stefano P Trasatti. γ-alumina as a
+support for catalysts: a review of fundamental aspects.
+European journal of inorganic chemistry, 2005(17):3393–
+3403, 2005.
+[31] Steve Plimpton.
+Fast parallel algorithms for short-
+range
+molecular
+dynamics.
+Journal
+of
+Computa-
+tional Physics, 117(1):1 – 19, 1995.
+ISSN 0021-9991.
+doi:https://doi.org/10.1006/jcph.1995.1039.
+URL
+http://www.sciencedirect.com/science/article/pii/S002199918571039X.
+[32] N. Salles, O. Politano, E. Amzallag, and R. T´etot.
+Molecular
+dynamics
+study
+of
+high-pressure
+alu-
+mina
+polymorphs
+with
+a
+tight-binding
+variable-
+charge
+model.
+Computational
+Materials
+Science,
+111:181
+–
+189,
+2016.
+ISSN
+0927-0256.
+doi:
+https://doi.org/10.1016/j.commatsci.2015.09.017.
+URL
+http://www.sciencedirect.com/science/article/pii/S0927025615005868.
+[33] William L. Hamilton, Rex Ying, and Jure Leskovec. In-
+ductive representation learning on large graphs. In NIPS,
+2017.
+[34] David
+E.
+Rumelhart,
+Geoffrey
+E.
+Hinton,
+and
+Ronald
+J.
+Williams.
+Learning
+Representations
+by
+Back-propagating
+Errors.
+Nature,
+323(6088):
+533–536,
+1986.
+doi:10.1038/323533a0.
+URL
+http://www.nature.com/articles/323533a0.
+[35] Justin S Smith, Olexandr Isayev, and Adrian E Roit-
+berg. Ani-1: an extensible neural network potential with
+dft accuracy at force field computational cost. Chemical
+science, 8(4):3192–3203, 2017.
+[36] J¨org Behler, Roman Martoˇn´ak, Davide Donadio, and
+Michele Parrinello.
+Pressure-induced phase transitions
+in silicon studied by neural network-based metadynamics
+simulations. physica status solidi (b), 245(12):2618–2629,
+2008.
+[37] Johan Klarbring and Sergei I Simak. Phase stability of
+dynamically disordered solids from first principles. Phys-
+ical review letters, 121(22):225702, 2018.
+[38] J¨org Behler and Michele Parrinello. Generalized neural-
+network representation of high-dimensional potential-
+energy surfaces. Physical review letters, 98(14):146401,
+2007.
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf,len=666
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='12059v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='mtrl-sci]  28 Jan 2023  Potential energy surface prediction of Alumina polymorphs using graph neural  network  Soumya Sanyal,1, ∗ Arun Kumar Sagotra,2, † Narendra Kumar,3 Sharad Rathi,4 Mohana  Krishna,4 Nagesh Somayajula,5 Duraivelan Palanisamy,6 Ram R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Ratnakar,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='7 Suchismita  Sanyal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='6 Partha Talukdar,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' ∗ Umesh Waghmare,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='3 and Janakiraman Balachandran2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' †  1University of Southern California,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' USA  2  3Jawaharlal Nehru Centre for Advanced Scientific Research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Bangalore  4Vellore Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Vellore  5Centura,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Colarado,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' USA  6Shell Technology Centre Bangalore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Bangalore  7Shell Technology Centre Houston,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' USA  8Google Research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Bangalore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' India  The process of design and discovery of new  materials  can  be  significantly  expedited  and  simplified if we can learn effectively from avail-  able data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Deep learning (DL) approaches have  recently received a lot of interest for their ability  to speed up the design of novel materials by pre-  dicting material properties with precision close  to experiments and ab-initio calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  The  application of deep learning to predict materials  properties measured by experiments are valuable  yet challenging due to the limited amount of ex-  perimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Most of the existing approaches  to predict properties from computational data  have also been directed towards specific material  properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  In this work, we extend this ap-  proach, by proposing Landscape Crystal Graph  Convolution Network(LCGCN), an accurate and  transferable deep learning framework based on  graph convolutional networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  LCGCN directly  learns the potential energy surface (PES) from  atomic configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' This approach can enable  transferable models that can predict different  material properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  We apply this framework  to bulk crystals (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  Al2O3), and test it by  calculating potential energy surfaces at different  temperatures  and  across  different  phases  of  crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  INTRODUCTION  The development of next generation technologies in  energy, chemicals, electronics and transportation indus-  tries are closely tied to the development of new materials  with the desired properties [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' For example in case of  ∗ Work done while at Indian Institute of Science Bangalore  † Corresponding  Author  and  Work  done  while  at  Shell  Technology  Centre  Bangalore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  arunsagotra91@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='com and janakiraman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='balachandran@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='com  energy, the need of the hour is the development of in-  expensive and environmentally safe materials that can  generate and store electrons with very high density and  efficiency while operating at high power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' In case of chemi-  cals, there is a critical need to develop new materials with  better catalytic performance [2], as well as those which  can make chemicals from alternative feedstock such as  recycled plastics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  Historically, discovery of materials has been driven by  empiricism, through the knowledge and intuition of re-  searchers working on these problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Over the past few  decades, they are being augmented by the insights and  knowledge derived from physics based models, in partic-  ular, atomistic models and simulations that can predict  the various properties of interest for a wide range of ma-  terials [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' However, these physics based models are  computationally expensive and as a result, only aid us  to explore a small subset of the materials search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  Further, within this approach, there is no formal way  to transfer information and knowledge from one system  to another, and from one application to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Data  driven models based on statistics and machine learning  can help in speeding up the materials discovery process  by providing a framework where algorithms can learn  from existing data obtained from models and experi-  ments on a subset of materials and predict the properties  and functionalities of new materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' In order to facilitate  such developments, various research groups from across  the world have also provided free public access to the  datasets of material structures [5–9] and properties ob-  tained predominantly from atomistic simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  These datasets and other datasets obtained from ex-  periments have been used in the recent literature to de-  velop data driven models to predict material proper-  ties [10–13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  Most of these models are based on hand  crafted descriptors that are suitable only for certain sub-  set of materials and for targeted properties of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  However, a transferable model needs to operate on raw  data to remove these biases introduced through hand  crafted descriptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' In case of materials and molecules,  such raw input data are the atoms and their position  coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  This basic materials information can be  2  naturally expressed in the form of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Graph Neu-  ral Networks (GNNs) are a class of deep learning mod-  els that have been successfully used to model graph-  structured data [14–16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' GNNs can learn to effectively  represent a molecular/crystal graph from raw data of  atom/bond features and use neural networks to perform  regression and classification tasks [17–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Recently, Xie  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' [13] developed crystal graph convolution neural  network (CGCNN) that was able to take this basic struc-  tural and chemical information of a crystal as graph to  perform graph convolutions on these graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' They were  able to obtain excellent results to predict properties of a  wide range of crystals, in their ground-state equilibrium  structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Later, Sanyal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' [21] integrated CGCNN  with multi-task learning to further improve the predic-  tion accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  OUTLINE OF THE WORK  Although CGCNN has been used to predict equilib-  rium properties of materials, many thermodynamic, ki-  netic and transport properties at non-zero temperatures  are determined only after incorporating the phononic  contributions that distort the crystal and its atomic com-  ponents from the equilibrium position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' These distortion  and the resultant phononic contributions depend upon  the potential energy surface (PES) associated with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  PES is a highly non-linear complex function of the chem-  ical composition, symmetry and atomic arrangement of  the crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Even a small change in these parameters  can significantly change the PES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' The PES is the input  to various statistical approaches such as Monte Carlo and  molecular dynamics simulations to predict the phononic  degrees of freedom and their influence on various material  properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  Conventionally PES are fitted to explicit function  forms from ab-initio simulation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' However, over  the past few years gaussian process based machine learn-  ing has been applied to develop interatomic potentials  for various materials ranging from elemental (amorphous  carbon, graphene, boron) [22–24] to multi-elemental  (methane) [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Although very successful, development of  Gaussian Approximation Potential (GAP) [26] requires  careful design of experiments to simulate small systems  whose inputs are needed to fit the interatomic potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  Further, there are also challenges in transfering the po-  tential to a new system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  These shortcomings can potentially be overcome by  GNN based deep learning approach, in which a model  learns from the raw materials data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  These models  can possibily lead to better transferability across dif-  ferent materials composition, crystal symmetry, atomic  arrangements and environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' In this work, we pro-  vide an initial demonstration of such transferability and  propose Landscape Crystal Graph Convolution  Network (LCGCN), an extension of CGCNN architec-  ture, to predict the PES of alumina (Al2O3), a material  that serves as the base material for various adsorbent and  catalytic applications [27–30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  Specifically, we demon-  strate the generalizability of this approach by developing  LCGCN models that not only accurately predict PES,  but are also transferable across scales, environmental  conditions (temperature) and different crystallographic  polymorphs, trigonal phase (space group R3c), cubic  phase (space group Ia3), Orthorhombic (space groups  Pbca and Pna21) of alumina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  METHODOLOGY  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  Molecular Dynamics  The data required to train, validate and test the  CGCNN models were obtained by performing Molecu-  lar Dynamics (MD) simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  We have performed  MD simulations under (N, V, T ) conditions with the  LAMMPS code [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' In these simulations the temper-  ature is kept fluctuating around a setpoint value by us-  ing Nos´e - Hoover thermostats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' We employ simulation  boxes, typically varying from 30 to 480 atoms, and apply  periodic boundary conditions along the three Cartesian  directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Newton’s equations of motion are integrated  using the customary Verlet’s algorithm with a time-step  length of 10−3 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' The typical duration of a MD run is  of 10 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' We used interatomic potentials of the Second-  Moment tight-Binding-QEq (SMTB-Q) form [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  In this work, employing the above MD protocol, we  simulate Al2O3 systems comprising of different sizes, en-  vironmental temperatures and polymorphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  We pick  10,000 configurations of these systems after thermal equi-  libriation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Unless specified otherwise, the 10,000 config-  urations were divided into a ratio of 6:2:2 for training,  validation and testing of the LCGCN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  Landscape Crystal Graph Convolution  Network(LCGCN)  CGCNN [13] is a graph neural network which focuses  on building a crystal graph from a given crystal struc-  ture and utilizes graph convolution networks (GCNs) to  model the crystal graph and predict their properties with  accuracy comparable to ab initio physics models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  We  propose Landscape Crystal Graph Convolution  Network (LCGCN), an extension of CGCNN, that can  be used to predict the potential energy surface (PES) of  crystal structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Next, we discuss the details of pro-  posed method LCGCN in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  A  crystal graph G  is an undirected  multigraph  defined  by  nodes  representing  atoms  and  edges  representing  bonds  in  a  crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  Formally,  let  G=(A, E, V, U) denote a crystal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  Here A rep-  resents the set of atoms in the crystal structure,  E={(i, j)k: kth bond between atom i and j}, is the set  of undirected edges denoting the bonds and |A|=N is  3  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Schematic representation of using LCGCN to predict potential energy of distorted alumina crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Initially, the  crystal structure is represented as a graph with nodes and edges representing the atoms and bonds in the crystal respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  Here, vi, vj and uij denote the node, neighbor and edge embeddings respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Then, CGCNN is used to learn a crystal  level embedding vG which is used to predict the potential energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Please refer to Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' III B for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  the number of atoms in the crystal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' vi ∈ V con-  tains the features of the ith atom which encodes various  properties of the atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' u(i,j)k ∈ U is the feature vector  for the kth bond between atoms i and j that captures  properties of the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' To capture the influence of local  neighborhood in the crystal structure, a graph convolu-  tion formulation similar to CGCNN [13] is used defined  as follows,  v(t+1)  i  = v(t)  i  +  �  j,k  σ(z(t)  (i,j)kW (t)  c  +b(t)  c )⊙g(z(t)  (i,j)kW (t)  s  +b(t)  s ),  where z(t)  (i,j)k = v(t)  i  ⊕v(t)  j ⊕u(i,j)k denotes the concate-  nation of atom and bond feature vectors of the neighbors  of ith atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' ⊙ denotes element-wise multiplication and  σ denotes a sigmoid function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  The σ(·) factor acts as  a learned weight matrix to incorporate different interac-  tion strengths between neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' W (t)  c , W (t)  s , b(t)  c  and  b(t)  s  are the convolution weight matrix, self weight ma-  trix, convolution bias and self bias of the t-th layer of  GCN respectively, and g(·) is the activation function for  introducing non-linear coupling between layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  Graph pooling [33] is a technique used to learn graph  representations given the learnt node and edge features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  The learnt atom features are pooled to get a vector rep-  resentation of the crystal (vG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' The crystal representa-  tion vG is then fed to a network of fully-connected layers  with non-linearities that learns to predict a property of  the crystal in a supervised manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' The workflow is de-  picted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' To train the network, we calculate the  loss using mean squared error between the true values  obtained using MD simulations and the model predic-  tions followed by back-propagation [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' The accuracy of  the model is quantified through mean absolute error per  atom (MAE/atom) by comparing the model predictions  on test configurations against the true values obtained  using MD simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  Pooling Operator  Formulation  Average  1  N  �N  i=1 vi  Max  max(vi),  ∀i ∈ [1, N]  FC-Max  max(viWpool + bpool),  ∀i ∈ [1, N]  TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Formulation of different graph pooling operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  Here, vi denotes the atom representation and max is an ele-  mentwise max operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Wpool and bpool are learnable pool-  ing parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Please refer to Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' IV C for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  RESULTS AND DISCUSSION  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  Effect of different pooling operators  In this section, we explore some of the existing graph  pooling operators - Average, Max and FC-max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  In  Average pooling, the graph embedding is defined as  the mean of all the node embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  Similarly, for  Max pooling, it is defined as the elementwise max of  all the node embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Hamilton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' [33] proposed  FC-max, a learnable pooling operator, which transforms  node embeddings using a fully-connected neural network  and then applies Max pooling operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' The formula-  tion of these pooling operators are defined in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  To analyze the effect of pooling, we perform the experi-  ment where we use alumina trigonal polymorph data for  training and predict performance on other alumina poly-  morphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' The results are shown in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' We observe  that FC-max performs the best among all of the pooling  operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' And hence, we use FC-max as the pooling  operator in all further analysis of our LCGCN architec-  ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  X1  X2  Predicted  LCGCN  FC  Build Graph  Value  Perturb1  0  0  0  0  0  0  Crystal Structure  Predicted  FC  LCGCN  Build Grap  Value  Final Fully  Potential Energy  Perturbed Crystal  Connected Layer  Crystal Graphs  Crystal  Structures  Embeddings4  Pooling Operator  Trigonal Orthorhombic (Pbca) Orthorhombic (Pna21)  Average  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='0009  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='0349  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='1796  Max  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='0007  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='4256  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='0789  FC-max  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='0007  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='0112  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='047  TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' MAE/atom of LCGCN model for different pooling operators on test data of 240 atom alumina orthorhombic (Pbca)  and Orthorhombic (Pna21) polymorph trained on trigonal data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Please refer to Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' IV A for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  CGCNN  LCGCN  Phase  Error/atom  Relative Error  Error/atom  Relative Error  Trigonal  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='0466  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='0018  0  Orthorhombic (Pbca)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='0208  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='0258  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='0001  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='0019  Orthorhombic (Pna21)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='0302  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='0164  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='0031  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='0049  TABLE III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Error/atom (eV/atom) and Relative Error (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' trigonal alumina phase) of LCGCN and CGCNN model on test  dataset of different polymorphs using models trained on setups as defined in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' IV B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' We observe that LCGCN consistently  outperforms CGCNN for all alumina phase equilibrium predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  Prediction of energy at equilibrium  Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' [13] train their CGCNN model to predict  formation energy per atom (∆Ef) using a large number  of equilibrium crystal structures obtained from the Ma-  terials Project [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Since (∆Ef) for a large number of  crystals can span across a wide range (approximately 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='5  eV/atom), such a model will not be able to accurately  predict the small energy differences across different poly-  morphs of the same composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' However, in this work  we train LCGCN model to predict the PES of various  polymorphs of alumina through training on distorted in-  put structures obtained from MD simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' The re-  sults are shown in table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Due to this training data, we  expect our model to perform better in predicting the en-  ergy differences of the alumina polymorphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' The errors  of predictions from the two approaches are shown in ta-  ble III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' We report both ’signed’ error and relative error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  For relative error, we choose the energy of the trigonal  phase as the reference value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' As expected, LCGCN sig-  nificantly outperforms the previous CGCNN model sug-  gesting that LCGCN is well-equipped than CGCNN to  predict small energy changes typically seen in distortions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  In the next upcoming subsections, we will discuss the  PES predictions of alumina across supercell sizes, tem-  perature and polymorphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  Prediction of potential energy  The LCGCN model was used to predict the potential  energy of the trigonal alumina polymorph of different su-  percell sizes – namely 120 and 480 atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' The data to  train, validate and test was obtained from MD simual-  tions as described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' III A under a temperature ramp  of 300K-500K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' We quantify the error by reporting mean  absolute error per atom (MAE/atom) (eV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' The results  for the two different supercells are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' As  expected, increasing the amount of training data reduces  the error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' We observe that with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='6 times the total alu-  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='6  T  raining Dataset Size (x10000)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='0006  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='0008  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='0010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='0012  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='0014  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='0016  MAE/atom (eV)  120 atoms  480 atoms  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Plotting test MAE/atom of LCGCN model in pre-  dicting potential energy of 120 and 480 atoms systems of  alumina trigonal polymorph with increasing size of training  dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Please refer to Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' IV C for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  mina data (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  6000 samples) for training, our model  is able to achieve an MAE/atom which is close to the  accuracy of DFT (1 meV/atom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  Transferable LCGCN models across scale  In many scenarios, the property of interest can only  be obtained from performing large scale atomistic and  quantum mechanical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' However, the scaling  of such simulations with system size can be non-linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  For example in case of DFT, the cpu time of a simula-  tion increases as O(N 3) where N is the number of atoms  that limits the system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Hence, an ML model that  is transferable from smaller scale systems to larger scale  systems are of importance in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Firstly, we note  that the cpu time for inference using our model increases  as O(N) with the number of nodes N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' We perform a  time analysis of the LCGCN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' For this, we use a  5  Training  Validation  Testing  120 atoms  480 atoms  480 atoms  480 atoms  MAE/atom  1000  2400  600  2000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='0017  3000  2400  600  2000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='0017  6000  2400  600  2000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='0011  10000  2400  600  2000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='0011  TABLE IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' MAE/atom of LCGCN model on test data of 480 atoms with increasing training dataset of 120 atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Please  refer to Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' IV D for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  Training  Validation  Testing  120 atoms  480 atoms  120 atoms  480 atoms  480 atoms  MAE/atom  2400  0  600  0  2000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='0023  2400  800  600  200  2000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='0014  2400  1600  600  400  2000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='0013  2400  2400  600  600  2000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='0011  TABLE V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' MAE/atom of LCGCN model on test data of 480 atoms with increasing training data of 480 atom configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  Please refer to Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' IV D for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  100  200  300  400  500  600  Number of Atoms  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='020  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='030  Time (Seconds)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Time analysis of the LCGCN code with respect to  the CPU run, as a function of number of atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  pre-trained LCGCN model to predict potential energy  for a set of 100 configurations of 120 atom alumina trig-  onal polymorph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Next, we repeat this for 240, 360, 480  and 600 atoms supercells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' The average cpu time for each  setup is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' We observe that LCGCN is  significantly faster than DFT and MD simulations, and  in contrast to MD and DFT the LCGCN scales linearly  with the system size and easily parallelized [35, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  Next, we demonstrate how increasing the training data  of the 120 atoms supercell of alumina trigonal polymorph  can reduce the test error of predicting the potential en-  ergy of the 480 atom supercell configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' For this,  we consider a setup where there are a fixed 2400 con-  figurations of 480 atoms for model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  Also, we  have 10000 configurations of 120 atoms that we progres-  sively add to the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' As observed from Table  IV, an increase in the number of 120 atom configura-  tions in training data leads to a reduction in the test er-  ror for the 480 atom configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' After adding all the  10000 configurations of 120 atoms in training data, the  error in MAE/atom for 480 atom configuration is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='0011  eV/atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' This is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' In a second set of ex-  periments, the data of 120 atoms is fixed in training set,  while the number of 480 atoms is increased in training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  The results are reported in Table V and depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  4b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' We find that an increase in the 480 atom configura-  tion in training set leads to significant improvements in  model performance on predicting potential energy for 480  atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' From these set of experiments we can conclude  that although it is not trivial for LCGCN to generalize,  although with comparitively higher error to predict PES  of the larger supercell (480 atoms) even when these larger  supercell configurations are not included in the training  data (1st row of Table V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' However, this error goes down  significantly even if a small amount of the 480 atoms con-  figurations are included into the training data (4th row  of Table V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  Transferable LCGCN models across  environmental conditions  Envrionmental conditions, in particular temperature,  play an important role in determining the dynamics of  the materials and the resultant phase stability and mate-  rial properties [1, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Higher temperature molecular dy-  namics has been employed as a tool in different problems  to sample the configurations rapidly and to predict the  resultant low energy configuration and their correspond-  ing properties [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' In this section, we analyze LCGCN’s  transferability in predicting potential energies of configu-  rations of trigonal alumina across different environmental  conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' We specifically focus on variation of temper-  ature and test the effectiveness of the model in three  temperature values for alumina - 300K, 500K and 800K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  We pretrain a LCGCN model using all the data (10,000  configurations) from one temperature and predict poten-  tial energy at a different temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' The results are  shown in Table VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  6  2000  4000  6000  8000  10000  T  raining Samples (120 atoms)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='0011  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='0012  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='0013  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='0014  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='0015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='0016  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='0017  MAE/atom (eV)  (a)  0  500  1000  1500  2000  2500  T  raining Samples (480 atoms)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='0012  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='0014  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='0016  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='0018  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='0020  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='0022  MAE/atom (eV)  (b)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' (a) Test MAE/atom of LCGCN model in predicting  potential energy of 480 atoms of alumina trigonal polymorph  as a function of increasing training dataset of 120 atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Re-  fer Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' IV D for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' (b) Test MAE/atom of LCGCN  model in predicting potential energy of 480 atoms of alumina  trigonal polymorph as a function of increasing number of 480  atom configurations in training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Refer to Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' IV D  for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  In α-Al2O3, two types of Al-O bonds (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='87 angstrom  and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='98 angstrom) are present at 0 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' But under clas-  sical MD simulation, these two bonds vary drastically at  a given temperature T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' At low temperature (say 300 K),  bond-bond distribution function shows two peaks (at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='85  and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='03 angstrom) while at high temperatures (500 K &  800 K), the peak corresponding to longer (weaker) Al-O  bond (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='03 angstrom peak in 300 K configuration) gets  disappeared, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' 5a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' It means that weakly bonded  Al and O got separated much compared to strongly  bonded Al and O due to thermal stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  Next, we observe the effect of mixing the data across  two different temperatures (resulting in a total of 20,000  configurations split into training, validation and test set)  in predicting energies for data at an unseen temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  The results are shown in Table VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' We see when the  model is fed with a mixture of two temperatures data for  training the prediction error reduces significantly for test  data at unseen temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Specifically, we observe a  significant decrease in the MAE for prediction at 800K,  when data for both 300K and 500K are mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' In order  to delineate if the improvement in model prediction is due  to training data volume, or due to data from two different  Temp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  300K  500K  800K  300K  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='0003  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='0212  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='0766  500K  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='0151  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='0004  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='0254  800K  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='0372  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='0192  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='0007  TABLE VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' MAE/atom of LCGCN model on test data of 240  atom alumina trigonal polymorph at three different temper-  atures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Row temperature indicates that model was trained  using the data of that particular temperature and the column  temperature denotes that the trained model was used to pre-  dict potential energy at that temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Please refer to Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  IV E for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  Mixture  Target  MAE/atom  300K & 500K  800K  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='02  300K & 800K  500K  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='0048  500K & 800K  300K  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='0035  300K & 500K(5000 each)  800K  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='0169  TABLE VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' MAE/atom of LCGCN model on test data of  240 atom alumina trigonal polymorph at a target tempera-  ture using a mixture of data for two different temperatures in  training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Please refer to Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' IV E for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  temperatures, we train a model with 5000 configurations  of 300K and 500K each and test the model on data from  800K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' The resulting MAE/atom (4th row of Table VII)  is still significantly lower than predictions at 800K using  only 300K or 500K data for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' This suggests that  the model is able to learn better when it is trained with  heterogeneous data from two different temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  Transferable LCGCN models across different  Polymorphs  One of the major challenges in materials and chem-  istry arises from the difficulty in transfering information,  model and knowledge across different crystal structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  A model that can learn properties such as potential en-  ergy, from equilibrium and distorted configurations of  some crystal structures and predict for new crystal struc-  tures with minimal data would be a very useful tool in  accelerating materials discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' As a preliminary step  toward this goal, in this work, we study the transfer-  ability of LCGCN models across different polymorphs  of alumina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' We consider three phases of alumina - or-  thorhombic (Pbca and Pna21) and trigonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' For the first  set of experiments, we predict the potential energy of  the different polymorphs based on the training data from  another polymorph (aka zero shot learning).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Similar to  temperature study, here we use all the data pertaining to  a polymorph (10,000 configurations) to train the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  The results are shown in Table VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  To understand the trends in the transferability of  LCGCN across different polymorphs, we plotted the  Al − O bond distribution and configurational energies  for the three alumina polymorphs as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' 5b  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' We observe that, the more the overlap be-  7  Phase  Orthorhombic (Pbca) Trigonal Orthorhombic (Pna21)  Orthorhombic (Pbca)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='0007  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='0265  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='037  Trigonal  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='0112  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='0007  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='047  Orthorhombic (Pna21)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='9689  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='1542  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='0006  TABLE VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' MAE/atom of LCGCN model on test data of 240 atom alumina for three different polymorphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' The row phase  denotes that the model was trained using the data for that particular polymorph and column phase indicates that the trained  model was used to predict the potential energy for that polymorph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Refer to Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' IV F for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='8  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='4  Bond Length  (Å)  0  2  4  6  8  Count  (a)  300 K  500 K  800 K  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='8  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='4  Bond Length (Å)  0  2  4  6  8  Count  (b)  Orthorhombic (Pbca)  Orthorhombic (Pna21)  T  rigonal  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' (a) Al-O bond distribution of α-Al2O3 at 300 K ,  500 K and 800 K temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' (b)Al-O bond distribution of  Alumina polymorphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' (c) Density of energy distribution of  Alumina polymorphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  Mixture  MAE/atom  Trigonal + Orthorhombic (Pna21)(5000 each)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='1804  Trigonal + Orthorhombic (Pna21)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='1312  TABLE IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' MAE/atom of LCGCN model on test data of  240 atom alumina othorhombic (Pbca) polymorph using a  mixture of data for different alumina polymorphs in training  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Please refer to Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' IV F for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  tween Al − O bond distribution of different polymorphs,  the better transferability, and hence lower the error in  LCGCN’s prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' For example, the overlap between  the Orthorhombic (Pbca) and Trigonal polymorphs is  greater than that between the Orthorhombic (Pbca) and  Orthorhombic (Pna21) polymorphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  As seen in Table  VIII, our LCGCN model, when trained on Orthorhom-  bic (Pbca) data, predicts the Trigonal polymorph with  a lower error compared to when it is trained on Or-  thorhombic (Pna21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' This is likely due to the fact that  the LCGCN model’s accuracy is highly dependent on  bond length, and transferability is better when predict-  ing systems with similar bond lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  In order to understand how the model accuracy im-  proves when trained with heterogeneous data, we train  the model with data from multiple polymorphs to pre-  dict potential energy of orthorhombic (Pbca) polymorph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  We choose orthorhombic (Pbca) for our analysis because  the error in PES prediction was the highest as shown in  Table VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' The results for this analysis is shown in Ta-  ble IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' We observe that if we restrict the training data  to a total of 10,000 configurations (each polymorph has  5,000 data points), the error in prediction is about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='1804  eV/atom suggesting that training on diverse dataset can  improve the transferability of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' However, if we  incorporate all the data from the two polymorphs (20,000  configurations in total), the model error further reduces  to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='1312 eV/atom suggesting that the transferability of  the model can be improved by increasing both the volume  and variety of training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  CONCLUSION  In this work, we proposed LCGCN, a graph neural  network framework to directly predict high-dimensional  potential energy surface for alumina (Al2O3), a base ma-  terial for various adsorbent and catalytic applications,  from atomic configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  This approach enables to  bypass calculating the derivatives of the PES which are  often the computational bottlenecks of existing ab-initio  methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  We  further  demonstrated  the  transferability  of  LCGCN across scales, environmental conditions and dif-  ferent alumina polymorphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Finally, We validated that,  with appropriate conditioning, LCGCN can obtain ac-  curacies that are close to DFT (1 meV/atom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Therefore,  we anticipate that LCGCN will significantly accelerate  the atomistic studies of other materials with its scala-  bility, versatility, and excellent potential energy surface  prediction accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  DATA AVAILABILITY  The data that support the findings of this study are  available from the corresponding author (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='S)  upon reasonable request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  8  ACKNOWLEDGMENTS  The authors would like to acknowledge Shell, JNCASR  and IISc for their support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  AUTHOR CONTRIBUTIONS  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='B, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='W and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='T conceived the study and planned  the research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' performed the MD sim-  ulations to generate the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='S , M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' K, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' R and  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='S implemented and optimized the LCGCN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  All authors discussed the results and their implications  and contributed to the writing of the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  COMPETING INTERESTS  The authors declare no competing interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
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+page_content='  [3] National Science and Technology Council (US).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Materi-  als genome initiative for global competitiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
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+page_content='  [4] Narendra Kumar, Padmini Rajagopalan, Praveen Panka-  jakshan,  Arnab  Bhattacharyya,  Suchismita  Sanyal,  Janakiraman Balachandran, and Umesh V Waghmare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
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+page_content='  [5] Anubhav Jain, Shyue Ping Ong, Geoffroy Hautier, Wei  Chen,  William Davidson Richards,  Stephen Dacek,  Shreyas Cholia,  Dan Gunter,  David Skinner,  Ger-  brand Ceder, and Kristin a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
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+page_content='  [9] James E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Saal, Scott Kirklin, Muratahan Aykol, Bryce  Meredig, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Wolverton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Materials Design and Dis-  covery with High-Throughput Density Functional The-  ory: The Open Quantum Materials Database (OQMD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
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+page_content='1007/s11837-013-0755-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  [10] Ghanshyam  Pilania,  Chenchen  Wang,  Xun  Jiang,  Sanguthevar Rajasekaran, and Ramamurthy Ramprasad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
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+page_content='  [11] Logan Ward, Ankit Agrawal, Alok Choudhary, and  Christopher Wolverton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
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+page_content='  [14] Joan Bruna, Wojciech Zaremba, Arthur Szlam, and Yann  LeCun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
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+page_content=' In International Conference on Learn-  ing Representations (ICLR), 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  [15] Michael M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Bronstein, Joan Bruna, Yann LeCun, Arthur  Szlam, and Pierre Vandergheynst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
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+page_content='  Mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=', 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  [16] Thomas N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Kipf and Max Welling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
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+page_content='  [17] David K Duvenaud, Dougal Maclaurin, Jorge Ipar-  raguirre,  Rafael  Bombarell,  Timothy  Hirzel,  Alan  Aspuru-Guzik, and Ryan P Adams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
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+page_content=',  2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  9  [18] Steven Kearnes, Kevin McCloskey, Marc Berndl, Vijay  Pande, and Patrick Riley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
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+page_content=' Journal of Computer-Aided  Molecular Design (CAMD), 30(8):595–608, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  [19] Justin Gilmer, Samuel S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Schoenholz, Patrick F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
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+page_content=' Dahl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
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+page_content=' B, 97:054303,  Feb 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
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+page_content=',  120:156001,  Apr 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
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+page_content='  [25] Max  Veit,  Sandeep  Kumar  Jain,  Satyanarayana  Bonakala,  Indranil Rudra,  Detlef Hohl,  and G´abor  Cs´anyi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
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+page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='136403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  [27] J´ozsef Hlavay and Kl´ara Poly´ak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Determination of sur-  face properties of iron hydroxide-coated alumina adsor-  bent prepared for removal of arsenic from drinking water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  Journal of Colloid and Interface Science, 284(1):71–77,  2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  [28] Vinod K Gupta, Shilpi Agarwal, and Tawfik A Saleh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  Synthesis and characterization of alumina-coated carbon  nanotubes and their application for lead removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Journal  of hazardous materials, 185(1):17–23, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  [29] Herman Pines and Werner O Haag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Alumina: catalyst  and support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' alumina, its intrinsic acidity and catalytic  activity1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Journal of the American Chemical Society, 82  (10):2471–2483, 1960.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  [30] Monica Trueba and Stefano P Trasatti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' γ-alumina as a  support for catalysts: a review of fundamental aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  European journal of inorganic chemistry, 2005(17):3393–  3403, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  [31] Steve Plimpton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  Fast parallel algorithms for short-  range  molecular  dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  Journal  of  Computa-  tional Physics, 117(1):1 – 19, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  ISSN 0021-9991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  doi:https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='1006/jcph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='1039.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  URL  http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='sciencedirect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='com/science/article/pii/S002199918571039X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  [32] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Salles, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Politano, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Amzallag, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' T´etot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  Molecular  dynamics  study  of  high-pressure  alu-  mina  polymorphs  with  a  tight-binding  variable-  charge  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  Computational  Materials  Science,  111:181  –  189,  2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  ISSN  0927-0256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  doi:  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='commatsci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  URL  http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='sciencedirect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='com/science/article/pii/S0927025615005868.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  [33] William L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Hamilton, Rex Ying, and Jure Leskovec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' In-  ductive representation learning on large graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' In NIPS,  2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  [34] David  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  Rumelhart,  Geoffrey  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  Hinton,  and  Ronald  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  Williams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  Learning  Representations  by  Back-propagating  Errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  Nature,  323(6088):  533–536,  1986.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='1038/323533a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  URL  http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='com/articles/323533a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  [35] Justin S Smith, Olexandr Isayev, and Adrian E Roit-  berg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Ani-1: an extensible neural network potential with  dft accuracy at force field computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Chemical  science, 8(4):3192–3203, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  [36] J¨org Behler, Roman Martoˇn´ak, Davide Donadio, and  Michele Parrinello.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  Pressure-induced phase transitions  in silicon studied by neural network-based metadynamics  simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' physica status solidi (b), 245(12):2618–2629,  2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  [37] Johan Klarbring and Sergei I Simak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Phase stability of  dynamically disordered solids from first principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Phys-  ical review letters, 121(22):225702, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content='  [38] J¨org Behler and Michele Parrinello.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Generalized neural-  network representation of high-dimensional potential-  energy surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
+page_content=' Physical review letters, 98(14):146401,  2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFLT4oBgHgl3EQfXC8y/content/2301.12059v1.pdf'}
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+Why Capsule Neural Networks Do Not Scale:
+Challenging the Dynamic Parse-Tree Assumption
+Matthias Mitterreiter,1,4 Marcel Koch,2 Joachim Giesen,1 S¨oren Laue3
+1Friedrich-Schiller-University Jena, Germany
+2Ernst Abbe University of Applied Sciences Jena, Germany
+3Technical University Kaiserslautern, Germany
+4Data Assessment Solutions GmbH, Hannover, Germany
+matthias.mitterreiter@uni-jena.de, marcel.koch@eah-jena.de, joachim.giesen@uni-jena.de, laue@cs.uni-kl.de
+Abstract
+Capsule neural networks replace simple, scalar-valued neu-
+rons with vector-valued capsules. They are motivated by
+the pattern recognition system in the human brain, where
+complex objects are decomposed into a hierarchy of sim-
+pler object parts. Such a hierarchy is referred to as a parse-
+tree. Conceptually, capsule neural networks have been de-
+fined to realize such parse-trees. The capsule neural network
+(CapsNet), by Sabour, Frosst, and Hinton, is the first ac-
+tual implementation of the conceptual idea of capsule neural
+networks. CapsNets achieved state-of-the-art performance on
+simple image recognition tasks with fewer parameters and
+greater robustness to affine transformations than compara-
+ble approaches. This sparked extensive follow-up research.
+However, despite major efforts, no work was able to scale
+the CapsNet architecture to more reasonable-sized datasets.
+Here, we provide a reason for this failure and argue that it
+is most likely not possible to scale CapsNets beyond toy ex-
+amples. In particular, we show that the concept of a parse-
+tree, the main idea behind capsule neuronal networks, is not
+present in CapsNets. We also show theoretically and experi-
+mentally that CapsNets suffer from a vanishing gradient prob-
+lem that results in the starvation of many capsules during
+training.
+1
+Introduction
+The concept of capsules (Hinton, Krizhevsky, and Wang
+2011) describes a hypothetical system that parses a complex
+image scene into a hierarchy of visual entities that stand in
+part-whole relationship to each other (Hinton, Ghahramani,
+and Teh 1999). A capsule is conceptually defined as a highly
+informative, compact representation of a visual entity or ob-
+ject within an image. The idea of capsules is motivated by
+the pattern recognition system in the visual cortex of the hu-
+man brain (Sabour, Frosst, and Hinton 2017). There is some
+psychological evidence that the human object recognition
+system assigns hierarchical structural descriptions to com-
+plex objects by decomposing them into parts (Hinton 1979).
+The theory of recognition by components (Biederman 1987)
+proposes that a relatively small set of simple 3D shapes,
+called geons, can be assembled in various arrangements to
+Copyright © 2023, Association for the Advancement of Artificial
+Intelligence (www.aaai.org). All rights reserved.
+form virtually any complex object, which can then be recog-
+nized by decomposition into its respective parts (Biederman
+1987).
+A capsule may represent a visual entity by encapsulating
+its properties, also known as instantiation parameters, such
+as position, size, orientation, deformation, texture, or hue.
+A multi-level assembly of such capsules represents a com-
+plex image scene, where lower-level capsules model less
+abstract objects or object parts, and higher-level capsules
+model complex and composite objects. Lower-level capsules
+are connected to higher-level capsules if the corresponding
+entities are in a part-whole relationship. For a composite ob-
+ject, the hierarchy of capsules defines a syntactic structure
+like a parse-tree defines the syntactic structure of a sentence.
+Therefore, the hierarchy of capsules is also referred to as
+parse-tree. If an object or object part is present in an image,
+its respective capsule will be present within the parse-tree.
+Ideally, the parse-tree is invariant under affine transfor-
+mations as well as changes of viewpoint. That is, a slightly
+modified viewpoint on a visual entity should not change
+a capsule’s presence within the parse-tree. Such parse-
+trees would be highly efficient distributed representations
+of image scenes (Sabour, Frosst, and Hinton 2017; Hin-
+ton, Ghahramani, and Teh 1999). Also, explainable machine
+learning can profit from interpretable capsules that stand for
+dedicated visual entities, and the discrete nature of trees may
+connect deep learning with a symbolic approach to AI. Fur-
+thermore, capsules can be related to inverse graphics, and
+there is hope that they can lead to debuggable, parameter ef-
+ficient, and interpretable models with a broad range of appli-
+cations for all kinds of image-related tasks like image clas-
+sification or segmentation.
+However, capsules are only conceptually defined, and the
+difficulty is finding an implementation with all the highly-
+desirable properties from above. The capsule neural net-
+work (CapsNet) by Sabour, Frosst, and Hinton (2017) aims
+at such an implementation of the conceptual capsule idea.
+It was specifically designed to surpass convolutional neu-
+ral networks (ConvNets) (LeCun et al. 1989) as the latter
+were found to suffer from several limitations, including a
+lack of robustness to affine transformations and change of
+viewpoint, the susceptibility to adversarial attacks, exponen-
+tial inefficiencies, and a general lack of interpretability in the
+network’s decision-making process. Considering these limi-
+arXiv:2301.01583v1  [cs.CV]  4 Jan 2023
+
+tations, the parse-tree sounds particularly appealing with all
+its advantages.
+Contributions.
+Here, our aim is a thorough investigation
+of the question, whether the CapsNets implementation as
+proposed by Sabour, Frosst, and Hinton (2017) realizes all
+the conceptual ideas that make capsule networks so appeal-
+ing. We summarize this in two key assumptions. The first
+key assumption is that the CapsNet learns to associate a
+capsule with a dedicated visual entity within an input im-
+age (Sabour, Frosst, and Hinton 2017). The second key as-
+sumption is that the CapsNet’s capsules can be organized
+hierarchically in a parse-tree that encodes part-whole rela-
+tionships. We test both assumptions experimentally and have
+to reject them. We show that the CapsNet does not exhibit
+any sign of an emerging parse-tree. Thus, the CapsNet im-
+plementation cannot provide the theoretical benefits of cap-
+sule networks like invariance under affine transformations
+and change of viewpoint. Furthermore, we provide a theo-
+retical analysis, exposing a vanishing gradient problem, that
+supports our experimental findings.
+2
+Related Work
+Early references to the hierarchy of parts appear already
+in (Hinton 1979). The idea of parsing images into parse-
+trees was proposed by Hinton, Ghahramani, and Teh (1999)
+and the concept of capsules was established in (Hinton,
+Krizhevsky, and Wang 2011). An important addition by
+the CapsNet (Sabour, Frosst, and Hinton 2017) was the
+routing-by-agreement (RBA) algorithm that creates cap-
+sule parse-trees from images. With its introduction, the
+CapsNet demonstrated state-of-the-art classification accu-
+racy on MNIST (LeCun et al. 1998) with fewer parame-
+ters and stronger robustness to affine transformations than
+the ConvNet baseline, which sparked a flood of follow-
+up research. This includes different routing mechanisms,
+such as EM-Routing (Hinton, Sabour, and Frosst 2018),
+Self-Routing (Hahn, Pyeon, and Kim 2019), Variational
+Bayes Routing (Ribeiro, Leontidis, and Kollias 2020), Re-
+ceptor Skeleton (Chen et al. 2021) and attention-based rout-
+ing (Ahmed and Torresani 2019; Tsai et al. 2020; Mazzia,
+Salvetti, and Chiaberge 2021; Gu 2021). Wang and Liu
+(2018) reframed the routing algorithm in (Sabour, Frosst,
+and Hinton 2017) as an optimization problem, and Rawl-
+inson, Ahmed, and Kowadlo (2018) introduced an unsu-
+pervised learning scheme for CapsNets. Other work re-
+placed the capsule vector representations by matrices (Hin-
+ton, Sabour, and Frosst 2018) or tensors (Rajasegaran et al.
+2019), or added classic ConvNet features to the general rout-
+ing mechanisms, such as dropout (Xiang et al. 2018) or skip-
+connections (Rajasegaran et al. 2019). The GLOM archi-
+tecture, which was proposed by (Hinton 2021), suggests a
+routing-free approach for creating parse-trees from images,
+but has not been implemented yet. Furthermore, other pub-
+lications focus on learning better first layer capsules (Prime-
+Caps), such as the Stacked Capsule Autoencoders (Kosiorek
+et al. 2019) and Flow Capsules (Sabour et al. 2021).
+However, after a while it turned out that the CapsNet
+falls short of the anticipated benefits and promises of the
+capsule idea. To this date, CapsNets do not scale beyond
+small-scale datasets. Works that empirically report scaling
+issues include (Xi, Bing, and Jin 2017; Paik, Kwak, and Kim
+2019). Although the CapsNet was introduced in the realm
+of computer vision, the best performing capsule implemen-
+tation (Ahmed and Torresani 2019) achieves only 60.07%
+top-1 image classification accuracy on ImageNet (Deng
+et al. 2009), far behind state-of-the-art transformer-based ap-
+proaches (Wortsman et al. 2022) and ConvNets (Pham et al.
+2021) with 90.88% and 90.02% accuracy respectively. The
+original CapsNet itself has not been demonstrated to work
+on ImageNet.
+Further negative results regarding CapsNets emerged,
+questioning the promised benefits and technical progress al-
+together. Paik, Kwak, and Kim (2019) observed that increas-
+ing the depth of various CapsNet variants did not improve
+accuracy, and routing algorithms, the core components of
+capsule implementations, do not provide any benefit regard-
+ing accuracy in image classification. Michels et al. (2019),
+and Gu, Wu, and Tresp (2021) showed that CapsNets can be
+as easily fooled as ConvNets when it comes to adversarial at-
+tacks. Gu, Tresp, and Hu (2021) showed that the individual
+parts of the CapsNet have contradictory effects on the per-
+formance on different tasks and conclude that with the right
+baseline, CapsNets are not generally superior to ConvNets.
+Finally, Gu and Tresp (2020) showed that removing the dy-
+namic routing component improves affine robustness, and
+Rawlinson, Ahmed, and Kowadlo (2018) show that capsules
+do not specialize without supervision.
+Here, we explain these shortcomings, which can be at-
+tributed to a lack of an emerging parse-tree.
+3
+The Capsule Neural Network
+The CapsNet implements capsules as parameter vectors. An
+illustration of the CapsNet architecture, which consists of a
+multi-layer hierarchy of capsules, is shown in Figure 1. In
+the following, we introduce basic notations and definitions,
+the generic CapsNet architecture, and a loss function for
+training CapsNets. Furthermore, we discuss how CapsNets
+implement the crucial concept of a parse-tree.
+R
+R
+R
+CNN
+CNN backbone
+capsule
+routing
+FC decoder
+Figure 1: A generic CapsNet architecture.
+3.1
+Notation
+Capsules.
+The matrix U l ∈ Rnl×dl holds nl normalized
+capsules of dimension dl for layer l ∈ {1, 2, . . . , ℓ}. The
+i-th capsule in U l is the vector ul
+(i,:) ∈ Rdl, and ul
+(i,j) ∈ R
+
+is the j-th entry of capsule i on layer l.
+Transformation
+matrices.
+The
+tensor
+W l
+∈
+Rnl+1×nl×dl+1×dl
+holds
+transformation
+matrices
+W l
+(j,i,:,:) ∈ Rdl+1×dl. The transformation matrix W l
+(j,i,:,:)
+maps the i-th capsule of layer l to its unnormalized contri-
+bution to the j-the capsule of layer l + 1.
+Coupling coefficients.
+The matrix Cl ∈ Rnl×nl+1 holds
+coupling coefficients for the connections of capsules from
+layer l to layer l + 1. The entry cl
+(i,j) ∈ [0, 1] specifies
+the coupling strength between capsule i on layer l and
+capsule j on layer l + 1. The coupling coefficients satisfy
+�nl+1
+j=1 cl
+(i,j) = 1 for all i ∈ {1, 2, . . . , nl}.
+Squashing function.
+The squashing function normal-
+izes the length of a capsule vector u
+∈
+Rd into the
+range [0, 1). Here, we use a slightly modified squashing
+function (Mazzia, Salvetti, and Chiaberge 2021),
+g(u) =
+�
+1 −
+1
+exp(∥u∥2)
+�
+u
+∥u∥2
+(1)
+that behaves similarly to the original squashing function pro-
+posed by Sabour, Frosst, and Hinton (2017), but is more sen-
+sitive to small changes near zero (Xi, Bing, and Jin 2017),
+which is required to stack multiple layers of capsules.
+3.2
+Architecture
+First, the backbone network extracts features from an in-
+put image into a feature matrix in Rn1×d1. The feature
+matrix is then normalized by applying the squashing func-
+tion to each row, which constitutes the first capsule layer in
+U 1 ∈ Rn1×d1. The capsules in U 1 are also called Prime-
+Caps. Starting from the PrimeCaps, consecutive layers of
+capsules are computed as follows: First, the linear contri-
+bution of capsule i on layer l to capsule j on layer l + 1 is
+computed as
+ˆul+1
+(i,j,:) = W l
+(j,i,:,:)ul
+(i,:),
+(2)
+where the entries in the matrix ˆU l+1
+(i) , which holds the vectors
+ˆul+1
+(i,j,:), are called votes from the i-th capsule on layer l. An
+upper layer capsule ul+1
+(j,:) is the squashed, weighted sum over
+all votes from lower layer capsules ul
+(i,:), that is,
+ul+1
+(j,:) = g
+�
+�
+nl
+�
+i=1
+cl
+(i,j) · ˆul+1
+(i,j,:)
+�
+� ,
+where the the coupling coefficients cl
+(i,j) are dynamically
+computed, that is, individually for every input image, by the
+Routing-by-agreement Algorithm (RBA), see Algorithm 1.
+The number of output capsules on the last layer ℓ is set
+to match the number of classes in the respective dataset. Fi-
+nally, the fully-connected decoder network reconstructs the
+input image from the capsules on layer ℓ.
+3.3
+Training
+The parameters in the backbone network, in the reconstruc-
+tion network, as well as the transformation tensors W l and
+Algorithm 1: Routing-by-agreement (RBA)
+Input: votes ˆu, number of iterations r, routing priors b
+Output: coupling coefficients c
+1: for r iterations do
+2:
+c(i,j) ←
+exp(b(i,j))
+�
+k exp(b(i,k))
+3:
+v(j,:) ← g
+��
+i c(i,j)ˆu(i,j,:)
+�
+4:
+b(i,j) ← b(i,j) + ⟨ˆu(i,j,:), v(j,:)⟩
+5: end for
+1
+3
+5
+7
+9
+11
+13
+15
+1
+2
+3
+4
+5
+1
+3
+5
+7
+9
+11
+13
+15
+1
+2
+3
+4
+5
+1
+3
+5
+7
+9
+11
+13
+15
+1
+2
+3
+4
+5
+1
+3
+5
+7
+9
+11
+13
+15
+1
+2
+3
+4
+5
+Figure 2: Fuzzy parse-trees for images from the AffNIST
+dataset for a model with five capsule layers, 16 capsules on
+each intermediate layer, and ten on the last layer. The figure
+shows the coupling coefficients as connections between cap-
+sules (left), the capsule norms/activations (middle), and the
+input image (right). The blue tone of the edges is darker for
+greater coupling coefficients.
+the RBA routing priors bl
+(i,j) are all learned by minimiz-
+ing a weighted sum of a supervised classification loss Lm
+and an unsupervised reconstruction loss Lr, that is, L =
+Lm + α · Lr, with α > 0. The classification loss function
+Lm =
+nℓ
+�
+j=1
+tj · max{0, m+ − ∥uℓ
+(j,:)∥2}2
++ λ · (1 − tj) · max{0, ∥uℓ
+(j,:)∥2 − m−}2
+(3)
+is only applied to the output capsules. Here m+, m− > 0
+and λ > 0 are regularization parameters, and tj is 1 if an
+object of the j-th class is present in the input image, and 0
+otherwise. Output capsules that correspond to classes not
+present in the input image are masked by zeros. The recon-
+struction loss function is applied to the output of the recon-
+struction network and sums the distances between the recon-
+struction and the pixel intensities in the input image.
+3.4
+Parse-Trees
+The parse-tree is the most important concept that allows us
+to understand and interpret capsules and their connections.
+The CapsNet defines a parse-tree, where capsules represent
+nodes, and coupling coefficients represent fuzzy edges.
+
+81
+3
+5
+7
+9
+11
+13
+15
+1
+2
+3
+4
+5
+(a) mean couplings
+1
+3
+5
+7
+9
+11
+13
+15
+1
+2
+3
+4
+5
+(b) std. couplings
+1
+3
+5
+7
+9
+11
+13
+15
+1
+2
+3
+4
+5
+(c) mean activations
+1
+3
+5
+7
+9
+11
+13
+15
+1
+2
+3
+4
+5
+(d) std. activations
+1
+3
+5
+7
+9
+11
+13
+15
+1
+2
+3
+4
+5
+(e) dead capsules
+Figure 3: Parse-tree statistics for the complete AffNIST validation dataset for a five-layer CapsNet model with 16/10 capsules.
+The mean (a) and the standard deviation (b) of the coupling coefficient matrices for each layer are visualized as connections
+between capsules. Higher coupling coefficients have a darker blue tone. The capsule norms’ mean (c) and standard deviation (d)
+are visualized by bars. Dead capsules (e) are highlighted with a red bar.
+The capsules magnitude, that is, the norm of the parameter
+vector, which is always in [0, 1) after applying the squash-
+ing function, represents the probability that the correspond-
+ing entity is present in an input image. The capsules direc-
+tion represents instantiation parameters of the entity like its
+position, size, or orientation. Changing the viewpoint on an
+entity does not affect its presence, but only its instantiation
+parameters. Therefore, the respective capsule’s magnitude
+should be unaffected, whereas its direction can change.
+In the CapsNet, the dynamic part-whole relationships are
+implemented by coupling coefficients between capsules on
+consecutive layers. The coupling coefficients in the routing
+layers are computed dynamically by the RBA Algorithm 1.
+Taking the row-wise softmax ensures that the coupling coef-
+ficients in cl
+(i,:) are positive and sum up to one. Therefore,
+we can view the coupling coefficients as fuzzy edges that
+connect capsules ul
+(i,:) and ul+1
+(j,:) with probability cl
+(i,j). The
+multi-layer hierarchy of capsule nodes, connected by fuzzy
+edges, defines the parse-tree analogously to a probabilistic
+context-free grammar. Examples are shown in Figure 2.
+4
+Challenging the Parse-Tree Assumption
+As mentioned in the introduction, there are two key as-
+sumptions regarding the parse-tree: (1) The nodes of the
+parse-tree, the activated capsules, are viewpoint-invariant
+representations of visual entities present in the input image.
+(2) Lower-level capsules represent object parts, higher-level
+capsules represent composite objects, and part-whole rela-
+tionships are represented by the edges of the parse tree, that
+is, by the coupling coefficients. In the following, we are go-
+ing to challenge both assumptions.
+If Assumption (2) holds, then the parse-tree computed by
+a CapsNet is a part-whole representation of the image scene,
+and the routing dynamics defined by the coupling coeffi-
+cients is specific to the input image. We conduct experiments
+challenging this assumption in Section 4.1.
+If Assumption (1) holds, then affine transformations of
+an image only change the direction of a parameter vector
+of a capsule, but not its magnitude. Hence, we take an im-
+age, transform it affinely, and analyze the resulting capsule
+activations. These experiments can be found in Section 4.2.
+Furthermore, we examine the capsule activation in general
+in Section 4.3.
+Experimental
+Setup
+We
+use
+the
+AffNIST
+bench-
+mark (Sabour, Frosst, and Hinton 2017) to assess a
+model’s robustness to affine transformations, and we use the
+CIFAR10 dataset (Krizhevsky 2009) to test a model’s per-
+formance on complex image scenes. We conduct extensive
+experiments using a total of 121 different model architec-
+tures of various scales, featuring different numbers of rout-
+ing layers and different numbers and capsule dimensions.
+Shallow models resemble the original CapsNet implemen-
+tation while deeper models allow for a more semantically
+expressive parse-tree.
+We refer to the appendix for detailed architecture and
+dataset descriptions, training procedures, and full results for
+all models used in our experiments.
+4.1
+Routing Dynamics
+We measure the diversity of parse-trees, that is, the routing
+dynamics, by assessing the diversity of routing targets for a
+single capsule u. For k input images, let C ∈ Rk×n hold
+all the coupling coefficients that connect u to capsules on
+the next layer, which contains n target capsules. We use the
+standard deviation std(c(:,i)) with respect to all input im-
+ages as a measure for the routing diversity of u to the i-
+th capsule on the next layer. A routing prn is called per-
+fect if it always routes to exactly one capsule on the next
+layer and routes to all n capsules on the next layer equally
+likely. The standard deviation of a perfect routing computes
+to std(prn) =
+��
+1 − 1
+n
+� 1
+n. We use a perfect routing to
+define the dynamic routing coefficient (dyr) for capsule u
+as
+dyr(u) = 1
+n
+n
+�
+i=1
+std(c(:,i))
+std(prn)
+The expected number of target capsules (dys) for u is
+dys(u) = n · dyr(u). For a whole layer, we define the co-
+efficients dyr and dys as the mean over all capsules of this
+layer.
+
+Results
+In the following, we report the routing statistics
+for a CapsNet architecture with four routing layers, 16 cap-
+sules per layer in the first four layers, and ten capsules in
+the last layer. We set the capsule dimension to eight and
+train multiple models on the AffNIST dataset until a tar-
+get accuracy of 99.2% is reached. The routing statistics for
+the models are summarized in Table 1 and the correspond-
+ing coupling coefficients of a single model are visualized
+in Figure 3. The dys values below two for Layers 2 and 3
+indicate low routing dynamics. A route is mostly predeter-
+mined once a capsule is activated; hence, the routing is al-
+most static. Only the last layer exhibits higher routing dy-
+namics, which can be attributed to the supervisory effect of
+the classification loss Lm. Since the routing is almost static,
+we conclude that the parse-trees do not encode the informa-
+tion necessary for a distributed representation of diverse im-
+age scenes, violating Assumption (2). As can be seen from
+Tables 7 to 12 in the supplement, the results look similar
+for all models trained on AffNIST. The more complex data
+set CIFAR10 exhibits even worse routing dynamics; see Ta-
+ble 13 and Figure 16 in the appendix.
+Layer
+Capsules Alive
+Routing Dynamics
+Rate (dyr)
+Mean (dys)
+1
+16.00 ± 0.00
+0.30 ± 0.00
+4.50 ± 0.17
+2
+14.90 ± 0.70
+0.25 ± 0.01
+1.72 ± 0.11
+3
+7.00 ± 0.63
+0.30 ± 0.02
+1.79 ± 0.16
+4
+5.90 ± 0.70
+0.38 ± 0.04
+3.78 ± 0.38
+output
+10.00 ± 0.00
+Table 1: The routing statistics for a CapsNet with four rout-
+ing layers, 16/10 capsules per layer, and a capsule dimen-
+sion of eight. We separately train and evaluate ten models
+on AffNIST the same way and report the mean and standard
+deviation.
+4.2
+Viewpoint Invariance
+We investigate to which degree capsule activations are in-
+variant to affine transformations of the input images. Let
+x(1) and x(2) show the same visual entity though differently
+instantiated. For one specific capsule, let u(1), u(2) ∈ Rd
+be the corresponding capsule responses for the two images.
+For a viewpoint invariant parse-tree, it holds that
+��u(1)��
+2 =
+��u(2)��
+2, since ∥u∥2 measures the probability that the visual
+entity is present in the image. We repeat this process for a set
+of k input images and collect the corresponding capsule ac-
+tivations in the two vectors v(1) and v(2) ∈ Rk. We compute
+the empirical cross-correlation between these two vectors as
+(v(1))⊤v(2)
+∥v(1)∥2·∥v(2)∥2
+. For one layer, we compute the average of
+this value over all capsules of this layer. For a viewpoint-
+invariant parse-tree, this value should be one.
+Results
+We observe that the capsule activation correlation
+decreases for increasingly stronger affine transformations,
+see Figure 4. This observation holds for all intermediate cap-
+sule layers, and all tested affine transformations. See also
+0
+20
+40
+60
+80
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+layer 1
+layer 2
+layer 3
+layer 4
+(a) rotation
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+layer 1
+layer 2
+layer 3
+layer 4
+(b) scaling
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+0.40
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+layer 1
+layer 2
+layer 3
+layer 4
+(c) horizontal translation
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+0.40
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+layer 1
+layer 2
+layer 3
+layer 4
+(d) vertical translation
+Figure 4: The capsule activation correlations for each layer
+with respect to increasing degree of affine transformations.
+1
+3
+5
+7
+9
+11
+13
+15
+1
+2
+3
+4
+5
+1
+3
+5
+7
+9
+11
+13
+15
+1
+2
+3
+4
+5
+1
+3
+5
+7
+9
+11
+13
+15
+1
+2
+3
+4
+5
+1
+3
+5
+7
+9
+11
+13
+15
+1
+2
+3
+4
+5
+Figure 5: Similar input image, different parse tree.
+Figure 5 for a qualitative example showing two different
+parse-trees that are expected to be identical. We conclude
+that the parse-tree is not invariant under affine transforma-
+tions of the input image, violating Assumption (1). Further-
+more, since already the activations of the PrimeCaps do not
+exhibit viewpoint-invariance, we believe that capsules need
+a different backbone that gives rise to better PrimeCaps.
+4.3
+Capsule Activation
+In this section, we analyze the capsule activation and thus the
+parse-tree nodes. For a layer with n capsules of dimension
+d, let U ∈ Rk×n×d hold the n capsule responses of dimen-
+sion d for k input images. We denote the entries of U by the
+lower letter u with corresponding lower indices. We define
+
+Capsule Layer
+Capsule Norms
+Capsule Activation
+Capsule Deaths
+Mean (cnm)
+Sum (cns)
+Rate (car)
+Sum (cas)
+Rate (cdr)
+Sum (cds)
+1
+0.95 ± 0.00
+15.25 ± 0.06
+1.00 ± 0.00
+16.00 ± 0.00
+0.00 ± 0.00
+0.00 ± 0.00
+2
+0.32 ± 0.01
+5.12 ± 0.18
+0.70 ± 0.03
+11.17 ± 0.48
+0.07 ± 0.04
+1.10 ± 0.70
+3
+0.18 ± 0.01
+2.83 ± 0.08
+0.35 ± 0.03
+5.57 ± 0.42
+0.56 ± 0.04
+9.00 ± 0.63
+4
+0.12 ± 0.00
+1.90 ± 0.07
+0.22 ± 0.02
+3.51 ± 0.27
+0.63 ± 0.04
+10.10 ± 0.70
+5
+0.15 ± 0.01
+1.48 ± 0.05
+0.30 ± 0.03
+3.05 ± 0.00
+0.00 ± 0.00
+0.00 ± 0.00
+Table 2: Capsule activation statistics for a CapsNet with five capsule layers, 16/10 capsules per layer, and a capsule dimension
+of eight. We separately train and evaluate ten models on AffNIST the same way and report the mean and standard deviation.
+the capsule norm sum (cns) as the sum of capsule norms in
+the respective layer averaged over all input images. For com-
+paring layers with different numbers of capsules, we define
+the capsule norm mean (cnm), which is the cns adjusted
+for the number of capsules that are present in the layer:
+cns(U) = 1
+k
+k
+�
+i=1
+n
+�
+j=1
+��u(i,j,:)
+��
+2 ,
+cnm(U) = cns(U)
+n
+We say a capsule j is active for input image i, if its norm
+exceeds a certain threshold ε, that is,
+1active(u(i,j,:)) =
+�1
+��u(i,j,:)
+��
+2 ≥ ε
+0
+otherwise.
+Furthermore, we define the sum of active capsules (cas) as
+the mean sum of active capsules per layer and the rate of
+activate capsules as the cas adjusted for the number of cap-
+sules:
+cas(U) = 1
+k
+k
+�
+i=1
+n
+�
+j=1
+1active(u(i,j,:)),
+car(U) = cas(U)
+n
+We say that a capsule j is dead if the mean µ and the
+standard deviation σ of its norm over the k input images are
+below certain thresholds, that is,
+1dead(u(:,j,:)) =
+�1
+µ(u(:,j,:)) ≤ ε1 and σ(u(:,j,:)) ≤ ε2
+0
+otherwise.
+We compute the sum of dead capsules (cds) and the rate of
+dead capsules (cdr) as
+cds(U) =
+n
+�
+j=1
+1dead
+�
+u(:,j,:)
+�
+,
+cdr(U) = cds(U)
+n
+Results
+Table 2 summarizes the capsule activation statis-
+tics for the AffNIST experiment. As expected, the agreement
+of lower layer capsules, enforced by the RBA algorithm, re-
+sults in a declining number of active capsules in the upper
+layers, as is witnessed by decreasing cas values. As a result,
+the overall sum of norms per layer drops, as can be seen in
+the cns values. Surprisingly, there is no sign of sparse acti-
+vation within the PrimeCaps. All PrimeCaps are consistently
+active, as seen from the car values. This implies that Prime-
+Caps do not represent parts that are present in one image and
+not present in another, questioning the underlying assump-
+tion of distributed representation learning. It is another indi-
+cation that the backbone does not deliver the representations
+required for PrimeCaps.
+Furthermore, we observe that the number of dead capsules
+cdr increases with the depth of the model. For instance, 63%
+of the capsules on Layer 4 in the AffNIST experiment are
+dead. Figure 3e highlights the dead capsules. In other ex-
+periments, this value increases up to 84%, see Table 20 in
+the appendix. This has the following implications: First, the
+depth of a CapsNet is limited as the number of dead cap-
+sules rises with the number of layers. Second, the parse-tree
+cannot carry separate semantic information for each class
+if the number of active capsules is less than the number of
+classes. Third, dead capsules limit the capacity of a Cap-
+sNet as their respective parameters are not in use. This ex-
+plains why baseline models trained with uniform routing
+perform better than models trained with RBA. Uniform rout-
+ing, which uses all parameters, achieves better classification
+accuracies; see Tables 5 and 14 in the appendix. In uniform
+routing, all entries in the coupling coefficient matrix are set
+to the same fixed value. Our results stand in line with prior
+work (Paik, Kwak, and Kim 2019; Gu and Tresp 2020; Gu,
+Tresp, and Hu 2021) that also observed a negative impact of
+routing on model performance.
+Theoretical Analysis
+In order to theoretically explain the
+dynamics of the activation of the capsules during training,
+we analyze the gradient of the loss function. We have the
+following theorem:
+Theorem 1 Let Lm be the margin loss function. The gradi-
+ent of a single capsule ul
+(j,:) of the upper-most layer l eval-
+uates to:
+∂Lm
+∂ul
+(j,:)
+=
+�
+−tj max(0, m+ −
+���ul
+(j,:)
+���
+2)
++ λ(1 − tj) max(0,
+���ul
+(j,:)
+���
+2 − m−)
+�
+·
+2ul
+(j,:)
+���ul
+(j,:)
+���
+2
+The theorem follows directly from the definition of the clas-
+sification loss function, Equation (3). The gradient is inde-
+pendent of the magnitude
+���ul
+(j,:)
+��� of the capsule activation.
+Hence, as long as the loss function is not zero, the gradient is
+large enough to force the capsules to either become active or
+inactive, depending on the label of the data point. Hence, all
+
+capsules on the upper-most layer will be active for the cor-
+responding data points. This is in stark contrast to capsules
+that are not on the upper-most layer. Here, it can happen that
+a capsule becomes dead during training. We observe, that
+once a capsule is dead, it never becomes active again, re-
+sulting in a starvation of capsules. The following theorem
+asserts this behavior.
+Theorem 2 Let Lm be the margin loss function. The gra-
+dient of a single capsule ul
+(i,:), that does not belong to the
+upper-most layer evaluates to:
+∂Lm
+∂ul
+(i,:)
+=
+�
+j
+∂Lm
+ˆul+1
+(i,j,:)
+· W l
+(j,i,:,:)
+The gradients of the corresponding weight matrices W l
+(j,i,:,:)
+evaluate to:
+∂Lm
+∂W l
+(j,i,:,:)
+=
+∂Lm
+∂ˆul+1
+(i,j,:)
+· ul
+(i,:)
+The theorem follows from Equation (2). It states that the gra-
+dient of the weight matrix scales with the activation of the
+corresponding capsule, and the gradient of the capsule scales
+with the magnitude of the weight matrix. Hence, once both
+are small, they will not change sufficiently. In the limit, i.e.,
+of magnitude zero, they will never change. In other terms,
+once a capsule becomes dead, it never becomes active again.
+Figure 10 in the appendix clearly show this behavior for the
+gradient of the capsule activation and Figures 11-14 in the
+appendix show this behavior for the gradient of the weight
+matrices. Dead capsules do not participate in the routing and
+are not part of any parse-tree. Also, note that the supervised
+loss forces the upper-most layers to be active. However, cap-
+sules can become dead on the intermediate layers where no
+supervised loss is directly present.
+5
+Comparing RBA with Self-Attention
+We compare routing-by-agreement with the multi-head
+self-attention (MHSA) mechanism used in transform-
+ers (Vaswani et al. 2017) and more task-related vision trans-
+formers (Dosovitskiy et al. 2021). Like RBA, the MHSA
+mechanism operates on vectors and uses the softmax func-
+tion to compute the normalized attention matrix, similar to
+the coupling coefficient matrix in RBA. However, contrary
+to RBA, which computes the softmax row-wise, MHSA
+enforces the softmax column-wise and, as a result, does
+not suffer from the previously discussed vanishing gradi-
+ent problem. However, MHSA is not considered routing and
+does not intend to implement a parse-tree. Considering space
+and time complexity, we observe that RBA is extremely ex-
+pensive compared to MHSA; see Table 3. This fact may con-
+tribute to the substantial interest in models relying on MHSA
+rather than RBA.
+6
+Broader Impact and Limitations
+In this work, we focus on the original CapsNet with RBA
+routing since it is the predominant implementation of the
+Space
+Time
+MHSA
+O(d2)
+O(n2 · d + n · d2)
+RBA
+O(n2 · d2)
+O(n2 · d2)
+Table 3: Comparing space and time complexity of routing-
+by-agreement and multi-head self-attention for a routing
+layer with n input and output vectors of dimension d.
+capsule idea. An exhaustive investigation, including all cap-
+sule variants and follow-up models, is difficult because
+the absence of a formal definition of capsules makes the
+topic hard to cover. Different approaches from the vast
+literature are technically diverse. As a result, whether a
+follow-up work implements the concept of capsules is not
+easy to judge. However, our claims are general enough to
+cover many implementations. The softmax-based routing
+approach is part of many capsule implementations, see for
+instance (Xiang et al. 2018; Zhou et al. 2019; Mazzia, Sal-
+vetti, and Chiaberge 2021), and we expect that they face sim-
+ilar issues.
+7
+Conclusion
+The core concept of capsules is the part-whole hierarchy of
+an image represented by a parse-tree. While this concept
+has appealing properties like robustness under affine trans-
+formations, interpretability, and parameter efficiency, so far,
+implementations of the capsule concept have not taken over
+yet. Instead, some of their properties were questioned in re-
+cent work. Here, we have shown that the core idea of a parse-
+tree does not emerge in the CapsNet implementation. Fur-
+thermore, starvation of capsules caused by a vanishing gradi-
+ent limits their capacity and depth. Our observations explain
+recently reported shortcomings, especially that CapsNets do
+not scale beyond small datasets. Hence, the CapsNet is not a
+sufficient implementation of the capsule idea.
+8
+Acknowledgments
+This work was supported by the German Science Founda-
+tion (DFG) grant (GI-711/5-1) within the priority program
+(SPP 1736) Algorithms for Big Data and by the Carl Zeiss
+Foundation within the project ”Interactive Inference”.
+References
+Ahmed, K.; and Torresani, L. 2019.
+STAR-Caps: Cap-
+sule Networks with Straight-Through Attentive Routing.
+In Advances in Neural Information Processing Systems,
+(NeurIPS).
+Biederman, I. 1987. Recognition-by-components: A The-
+ory of Human Image Understanding. Psychological review,
+94(2): 115.
+Chen, J.; Yu, H.; Qian, C.; Chen, D. Z.; and Wu, J. 2021. A
+Receptor Skeleton for Capsule Neural Networks. In Meila,
+M.; and Zhang, T., eds., International Conference on Ma-
+chine Learning, (ICML).
+Deng, J.; Dong, W.; Socher, R.; Li, L.-J.; Li, K.; and Fei-
+Fei, L. 2009. Imagenet: A Large-scale Hierarchical Image
+
+Database. In Conference on Computer Vision and Pattern
+Recognition, (CVPR).
+Dosovitskiy, A.; Beyer, L.; Kolesnikov, A.; Weissenborn,
+D.; Zhai, X.; Unterthiner, T.; Dehghani, M.; Minderer, M.;
+Heigold, G.; Gelly, S.; Uszkoreit, J.; and Houlsby, N. 2021.
+An Image is Worth 16x16 Words: Transformers for Image
+Recognition at Scale. In International Conference on Learn-
+ing Representations, (ICLR).
+Gu, J. 2021. Interpretable Graph Capsule Networks for Ob-
+ject Recognition. In Conference on Artificial Intelligence,
+(AAAI).
+Gu, J.; and Tresp, V. 2020. Improving the Robustness of
+Capsule Networks to Image Affine Transformations.
+In
+Conference on Computer Vision and Pattern Recognition,
+(CVPR).
+Gu, J.; Tresp, V.; and Hu, H. 2021. Capsule Network Is Not
+More Robust Than Convolutional Network. In Conference
+on Computer Vision and Pattern Recognition, (CVPR).
+Gu, J.; Wu, B.; and Tresp, V. 2021. Effective and Efficient
+Vote Attack on Capsule Networks. In International Confer-
+ence on Learning Representations, (ICLR).
+Hahn, T.; Pyeon, M.; and Kim, G. 2019. Self-Routing Cap-
+sule Networks. In Advances in Neural Information Process-
+ing Systems, (NeurIPS).
+Hinton, G. E. 1979. Some Demonstrations of the Effects of
+Structural Descriptions in Mental Imagery. Cognitive Sci-
+ence, 3(3): 231–250.
+Hinton, G. E. 2021. How to represent part-whole hierarchies
+in a neural network. arXiv, abs/2102.12627.
+Hinton, G. E.; Ghahramani, Z.; and Teh, Y. W. 1999. Learn-
+ing to Parse Images.
+In Advances in Neural Information
+Processing Systems, (NIPS).
+Hinton, G. E.; Krizhevsky, A.; and Wang, S. D. 2011. Trans-
+forming Auto-Encoders. In International Conference on Ar-
+tificial Neural Networks, (ICANN).
+Hinton, G. E.; Sabour, S.; and Frosst, N. 2018. Matrix Cap-
+sules with EM Routing.
+In International Conference on
+Learning Representations, (ICLR).
+Kosiorek, A. R.; Sabour, S.; Teh, Y. W.; and Hinton, G. E.
+2019. Stacked Capsule Autoencoders. In Advances in Neu-
+ral Information Processing Systems, (NeurIPS).
+Krizhevsky, A. 2009. Learning multiple layers of features
+from tiny images. Technical report, University of Toronto.
+Laue, S.; Mitterreiter, M.; and Giesen, J. 2018. Computing
+Higher Order Derivatives of Matrix and Tensor Expressions.
+In Neural Information Processing Systems (NeurIPS), 2755–
+2764.
+Laue, S.; Mitterreiter, M.; and Giesen, J. 2020. A Simple
+and Efficient Tensor Calculus. In Conference on Artificial
+Intelligence (AAAI), 4527–4534.
+LeCun, Y.; Boser, B. E.; Denker, J. S.; Henderson, D.;
+Howard, R. E.; Hubbard, W. E.; and Jackel, L. D. 1989.
+Backpropagation Applied to Handwritten Zip Code Recog-
+nition. Neural Computation, 1(4): 541–551.
+LeCun, Y.; Bottou, L.; Bengio, Y.; and Haffner, P. 1998.
+Gradient-based learning applied to document recognition.
+Proceedings of the IEEE, 86(11): 2278–2324.
+Mazzia, V.; Salvetti, F.; and Chiaberge, M. 2021. Efficient-
+CapsNet: capsule network with self-attention routing. Sci-
+entific reports, 11(1): 1–13.
+Michels, F.; Uelwer, T.; Upschulte, E.; and Harmeling, S.
+2019. On the Vulnerability of Capsule Networks to Adver-
+sarial Attacks. arXiv, abs/1906.03612.
+Paik, I.; Kwak, T.; and Kim, I. 2019.
+Capsule Networks
+Need an Improved Routing Algorithm. In Asian Conference
+on Machine Learning, (ACML).
+Pham, H.; Dai, Z.; Xie, Q.; and Le, Q. V. 2021. Meta Pseudo
+Labels.
+In Conference on Computer Vision and Pattern
+Recognition, (CVPR).
+Rajasegaran, J.; Jayasundara, V.; Jayasekara, S.; Jayasekara,
+H.; Seneviratne, S.; and Rodrigo, R. 2019. DeepCaps: Go-
+ing Deeper With Capsule Networks. In Conference on Com-
+puter Vision and Pattern Recognition, (CVPR).
+Rawlinson, D.; Ahmed, A.; and Kowadlo, G. 2018. Sparse
+Unsupervised Capsules Generalize Better.
+Ribeiro, F. D. S.; Leontidis, G.; and Kollias, S. D. 2020.
+Capsule Routing via Variational Bayes. In Conference on
+Artificial Intelligence, (AAAI).
+Sabour, S.; Frosst, N.; and Hinton, G. E. 2017. Dynamic
+Routing Between Capsules. In Advances in Neural Infor-
+mation Processing Systems, (NIPS).
+Sabour, S.; Tagliasacchi, A.; Yazdani, S.; Hinton, G. E.; and
+Fleet, D. J. 2021. Unsupervised Part Representation by Flow
+Capsules. In International Conference on Machine Learn-
+ing, (ICML).
+Tsai, Y. H.; Srivastava, N.; Goh, H.; and Salakhutdinov, R.
+2020. Capsules with Inverted Dot-Product Attention Rout-
+ing. In International Conference on Learning Representa-
+tions, (ICLR).
+Vaswani, A.; Shazeer, N.; Parmar, N.; Uszkoreit, J.; Jones,
+L.; Gomez, A. N.; Kaiser, L.; and Polosukhin, I. 2017. At-
+tention is All you Need. In Advances in Neural Information
+Processing Systems, (NIPS).
+Wang, D.; and Liu, Q. 2018. An Optimization View on Dy-
+namic Routing Between Capsules. In International Confer-
+ence on Learning Representations, (ICLR).
+Wortsman, M.; Ilharco, G.; Gadre, S. Y.; Roelofs, R.; Lopes,
+R. G.; Morcos, A. S.; Namkoong, H.; Farhadi, A.; Carmon,
+Y.; Kornblith, S.; and Schmidt, L. 2022. Model soups: av-
+eraging weights of multiple fine-tuned models improves ac-
+curacy without increasing inference time. In International
+Conference on Machine Learning, (ICML).
+Xi, E.; Bing, S.; and Jin, Y. 2017. Capsule Network Perfor-
+mance on Complex Data. arXiv, abs/1712.03480.
+Xiang, C.; Zhang, L.; Tang, Y.; Zou, W.; and Xu, C. 2018.
+MS-CapsNet: A Novel Multi-Scale Capsule Network. IEEE
+Signal Processing Letters, 25(12): 1850–1854.
+Zhou, Y.; Ji, R.; Su, J.; Sun, X.; and Chen, W. 2019. Dy-
+namic Capsule Attention for Visual Question Answering. In
+Conference on Artificial Intelligence, (AAAI).
+
+Why Capsule Neural Networks Do Not Scale:
+Challenging the Dynamic Parse-Tree Assumption
+Appendix
+This appendix provides additional materials which did not fit into the main paper. It is organized as follows:
+A
+Detailed descriptions of models, training procedures, and data sets.
+B
+Additional results from experiments testing viewpoint invariance
+(Section 4.2 in the main paper) for different models.
+C
+Additional results for different models on the AffNIST data set.
+D
+Exhaustive experiments concerning different aspects on the AffNIST dataset.
+E
+Detailed evaluation of a single CIFAR10 model.
+F
+Exhaustive experiments concerning different aspects on the CIFAR10 data set.
+A
+Model Architectures and Training Procedures
+A.1
+CapsNets for AffNIST
+The AffNIST dataset (Sabour, Frosst, and Hinton 2017) is derived from the classic MNIST data set (LeCun et al. 1998) as
+follows: First, all MNIST images are zero-padded to dimensions 40×40 and affinely transformed by random rotations up to
+20 degrees, random shearings up to 40 degrees, random scaling from 0.8 to 1.2, and random translations up to eight pixels
+in each direction. We use this dataset to assess a model’s robustness to affine transformations of the input data. For this, we
+train the model on the original MNIST train set, where the images are randomly placed on a 40×40 empty background. For
+a fair comparison of different models, training is stopped once a model reaches a target accuracy of 99.20% on the original
+MNIST validation set. The trained models are then evaluated on the AffNIST validation set. The difference between the MNIST
+validation set accuracy and the AffNIST validation set accuracy measures the model’s robustness to affine transformations.
+The general model architecture for all models trained on AffNIST largely follows the description in Section 3 of the main
+paper. We borrow the backbone proposed by Mazzia, Salvetti, and Chiaberge (2021), which utilizes four standard convolutional
+layers Conv(32,7,1), Conv(64,3,1), Conv(64,3,2), and Conv(n1 · d1,3,2), followed by a fifth depthwise convolutional layer
+Conv(n1·d1,7,1,n1·d1). The reconstruction network consists of three fully connected layers FC(512), FC(1024), and FC(40·40),
+all using the ReLU activation function. The hyper-parameters for the loss all are set to m+ = 0.9, m− = 1−m+ = 0.1, λ = 0.5
+and α = 0.392. We found the best number of iterations for the routing algorithm to be ten. We used the Adam optimizer with
+an initial learning rate of 10−3, an exponential learning rate decay of 0.97, and a weight decay regularizer with a value of 10−6.
+We used a batch size of 512.
+A.2
+CapsNets for CIFAR10
+The CapsNet architecture for the CIFAR10 classification task largely follows the architecture used for the AffNIST benchmark
+with slight modifications in the backbone to adjust for the smaller input image size. The backbone utilizes four standard convo-
+lutional layers Conv(32,7,1), Conv(64,3,1), Conv(128,3,2), and Conv(n1 · d1,3,2), followed by a fifth deepthwise convolutional
+layer Conv(n1 · d1,5,1,n1 · d1).
+We trained all CIFAR10 models for 100 epochs similarly to the AffNIST models, but with an initial learning rate of 10−4.
+For computing the performance metrics, we selected the best model regarding validation set accuracy.
+A.3
+The Original CapsNet for MNIST Digit Classification
+Here we describe the original CapsNet architecture for the MNIST classification task by Sabour, Frosst, and Hinton (2017). The
+backbone function consists of two consecutive convolutional layers, Conv(256,9,1) and Conv(256,9,2), followed by a single
+routing layer as generally described in Section 3 of the main paper. There are 1152 PrimeCaps of dimension eight and ten output
+capsules of dimension 16. The reconstruction network consists of three fully connected layers, namely FC(512), FC(1024), and
+FC(28·28), all using the ReLU activation function. The hyper-parameters for the loss are set to m+ = 0.9, m− = 1−m+ = 0.1,
+λ = 0.5 and α = 0.0005. The number of iterations for the routing algorithm is set to 3.
+
+A.4
+Notes on Backbone Architectures and PrimeCaps
+The role of the backbone is to extract meaningful features from the input images. The features constitute the PrimeCaps.
+Therefore, the backbone also controls the number and dimension of the PrimeCaps. The concept of capsules requires that a
+capsule is activated if its related conceptual entity is present within the input image (Sabour, Frosst, and Hinton 2017). For this
+reason, each capsule must have a full receptive field over the input image.
+The original CapsNet backbone, used on the MNIST classification task, consists of two consecutive convolutional layers.
+However, for larger input dimensions, e.g., AffNIST (40×40) or ImageNet (224×224), using only two convolutional layers is
+not practical since this would either result in a vast number of PrimeCaps or large capsule dimensions that make the models
+computationally infeasible. Table 4 lists the number of parameters in the routing layer for the original CapsNet model for
+different input image dimensions, depending on the dimension of the output capsules. As can be seen, the number of parameters
+increases quickly with the input dimension, and as a result, computational and space requirements also rise quickly; see Table 3
+in the main paper.
+Therefore, we borrow the backbone proposed by Mazzia, Salvetti, and Chiaberge (2021) because it allows us to control both
+the number and dimension of the PrimeCaps, and thus scales well to larger input dimensions since the number of parameters
+only grows moderately with the input dimension.
+However, we also evaluated the original CapsNet backbone in our investigations of viewpoint invariance. See Appendix B
+for more details.
+Dataset
+Dimension of a DigitCaps
+Name
+Classes
+Input Dimensions
+16
+32
+64
+128
+MNIST
+10
+(28x28x1)
+1.5
+3.0
+5.9
+11.8
+AFFNIST
+10
+(40x40x1)
+5.9
+11.8
+23.6
+47.2
+CIFAR10
+10
+(32x32x3)
+2.6
+5.3
+10.5
+21.0
+TinyImageNet
+200
+(64x64x3)
+472.3
+944.2
+1887.9
+3775.3
+ImageNet 224
+1000
+(224x224x3)
+44324.0
+88626.3
+177231.0
+354440.3
+Table 4: The number of parameters in the routing layer for the original CapsNet architecture for different input image sizes
+and output capsule dimensions. The number of output capsules is set to match the number of data set classes. The number of
+parameters is given in millions.
+
+B
+Viewpoint Invariance: Additional Results
+In this section, we report the results from additional experiments on the viewpoint invariance of the parse-tree. For experi-
+ment details, see Section 4.2 of the main paper. Figure 6 shows the capsule activation correlation for the original CapsNet
+model (Sabour, Frosst, and Hinton 2017). Since it was defined for an input image size of 28×28, we rescaled all the 40×40
+AffNIST images down to 28×28. Figure 7 shows the capsule activation correlation for a CapsNet with one routing layer, 16
+PrimeCaps and a capsule dimension of eight. Figure 8 shows the capsule activation correlation for an eight layer CapsNet with
+16 capsules per layer and a capsule dimension of 32. We observe a substantial decrease in capsule activation correlation for all
+models, similar to the results reported in the main paper.
+0
+20
+40
+60
+80
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+(a) rotation
+0
+10
+20
+30
+40
+50
+60
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+(b) shearing
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+(c) scaling
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+0.40
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+(d) horizontal translation
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+0.40
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+(e) vertical translation
+Figure 6: Capsule activation correlation for the original CapsNet (Appendix A.3). The correlation decreases with stronger
+affine transformation. Left to right: rotation, shearing, scaling, translation in x-direction, translation in y-direction. Shown are
+the means and the standard deviations from PrimeCaps activation over ten trained models.
+0
+20
+40
+60
+80
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+(a) rotation
+0
+10
+20
+30
+40
+50
+60
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+(b) shearing
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+(c) scaling
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+0.40
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+(d) horizontal translation
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+0.40
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+(e) vertical translation
+Figure 7: Capsule activation correlation for a CapsNet trained on AffNIST with a single routing layer. The correlation decreases
+with stronger affine transformation. Left to right: rotation, shearing, scaling, translation in x-direction, translation in y-direction.
+Shown are the means and the standard deviations from PrimeCaps activation over ten trained models.
+0
+20
+40
+60
+80
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+layer 1
+layer 2
+layer 3
+layer 4
+layer 5
+layer 6
+layer 7
+layer 8
+(a) rotation
+0
+10
+20
+30
+40
+50
+60
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+layer 1
+layer 2
+layer 3
+layer 4
+layer 5
+layer 6
+layer 7
+layer 8
+(b) shearing
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+layer 1
+layer 2
+layer 3
+layer 4
+layer 5
+layer 6
+layer 7
+layer 8
+(c) scaling
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+0.40
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+layer 1
+layer 2
+layer 3
+layer 4
+layer 5
+layer 6
+layer 7
+layer 8
+(d) horizontal translation
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+0.40
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+layer 1
+layer 2
+layer 3
+layer 4
+layer 5
+layer 6
+layer 7
+layer 8
+(e) vertical translation
+Figure 8: Capsule activation correlation for a CapsNet with eight routing layers, trained on AffNIST. The correlation decreases
+with stronger affine transformation. Left to right: rotation, shearing, scaling, translation in x-direction, translation in y-direction.
+Shown are the means and the standard deviations of capsule activation for all layers except the last over ten trained models.
+
+C
+AffNIST: Additional Results for the Model from the Main Paper
+In this section, we report additional results about the model architecture used in the main paper. The model architecture consists
+of five capsule layers. Each layer contains 16 capsules, except the last, where we set the number of capsules to match the number
+of digits, namely ten. We use a capsule dimension of eight. We trained a total of ten models with these settings. We trained the
+model as described in Section A. The activation and dynamics statistics of these ten models are reported in Tables 1 and 2 of
+the main paper. Furthermore, we selected one of these models to create Figures 2, 3, 4, 5 and Figure 9.
+Here, we provide additional results for these models. For the selected model, Figure 10 shows the maximum norm of all
+the capsule vectors tracked over the training process. We see that the norm of a capsule remains zero once it becomes zero.
+This observation is explained by Theorem 2 in the main paper. The norm of the gradients of all the capsule weight matrices,
+tracked over the training period is visualized in Figures 11 to 14. As can be seen, the weights referred to incoming and outgoing
+votes from and to dead capsules do not change since the norm of the gradients approaches zero, whereas, the gradient norms of
+weights from alive capsules to alive capsules show healthy updates.
+1
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+4
+5
+1
+3
+5
+7
+9
+11
+13
+15
+1
+2
+3
+4
+5
+Figure 9: The parse-trees for AffNIST validation set samples.
+
+8340
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+0
+1
+Figure 10: The maximal capsule norm/activation per batch over the training period of the model. The capsule layers are shown
+in the columns and the individual capsules per layer are shown in the rows.
+
+0
+0.00
+0.25
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
+0
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+1
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+250000
+25000
+15
+Figure 11: First routing layer: The gradient norms for the weight matrices
+∂Lm
+∂W 1
+(j,i,:,:) over the training period of the model with
+lower layer capsules i in the rows and upper layer capsules j in the columns.
+
+0
+0.0
+0.2
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
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+1
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+2
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+3
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+4
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+5
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+6
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+7
+0.0
+0.2
+8
+0.0
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+9
+0.0
+0.2
+10
+0.0
+0.2
+11
+0.0
+0.2
+12
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+13
+0.0
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+0
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+250000
+250000
+250000
+250000
+250000
+250000
+25000
+15
+Figure 12: Second routing layer: The gradient norms for the weight matrices
+∂Lm
+∂W 2
+(j,i,:,:) over the training period of the model
+with lower layer capsules i in the rows and upper layer capsules j in the columns.
+
+0
+0.0
+0.1
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
+0
+0.0
+0.1
+1
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+2
+0.0
+0.1
+3
+0.0
+0.1
+4
+0.0
+0.1
+5
+0.0
+0.1
+6
+0.0
+0.1
+7
+0.0
+0.1
+8
+0.0
+0.1
+9
+0.0
+0.1
+10
+0.0
+0.1
+11
+0.0
+0.1
+12
+0.0
+0.1
+13
+0.0
+0.1
+14
+0
+25000
+0.0
+0.1
+0
+250000
+250000
+250000
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+250000
+250000
+250000
+250000
+250000
+250000
+250000
+250000
+250000
+250000
+25000
+15
+Figure 13: Third routing layer: The gradient norms for the weight matrices
+∂Lm
+∂W 3
+(j,i,:,:) over the training period of the model
+with lower layer capsules i in the rows and upper layer capsules j in the columns.
+
+0
+0.0
+0.1
+1
+2
+3
+4
+5
+6
+7
+8
+9
+0
+0.0
+0.1
+1
+0.0
+0.1
+2
+0.0
+0.1
+3
+0.0
+0.1
+4
+0.0
+0.1
+5
+0.0
+0.1
+6
+0.0
+0.1
+7
+0.0
+0.1
+8
+0.0
+0.1
+9
+0.0
+0.1
+10
+0.0
+0.1
+11
+0.0
+0.1
+12
+0.0
+0.1
+13
+0.0
+0.1
+14
+0
+25000
+0.0
+0.1
+0
+250000
+250000
+250000
+250000
+250000
+250000
+250000
+250000
+25000
+15
+Figure 14: Final routing layer: The gradient norms for the weight matrices
+∂Lm
+∂W 4
+(j,i,:,:) over the training period of the model
+with lower layer capsules i in the rows and upper layer capsules j in the columns.
+
+D
+AffNIST: Exhaustive Experiments on Model Architectures
+In this section, we report the results of our exhaustive experiments on model architectures for the AffNIST benchmark. We
+trained all models following the training procedure as described in Appendix A. We used a total of 81 different architectures
+with varying numbers of routing layers {1, 2, 3, 4, 5, 6, 7, 8}, numbers of capsules per layer {15, 32, 64} as well as capsule
+dimension {8, 32, 64}. We chose these parameters to cover a broad range of settings, from simple one-layer models to complex
+models that used all the available GPU RAM (48GB). We report the accuracy on the AffNIST validation set in Figure 15.
+Table 5 lists the best overall models with architecture details, number of parameters and a uniform routing baselines. The best
+models per depth are given in Table 6 and the corresponding metrics and measurements are given in Tables 7, 8, 9, 10, 11, 12.
+1
+2
+3
+4
+5
+6
+7
+8
+# routing layers
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+16
+8
+acc mnist valid
+acc affnist valid
+acc mnist_train
+1
+2
+3
+4
+5
+6
+7
+8
+# routing layers
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+16
+acc mnist valid
+acc affnist valid
+acc mnist_train
+1
+2
+3
+4
+5
+6
+7
+8
+# routing layers
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+32
+acc mnist valid
+acc affnist valid
+acc mnist_train
+1
+2
+3
+4
+5
+6
+7
+8
+# routing layers
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+64
+acc mnist valid
+acc affnist valid
+acc mnist_train
+1
+2
+3
+4
+5
+6
+7
+8
+# routing layers
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+32
+acc mnist valid
+acc affnist valid
+acc mnist_train
+1
+2
+3
+4
+5
+6
+7
+8
+# routing layers
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+acc mnist valid
+acc affnist valid
+acc mnist_train
+1
+2
+3
+4
+5
+6
+7
+8
+# routing layers
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+acc mnist valid
+acc affnist valid
+acc mnist_train
+1
+2
+3
+4
+# routing layers
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+acc mnist valid
+acc affnist valid
+acc mnist_train
+1
+2
+3
+4
+5
+6
+7
+8
+# routing layers
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+64
+acc mnist valid
+acc affnist valid
+acc mnist_train
+1
+2
+3
+4
+5
+6
+7
+8
+# routing layers
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+acc mnist valid
+acc affnist valid
+acc mnist_train
+1
+2
+3
+4
+# routing layers
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+acc mnist valid
+acc affnist valid
+acc mnist_train
+1
+# routing layers
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+acc mnist valid
+acc affnist valid
+acc mnist_train
+Figure 15: Reported accuracies for the AffNIST experiments.
+
+Model Settings
+Parameters
+AffNIST Acc.
+#caps
+dim
+depth
+Routing
+Backbone
+RBA
+Uniform
+64
+8
+2
+35
+73
+0.88
+0.92
+16
+8
+1
+2
+16
+0.88
+0.92
+64
+8
+1
+8
+46
+0.87
+0.90
+64
+32
+2
+453
+587
+0.87
+0.90
+32
+64
+3
+872
+1006
+0.86
+0.89
+16
+16
+2
+11
+33
+0.86
+0.92
+32
+8
+2
+11
+33
+0.86
+0.90
+16
+8
+2
+4
+18
+0.86
+0.91
+64
+16
+2
+122
+192
+0.86
+0.90
+32
+16
+2
+35
+72
+0.85
+0.90
+32
+16
+1
+8
+46
+0.85
+0.89
+32
+8
+1
+4
+26
+0.85
+0.90
+64
+32
+1
+33
+167
+0.85
+0.88
+32
+64
+1
+33
+167
+0.85
+0.89
+64
+64
+1
+66
+329
+0.85
+0.88
+16
+64
+4
+331
+401
+0.85
+0.90
+32
+16
+4
+87
+125
+0.84
+0.92
+64
+32
+3
+873
+1007
+0.84
+0.92
+32
+64
+2
+452
+587
+0.84
+0.90
+32
+32
+2
+121
+192
+0.84
+0.89
+16
+32
+2
+34
+72
+0.84
+0.89
+Table 5: Overview of the best overall models on the AffNIST benchmark. For a model, we list the number of capsules per layer
+(#caps), the dimension of the capsules (dim), and the number of routing layers (depth). The number of backbone parameters
+and the sum of all routing layer parameters are listed separately in 10k. We give the validation accuracy for the model when
+trained with uniform routing and RBA.
+Model Settings
+Parameters
+AffNIST Acc.
+depth
+#caps
+dim
+Routing
+Backbone
+1
+16
+8
+2
+16
+0.88
+2
+64
+8
+35
+73
+0.88
+3
+32
+64
+872
+1006
+0.86
+4
+16
+64
+331
+401
+0.85
+5
+32
+16
+113
+151
+0.84
+6
+16
+32
+139
+177
+0.83
+Table 6: Overview of the best models per depth on the AffNIST benchmark. For a model, we list the number of capsules
+per layer (#caps), the dimension of the capsules (dim), and the number of routing layers (depth). The number of backbone
+parameters and the sum of all routing layer parameters are listed separately in 10k.
+Capsule Layer
+Capsule Norms
+Capsule Activation
+Capsule Deaths
+Mean (cnm)
+Sum (cns)
+Rate (car)
+Sum (cas)
+Rate (cdr)
+Sum (cds)
+1
+0.90
+14.40
+1.00
+16.00
+0.00
+0.00
+2
+0.18
+1.79
+0.42
+4.23
+0.00
+0.00
+(a)
+Routing Layer
+Capsules Alive
+Routing Dynamics
+From lower layer
+To higher layer
+Rate (dyr)
+Mean (dys)
+1
+16
+10
+0.31
+3.05
+(b)
+Table 7: Capsule activation and routing dynamics for the best model with one routing layer.
+
+Capsule Layer
+Capsule Norms
+Capsule Activation
+Capsule Deaths
+Mean (cnm)
+Sum (cns)
+Rate (car)
+Sum (cas)
+Rate (cdr)
+Sum (cds)
+1
+0.93
+59.73
+1.00
+64.00
+0.00
+0.00
+2
+0.08
+5.35
+0.14
+8.68
+0.75
+48.00
+3
+0.13
+1.31
+0.22
+2.16
+0.00
+0.00
+(a)
+Routing Layer
+Capsules Alive
+Routing Dynamics
+From lower layer
+To higher layer
+Rate (dyr)
+Mean (dys)
+1
+64
+16
+0.14
+2.27
+2
+16
+10
+0.26
+2.63
+(b)
+Table 8: Capsule activation and routing dynamics for the best model with two routing layers.
+Capsule Layer
+Capsule Norms
+Capsule Activation
+Capsule Deaths
+Mean (cnm)
+Sum (cns)
+Rate (car)
+Sum (cas)
+Rate (cdr)
+Sum (cds)
+1
+0.99
+31.76
+1.00
+32.00
+0.00
+0.00
+2
+0.21
+6.71
+0.37
+11.81
+0.22
+7.00
+3
+0.07
+2.18
+0.09
+3.02
+0.72
+23.00
+4
+0.13
+1.25
+0.16
+1.58
+0.00
+0.00
+(a)
+Routing Layer
+Capsules Alive
+Routing Dynamics
+From lower layer
+To higher layer
+Rate (dyr)
+Mean (dys)
+1
+32
+25
+0.41
+10.18
+2
+25
+9
+0.20
+1.80
+3
+9
+10
+0.38
+3.80
+(b)
+Table 9: Capsule activation and routing dynamics for the best model with three routing layers.
+
+Capsule Layer
+Capsule Norms
+Capsule Activation
+Capsule Deaths
+Mean (cnm)
+Sum (cns)
+Rate (car)
+Sum (cas)
+Rate (cdr)
+Sum (cds)
+1
+0.99
+15.92
+1.00
+16.00
+0.00
+0.00
+2
+0.41
+6.60
+0.69
+10.98
+0.00
+0.00
+3
+0.18
+2.83
+0.25
+3.96
+0.50
+8.00
+4
+0.10
+1.52
+0.14
+2.24
+0.56
+9.00
+5
+0.12
+1.23
+0.17
+1.71
+0.00
+0.00
+(a)
+Routing Layer
+Capsules Alive
+Routing Dynamics
+From lower layer
+To higher layer
+Rate (dyr)
+Mean (dys)
+1
+16
+16
+0.49
+7.78
+2
+16
+8
+0.31
+2.51
+3
+8
+7
+0.34
+2.41
+4
+7
+10
+0.37
+3.66
+(b)
+Table 10: Capsule activation and routing dynamics for the best model with four routing layers.
+Capsule Layer
+Capsule Norms
+Capsule Activation
+Capsule Deaths
+Mean (cnm)
+Sum (cns)
+Rate (car)
+Sum (cas)
+Rate (cdr)
+Sum (cds)
+1
+0.96
+30.83
+1.00
+32.00
+0.00
+0.00
+2
+0.17
+5.29
+0.27
+8.79
+0.34
+11.00
+3
+0.07
+2.14
+0.08
+2.56
+0.72
+23.00
+4
+0.05
+1.50
+0.06
+2.05
+0.91
+29.00
+5
+0.05
+1.45
+0.08
+2.55
+0.75
+24.00
+6
+0.14
+1.39
+0.19
+1.89
+0.00
+0.00
+(a)
+Routing Layer
+Capsules Alive
+Routing Dynamics
+From lower layer
+To higher layer
+Rate (dyr)
+Mean (dys)
+1
+32
+21
+0.28
+5.81
+2
+21
+9
+0.20
+1.82
+3
+9
+3
+0.19
+0.57
+4
+3
+8
+0.20
+1.63
+5
+8
+10
+0.31
+3.10
+(b)
+Table 11: Capsule activation and routing dynamics for the best model with five routing layers.
+
+Capsule Layer
+Capsule Norms
+Capsule Activation
+Capsule Deaths
+Mean (cnm)
+Sum (cns)
+Rate (car)
+Sum (cas)
+Rate (cdr)
+Sum (cds)
+1
+0.98
+15.75
+1.00
+16.00
+0.00
+0.00
+2
+0.27
+4.36
+0.45
+7.13
+0.19
+3.00
+3
+0.14
+2.18
+0.18
+2.89
+0.62
+10.00
+4
+0.09
+1.50
+0.12
+1.92
+0.69
+11.00
+5
+0.09
+1.41
+0.13
+2.08
+0.56
+9.00
+6
+0.09
+1.42
+0.15
+2.34
+0.56
+9.00
+7
+0.14
+1.41
+0.24
+2.40
+0.00
+0.00
+(a)
+Routing Layer
+Capsules Alive
+Routing Dynamics
+From lower layer
+To higher layer
+Rate (dyr)
+Mean (dys)
+1
+16
+13
+0.40
+5.26
+2
+13
+6
+0.23
+1.39
+3
+6
+5
+0.25
+1.26
+4
+5
+7
+0.26
+1.80
+5
+7
+7
+0.31
+2.17
+6
+7
+10
+0.35
+3.46
+(b)
+Table 12: Capsule activation and routing dynamics for the best model with six routing layers.
+
+E
+CIFAR10: Complete Results for a Single Model
+1
+5
+9
+13
+17
+21
+25
+29
+1
+2
+3
+4
+5
+(a) Mean Couplings
+1
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+9
+13
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+21
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+29
+1
+2
+3
+4
+5
+(b) Std Couplings
+1
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+1
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+3
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+5
+(c) Mean Activations
+1
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+1
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+(d) Std Activations
+1
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+25
+29
+1
+2
+3
+4
+5
+(e) Dead Capsules
+Figure 16: Parse-tree statistics for the complete CIFAR10 validation dataset for a five-layer CapsNet model with 32/10 capsules.
+The mean (a) and the standard deviation (b) of the coupling coefficient matrices for each layer are visualized as connections
+between capsules. Higher coupling coefficients have a darker blue tone. The capsule norms’ mean (c) and standard deviation (d)
+are visualized by bars. Dead capsules (e) are highlighted with a red bar.
+Capsule Layer
+Capsule Norms
+Capsule Activation
+Capsule Deaths
+Mean (cnm)
+Sum (cns)
+Rate (car)
+Sum (cas)
+Rate (cdr)
+Sum (cds)
+1
+0.98
+31.49
+1.00
+32.00
+0.00
+0.00
+2
+0.20
+6.55
+0.35
+11.30
+0.53
+17.00
+3
+0.12
+3.69
+0.16
+5.16
+0.81
+26.00
+4
+0.08
+2.52
+0.10
+3.36
+0.81
+26.00
+5
+0.20
+2.02
+0.68
+6.79
+0.00
+0.00
+(a)
+Routing Layer
+Capsules Alive
+Routing Dynamics
+From lower layer
+To higher layer
+Rate (dyr)
+Mean (dys)
+1
+32
+15
+0.19
+2.90
+2
+15
+6
+0.23
+1.37
+3
+6
+6
+0.27
+1.61
+4
+6
+10
+0.26
+2.60
+(b)
+Table 13: Capsule activation and routing dynamics.
+
+1
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+1
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+3
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+1
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+1
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+1
+2
+3
+4
+5
+Figure 17: The parse-trees for CIFAR10 validation samples.
+
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+Figure 18: The maximal capsule norm/activation per batch over the training period of the model with the capsule layers shown
+in columns and the individual capsules per layer shown in rows.
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+0 25000
+0 25000
+0 25000
+0 25000
+0 25000
+0 25000
+0 25000
+0 25000
+0 25000
+31
+Figure 19: First routing layer: The gradient norms for the weight matrices
+∂Lm
+∂W 1
+(j,i,:,:) over the training period of the model with
+lower layer capsules i in rows and upper layer capsules j in columns.
+
+0
+0.0
+0.2
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
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+14
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+18
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+0 25000
+0 25000
+0 25000
+0 25000
+0 25000
+31
+Figure 20: Second routing layer: The gradient norms for the weight matrices
+∂Lm
+∂W 2
+(j,i,:,:) over the training period of the model
+with lower layer capsules i in rows and upper layer capsules j in columns.
+
+0
+0.0
+0.1
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
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+17
+18
+19
+20
+21
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+24
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+0
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+0 25000
+0 25000
+0 25000
+0 25000
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+31
+Figure 21: Third routing layer: The gradient norms for the weight matrices
+∂Lm
+∂W 3
+(j,i,:,:) over the training period of the model
+with lower layer capsules i in rows and upper layer capsules j in columns.
+
+0
+0.0
+0.1
+1
+2
+3
+4
+5
+6
+7
+8
+9
+0
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+0 25000
+0 25000
+0 25000
+0 25000
+0 25000
+0 25000
+0 25000
+0 25000
+31
+Figure 22: Final routing layer: The gradient norms for the weight matrices
+∂Lm
+∂W 4
+(j,i,:,:) over the training period of the model
+with lower layer capsules i in rows and upper layer capsules j in columns.
+
+F
+CIFAR10: Exhaustive Experiments on Model Architectures
+In this section, we report the results of our exhaustive experiments on model architectures for the CIFAR10 image classification
+task. We trained all models following the training procedure as described in Appendix A. We used a total of 40 different
+architectures with varying numbers of routing layers {1, 2, 3, 4, 5, 6, 7, 8}, capsules per layer {32, 64, 128} as well as capsule
+dimensions {16, 32, 64}. We chose these parameters to cover a broad range of settings, from simple one-layer models to
+complex models that used all the available GPU RAM. We report the best achieved accuracies in Figure 23. Table 14 lists the
+best overall models with architecture details, number of parameters and a uniform routing baselines. The best models per depth
+are given in Table 15 and the corresponding metrics and measurements are given in Tables 16, 17, 18, 19, 20, 21.
+1
+2
+3
+4
+5
+6
+7
+8
+# routing layers
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+32
+16
+acc valid
+acc train
+1
+2
+3
+4
+5
+6
+7
+8
+# routing layers
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+32
+acc valid
+acc train
+1
+2
+3
+4
+# routing layers
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+64
+acc valid
+acc train
+1
+2
+3
+4
+5
+6
+7
+8
+# routing layers
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+64
+acc valid
+acc train
+1
+2
+3
+4
+# routing layers
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+acc valid
+acc train
+1
+# routing layers
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+acc valid
+acc train
+1
+2
+3
+4
+# routing layers
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+128
+acc valid
+acc train
+1
+# routing layers
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+acc valid
+acc train
+1
+# routing layers
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+acc valid
+acc train
+Figure 23: Reported accuracies for the CIFAR10 image classification experiments.
+
+Model Settings
+Parameters
+Valid Acc.
+#caps
+dim
+depth
+Routing
+Backbone
+RBA
+Uniform
+32
+64
+2
+452
+704
+0.79
+0.79
+128
+16
+2
+454
+706
+0.78
+0.76
+128
+16
+1
+33
+285
+0.78
+0.76
+64
+32
+2
+453
+704
+0.78
+0.77
+128
+32
+1
+66
+559
+0.77
+0.76
+32
+32
+2
+121
+252
+0.77
+0.76
+64
+16
+1
+16
+147
+0.77
+0.75
+32
+64
+3
+872
+1123
+0.77
+0.77
+64
+32
+1
+33
+284
+0.77
+0.74
+128
+64
+1
+131
+1108
+0.76
+0.75
+64
+16
+2
+122
+252
+0.76
+0.73
+32
+16
+1
+8
+78
+0.76
+0.74
+64
+64
+1
+66
+559
+0.76
+0.75
+64
+32
+3
+873
+1124
+0.76
+0.2
+32
+16
+2
+35
+105
+0.75
+0.73
+32
+32
+1
+16
+147
+0.75
+0.73
+32
+64
+1
+33
+284
+0.74
+0.68
+32
+32
+3
+226
+357
+0.71
+0.62
+32
+64
+4
+1291
+1543
+0.69
+0.62
+32
+32
+4
+331
+462
+0.66
+0.13
+32
+16
+3
+61
+131
+0.66
+0.21
+Table 14: Overview of the best overall models on the CIFAR10 image classification task. For a model, we list the number
+of capsules per layer (#caps), the dimension of the capsules (dim), and the number of routing layers (depth). The number of
+backbone parameters and the sum of all routing layer parameters are listed separately in 10k. We give the validation accuracy
+for the model when trained with uniform routing and with RBA.
+Model Settings
+Parameters
+Valid Acc.
+depth
+#caps
+dim
+Routing
+Backbone
+1
+128
+16
+33
+285
+0.78
+2
+32
+64
+452
+704
+0.79
+3
+32
+64
+872
+1123
+0.77
+4
+32
+64
+1291
+1543
+0.69
+5
+32
+32
+436
+567
+0.49
+6
+32
+32
+541
+672
+0.43
+Table 15: Overview of the best models per depth on the CIFAR10 image classification task. For a model, we list the number
+of capsules per layer (#caps), the dimension of the capsules (dim), and the number of routing layers (depth). The number of
+backbone parameters and the sum of all routing layer parameters are listed separately in 10k.
+Capsule Layer
+Capsule Norms
+Capsule Activation
+Capsule Deaths
+Mean (cnm)
+Sum (cns)
+Rate (car)
+Sum (cas)
+Rate (cdr)
+Sum (cds)
+1
+0.90
+115.56
+1.00
+128.00
+0.00
+0.00
+2
+0.21
+2.10
+0.78
+7.78
+0.00
+0.00
+(a)
+Routing Layer
+Capsules Alive
+Routing Dynamics
+From lower layer
+To higher layer
+Rate (dyr)
+Mean (dys)
+1
+128
+10
+0.06
+0.62
+(b)
+Table 16: Capsule activation and routing dynamics for the best model with one routing layer.
+
+Capsule Layer
+Capsule Norms
+Capsule Activation
+Capsule Deaths
+Mean / Capsule
+Mean / Layer
+Rate / Layer
+Mean / Layer
+Rate / Layer
+Mean / Layer
+1
+0.99
+31.76
+1.00
+32.00
+0.00
+0.00
+2
+0.18
+5.92
+0.63
+20.10
+0.47
+15.00
+3
+0.18
+1.82
+0.40
+3.96
+0.00
+0.00
+(a)
+Routing Layer
+Capsules Alive
+Routing Dynamics
+From lower layer
+To higher layer
+Rate (dyr)
+Mean (dys)
+1
+32
+17
+0.19
+3.25
+2
+17
+10
+0.17
+1.73
+(b)
+Table 17: Capsule activation and routing dynamics for the best model with two routing layers.
+Capsule Layer
+Capsule Norms
+Capsule Activation
+Capsule Deaths
+Mean (cnm)
+Sum (cns)
+Rate (car)
+Sum (cas)
+Rate (cdr)
+Sum (cds)
+1
+0.99
+31.75
+1.00
+32.00
+0.00
+0.00
+2
+0.24
+7.80
+0.56
+17.89
+0.22
+7.00
+3
+0.10
+3.12
+0.18
+5.81
+0.72
+23.00
+4
+0.17
+1.75
+0.39
+3.93
+0.00
+0.00
+(a)
+Routing Layer
+Capsules Alive
+Routing Dynamics
+From lower layer
+To higher layer
+Rate (dyr)
+Mean (dys)
+1
+32
+25
+0.26
+6.49
+2
+25
+9
+0.15
+1.36
+3
+9
+10
+0.27
+2.66
+(b)
+Table 18: Capsule activation and routing dynamics for the best model with three routing layers.
+
+Capsule Layer
+Capsule Norms
+Capsule Activation
+Capsule Deaths
+Mean (cnm)
+Sum (cns)
+Rate (car)
+Sum (cas)
+Rate (cdr)
+Sum (cds)
+1
+0.99
+31.57
+1.00
+32.00
+0.00
+0.00
+2
+0.23
+7.39
+0.35
+11.17
+0.53
+17.00
+3
+0.14
+4.43
+0.27
+8.79
+0.66
+21.00
+4
+0.08
+2.67
+0.16
+5.08
+0.78
+25.00
+5
+0.19
+1.95
+0.61
+6.10
+0.00
+0.00
+(a)
+Routing Layer
+Capsules Alive
+Routing Dynamics
+From lower layer
+To higher layer
+Rate (dyr)
+Mean (dys)
+1
+32
+15
+0.30
+4.45
+2
+15
+11
+0.20
+2.21
+3
+11
+7
+0.13
+0.93
+4
+7
+10
+0.26
+2.63
+(b)
+Table 19: Capsule activation and routing dynamics for the best model with four routing layers.
+Capsule Layer
+Capsule Norms
+Capsule Activation
+Capsule Deaths
+Mean (cnm)
+Sum (cns)
+Rate (car)
+Sum (cas)
+Rate (cdr)
+Sum (cds)
+1
+0.98
+31.41
+1.00
+32.00
+0.00
+0.00
+2
+0.12
+3.77
+0.17
+5.47
+0.75
+24.00
+3
+0.10
+3.04
+0.15
+4.92
+0.81
+26.00
+4
+0.08
+2.71
+0.11
+3.62
+0.84
+27.00
+5
+0.07
+2.36
+0.10
+3.12
+0.81
+26.00
+6
+0.22
+2.19
+0.82
+8.17
+0.00
+0.00
+(a)
+Routing Layer
+Capsules Alive
+Routing Dynamics
+From lower layer
+To higher layer
+Rate (dyr)
+Mean (dys)
+1
+32
+8
+0.19
+1.55
+2
+8
+6
+0.16
+0.93
+3
+6
+5
+0.21
+1.06
+4
+5
+6
+0.14
+0.84
+5
+6
+10
+0.21
+2.12
+(b)
+Table 20: Capsule activation and routing dynamics for the best model with five routing layers.
+
+Capsule Layer
+Capsule Norms
+Capsule Activation
+Capsule Deaths
+Mean (cnm)
+Sum (cns)
+Rate (car)
+Sum (cas)
+Rate (cdr)
+Sum (cds)
+1
+0.96
+30.84
+1.00
+32.00
+0.00
+0.00
+2
+0.17
+5.28
+0.20
+6.44
+0.72
+23.00
+3
+0.11
+3.60
+0.18
+5.75
+0.72
+23.00
+4
+0.09
+2.80
+0.12
+3.79
+0.84
+27.00
+5
+0.08
+2.55
+0.09
+2.78
+0.84
+27.00
+6
+0.07
+2.20
+0.09
+2.97
+0.84
+27.00
+7
+0.22
+2.24
+0.83
+8.27
+0.00
+0.00
+(a)
+Routing Layer
+Capsules Alive
+Routing Dynamics
+From lower layer
+To higher layer
+Rate (dyr)
+Mean (dys)
+1
+32
+9
+0.25
+2.24
+2
+9
+9
+0.18
+1.63
+3
+9
+5
+0.22
+1.12
+4
+5
+5
+0.25
+1.25
+5
+5
+5
+0.20
+0.99
+6
+5
+10
+0.18
+1.78
+(b)
+Table 21: Capsule activation and routing dynamics for the best model with six routing layers.
+
diff --git a/cNAzT4oBgHgl3EQfnf3Z/content/tmp_files/load_file.txt b/cNAzT4oBgHgl3EQfnf3Z/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..e80f3fc3ffb15da495a801b27a3d9efe351c0234
--- /dev/null
+++ b/cNAzT4oBgHgl3EQfnf3Z/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf,len=3760
+page_content='Why Capsule Neural Networks Do Not Scale:  Challenging the Dynamic Parse-Tree Assumption  Matthias Mitterreiter,1,4 Marcel Koch,2 Joachim Giesen,1 S¨oren Laue3  1Friedrich-Schiller-University Jena, Germany  2Ernst Abbe University of Applied Sciences Jena, Germany  3Technical University Kaiserslautern, Germany  4Data Assessment Solutions GmbH, Hannover, Germany  matthias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='mitterreiter@uni-jena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='de, marcel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='koch@eah-jena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='de, joachim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='giesen@uni-jena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='de, laue@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='uni-kl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='de  Abstract  Capsule neural networks replace simple, scalar-valued neu-  rons with vector-valued capsules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' They are motivated by  the pattern recognition system in the human brain, where  complex objects are decomposed into a hierarchy of sim-  pler object parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Such a hierarchy is referred to as a parse-  tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Conceptually, capsule neural networks have been de-  fined to realize such parse-trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' The capsule neural network  (CapsNet), by Sabour, Frosst, and Hinton, is the first ac-  tual implementation of the conceptual idea of capsule neural  networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' CapsNets achieved state-of-the-art performance on  simple image recognition tasks with fewer parameters and  greater robustness to affine transformations than compara-  ble approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' This sparked extensive follow-up research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  However, despite major efforts, no work was able to scale  the CapsNet architecture to more reasonable-sized datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Here, we provide a reason for this failure and argue that it  is most likely not possible to scale CapsNets beyond toy ex-  amples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' In particular, we show that the concept of a parse-  tree, the main idea behind capsule neuronal networks, is not  present in CapsNets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' We also show theoretically and experi-  mentally that CapsNets suffer from a vanishing gradient prob-  lem that results in the starvation of many capsules during  training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  1  Introduction  The concept of capsules (Hinton, Krizhevsky, and Wang  2011) describes a hypothetical system that parses a complex  image scene into a hierarchy of visual entities that stand in  part-whole relationship to each other (Hinton, Ghahramani,  and Teh 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' A capsule is conceptually defined as a highly  informative, compact representation of a visual entity or ob-  ject within an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' The idea of capsules is motivated by  the pattern recognition system in the visual cortex of the hu-  man brain (Sabour, Frosst, and Hinton 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' There is some  psychological evidence that the human object recognition  system assigns hierarchical structural descriptions to com-  plex objects by decomposing them into parts (Hinton 1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  The theory of recognition by components (Biederman 1987)  proposes that a relatively small set of simple 3D shapes,  called geons, can be assembled in various arrangements to  Copyright © 2023, Association for the Advancement of Artificial  Intelligence (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='aaai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' All rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  form virtually any complex object, which can then be recog-  nized by decomposition into its respective parts (Biederman  1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  A capsule may represent a visual entity by encapsulating  its properties, also known as instantiation parameters, such  as position, size, orientation, deformation, texture, or hue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  A multi-level assembly of such capsules represents a com-  plex image scene, where lower-level capsules model less  abstract objects or object parts, and higher-level capsules  model complex and composite objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Lower-level capsules  are connected to higher-level capsules if the corresponding  entities are in a part-whole relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' For a composite ob-  ject, the hierarchy of capsules defines a syntactic structure  like a parse-tree defines the syntactic structure of a sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Therefore, the hierarchy of capsules is also referred to as  parse-tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' If an object or object part is present in an image,  its respective capsule will be present within the parse-tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Ideally, the parse-tree is invariant under affine transfor-  mations as well as changes of viewpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' That is, a slightly  modified viewpoint on a visual entity should not change  a capsule’s presence within the parse-tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Such parse-  trees would be highly efficient distributed representations  of image scenes (Sabour, Frosst, and Hinton 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Hin-  ton, Ghahramani, and Teh 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Also, explainable machine  learning can profit from interpretable capsules that stand for  dedicated visual entities, and the discrete nature of trees may  connect deep learning with a symbolic approach to AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Fur-  thermore, capsules can be related to inverse graphics, and  there is hope that they can lead to debuggable, parameter ef-  ficient, and interpretable models with a broad range of appli-  cations for all kinds of image-related tasks like image clas-  sification or segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  However, capsules are only conceptually defined, and the  difficulty is finding an implementation with all the highly-  desirable properties from above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' The capsule neural net-  work (CapsNet) by Sabour, Frosst, and Hinton (2017) aims  at such an implementation of the conceptual capsule idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  It was specifically designed to surpass convolutional neu-  ral networks (ConvNets) (LeCun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' 1989) as the latter  were found to suffer from several limitations, including a  lack of robustness to affine transformations and change of  viewpoint, the susceptibility to adversarial attacks, exponen-  tial inefficiencies, and a general lack of interpretability in the  network’s decision-making process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Considering these limi-  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='01583v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='CV]  4 Jan 2023  tations, the parse-tree sounds particularly appealing with all  its advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Here, our aim is a thorough investigation  of the question, whether the CapsNets implementation as  proposed by Sabour, Frosst, and Hinton (2017) realizes all  the conceptual ideas that make capsule networks so appeal-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' We summarize this in two key assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' The first  key assumption is that the CapsNet learns to associate a  capsule with a dedicated visual entity within an input im-  age (Sabour, Frosst, and Hinton 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' The second key as-  sumption is that the CapsNet’s capsules can be organized  hierarchically in a parse-tree that encodes part-whole rela-  tionships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' We test both assumptions experimentally and have  to reject them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' We show that the CapsNet does not exhibit  any sign of an emerging parse-tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Thus, the CapsNet im-  plementation cannot provide the theoretical benefits of cap-  sule networks like invariance under affine transformations  and change of viewpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Furthermore, we provide a theo-  retical analysis, exposing a vanishing gradient problem, that  supports our experimental findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  2  Related Work  Early references to the hierarchy of parts appear already  in (Hinton 1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' The idea of parsing images into parse-  trees was proposed by Hinton, Ghahramani, and Teh (1999)  and the concept of capsules was established in (Hinton,  Krizhevsky, and Wang 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' An important addition by  the CapsNet (Sabour, Frosst, and Hinton 2017) was the  routing-by-agreement (RBA) algorithm that creates cap-  sule parse-trees from images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' With its introduction, the  CapsNet demonstrated state-of-the-art classification accu-  racy on MNIST (LeCun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' 1998) with fewer parame-  ters and stronger robustness to affine transformations than  the ConvNet baseline, which sparked a flood of follow-  up research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' This includes different routing mechanisms,  such as EM-Routing (Hinton, Sabour, and Frosst 2018),  Self-Routing (Hahn, Pyeon, and Kim 2019), Variational  Bayes Routing (Ribeiro, Leontidis, and Kollias 2020), Re-  ceptor Skeleton (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' 2021) and attention-based rout-  ing (Ahmed and Torresani 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Tsai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Mazzia,  Salvetti, and Chiaberge 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Gu 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Wang and Liu  (2018) reframed the routing algorithm in (Sabour, Frosst,  and Hinton 2017) as an optimization problem, and Rawl-  inson, Ahmed, and Kowadlo (2018) introduced an unsu-  pervised learning scheme for CapsNets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Other work re-  placed the capsule vector representations by matrices (Hin-  ton, Sabour, and Frosst 2018) or tensors (Rajasegaran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  2019), or added classic ConvNet features to the general rout-  ing mechanisms, such as dropout (Xiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' 2018) or skip-  connections (Rajasegaran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' The GLOM archi-  tecture, which was proposed by (Hinton 2021), suggests a  routing-free approach for creating parse-trees from images,  but has not been implemented yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Furthermore, other pub-  lications focus on learning better first layer capsules (Prime-  Caps), such as the Stacked Capsule Autoencoders (Kosiorek  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' 2019) and Flow Capsules (Sabour et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  However, after a while it turned out that the CapsNet  falls short of the anticipated benefits and promises of the  capsule idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' To this date, CapsNets do not scale beyond  small-scale datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Works that empirically report scaling  issues include (Xi, Bing, and Jin 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Paik, Kwak, and Kim  2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Although the CapsNet was introduced in the realm  of computer vision, the best performing capsule implemen-  tation (Ahmed and Torresani 2019) achieves only 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='07%  top-1 image classification accuracy on ImageNet (Deng  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' 2009), far behind state-of-the-art transformer-based ap-  proaches (Wortsman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' 2022) and ConvNets (Pham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  2021) with 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='88% and 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='02% accuracy respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' The  original CapsNet itself has not been demonstrated to work  on ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Further negative results regarding CapsNets emerged,  questioning the promised benefits and technical progress al-  together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Paik, Kwak, and Kim (2019) observed that increas-  ing the depth of various CapsNet variants did not improve  accuracy, and routing algorithms, the core components of  capsule implementations, do not provide any benefit regard-  ing accuracy in image classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Michels et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' (2019),  and Gu, Wu, and Tresp (2021) showed that CapsNets can be  as easily fooled as ConvNets when it comes to adversarial at-  tacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Gu, Tresp, and Hu (2021) showed that the individual  parts of the CapsNet have contradictory effects on the per-  formance on different tasks and conclude that with the right  baseline, CapsNets are not generally superior to ConvNets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Finally, Gu and Tresp (2020) showed that removing the dy-  namic routing component improves affine robustness, and  Rawlinson, Ahmed, and Kowadlo (2018) show that capsules  do not specialize without supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Here, we explain these shortcomings, which can be at-  tributed to a lack of an emerging parse-tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  3  The Capsule Neural Network  The CapsNet implements capsules as parameter vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' An  illustration of the CapsNet architecture, which consists of a  multi-layer hierarchy of capsules, is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' In  the following, we introduce basic notations and definitions,  the generic CapsNet architecture, and a loss function for  training CapsNets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Furthermore, we discuss how CapsNets  implement the crucial concept of a parse-tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  R  R  R  CNN  CNN backbone  capsule  routing  FC decoder  Figure 1: A generic CapsNet architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='1  Notation  Capsules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  The matrix U l ∈ Rnl×dl holds nl normalized  capsules of dimension dl for layer l ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' , ℓ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' The  i-th capsule in U l is the vector ul  (i,:) ∈ Rdl, and ul  (i,j) ∈ R  is the j-th entry of capsule i on layer l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Transformation  matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  The  tensor  W l  ∈  Rnl+1×nl×dl+1×dl  holds  transformation  matrices  W l  (j,i,:,:) ∈ Rdl+1×dl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' The transformation matrix W l  (j,i,:,:)  maps the i-th capsule of layer l to its unnormalized contri-  bution to the j-the capsule of layer l + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Coupling coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  The matrix Cl ∈ Rnl×nl+1 holds  coupling coefficients for the connections of capsules from  layer l to layer l + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' The entry cl  (i,j) ∈ [0, 1] specifies  the coupling strength between capsule i on layer l and  capsule j on layer l + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' The coupling coefficients satisfy  �nl+1  j=1 cl  (i,j) = 1 for all i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' , nl}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Squashing function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  The squashing function normal-  izes the length of a capsule vector u  ∈  Rd into the  range [0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Here, we use a slightly modified squashing  function (Mazzia, Salvetti, and Chiaberge 2021),  g(u) =  �  1 −  1  exp(∥u∥2)  �  u  ∥u∥2  (1)  that behaves similarly to the original squashing function pro-  posed by Sabour, Frosst, and Hinton (2017), but is more sen-  sitive to small changes near zero (Xi, Bing, and Jin 2017),  which is required to stack multiple layers of capsules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='2  Architecture  First, the backbone network extracts features from an in-  put image into a feature matrix in Rn1×d1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' The feature  matrix is then normalized by applying the squashing func-  tion to each row, which constitutes the first capsule layer in  U 1 ∈ Rn1×d1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' The capsules in U 1 are also called Prime-  Caps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Starting from the PrimeCaps, consecutive layers of  capsules are computed as follows: First, the linear contri-  bution of capsule i on layer l to capsule j on layer l + 1 is  computed as  ˆul+1  (i,j,:) = W l  (j,i,:,:)ul  (i,:),  (2)  where the entries in the matrix ˆU l+1  (i) , which holds the vectors  ˆul+1  (i,j,:), are called votes from the i-th capsule on layer l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' An  upper layer capsule ul+1  (j,:) is the squashed, weighted sum over  all votes from lower layer capsules ul  (i,:), that is,  ul+1  (j,:) = g  �  �  nl  �  i=1  cl  (i,j) · ˆul+1  (i,j,:)  �  � ,  where the the coupling coefficients cl  (i,j) are dynamically  computed, that is, individually for every input image, by the  Routing-by-agreement Algorithm (RBA), see Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  The number of output capsules on the last layer ℓ is set  to match the number of classes in the respective dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Fi-  nally, the fully-connected decoder network reconstructs the  input image from the capsules on layer ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='3  Training  The parameters in the backbone network,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' in the reconstruc-  tion network,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' as well as the transformation tensors W l and  Algorithm 1: Routing-by-agreement (RBA)  Input: votes ˆu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' number of iterations r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' routing priors b  Output: coupling coefficients c  1: for r iterations do  2:  c(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='j) ←  exp(b(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='j))  �  k exp(b(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='k))  3:  v(j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=':) ← g  ��  i c(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='j)ˆu(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=':)  �  4:  b(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='j) ← b(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='j) + ⟨ˆu(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=':),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' v(j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=':)⟩  5: end for  1  3  5  7  9  11  13  15  1  2  3  4  5  1  3  5  7  9  11  13  15  1  2  3  4  5  1  3  5  7  9  11  13  15  1  2  3  4  5  1  3  5  7  9  11  13  15  1  2  3  4  5  Figure 2: Fuzzy parse-trees for images from the AffNIST  dataset for a model with five capsule layers,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' 16 capsules on  each intermediate layer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' and ten on the last layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' The figure  shows the coupling coefficients as connections between cap-  sules (left), the capsule norms/activations (middle), and the  input image (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' The blue tone of the edges is darker for  greater coupling coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  the RBA routing priors bl  (i,j) are all learned by minimiz-  ing a weighted sum of a supervised classification loss Lm  and an unsupervised reconstruction loss Lr, that is, L =  Lm + α · Lr, with α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' The classification loss function  Lm =  nℓ  �  j=1  tj · max{0, m+ − ∥uℓ  (j,:)∥2}2  + λ · (1 − tj) · max{0, ∥uℓ  (j,:)∥2 − m−}2  (3)  is only applied to the output capsules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Here m+, m− > 0  and λ > 0 are regularization parameters, and tj is 1 if an  object of the j-th class is present in the input image, and 0  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Output capsules that correspond to classes not  present in the input image are masked by zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' The recon-  struction loss function is applied to the output of the recon-  struction network and sums the distances between the recon-  struction and the pixel intensities in the input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='4  Parse-Trees  The parse-tree is the most important concept that allows us  to understand and interpret capsules and their connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  The CapsNet defines a parse-tree, where capsules represent  nodes, and coupling coefficients represent fuzzy edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  81  3  5  7  9  11  13  15  1  2  3  4  5  (a) mean couplings  1  3  5  7  9  11  13  15  1  2  3  4  5  (b) std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' couplings  1  3  5  7  9  11  13  15  1  2  3  4  5  (c) mean activations  1  3  5  7  9  11  13  15  1  2  3  4  5  (d) std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' activations  1  3  5  7  9  11  13  15  1  2  3  4  5  (e) dead capsules  Figure 3: Parse-tree statistics for the complete AffNIST validation dataset for a five-layer CapsNet model with 16/10 capsules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  The mean (a) and the standard deviation (b) of the coupling coefficient matrices for each layer are visualized as connections  between capsules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Higher coupling coefficients have a darker blue tone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' The capsule norms’ mean (c) and standard deviation (d)  are visualized by bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Dead capsules (e) are highlighted with a red bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  The capsules magnitude, that is, the norm of the parameter  vector, which is always in [0, 1) after applying the squash-  ing function, represents the probability that the correspond-  ing entity is present in an input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' The capsules direc-  tion represents instantiation parameters of the entity like its  position, size, or orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Changing the viewpoint on an  entity does not affect its presence, but only its instantiation  parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Therefore, the respective capsule’s magnitude  should be unaffected, whereas its direction can change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  In the CapsNet, the dynamic part-whole relationships are  implemented by coupling coefficients between capsules on  consecutive layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' The coupling coefficients in the routing  layers are computed dynamically by the RBA Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Taking the row-wise softmax ensures that the coupling coef-  ficients in cl  (i,:) are positive and sum up to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Therefore,  we can view the coupling coefficients as fuzzy edges that  connect capsules ul  (i,:) and ul+1  (j,:) with probability cl  (i,j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' The  multi-layer hierarchy of capsule nodes, connected by fuzzy  edges, defines the parse-tree analogously to a probabilistic  context-free grammar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Examples are shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  4  Challenging the Parse-Tree Assumption  As mentioned in the introduction, there are two key as-  sumptions regarding the parse-tree: (1) The nodes of the  parse-tree, the activated capsules, are viewpoint-invariant  representations of visual entities present in the input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  (2) Lower-level capsules represent object parts, higher-level  capsules represent composite objects, and part-whole rela-  tionships are represented by the edges of the parse tree, that  is, by the coupling coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' In the following, we are go-  ing to challenge both assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  If Assumption (2) holds, then the parse-tree computed by  a CapsNet is a part-whole representation of the image scene,  and the routing dynamics defined by the coupling coeffi-  cients is specific to the input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' We conduct experiments  challenging this assumption in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  If Assumption (1) holds, then affine transformations of  an image only change the direction of a parameter vector  of a capsule, but not its magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Hence, we take an im-  age, transform it affinely, and analyze the resulting capsule  activations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' These experiments can be found in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Furthermore, we examine the capsule activation in general  in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Experimental  Setup  We  use  the  AffNIST  bench-  mark (Sabour, Frosst, and Hinton 2017) to assess a  model’s robustness to affine transformations, and we use the  CIFAR10 dataset (Krizhevsky 2009) to test a model’s per-  formance on complex image scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' We conduct extensive  experiments using a total of 121 different model architec-  tures of various scales, featuring different numbers of rout-  ing layers and different numbers and capsule dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Shallow models resemble the original CapsNet implemen-  tation while deeper models allow for a more semantically  expressive parse-tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  We refer to the appendix for detailed architecture and  dataset descriptions, training procedures, and full results for  all models used in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='1  Routing Dynamics  We measure the diversity of parse-trees, that is, the routing  dynamics, by assessing the diversity of routing targets for a  single capsule u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' For k input images, let C ∈ Rk×n hold  all the coupling coefficients that connect u to capsules on  the next layer, which contains n target capsules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' We use the  standard deviation std(c(:,i)) with respect to all input im-  ages as a measure for the routing diversity of u to the i-  th capsule on the next layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' A routing prn is called per-  fect if it always routes to exactly one capsule on the next  layer and routes to all n capsules on the next layer equally  likely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' The standard deviation of a perfect routing computes  to std(prn) =  ��  1 − 1  n  � 1  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' We use a perfect routing to  define the dynamic routing coefficient (dyr) for capsule u  as  dyr(u) = 1  n  n  �  i=1  std(c(:,i))  std(prn)  The expected number of target capsules (dys) for u is  dys(u) = n · dyr(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' For a whole layer, we define the co-  efficients dyr and dys as the mean over all capsules of this  layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Results  In the following, we report the routing statistics  for a CapsNet architecture with four routing layers, 16 cap-  sules per layer in the first four layers, and ten capsules in  the last layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' We set the capsule dimension to eight and  train multiple models on the AffNIST dataset until a tar-  get accuracy of 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='2% is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' The routing statistics for  the models are summarized in Table 1 and the correspond-  ing coupling coefficients of a single model are visualized  in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' The dys values below two for Layers 2 and 3  indicate low routing dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' A route is mostly predeter-  mined once a capsule is activated;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' hence, the routing is al-  most static.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Only the last layer exhibits higher routing dy-  namics, which can be attributed to the supervisory effect of  the classification loss Lm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Since the routing is almost static,  we conclude that the parse-trees do not encode the informa-  tion necessary for a distributed representation of diverse im-  age scenes, violating Assumption (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' As can be seen from  Tables 7 to 12 in the supplement, the results look similar  for all models trained on AffNIST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' The more complex data  set CIFAR10 exhibits even worse routing dynamics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' see Ta-  ble 13 and Figure 16 in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Layer  Capsules Alive  Routing Dynamics  Rate (dyr)  Mean (dys)  1  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='00  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='50 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='17  2  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='90 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='70  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='01  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='72 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='11  3  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='63  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='02  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='79 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='16  4  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='90 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='70  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='38 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='04  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='78 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='38  output  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='00  Table 1: The routing statistics for a CapsNet with four rout-  ing layers, 16/10 capsules per layer, and a capsule dimen-  sion of eight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' We separately train and evaluate ten models  on AffNIST the same way and report the mean and standard  deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='2  Viewpoint Invariance  We investigate to which degree capsule activations are in-  variant to affine transformations of the input images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Let  x(1) and x(2) show the same visual entity though differently  instantiated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' For one specific capsule, let u(1), u(2) ∈ Rd  be the corresponding capsule responses for the two images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  For a viewpoint invariant parse-tree, it holds that  ��u(1)��  2 =  ��u(2)��  2, since ∥u∥2 measures the probability that the visual  entity is present in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' We repeat this process for a set  of k input images and collect the corresponding capsule ac-  tivations in the two vectors v(1) and v(2) ∈ Rk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' We compute  the empirical cross-correlation between these two vectors as  (v(1))⊤v(2)  ∥v(1)∥2·∥v(2)∥2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' For one layer, we compute the average of  this value over all capsules of this layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' For a viewpoint-  invariant parse-tree, this value should be one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Results  We observe that the capsule activation correlation  decreases for increasingly stronger affine transformations,  see Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' This observation holds for all intermediate cap-  sule layers, and all tested affine transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' See also  0  20  40  60  80  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
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+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
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+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='0  layer 1  layer 2  layer 3  layer 4  (a) rotation  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
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+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='0  layer 1  layer 2  layer 3  layer 4  (d) vertical translation  Figure 4: The capsule activation correlations for each layer  with respect to increasing degree of affine transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  1  3  5  7  9  11  13  15  1  2  3  4  5  1  3  5  7  9  11  13  15  1  2  3  4  5  1  3  5  7  9  11  13  15  1  2  3  4  5  1  3  5  7  9  11  13  15  1  2  3  4  5  Figure 5: Similar input image, different parse tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Figure 5 for a qualitative example showing two different  parse-trees that are expected to be identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' We conclude  that the parse-tree is not invariant under affine transforma-  tions of the input image, violating Assumption (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Further-  more, since already the activations of the PrimeCaps do not  exhibit viewpoint-invariance, we believe that capsules need  a different backbone that gives rise to better PrimeCaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='3  Capsule Activation  In this section, we analyze the capsule activation and thus the  parse-tree nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' For a layer with n capsules of dimension  d, let U ∈ Rk×n×d hold the n capsule responses of dimen-  sion d for k input images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' We denote the entries of U by the  lower letter u with corresponding lower indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' We define  Capsule Layer  Capsule Norms  Capsule Activation  Capsule Deaths  Mean (cnm)  Sum (cns)  Rate (car)  Sum (cas)  Rate (cdr)  Sum (cds)  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='95 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
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+page_content='00  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
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+page_content='70  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
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+page_content='57 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='42  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
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+page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='63  4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
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+page_content='90 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='07  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='02  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='51 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='27  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='63 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='04  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='10 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='70  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='01  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='48 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='03  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='00  Table 2: Capsule activation statistics for a CapsNet with five capsule layers, 16/10 capsules per layer, and a capsule dimension  of eight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' We separately train and evaluate ten models on AffNIST the same way and report the mean and standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  the capsule norm sum (cns) as the sum of capsule norms in  the respective layer averaged over all input images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' For com-  paring layers with different numbers of capsules, we define  the capsule norm mean (cnm), which is the cns adjusted  for the number of capsules that are present in the layer:  cns(U) = 1  k  k  �  i=1  n  �  j=1  ��u(i,j,:)  ��  2 ,  cnm(U) = cns(U)  n  We say a capsule j is active for input image i, if its norm  exceeds a certain threshold ε, that is,  1active(u(i,j,:)) =  �1  ��u(i,j,:)  ��  2 ≥ ε  0  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Furthermore, we define the sum of active capsules (cas) as  the mean sum of active capsules per layer and the rate of  activate capsules as the cas adjusted for the number of cap-  sules:  cas(U) = 1  k  k  �  i=1  n  �  j=1  1active(u(i,j,:)),  car(U) = cas(U)  n  We say that a capsule j is dead if the mean µ and the  standard deviation σ of its norm over the k input images are  below certain thresholds, that is,  1dead(u(:,j,:)) =  �1  µ(u(:,j,:)) ≤ ε1 and σ(u(:,j,:)) ≤ ε2  0  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  We compute the sum of dead capsules (cds) and the rate of  dead capsules (cdr) as  cds(U) =  n  �  j=1  1dead  �  u(:,j,:)  �  ,  cdr(U) = cds(U)  n  Results  Table 2 summarizes the capsule activation statis-  tics for the AffNIST experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' As expected, the agreement  of lower layer capsules, enforced by the RBA algorithm, re-  sults in a declining number of active capsules in the upper  layers, as is witnessed by decreasing cas values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' As a result,  the overall sum of norms per layer drops, as can be seen in  the cns values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Surprisingly, there is no sign of sparse acti-  vation within the PrimeCaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' All PrimeCaps are consistently  active, as seen from the car values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' This implies that Prime-  Caps do not represent parts that are present in one image and  not present in another, questioning the underlying assump-  tion of distributed representation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' It is another indi-  cation that the backbone does not deliver the representations  required for PrimeCaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Furthermore, we observe that the number of dead capsules  cdr increases with the depth of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' For instance, 63%  of the capsules on Layer 4 in the AffNIST experiment are  dead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Figure 3e highlights the dead capsules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' In other ex-  periments, this value increases up to 84%, see Table 20 in  the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' This has the following implications: First, the  depth of a CapsNet is limited as the number of dead cap-  sules rises with the number of layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Second, the parse-tree  cannot carry separate semantic information for each class  if the number of active capsules is less than the number of  classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Third, dead capsules limit the capacity of a Cap-  sNet as their respective parameters are not in use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' This ex-  plains why baseline models trained with uniform routing  perform better than models trained with RBA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Uniform rout-  ing, which uses all parameters, achieves better classification  accuracies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' see Tables 5 and 14 in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' In uniform  routing, all entries in the coupling coefficient matrix are set  to the same fixed value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Our results stand in line with prior  work (Paik, Kwak, and Kim 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Gu and Tresp 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Gu,  Tresp, and Hu 2021) that also observed a negative impact of  routing on model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Theoretical Analysis  In order to theoretically explain the  dynamics of the activation of the capsules during training,  we analyze the gradient of the loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' We have the  following theorem:  Theorem 1 Let Lm be the margin loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' The gradi-  ent of a single capsule ul  (j,:) of the upper-most layer l eval-  uates to:  ∂Lm  ∂ul  (j,:)  =  �  −tj max(0, m+ −  ���ul  (j,:)  ���  2)  + λ(1 − tj) max(0,  ���ul  (j,:)  ���  2 − m−)  � 2ul  (j,:)  ���ul  (j,:)  ���  2  The theorem follows directly from the definition of the clas-  sification loss function, Equation (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' The gradient is inde-  pendent of the magnitude  ���ul  (j,:)  ��� of the capsule activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Hence, as long as the loss function is not zero, the gradient is  large enough to force the capsules to either become active or  inactive, depending on the label of the data point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Hence, all  capsules on the upper-most layer will be active for the cor-  responding data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' This is in stark contrast to capsules  that are not on the upper-most layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Here, it can happen that  a capsule becomes dead during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' We observe, that  once a capsule is dead, it never becomes active again, re-  sulting in a starvation of capsules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' The following theorem  asserts this behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Theorem 2 Let Lm be the margin loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' The gra-  dient of a single capsule ul  (i,:), that does not belong to the  upper-most layer evaluates to:  ∂Lm  ∂ul  (i,:)  =  �  j  ∂Lm  ˆul+1  (i,j,:)  W l  (j,i,:,:)  The gradients of the corresponding weight matrices W l  (j,i,:,:)  evaluate to:  ∂Lm  ∂W l  (j,i,:,:)  =  ∂Lm  ∂ˆul+1  (i,j,:)  ul  (i,:)  The theorem follows from Equation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' It states that the gra-  dient of the weight matrix scales with the activation of the  corresponding capsule, and the gradient of the capsule scales  with the magnitude of the weight matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Hence, once both  are small, they will not change sufficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' In the limit, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=',  of magnitude zero, they will never change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' In other terms,  once a capsule becomes dead, it never becomes active again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Figure 10 in the appendix clearly show this behavior for the  gradient of the capsule activation and Figures 11-14 in the  appendix show this behavior for the gradient of the weight  matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Dead capsules do not participate in the routing and  are not part of any parse-tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Also, note that the supervised  loss forces the upper-most layers to be active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' However, cap-  sules can become dead on the intermediate layers where no  supervised loss is directly present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  5  Comparing RBA with Self-Attention  We compare routing-by-agreement with the multi-head  self-attention (MHSA) mechanism used in transform-  ers (Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' 2017) and more task-related vision trans-  formers (Dosovitskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Like RBA, the MHSA  mechanism operates on vectors and uses the softmax func-  tion to compute the normalized attention matrix, similar to  the coupling coefficient matrix in RBA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' However, contrary  to RBA, which computes the softmax row-wise, MHSA  enforces the softmax column-wise and, as a result, does  not suffer from the previously discussed vanishing gradi-  ent problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' However, MHSA is not considered routing and  does not intend to implement a parse-tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Considering space  and time complexity, we observe that RBA is extremely ex-  pensive compared to MHSA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' see Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' This fact may con-  tribute to the substantial interest in models relying on MHSA  rather than RBA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  6  Broader Impact and Limitations  In this work, we focus on the original CapsNet with RBA  routing since it is the predominant implementation of the  Space  Time  MHSA  O(d2)  O(n2 · d + n · d2)  RBA  O(n2 · d2)  O(n2 · d2)  Table 3: Comparing space and time complexity of routing-  by-agreement and multi-head self-attention for a routing  layer with n input and output vectors of dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  capsule idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' An exhaustive investigation, including all cap-  sule variants and follow-up models, is difficult because  the absence of a formal definition of capsules makes the  topic hard to cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Different approaches from the vast  literature are technically diverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' As a result, whether a  follow-up work implements the concept of capsules is not  easy to judge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' However, our claims are general enough to  cover many implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' The softmax-based routing  approach is part of many capsule implementations, see for  instance (Xiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Mazzia, Sal-  vetti, and Chiaberge 2021), and we expect that they face sim-  ilar issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  7  Conclusion  The core concept of capsules is the part-whole hierarchy of  an image represented by a parse-tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' While this concept  has appealing properties like robustness under affine trans-  formations, interpretability, and parameter efficiency, so far,  implementations of the capsule concept have not taken over  yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Instead, some of their properties were questioned in re-  cent work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Here, we have shown that the core idea of a parse-  tree does not emerge in the CapsNet implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Fur-  thermore, starvation of capsules caused by a vanishing gradi-  ent limits their capacity and depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Our observations explain  recently reported shortcomings, especially that CapsNets do  not scale beyond small datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Hence, the CapsNet is not a  sufficient implementation of the capsule idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  8  Acknowledgments  This work was supported by the German Science Founda-  tion (DFG) grant (GI-711/5-1) within the priority program  (SPP 1736) Algorithms for Big Data and by the Carl Zeiss  Foundation within the project ”Interactive Inference”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  References  Ahmed, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' and Torresani, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  STAR-Caps: Cap-  sule Networks with Straight-Through Attentive Routing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  In Advances in Neural Information Processing Systems,  (NeurIPS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Biederman, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' 1987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Recognition-by-components: A The-  ory of Human Image Understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Psychological review,  94(2): 115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Chen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Yu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Qian, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Chen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' and Wu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' A  Receptor Skeleton for Capsule Neural Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' In Meila,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' and Zhang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=', eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=', International Conference on Ma-  chine Learning, (ICML).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Deng, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Dong, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Socher, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Li, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Li, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' and Fei-  Fei, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Imagenet: A Large-scale Hierarchical Image  Database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' In Conference on Computer Vision and Pattern  Recognition, (CVPR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Dosovitskiy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Beyer, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Kolesnikov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Weissenborn,  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Zhai, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Unterthiner, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Dehghani, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Minderer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Heigold, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Gelly, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Uszkoreit, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' and Houlsby, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  An Image is Worth 16x16 Words: Transformers for Image  Recognition at Scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' In International Conference on Learn-  ing Representations, (ICLR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Gu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Interpretable Graph Capsule Networks for Ob-  ject Recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' In Conference on Artificial Intelligence,  (AAAI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Gu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' and Tresp, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Improving the Robustness of  Capsule Networks to Image Affine Transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  In  Conference on Computer Vision and Pattern Recognition,  (CVPR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Gu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Tresp, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' and Hu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Capsule Network Is Not  More Robust Than Convolutional Network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' In Conference  on Computer Vision and Pattern Recognition, (CVPR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Gu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Wu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' and Tresp, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Effective and Efficient  Vote Attack on Capsule Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' In International Confer-  ence on Learning Representations, (ICLR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Hahn, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Pyeon, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' and Kim, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Self-Routing Cap-  sule Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' In Advances in Neural Information Process-  ing Systems, (NeurIPS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Hinton, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' 1979.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Some Demonstrations of the Effects of  Structural Descriptions in Mental Imagery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Cognitive Sci-  ence, 3(3): 231–250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Hinton, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' How to represent part-whole hierarchies  in a neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' arXiv, abs/2102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='12627.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Hinton, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Ghahramani, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' and Teh, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Learn-  ing to Parse Images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  In Advances in Neural Information  Processing Systems, (NIPS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Hinton, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Krizhevsky, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' and Wang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Trans-  forming Auto-Encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' In International Conference on Ar-  tificial Neural Networks, (ICANN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Hinton, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Sabour, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' and Frosst, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Matrix Cap-  sules with EM Routing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  In International Conference on  Learning Representations, (ICLR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Kosiorek, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Sabour, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Teh, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' and Hinton, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Stacked Capsule Autoencoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' In Advances in Neu-  ral Information Processing Systems, (NeurIPS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Krizhevsky, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Learning multiple layers of features  from tiny images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Technical report, University of Toronto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Laue, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Mitterreiter, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' and Giesen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Computing  Higher Order Derivatives of Matrix and Tensor Expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  In Neural Information Processing Systems (NeurIPS), 2755–  2764.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Laue, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Mitterreiter, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' and Giesen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' A Simple  and Efficient Tensor Calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' In Conference on Artificial  Intelligence (AAAI), 4527–4534.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  LeCun, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Boser, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Denker, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Henderson, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Howard, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Hubbard, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' and Jackel, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
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+page_content=' Neural Computation, 1(4): 541–551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  LeCun, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Bottou, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Bengio, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' and Haffner, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Gradient-based learning applied to document recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Proceedings of the IEEE, 86(11): 2278–2324.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Mazzia, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Salvetti, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' and Chiaberge, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Efficient-  CapsNet: capsule network with self-attention routing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Sci-  entific reports, 11(1): 1–13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Michels, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Uelwer, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Upschulte, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' and Harmeling, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' On the Vulnerability of Capsule Networks to Adver-  sarial Attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' arXiv, abs/1906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='03612.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Paik, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Kwak, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' and Kim, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Capsule Networks  Need an Improved Routing Algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' In Asian Conference  on Machine Learning, (ACML).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Pham, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Dai, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Xie, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' and Le, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Meta Pseudo  Labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  In Conference on Computer Vision and Pattern  Recognition, (CVPR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Rajasegaran, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Jayasundara, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Jayasekara, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Jayasekara,  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Seneviratne, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' and Rodrigo, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' DeepCaps: Go-  ing Deeper With Capsule Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' In Conference on Com-  puter Vision and Pattern Recognition, (CVPR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Rawlinson, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Ahmed, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' and Kowadlo, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Sparse  Unsupervised Capsules Generalize Better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Ribeiro, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Leontidis, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' and Kollias, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Capsule Routing via Variational Bayes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' In Conference on  Artificial Intelligence, (AAAI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Sabour, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Frosst, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' and Hinton, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Dynamic  Routing Between Capsules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' In Advances in Neural Infor-  mation Processing Systems, (NIPS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Sabour, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Tagliasacchi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Yazdani, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Hinton, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' and  Fleet, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Unsupervised Part Representation by Flow  Capsules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' In International Conference on Machine Learn-  ing, (ICML).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Tsai, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Srivastava, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Goh, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' and Salakhutdinov, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Capsules with Inverted Dot-Product Attention Rout-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' In International Conference on Learning Representa-  tions, (ICLR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Vaswani, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Shazeer, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Parmar, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Uszkoreit, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Jones,  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Gomez, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Kaiser, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' and Polosukhin, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' At-  tention is All you Need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' In Advances in Neural Information  Processing Systems, (NIPS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Wang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' and Liu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' An Optimization View on Dy-  namic Routing Between Capsules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' In International Confer-  ence on Learning Representations, (ICLR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Wortsman, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Ilharco, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Gadre, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Roelofs, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Lopes,  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Morcos, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Namkoong, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Farhadi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Carmon,  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Kornblith, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' and Schmidt, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Model soups: av-  eraging weights of multiple fine-tuned models improves ac-  curacy without increasing inference time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' In International  Conference on Machine Learning, (ICML).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Xi, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Bing, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' and Jin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Capsule Network Perfor-  mance on Complex Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' arXiv, abs/1712.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='03480.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Xiang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Zhang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Tang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Zou, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' and Xu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  MS-CapsNet: A Novel Multi-Scale Capsule Network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' IEEE  Signal Processing Letters, 25(12): 1850–1854.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Zhou, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Ji, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Su, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Sun, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' and Chen, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Dy-  namic Capsule Attention for Visual Question Answering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' In  Conference on Artificial Intelligence, (AAAI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Why Capsule Neural Networks Do Not Scale:  Challenging the Dynamic Parse-Tree Assumption  Appendix  This appendix provides additional materials which did not fit into the main paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' It is organized as follows:  A  Detailed descriptions of models, training procedures, and data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  B  Additional results from experiments testing viewpoint invariance  (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='2 in the main paper) for different models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  C  Additional results for different models on the AffNIST data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  D  Exhaustive experiments concerning different aspects on the AffNIST dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  E  Detailed evaluation of a single CIFAR10 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  F  Exhaustive experiments concerning different aspects on the CIFAR10 data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  A  Model Architectures and Training Procedures  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='1  CapsNets for AffNIST  The AffNIST dataset (Sabour, Frosst, and Hinton 2017) is derived from the classic MNIST data set (LeCun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' 1998) as  follows: First, all MNIST images are zero-padded to dimensions 40×40 and affinely transformed by random rotations up to  20 degrees, random shearings up to 40 degrees, random scaling from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='8 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='2, and random translations up to eight pixels  in each direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' We use this dataset to assess a model’s robustness to affine transformations of the input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' For this, we  train the model on the original MNIST train set, where the images are randomly placed on a 40×40 empty background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' For  a fair comparison of different models, training is stopped once a model reaches a target accuracy of 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='20% on the original  MNIST validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' The trained models are then evaluated on the AffNIST validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' The difference between the MNIST  validation set accuracy and the AffNIST validation set accuracy measures the model’s robustness to affine transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  The general model architecture for all models trained on AffNIST largely follows the description in Section 3 of the main  paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' We borrow the backbone proposed by Mazzia, Salvetti, and Chiaberge (2021), which utilizes four standard convolutional  layers Conv(32,7,1), Conv(64,3,1), Conv(64,3,2), and Conv(n1 · d1,3,2), followed by a fifth depthwise convolutional layer  Conv(n1·d1,7,1,n1·d1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' The reconstruction network consists of three fully connected layers FC(512), FC(1024), and FC(40·40),  all using the ReLU activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' The hyper-parameters for the loss all are set to m+ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='9, m− = 1−m+ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='1, λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='5  and α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='392.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' We found the best number of iterations for the routing algorithm to be ten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' We used the Adam optimizer with  an initial learning rate of 10−3, an exponential learning rate decay of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='97, and a weight decay regularizer with a value of 10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  We used a batch size of 512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='2  CapsNets for CIFAR10  The CapsNet architecture for the CIFAR10 classification task largely follows the architecture used for the AffNIST benchmark  with slight modifications in the backbone to adjust for the smaller input image size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' The backbone utilizes four standard convo-  lutional layers Conv(32,7,1), Conv(64,3,1), Conv(128,3,2), and Conv(n1 · d1,3,2), followed by a fifth deepthwise convolutional  layer Conv(n1 · d1,5,1,n1 · d1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  We trained all CIFAR10 models for 100 epochs similarly to the AffNIST models, but with an initial learning rate of 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  For computing the performance metrics, we selected the best model regarding validation set accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='3  The Original CapsNet for MNIST Digit Classification  Here we describe the original CapsNet architecture for the MNIST classification task by Sabour, Frosst, and Hinton (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' The  backbone function consists of two consecutive convolutional layers, Conv(256,9,1) and Conv(256,9,2), followed by a single  routing layer as generally described in Section 3 of the main paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' There are 1152 PrimeCaps of dimension eight and ten output  capsules of dimension 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' The reconstruction network consists of three fully connected layers, namely FC(512), FC(1024), and  FC(28·28), all using the ReLU activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' The hyper-parameters for the loss are set to m+ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='9, m− = 1−m+ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='1,  λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='5 and α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='0005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' The number of iterations for the routing algorithm is set to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='4  Notes on Backbone Architectures and PrimeCaps  The role of the backbone is to extract meaningful features from the input images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' The features constitute the PrimeCaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Therefore, the backbone also controls the number and dimension of the PrimeCaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' The concept of capsules requires that a  capsule is activated if its related conceptual entity is present within the input image (Sabour, Frosst, and Hinton 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' For this  reason, each capsule must have a full receptive field over the input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  The original CapsNet backbone, used on the MNIST classification task, consists of two consecutive convolutional layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  However, for larger input dimensions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=', AffNIST (40×40) or ImageNet (224×224), using only two convolutional layers is  not practical since this would either result in a vast number of PrimeCaps or large capsule dimensions that make the models  computationally infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Table 4 lists the number of parameters in the routing layer for the original CapsNet model for  different input image dimensions, depending on the dimension of the output capsules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' As can be seen, the number of parameters  increases quickly with the input dimension, and as a result, computational and space requirements also rise quickly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' see Table 3  in the main paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Therefore, we borrow the backbone proposed by Mazzia, Salvetti, and Chiaberge (2021) because it allows us to control both  the number and dimension of the PrimeCaps, and thus scales well to larger input dimensions since the number of parameters  only grows moderately with the input dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  However, we also evaluated the original CapsNet backbone in our investigations of viewpoint invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' See Appendix B  for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Dataset  Dimension of a DigitCaps  Name  Classes  Input Dimensions  16  32  64  128  MNIST  10  (28x28x1)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='0  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='9  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='8  AFFNIST  10  (40x40x1)  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='9  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='8  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='6  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='2  CIFAR10  10  (32x32x3)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='6  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='3  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='5  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='0  TinyImageNet  200  (64x64x3)  472.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='3  944.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='2  1887.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='9  3775.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='3  ImageNet 224  1000  (224x224x3)  44324.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='0  88626.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='3  177231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='0  354440.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='3  Table 4: The number of parameters in the routing layer for the original CapsNet architecture for different input image sizes  and output capsule dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' The number of output capsules is set to match the number of data set classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' The number of  parameters is given in millions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  B  Viewpoint Invariance: Additional Results  In this section, we report the results from additional experiments on the viewpoint invariance of the parse-tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' For experi-  ment details, see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='2 of the main paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Figure 6 shows the capsule activation correlation for the original CapsNet  model (Sabour, Frosst, and Hinton 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Since it was defined for an input image size of 28×28, we rescaled all the 40×40  AffNIST images down to 28×28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Figure 7 shows the capsule activation correlation for a CapsNet with one routing layer, 16  PrimeCaps and a capsule dimension of eight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Figure 8 shows the capsule activation correlation for an eight layer CapsNet with  16 capsules per layer and a capsule dimension of 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' We observe a substantial decrease in capsule activation correlation for all  models, similar to the results reported in the main paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
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+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='0  layer 1  layer 2  layer 3  layer 4  layer 5  layer 6  layer 7  layer 8  (e) vertical translation  Figure 8: Capsule activation correlation for a CapsNet with eight routing layers, trained on AffNIST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' The correlation decreases  with stronger affine transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Left to right: rotation, shearing, scaling, translation in x-direction, translation in y-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Shown are the means and the standard deviations of capsule activation for all layers except the last over ten trained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  C  AffNIST: Additional Results for the Model from the Main Paper  In this section, we report additional results about the model architecture used in the main paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' The model architecture consists  of five capsule layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Each layer contains 16 capsules, except the last, where we set the number of capsules to match the number  of digits, namely ten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' We use a capsule dimension of eight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' We trained a total of ten models with these settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' We trained the  model as described in Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' The activation and dynamics statistics of these ten models are reported in Tables 1 and 2 of  the main paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Furthermore, we selected one of these models to create Figures 2, 3, 4, 5 and Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Here, we provide additional results for these models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' For the selected model, Figure 10 shows the maximum norm of all  the capsule vectors tracked over the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' We see that the norm of a capsule remains zero once it becomes zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  This observation is explained by Theorem 2 in the main paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' The norm of the gradients of all the capsule weight matrices,  tracked over the training period is visualized in Figures 11 to 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' As can be seen, the weights referred to incoming and outgoing  votes from and to dead capsules do not change since the norm of the gradients approaches zero, whereas, the gradient norms of  weights from alive capsules to alive capsules show healthy updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
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+page_content='1  0  250000  250000  250000  250000  250000  250000  250000  250000  25000  15  Figure 14: Final routing layer: The gradient norms for the weight matrices  ∂Lm  ∂W 4  (j,i,:,:) over the training period of the model  with lower layer capsules i in the rows and upper layer capsules j in the columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  D  AffNIST: Exhaustive Experiments on Model Architectures  In this section, we report the results of our exhaustive experiments on model architectures for the AffNIST benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' We  trained all models following the training procedure as described in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' We used a total of 81 different architectures  with varying numbers of routing layers {1, 2, 3, 4, 5, 6, 7, 8}, numbers of capsules per layer {15, 32, 64} as well as capsule  dimension {8, 32, 64}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' We chose these parameters to cover a broad range of settings, from simple one-layer models to complex  models that used all the available GPU RAM (48GB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' We report the accuracy on the AffNIST validation set in Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Table 5 lists the best overall models with architecture details, number of parameters and a uniform routing baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' The best  models per depth are given in Table 6 and the corresponding metrics and measurements are given in Tables 7, 8, 9, 10, 11, 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
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+page_content='0  64  acc mnist valid  acc affnist valid  acc mnist_train  1  2  3  4  5  6  7  8  # routing layers  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
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+page_content='  Model Settings  Parameters  AffNIST Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
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+page_content='90  32  32  2  121  192  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
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+page_content='89  16  32  2  34  72  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
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+page_content='89  Table 5: Overview of the best overall models on the AffNIST benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' For a model, we list the number of capsules per layer  (#caps), the dimension of the capsules (dim), and the number of routing layers (depth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' The number of backbone parameters  and the sum of all routing layer parameters are listed separately in 10k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' We give the validation accuracy for the model when  trained with uniform routing and RBA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Model Settings  Parameters  AffNIST Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  depth  #caps  dim  Routing  Backbone  1  16  8  2  16  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
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+page_content='86  4  16  64  331  401  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='85  5  32  16  113  151  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='84  6  16  32  139  177  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='83  Table 6: Overview of the best models per depth on the AffNIST benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' For a model, we list the number of capsules  per layer (#caps), the dimension of the capsules (dim), and the number of routing layers (depth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' The number of backbone  parameters and the sum of all routing layer parameters are listed separately in 10k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Capsule Layer  Capsule Norms  Capsule Activation  Capsule Deaths  Mean (cnm)  Sum (cns)  Rate (car)  Sum (cas)  Rate (cdr)  Sum (cds)  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
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+page_content='  Capsule Layer  Capsule Norms  Capsule Activation  Capsule Deaths  Mean (cnm)  Sum (cns)  Rate (car)  Sum (cas)  Rate (cdr)  Sum (cds)  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
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+page_content='  Capsule Layer  Capsule Norms  Capsule Activation  Capsule Deaths  Mean (cnm)  Sum (cns)  Rate (car)  Sum (cas)  Rate (cdr)  Sum (cds)  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
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+page_content='  E  CIFAR10: Complete Results for a Single Model  1  5  9  13  17  21  25  29  1  2  3  4  5  (a) Mean Couplings  1  5  9  13  17  21  25  29  1  2  3  4  5  (b) Std Couplings  1  5  9  13  17  21  25  29  1  2  3  4  5  (c) Mean Activations  1  5  9  13  17  21  25  29  1  2  3  4  5  (d) Std Activations  1  5  9  13  17  21  25  29  1  2  3  4  5  (e) Dead Capsules  Figure 16: Parse-tree statistics for the complete CIFAR10 validation dataset for a five-layer CapsNet model with 32/10 capsules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  The mean (a) and the standard deviation (b) of the coupling coefficient matrices for each layer are visualized as connections  between capsules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Higher coupling coefficients have a darker blue tone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
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+page_content='1  0 25000  0 25000  0 25000  0 25000  0 25000  0 25000  0 25000  0 25000  0 25000  31  Figure 22: Final routing layer: The gradient norms for the weight matrices  ∂Lm  ∂W 4  (j,i,:,:) over the training period of the model  with lower layer capsules i in rows and upper layer capsules j in columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  F  CIFAR10: Exhaustive Experiments on Model Architectures  In this section, we report the results of our exhaustive experiments on model architectures for the CIFAR10 image classification  task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' We trained all models following the training procedure as described in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' We used a total of 40 different  architectures with varying numbers of routing layers {1, 2, 3, 4, 5, 6, 7, 8}, capsules per layer {32, 64, 128} as well as capsule  dimensions {16, 32, 64}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' We chose these parameters to cover a broad range of settings, from simple one-layer models to  complex models that used all the available GPU RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' We report the best achieved accuracies in Figure 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' Table 14 lists the  best overall models with architecture details, number of parameters and a uniform routing baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' The best models per depth  are given in Table 15 and the corresponding metrics and measurements are given in Tables 16, 17, 18, 19, 20, 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
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+page_content='0  acc valid  acc train  Figure 23: Reported accuracies for the CIFAR10 image classification experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Model Settings  Parameters  Valid Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
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+page_content='69  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='62  32  32  4  331  462  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='66  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='13  32  16  3  61  131  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='66  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='21  Table 14: Overview of the best overall models on the CIFAR10 image classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' For a model, we list the number  of capsules per layer (#caps), the dimension of the capsules (dim), and the number of routing layers (depth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' The number of  backbone parameters and the sum of all routing layer parameters are listed separately in 10k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' We give the validation accuracy  for the model when trained with uniform routing and with RBA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Model Settings  Parameters  Valid Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  depth  #caps  dim  Routing  Backbone  1  128  16  33  285  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='78  2  32  64  452  704  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='79  3  32  64  872  1123  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='77  4  32  64  1291  1543  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='69  5  32  32  436  567  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='49  6  32  32  541  672  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='43  Table 15: Overview of the best models per depth on the CIFAR10 image classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' For a model, we list the number  of capsules per layer (#caps), the dimension of the capsules (dim), and the number of routing layers (depth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content=' The number of  backbone parameters and the sum of all routing layer parameters are listed separately in 10k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Capsule Layer  Capsule Norms  Capsule Activation  Capsule Deaths  Mean (cnm)  Sum (cns)  Rate (car)  Sum (cas)  Rate (cdr)  Sum (cds)  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='90  115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='56  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='00  128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
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+page_content='00  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
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+page_content='78  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='78  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='00  (a)  Routing Layer  Capsules Alive  Routing Dynamics  From lower layer  To higher layer  Rate (dyr)  Mean (dys)  1  128  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='62  (b)  Table 16: Capsule activation and routing dynamics for the best model with one routing layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Capsule Layer  Capsule Norms  Capsule Activation  Capsule Deaths  Mean / Capsule  Mean / Layer  Rate / Layer  Mean / Layer  Rate / Layer  Mean / Layer  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
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+page_content='00  (a)  Routing Layer  Capsules Alive  Routing Dynamics  From lower layer  To higher layer  Rate (dyr)  Mean (dys)  1  32  17  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
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+page_content='73  (b)  Table 17: Capsule activation and routing dynamics for the best model with two routing layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
+page_content='  Capsule Layer  Capsule Norms  Capsule Activation  Capsule Deaths  Mean (cnm)  Sum (cns)  Rate (car)  Sum (cas)  Rate (cdr)  Sum (cds)  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfnf3Z/content/2301.01583v1.pdf'}
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+One-Time Universal Hashing Quantum Digital Signatures without Perfect Keys
+Bing-Hong Li,1 Yuan-Mei Xie,1 Xiao-Yu Cao,1 Chen-Long Li,1 Yao Fu,2, ∗ Hua-Lei Yin,1, † and Zeng-Bing Chen1, ‡
+1National Laboratory of Solid State Microstructures and School of Physics,
+Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China
+2Beijing National Laboratory for Condensed Matter Physics and Institute of Physics,
+Chinese Academy of Sciences, Beijing 100190, China
+(Dated: January 4, 2023)
+Quantum digital signatures (QDS), generating correlated bit strings among three remote parties
+for signatures through quantum law, can guarantee non-repudiation, authenticity and integrity of
+messages. Recently, one-time universal hashing QDS framework, exploiting the quantum asymmetric
+encryption and universal hash functions, was proposed to significantly improve the signature rate
+and ensure unconditional security by directly signing the hash value of long messages. However,
+similar to quantum key distribution, this framework utilizes keys with perfect secrecy via performing
+privacy amplification that introduces huge matrix operations, thus consuming large computational
+resources, causing time delays and increasing failure probability. Here, we prove that, different from
+private communication, imperfect quantum keys with limited information leakage can be used for
+digital signatures and authentication without compromising the security while having eight orders of
+magnitude improvement on signature rate for signing a megabit message compared with conventional
+single-bit schemes. Our work significantly reduces the time delay for data postprocessing and is
+compatible with any quantum key generation protocols.
+In our simulation, taking two-photon
+twin-field key generation protocol as an example, QDS can be practically implemented over a fiber
+distance of 600 km between the signer and receiver. Our work for the first time offers a cryptographic
+application of quantum keys with imperfect secrecy and paves a way for the practical and agile
+implementation of digital signatures in a future quantum network.
+I.
+INTRODUCTION
+Digital signatures are cryptographic primitives promis-
+ing data authenticity and integrity [1], especially for the
+non-repudiation of sensitive information. It has become
+an indispensable and essential technique in the global in-
+ternet due to its wide application especially in digital
+financial transactions, e-mail and digital currency. How-
+ever, the security of classical digital signatures, guaran-
+teed by public-key infrastructure [2–4], is threatened by
+rapidly developing algorithms [5, 6] and quantum com-
+puting [7].
+Different from classical digital signatures,
+quantum digital signatures (QDSs) can provide a higher
+level of security, information-theoretical security, by em-
+ploying the fundamental principles of quantum mechan-
+ics. That is, QDS can protect data integrity, authentic-
+ity and non-repudiation even if the attacker has unlim-
+ited computational power. The rudiment of the single-bit
+QDS scheme was introduced in 2001 [8], but it could not
+be implemented due to some impractical requirements
+such as high-dimensional single-photon states and quan-
+tum memories. Subsequently, there have been many de-
+velopments to remove these impractical requirements [9–
+11], making QDS closer to real implementation. Further-
+more, based on non-orthogonal encoding [12] and orthog-
+onal encoding [13], respectively, two independent single-
+bit QDS protocols without secure quantum channels were
+∗ yfu@iphy.ac.cn
+† hlyin@nju.edu.cn
+‡ zbchen@nju.edu.cn
+proposed and proved to be secure for the first time.
+These two protocols have triggered numerous achieve-
+ments of single-bit QDS theoretically [14–25] and exper-
+imentally [26–36].
+However, all these schemes still have several limita-
+tions.
+Protocols utilizing orthogonal encoding require
+additional symmetrization steps which results in extra
+secure channels [13]. That is, to guarantee information-
+theoretical security, quantum key distribution and one-
+time pad encryption are still required between two re-
+ceivers in orthogonal-type protocols [28, 29]. Single-bit
+QDS schemes based on non-orthogonal encoding [12, 20,
+23] are independent of additional quantum key distribu-
+tion channels, but the signature rate is sensitive to the
+misalignment error of the quantum channel.
+In addi-
+tion, all these schemes can only sign a one-bit message in
+each round. If one wants to sign a multi-bit message by
+single-bit QDS schemes, he needs to encode it into a new
+message string and sign the new string bit by bit [22, 37–
+41]. However, these solutions have not been completely
+proved information-theoretically secure with the quanti-
+fied failure probability, and the signature rate is very low
+and far from implementation for long messages with a lot
+of bits.
+Recently, an efficient QDS scheme has been proposed
+based on secret sharing, one-time pad, and one-time uni-
+versal hashing (OTUH) [42].
+Different from single-bit
+QDS protocols that require a long key string to sign
+a one-bit message, this OUTH-QDS protocol offers a
+method to directly sign the hash value of multi-bit mes-
+sages through one key string with information-theoretic
+security, and thus drastically improves the QDS effi-
+arXiv:2301.01132v1  [quant-ph]  3 Jan 2023
+
+2
+ciency.
+However, this framework requires perfect keys
+with full secrecy that is an expensive resource guaran-
+teed by complete procedure of quantum key distribution
+(QKD) or quantum secret sharing (QSS). That is, pri-
+vacy amplification steps are required, thereby adding to
+the complexity of the algorithm and causing unendurable
+time delays.
+Here, we point out that quantum keys with imperfect
+secrecy are sufficient for protecting authenticity and in-
+tegrity of messages in such a digital signature scheme.
+Based on this idea, we propose a new OTUH-QDS pro-
+tocol with imperfectly secret keys, utilizing only asym-
+metric quantum keys without full secrecy to sign multi-
+bit messages. We prove that our new scheme provides
+information-theoretic security for digital signature tasks
+and simulate the performance of our protocol. The result
+shows that our protocol outperforms other QDS schemes
+in terms of signature rate and transmission distance. For
+a practical case in which a megabit message is to be
+signed, the proposed scheme has a superior signature
+rate of nearly eight orders of magnitude, compared with
+single-bit QDS schemes. In addition, we show that our
+scheme can significantly reduce the computational costs
+and time delays of postprocessing due to the removal of
+privacy amplification. Our protocol is also robust against
+message size, security bound and finite-size effect. More-
+over, the proposed scheme is a general framework and
+can be applied to all existing QKD protocols. When uti-
+lizing the idea of two-photon twin-field QKD [43], one of
+the most efficient QKD protocols, to execute our work, a
+transmission distance of 650 km can be achieved with a
+signature rate of 0.01 times per second.
+To date, almost all quantum communication protocols
+such as QKD [44–54], QSS [55, 56] and quantum confer-
+ence key agreement [55, 57] aim at generating quantum
+keys with perfect secrecy and correctness. These keys are
+then used to finish the corresponding cryptographic tasks
+such as private communication, secret sharing and group
+encryption. In contrast, our proposed protocol offers a
+new approach to digital signature tasks that only requires
+keys with full correctness and imperfect secrecy through
+quantum optical communication. It is the first time that
+this kind of keys is applied to cryptographic tasks with
+information-theoretic security. We believe that our pro-
+posed solution can provide a feasible approach to the
+practical application of QDS and enlighten other applica-
+tions of quantum keys with imperfect secrecy in a future
+quantum network.
+The rest of this paper is organized as follows. In Sec. II
+we review OTUH-QDS scheme and introduce the moti-
+vation of this work. In Sec. III we propose our protocol
+with two approaches of universal hashing. In Sec. IV we
+give the security proof of authentication based on quan-
+tum keys with imperfect secrecy and then, the security
+analysis of our proposed QDS protocol. In Sec. V we dis-
+cuss the performance of our scheme and compare it with
+both single-bit QDS and OTUH-QDS schemes. Finally
+we conclude the paper in Sec. VI.
+II.
+PRELIMINARIES
+A.
+OTUH-QDS protocol
+In this section we review the schematic of OTUH-
+QDS [42]. The protocol can be segmented into the distri-
+bution stage and messaging stage, consistent with that
+in single-bit QDS that is introduced in Appendix A 1.
+Denote the length of message as m. The schematic of
+OTUH-QDS is shown in Fig. 1 (a).
+1.
+distribution stage
+Alice, Bob, and Charlie each have two key bit strings
+{Xa, Xb, Xc} with n bits and {Ya, Yb, Yc} with 2n
+bits, where the key bit strings meet the perfect correla-
+tion Xa = Xb ⊕ Xc and Ya = Yb ⊕ Yc, respectively. The
+distribution stage can be realized using quantum com-
+munication protocols, such as QKD and QSS. It need
+to be mentioned that single-bit QDS only requires the
+quantum part of QKD protocols, also called key genera-
+tion protocol (KGP). In OTUH-QDS, the users need to
+perform extra error correction and privacy amplification
+steps after KGP.
+2.
+messaging stage
+(i) Signing of Alice—First, Alice uses a local quantum
+random number, characterized by an n-bit string pa, to
+randomly generate an irreducible polynomial p(x) of de-
+gree n [58]. Second, she uses the initial vector (key bit
+string Xa) and irreducible polynomial (quantum random
+number pa) to generate a random linear feedback shift
+register-based (LFSR-based) Toeplitz matrix [59] Hnm,
+with n rows and m columns. Third, she uses a hash op-
+eration with Hash= Hnm · Doc to acquire an n-bit hash
+value of the m-bit document. Fourth, she exploits the
+hash value and the irreducible polynomial to constitute
+the 2n-bit digest Dig = (Hash||pa). Fifth, she encrypts
+the digest with her key bit string Ya to obtain the 2n-bit
+signature Sig = Dig ⊕ Ya using OTP. Finally, she uses
+the public channel to send the signature and document
+{Sig, Doc} to Bob.
+(ii) Verification of Bob—Bob uses the authentication
+classical channel to transmit the received {Sig, Doc}, as
+well as his key bit strings {Xb, Yb}, to Charlie. Then,
+Charlie uses the same authentication channel to forward
+his key bit strings {Xc, Yc} to Bob. Bob obtains two
+new key bit strings {KXb = Xb ⊕ Xc, KYb = Yb ⊕ Yc} by
+the XOR operation. Bob exploits KYb to obtain an ex-
+pected digest and bit string pb via XOR decryption. Bob
+utilizes the initial vector KXb and irreducible polynomial
+pb to establish an LFSR-based Toeplitz matrix. He uses
+a hash operation to acquire an n-bit hash value and then
+constitutes a 2n-bit actual digest. Bob will accept the
+signature if the actual digest is equal to the expected
+
+3
+Alice
+Bob
+Charlie
+error correction
+privacy amplification
+Alice
+Bob
+Charlie
+(a)
+(b)
+Distribution stage
+Messaging stage
+Alice
+Bob
+Charlie
+Alice
+Bob
+Charlie
+Distribution stage
+Messaging stage
+KGP 
+KGP 
+error correction
+privacy amplification
+KGP 
+KGP 
+error correction
+error correction
+classic channel
+quantum channel
+FIG. 1.
+(a) OTUH-QDS [42]. In the distribution stage, Alice, Bob and Charlie share key bit strings with perfect secret sharing
+relationship through key generation protocol (KGP), error correction and privacy amplification. In the messaging stage Alice
+generates the signature through AXU hashing, and sends the message and signature to Bob. Bob then sends his keys and
+received information to Charlie, who will later send his keys to Bob. Bob and Charlie use their own and received keys to infer
+Alice keys and then perform AXU hashing to verify the signature. (b) Schematic of our proposed protocol. In the distribution
+stage, the users only perform KGP and error correction to share keys with full correctness but some secrecy leakage. Their
+keys still hold secret sharing relationship. In the messaging stage the manipulation of classic information is analogous to that
+in OUTH-QDS.
+digest. Then, he informs Charlie of the result. Other-
+wise, Bob rejects the signature and announces to abort
+the protocol.
+(iii) Verification of Charlie—If Bob announces that he
+accepts the signature, Charlie then uses his original key
+and the key sent to Bob to create two new key bit strings
+{KXc = Xb ⊕ Xc, KYc = Yb ⊕ Yc}.
+Charlie employs
+KYc to acquire an expected digest and bit string pc via
+XOR decryption. Charlie uses a hash operation to ob-
+tain an n-bit hash value and then constitutes a 2n-bit
+actual digest, where the hash function is an LFSR-based
+Toeplitz matrix generated by initial vector KXc and ir-
+reducible polynomial pc. Charlie accepts the signature if
+the two digests are identical. Otherwise, Charlie rejects
+the signature.
+The core point of this protocol is to realize the perfect
+bits correlation of the three parties, building a completely
+asymmetric key relationship for them, and perform one-
+time almost XOR universal2 (AXU) hashing, specifically,
+LFSR-based Toeplitz hashing, to generate the signature.
+AXU hash functions is a special class of hash functions
+that can map the input value of arbitrary length into an
+almost random hash value with a fixed length [60]. The
+signature generated in OTUH-QDS is just the AXU hash
+value of the long message to be signed, where the AXU
+hash function is determined by just one string of Alice
+keys. After distribution stage, Alice, Bob and Charlie’s
+keys are completely secret and correct with the relation-
+ship of secret sharing. Bob (Charlie) can only obtain Al-
+ice’s keys after he receives keys of Charlie (Bob). Thus
+Bob can obtain no information of Alice’s keys which de-
+cides the AXU hash function before he transfer the mes-
+sage and signature to Charlie. Then Bob’s forging attack
+under this protocol is equivalent to Bob’s attack against
+an authentication scenario where Alice sends an authenti-
+cated message to Charlie. It has been proved that such a
+message authentication scheme based on AXU hashing
+is information-theoretically secure [59].
+Consequently,
+forging attack is protected by one-time AXU hash func-
+tions and key relationship among three parties. From the
+perspective of Alice, Bob and Charlie’s keys are totally
+symmetric when they verify the signature. Thus, Alice’s
+repudiation attack is also prevented.
+B.
+Motivation of this work
+Different from all single-bit QDS protocols that require
+a long key string to sign a one-bit message, OUTH-QDS
+offers a method to sign multi-bit messages through one
+
+4
+key string with information-theoretic security, and thus
+drastically improves the QDS efficiency. This advantage
+is essentially introduced by AXU hash functions, which is
+only proved to be information-theoretically secure under
+perfectly secret keys in previous studies. Thus, OTUH-
+QDS requires extra error correction and privacy amplifi-
+cation steps to realize the perfect bits correlation in the
+distribution stage, compared with single-bit QDS. These
+postprocessing steps especially privacy amplification re-
+quires calculations of a matrix with a comparable length
+of data size, which introduces heavy computational costs
+and unpleasant delays in practical scenarios. When the
+data size is large, the delays will become unendurable
+and constrain the practicality.
+The process of AXU hashing is equivalent to the sce-
+nario where the input value decides the function, map-
+ping the initial input keys into almost random output
+hash values. We notice that limited secrecy leakage of in-
+put value (keys) will be concealed in AXU hash value be-
+cause of its randomness. Thus, such imperfect keys with
+limited secrecy leakage will not undermine the authentic-
+ity of messages in a QDS scheme like OTUH-QDS. More-
+over, the integrity of messages is also not compromised.
+Based on this idea, in this paper we propose a solution
+for OTUH-QDS protocols with imperfectly secret keys.
+In other words, we implement QDS with quantum keys
+without privacy amplification.
+Because the additional
+computational cost and time delays of OTUH-QDS are
+mainly introduced by privacy amplification, this concept
+can effectively reduce the weaknesses of OTUH-QDS and
+lay a ground for the future implementation of QDS in a
+quantum internet.
+The schematic of our proposed protocol is shown in
+fig. 1 (b). In the distribution stage users only perform
+the error correction step after KGP, ensuring that their
+keys have no mismatches, and build a secret sharing re-
+lationship by Alice’s XOR operation. The final keys will
+be randomly divided into several n-bit groups for AXU
+hashing. Each of these groups of keys contains full cor-
+rectness and some secrecy leakage with an upper bound
+which can be estimated through finite-size analysis using
+experimental data in KGP. In the messaging stage, the
+rules of information exchange are consistent with that in
+OTUH-QDS. We will prove that the bit stings generated
+in our distribution stage are sufficient for AXU hashing
+and quantify the security bound in Sec. IV. In addition,
+we give two solutions based on different types of AXU
+hash functions.
+III.
+QDS PROTOCOL
+A schematic of setups of our QDS protocol is illus-
+trated below and shown in Fig. 2.
+A.
+Distribution stage
+Our proposal is a general framework in which KGP can
+be derived from any type of QKD protocol. As an ex-
+ample, we demonstrate our scheme based on two-photon
+twin-field (TP-TF) QKD [43]. In the distribution stage
+Alice–Bob and Alice–Charlie independently implement
+TP-TF KGP (TP-TFKGP for simplicity) to share key
+bit strings.
+1. Preparation. At each time bin i ∈ {1, 2, . . . , N}, Al-
+ice and Bob (Alice and Charlie) each independently pre-
+pare a weak coherent pulse |ei(θi
+x+ri
+xπ)�
+kix⟩ with prob-
+ability pkx, where the subscript x ∈ {a, b, c} repre-
+sents the user Alice, Bob or Charlie, the phase θi
+x
+∈ [0, 2π), classical bit ri
+x ∈ {0, 1}, intensity ki
+a(b) ∈
+{µa(b),
+νa(b),
+oa(b),
+ˆoa(b)} (represent signal,
+decoy,
+preserve-vacuum and declare-vacuum intensity, µa(b) >
+νa(b) > oa(b) = ˆoa(b) = 0) are chosen randomly. Then
+Alice and Bob (Alice and Charlie) send the correspond-
+ing pulses to the untrusted relay Eve through insecure
+quantum channels, respectively. In addition, they send a
+bright reference light to Eve to measure the phase noise
+difference φi
+ab (φi
+ac).
+2. Measurement. Eve performs interference measure-
+ments on every received pulse pair with a beam splitter
+and two detectors. If one and only one detector clicks,
+Eve publicly announces that she obtained a successful
+detection event and which detector clicked.
+3. Sifting. For successful detection events of Alice-Bob,
+those when neither Alice nor Bob chooses the declare-
+signal or decoy or declare-vacuum intensity, i.e., {µa, ob},
+{µa, µb}, {oa, µb} and {oa, ob} will be used for generat-
+ing data in the Z basis. For these events, Alice randomly
+matches a time bin i of intensity µa with another time
+bin j of intensity oa. Then she sets her bit value as 0 (1)
+if i < j (i > j) and informs the serial numbers i and j
+to Bob. In the corresponding time bins, if Bob chooses
+intensities kmin{i,j}
+b
+= µb (ob) and kmax{i,j}
+b
+= ob (µb), he
+sets his bit value as 0 (1). Bob announces to abort the
+event where ki
+b = kj
+b = ob or µb. To conclude, the pre-
+served events in the Z basis are sifted as {µaoa, obµb},
+{µaoa, µbob}, {oaµa, obµb} and {oaµa, µbob}.
+For
+other successful detection events, Alice and Bob commu-
+nicate their intensities and phase information with each
+other via an authenticated channel and later use them
+for decoy analysis.
+Define the global phase difference at time bin i as
+θi := θi
+a − θi
+b + φi
+ab. Alice and Bob keep detection events
+{νi
+a, νi
+b} only if θi ∈ [−δ, δ] ∪ [π − δ, π + δ]. They ran-
+domly select two retained detection events that satisfy
+��θi − θj�� = 0 or π, and then match these two events , de-
+noted as {νi
+aνj
+a, νi
+bνj
+b}. By calculating classical bits ri
+a⊕rj
+a
+and ri
+b ⊕ rj
+b, Alice and Bob extract a bit value in the X
+basis, respectively. Subsequently, in the Z basis, Bob al-
+ways flips his bit. In the X basis, Bob flips part of his
+bits to correctly correlate them with those of Alice. To
+be specific, when the global phase difference between two
+
+5
+Eve
+Charlie
+Alice
+Laser
+SPD
+BS
+PM
+AWG
+VOA
+PM
+AWG
+VOA
+Laser
+Bob
+PM
+AWG
+VOA
+Laser
+IM
+IM
+IM
+SPD
+Optical 
+switch
+FIG. 2. Schematic of the setup of our QDS protocol. The red line represents quantum optical channel in the distribution stage,
+and the black arrow line represents the information exchange through classic authenticated channel in the messaging stage. (i)
+In the distribution stage, Alice, Bob and Charlie utilize a narrow-linewidth continuous-wave laser, intensity modulator (IM),
+phase modulator (PM), arbitrary wave generator (AWG) and variable optical attenuator (VOA) to prepare a phase-randomized
+weak coherent source with different intensities and phases. The signals from Bob and Charlie will both go through an optical
+switch. An untrusted relay Eve performs interference measurement on the signals from Alice and an optical switch with a beam
+splitter (BS) and a single-photon detector (SPD). After sifting, parameter estimation and error correction, Alice can share bit
+strings with Bob and Charlie, respectively. (ii) In the messaging stage, Alice sends the desired message to Bob. Bob sends the
+message as well as his keys to Charlie. Charlie will then send his keys to Bob. Then Bob verifies the signature by his own and
+received keys. If he accepts the signature, he will inform Charlie and then Charlie will also verify the signature by his own and
+received keys. The signature is successful if both Bob and Charlie accept it.
+matching time bins is 0 (π) and the two clicking detectors
+announced by Eve are different (same), Bob will flip his
+bits. Otherwise, Bob will directly save his bits for later
+use.
+Similarly, Alice and Charlie will sift their successful
+detection events in the same way.
+4. Parameter estimation.
+Alice and Bob (Alice and
+Charlie) form the nZ-length raw key bit from the random
+bits under the Z basis. The remaining bits in the Z basis
+are used to estimate the bit error rate Ez. They also
+communicate all bit values in the X basis to obtain the
+total number of errors. The decoy-state method [61, 62]
+is used to estimate the number of vacuum events in the
+Z basis sz
+0µb, the count of single-photon pairs sz
+11, and
+the phase error rate of single-photon pairs φz
+11 on the Z
+basis.
+5. Error correction and examination. Alice and Bob
+(Alice and Charlie) distill final keys by utilizing an error
+correction algorithm with εcor-correctness. [63, 64] The
+size of the distilled key is still nZ. The unknown infor-
+mation of a possible attacker can be expressed as H. Al-
+ice then randomly disturbs the orders of the distilled key
+and publicizes the new order to Bob (Charlie) through
+the authenticated channel. Subsequently, Alice and Bob
+(Alice and Charlie) divide the final keys into several n-
+bit strings, each of which is used to perform a task in the
+messaging stage. This ensures that the attacker cannot
+know the position of each bit of one concrete group be-
+forehand. Thus the grouping process can be considered
+as a random sampling in our finite-size analysis.
+The
+maximum unknown information of an n-bit group to the
+attacker after error correction Hn can be estimated by
+Hn ≤ szn
+0µb + szn
+11
+�
+1 − H(φzn
+11 )
+�
+− nfH(Ez),
+(1)
+where f is the error correction efficiency, szn
+0µb and szn
+11 are
+the lower bounds of vacuum events and single-photon
+pairs, respectively, and φzn
+11 is the upper bound of the
+phase error rate of single-photon pairs in the n-bit string.
+B.
+Messaging stage
+Various AXU hash functions can be used in the mes-
+saging stage of our protocol by following the framework
+of Fig. 1 (b).
+To demonstrate the detailed procedure,
+we here present two specific approaches to the messag-
+ing stage utilizing LFSR based Toeplitz hashing and
+generalized division hashing, respectively. LFSR-based
+Toeplitz hashing is highly compatible with in hardware
+systems and generalized division hashing is more suit-
+able for achieving software systems. In a practical case
+
+6
+we choose either method of hashing depending on the
+different application environments of users.
+The mes-
+sage to be signed is denoted as M.
+For each M if
+using LFSR-based Toeplitz hashing Alice generates six
+bit strings XB, XC, YB, YC, ZB, ZC, each of length
+n. If choosing generalized division hashing in the mes-
+saging stage, Alice will only generate four bit strings
+XB, XC, YB, YC. The subscripts represent the partic-
+ipants performing KGP with Alice, where B represents
+Bob and C represents Charlie. Then Alice will generate
+Xa = Xb⊕Xc, Ya = Yb⊕Yc and Za = Zb⊕Zc as her own
+key strings. For the scheme with LFSR-based Toeplitz
+hashing the signature rate is R = nZ/3n, whereas for
+generalized division hashing there is R = nZ/2n.
+1.
+Utilizing LFSR-based Toeplitz hashing
+Definition 1. LFSR-based Toeplitz hash functions:
+LFSR-based Toeplitz hash functions can be expressed as
+hp,s(M) = Hnm · M, where p, s determines the function
+and M is the message in the form of an m-bit vector.
+The detailed process of generating LFSR-based Toeplitz
+hash function is as follows.
+A randomly selected irreducible polynomial of order
+n in the field GF(2), p(x), determines the construction
+of LFSR. p(x) = xn + pn−1xn−1 + ... + p1x + p0 can
+be characterized by its coefficients of order from 0 to
+n − 1, i.e., pa = (pn−1, pn−2, ..., p1, p0).
+For the ini-
+tial state s which is also represented as an n-bit vector
+s = (an, an−1, ..., a2, a1)T , the LFSR will be performed
+n times to generate n vectors. Specifically, it will shift
+down every element in the previous column, and add a
+new element to the top of the column.
+For example,
+the LFSR transforms s into s1 = (an+1, an, ..., a3, a2)T ,
+where an+1 = pa · s, and likewise, transforms s1 to s2.
+Then the m vectors s, s1, ..., sm−1 will together construct
+the Toeplitz matrix Hnm = (s, s1, ..., sm−1), and the hash
+value of the message is Hnm · M.
+(i) Alice obtains a string of random numbers through
+a quantum random number generator and uses it to ran-
+domly generate a monic irreducible polynomial in GF(2)
+of order n, denoted as p(x). p(x) can be characterized
+by its coefficients of order from 0 to n − 1, i.e., an n-bit
+string, denoted by pa. Details of generating p(x) can be
+found in Appendix B.
+(ii) Alice uses her key bit string Ya and p(x) to per-
+form LFSR-based Toeplitz hashing and generates an n-
+bit hash value Dig = HY,pa(M), and encrypts it by Za to
+obtain the final signature Sig = Dig ⊕ Za. In addition,
+Alice encrypts pa by the key set Xa to get the encrypted
+string p = pa ⊕ Xa.
+She then sends {Sig, p, M} to Bob through an au-
+thenticated classical channel.
+(iii) Bob transmits {Sig, p, M} as well as his key
+bit strings {Xb, Yb, Zb} to Charlie so as to inform
+Charlie that he has received the signature. Then Char-
+lie forwards his key bit strings {Xc, Yc, Zc} to Bob.
+These data are all transmitted through an authenti-
+cated channel.
+Bob obtains three new key bit strings
+KXb = Xb ⊕ Xc, KYb = Yb ⊕ Yc and KZb = Zb ⊕ Zc
+using the XOR operation. He exploits KXb and KZb to
+obtain the expected digest and string pb via XOR decryp-
+tion. He utilizes KYb and pb to establish an LFSR-based
+Toeplitz matrix and acquires an actual digest via a hash
+operation.
+Bob will accept the signature if the actual
+digest is equal to the expected digest. Then he informs
+Charlie of the result.
+(iv) If Bob announces that he accepts the signature,
+Charlie creates three new key bit strings KXc = Xb ⊕
+Xc, , KYc = Yb ⊕ Yc and KZc = Zb ⊕ Zc using his orig-
+inal key and the key sent by Bob. He employs KXc and
+KZc to acquire the expected digest and variable pc via
+XOR decryption.
+Charlie obtains an actual digest via
+hash operation, where the hash function is an LFSR-
+based Toeplitz matrix generated by KYc and pc. Charlie
+accepts the signature if the two digests are identical.
+2.
+Utilizing generalized division hashing
+Definition 2. Generalized division hash functions: The
+generalized division hash functions can be expressed as
+hP (M) = m(x)·xn/k mod P(x), where P(x) is a monic ir-
+reducible polynomial of order n/k in the field GF(2k), M
+is the message in the form of a polynomial of order m/k
+in GF(2k) with every k bits corresponding to a coefficient
+of the polynomial. The calculation is also performed in
+GF(2k). The final result is a polynomial of order n/k
+in field GF(2k), and can be transformed into an n-bit
+strings.
+Commonly, k is set as k = 2x for simplicity, where x is
+a positive integer. In our scheme, we select k = 23 = 8.
+(i) In this case, Alice chooses Xa = Xb ⊕ Xc and
+Ya = Yb ⊕ Yc as her own key sets. Alice first obtains
+a string of random numbers through a quantum random
+number generator and uses it to randomly generate a
+monic irreducible polynomial in GF(28) of order n/8, de-
+noted by P(x). Details of generating p(x) can be found
+in Appendix B. P(x) can be characterized by its coef-
+ficients of order from 0 to n/8 − 1. By encoding each
+coefficient into an 8-bit string, we can use an n-bit string
+to express P(x), denoted as Pa. Alice encrypts Pa by the
+key set Xa to obtain the encrypted string P = Pa ⊕ Xa
+(ii) Alice uses P(x) to perform the generalized divi-
+sion hashing [65] to obtain an n-bit hash value Dig =
+hPa(M). She encrypts Dig by Ya to obtain the signa-
+ture Sig = Dig⊕Ya and sends the message and signature
+{Sig, p, M} to Bob.
+step (iii) and (iv) are similar to those utilizing LFSR-
+based Toeplitz hashing. Bob and Charlie will exchange
+their key strings in turn through an authenticated chan-
+nel and examine their expected and received digests.
+Above is the whole procedure of our protocol. Note
+that the TP-TFKGP can be replaced by any other KGP
+
+7
+such as BB84-KGP or sending-or-not-sending (SNS)-
+KGP. Actually, in the distribution stage Alice shares bit
+strings with Bob and Charlie in the relationship of secret
+sharing. Thus, the distribution stage can also be per-
+formed based on QSS without the privacy amplification
+step.
+IV.
+SECURITY ANALYSIS
+Similar to OTUH-QDS, the core point of our protocol
+is the security of authentication based on AXU hash-
+ing, which directly protects the security of QDS against
+fogery [42]. However, the security of our protocol differs
+because of the information leakage during the distribu-
+tion stage. In this section we provide a more detailed
+security analysis of AXU hashing under imperfect keys
+with limited secrecy leakage, and then demonstrate the
+security of our protocol.
+A.
+Security of authentication based on hashing
+In digital signatures, hashing is used to perform the
+authentication task. Thus we first analyze the security
+of authentication based on ϵ-AXU hash functions.
+Definition 3. ϵ-AXU hash functions: The class of func-
+tions H = {h|h : M → T} is ϵ-AXU (ϵ-almost XOR
+universal) if and only if, for a randomly selected hash
+function h ∈ H and any m1, m2 ∈ M, t ∈ T, it holds
+that the probability of h(m1) ⊕ h(m2) = t is at most ϵ.
+The hash functions used in our scheme (i.e., LFSR-
+based Toeplitz hash functions and generalized division
+hash functions) are all linear. There is the relationship
+that h(m1) ⊕ h(m2) = h(m1 ⊕ m2). Thus, the definition
+of ϵ-AXU hash function is equivalent to:
+Definition 4. ϵ-AXU hash functions: The class of func-
+tions H = {h|h : M → T} is ϵ-AXU if and only if, for a
+randomly selected hash function h ∈ H and any m ∈ M,
+t ∈ T, the probability of h(m) = t is at most ϵ.
+Suppose the authentication scenario that an attacker
+captures a message and signature {M, Sig = h(M) ⊕ r}
+from the sender. h is randomly selected from the set of
+ϵ-AXU hash functions H, and r is secret keys. He can
+manage to forge if and only if he chooses a combination
+{m, t} with the relationship h(m) = t, and sends {M ⊕
+m, Sig ⊕ t} to the recipient. In this case the recipient
+will accept the message because of the relationship h(M⊕
+m) = h(M)⊕h(m) = Sig⊕t. It should be mentioned that
+m ̸= 0 due to the requirement for a valid forge. Through
+Def. 4, for message authentication schemes based on ϵ-
+AXU hashing, the failure probability is no more than
+ϵ [59].
+Previous proof only applies for perfect keys with full
+secrecy.
+Here we analyze the scenario with imperfect
+keys without perfect secrecy. The attacker is also to gen-
+erate a combination m, t, as analyzed above.
+He can
+perform various kinds of attacks. The worst one is to
+randomly generate m, t whose success probability is only
+2−n. We will leave out this attack in our analysis. Other
+attacks include guessing keys and recovering keys from
+signatures.
+First we quantify the guessing probability of the at-
+tacker when he is to guess an n-bit key string with limited
+secrecy leakage.Suppose an n-bit key string as X and the
+attacker’s system is B. We consider the general attack
+where attackers can perform any operations on the sys-
+tem of all quantum states, get a system ρx
+B and perform
+any positive operator-valued measure {Ex
+B}x on it. The
+probability that the attacker correctly guesses X when
+using an optimal strategy is denoted as Pguess(X|B). Ac-
+cording to the definition of min-entropy in Ref. [66],
+Pguess(X|B) = max
+{Ex
+B}x
+�
+x
+Px tr(Ex
+Bρx
+B) = 2−Hmin(X|B)ρ
+(2)
+where Hmin(X|B)ρ = Hn is the min-entropy of X and B.
+If X is generated in the distribution stage of our protocol,
+Hmin(X|B)ρ can be estimated by
+Hmin(X|B)ρ = Hn.
+(3)
+Before analyzing the attack strategy we first introduce
+a proposition of LFSR-based Toeplitz.
+Proposition 1. For the LFSR-based Toeplitz hash func-
+tion hp,s(M)= Hnm ·M, if p(x)|m(x), then hp,s(M) = 0.
+Proof.
+hp,s(M) can be transformed into hp,M(s) =
+�m−1
+i=0 Mi+1W i · s, where W is the n × n circulant ma-
+trix with the first row being p. Thus, p(x) is just the
+characteristic polynomial of W. Denote the characteris-
+tic value of W by λi, where λi is an element in GF(2n).
+Then there is the relationship p(λi) = 0.
+We can di-
+agonalize �m−1
+i=0 Mi+1W i, and the diagonal elements are
+m(λi). hp,s(M) = 0 if and only if there exists a zero diag-
+onal element, i.e., there exists a λi satisfying m(λi) = 0.
+If p(x)|m(x), because p(λi) = 0, then the condition
+m(λi) = 0 will be absolutely satisfied.
+1.
+Attack of guessing keys
+The LFSR-based Toeplitz hash function is hp,s(M)=
+Hnm·M, where Hnm is determined by the two bit strings
+p and s. Here we follow the terminology in the messaging
+stage of our protocol where p is actually pa encrypted by
+Xa, s is Ya and the hash value Dig is encrypted by Za.
+We show that only guessing Xa or in other words, only
+guessing pa is enough to perform an attack.
+Suppose the attacker obtain a string Xg as his estima-
+tion of Xa. He can decrypt it to obtain pg as his guessing
+of pa and transform pg into a polynomial pg(x). Then
+the attacker can easily generate a bit string m satisfy-
+ing pg(x)|m(x), and there is the relationship h(m) = 0 if
+
+8
+pg = pa (or equivalently Xg = Xa) according to Propo-
+sitions 1.
+Since m(x) is m-order and the polynomial
+is n-order, the attacker can select no more than m/n
+polynomials and multiply them to consist his choice of
+m(x). In other words, he can guess the string Xa for no
+more than m/n times. It must be considered that the
+attacker knows pa is irreducible, so he will only choose
+those guesses that satisfy pa is irreducible. The success
+probability of this optimized strategy can be expressed
+as
+P1 = m
+n · P(Xa = Xg|pg ∈ I),
+(4)
+where P(A|B) represents the probability of event A un-
+der the condition that event B occurs, and I is the set
+of all irreducible polynomials of order n in GF(2). The
+cardinal number of I, i.e., the number of all irreducible
+polynomials of order n in GF(2), is more than 2n−1/n.
+Thus P(pg ∈ I) ≤ (2n−1/n)/2n = 1/2n.
+It is obvi-
+ous that P(Xa = Xg, pg ∈ I) = P(Xa = Xg) because if
+Xa = Xg then pg = pa ∈ I. Then we can get the upper
+bound of the success probability of this kind of attack,
+denoted as ϵ1
+LFSR.
+P1 =m
+n · P(Xa = Xg)
+P(pg ∈ I)
+(5)
+≤m
+n · 2−Hn
+1
+2n
+(6)
+=m · 2−1−Hn = ϵ1
+LFSR
+(7)
+The attacker can also guess the strings Xa and Ya to
+obtain p and s so that he can guess a hash function and
+make a successful attack for certainty. Under this cir-
+cumstance his success probability is no more than ϵ1.
+P2 =P(Xa = Xg, Ya = Yg|pg ∈ I)
+≤P(Xa = Xg|pg ∈ I)
+≤ϵ1
+LFSR
+The generalized division hash function hP (M)= m(x)·
+xn/8 mod P(x) is determined only by P. We also follow
+the terminology in our protocol that P is actually Pa
+encrypted by Xa and the hash value Dig is encrypted by
+Ya. The attacker’s strategy is to guess a string Xg so that
+he can obtain Pg and then forge a message. Analogous
+to the analysis above, the upper bound of the success
+probability is
+ϵ1
+GDH = m
+n · 2−Hn
+4
+n
+= m · 22−Hn.
+(8)
+The only difference in calculation is that there are at
+least 2n−1/(n/8) irreducible polynomials of order n/8 in
+GF(28), so P(Pg ∈ I) ≤ (2n−1/(n/8))/2n = 4/n.
+2.
+Attack of recovering keys from signature
+The attacker can try to recover the desired keys from
+the captured signature. In both kinds of hashing the hash
+value is encrypted to generate the signature. Thus the
+attacker must first guess the corresponding key strings
+(Za in LFSR-based Toeplitz hashing or Ya in general-
+ized division hashing) and then perform the recovering
+algorithm. The success probability of this strategy is no
+more than that only guessing the bit string (P(Za = Zg)
+or P(Ya = Yg)) and is obviously no more than ϵ1
+LFSR or
+ϵ1
+GDH.
+In conclusion, the optimal strategy on LFSR-based
+Toeplitz hashing and generalized division hashing is to
+guess the key string that encrypts the polynomial. We
+can quantify the upper bound of failure probability of
+authentication based on both types of hashing with im-
+perfect keys of secrecy leakage.
+ϵLFSR =m · 2−1−Hn
+(9)
+ϵGDH =m · 22−Hn
+(10)
+B.
+Security of the QDS scheme
+In a QDS scheme, the security analysis contains three
+parts, robustness, repudiation, and forgery.
+1.
+Robustness.
+The honest run abortion occurs only when Alice and
+Bob (or Charlie) share different key bits after the distri-
+bution stage. In our protocol Alice and Bob (Charlie)
+perform error correction in the distribution stage. Thus,
+they share the identical final key, and the robustness
+bound is ϵrob = 2ϵcor, where ϵcor is the failure probability
+of the error correction protocol.
+2.
+Repudiation.
+Alice successfully repudiates when Bob accepts the
+message while Charlie rejects it.
+For Alice’s repudia-
+tion attacks, Bob and Charlie are both honest and sym-
+metric and have the same new key strings.
+They will
+make the same decision for the same message and sig-
+nature. In other words, when Bob rejects (accepts) the
+message, Charlie also rejects (accepts) it. Therefore, our
+QDS protocol is naturally immune to repudiation, i.e.,
+the repudiation bound ϵrep = 0.
+3.
+Forgery.
+Bob forges successfully when Charlie accepts the forged
+message forwarded by Bob. According to our protocol,
+Charlie accepts the message if and only if Charlie ob-
+tains the same result through one-time pad decryption
+and one-time AXU hash functions. Actually this is the
+
+9
+0
+100
+200
+300
+400
+500
+600
+700
+Distance (km)
+10-6
+10-4
+10-2
+100
+102
+104
+Signature rate (tps)
+This work with TP-TFKGP
+This work with BB84-KGP
+This work with SNS-KGP
+Ref. [13]
+Ref. [21]
+Ref. [24] with SNS-KGP
+FIG. 3.
+Signature rates of our protocol with TP-TFKGP,
+BB84-KGP, SNS-KGP, and decoy state BB84-QDS [13], SNS-
+QDS [21], SNS-QDS with random pairing [24] with the mes-
+sage size of 1 Kb. In our protocol we use generalized division
+hashing in messaging stage. The repetition rate of the laser
+is 1 GHz. The distances between Alice-Bob and Alice-Charlie
+are assumed to be the same. The data size N is 1013 and the
+security bound is 10−10.
+same as an authentication scenario where Bob is the at-
+tacker attempting to forge the information sent from Al-
+ice to Charlie. Therefore, the probability of a successful
+forgery can be determined by the failure probability of
+hashing, i.e., one chooses two distinct messages with iden-
+tical hash values. For the scheme utilizing LFSR-based
+Toeplitz hash ϵfor = m · 2−1−Hn, and for generalized di-
+vision hashing ϵfor = m · 22−Hn.
+The total security bound of QDS, i.e., the max-
+imum failure probability of the protocol,
+is ϵ
+=
+max{ϵrobb, ϵrep, ϵfor}
+V.
+DISCUSSION
+For two types of hashing, there is just differences in a
+constant 2/3 between two signature rates and a constant
+8 between two security parameters. For simplicity in this
+section we only analyze our protocol with generalized di-
+vision hashing hashing.
+TABLE I. Simulation parameters. ηd and pd are the detector
+efficiency and dark count rate, respectively.
+ed is the mis-
+alignment error rate. N is the data size. α is the attenuation
+coefficient of the fiber. f is the error correction efficiency. ϵ
+is the failure probability of QDS schemes.
+ηd
+pd
+ed
+N
+α
+f
+ϵ
+70%
+10−8
+0.02
+1013
+0.165
+1.1
+10−10
+0
+100
+200
+300
+400
+500
+600
+700
+Distance (km)
+10-10
+10-5
+100
+105
+Signature rate (tps)
+This work with TP-TFKGP
+This work with BB84-KGP
+This work with SNS-KGP
+Ref. [13]
+Ref. [21]
+Ref. [24] with SNS-KGP
+FIG. 4.
+Signature rates of our protocol with TP-TFKGP,
+BB84-KGP, SNS-KGP, and decoy state BB84-QDS [13], SNS-
+QDS [21], SNS-QDS with random pairing [24] with the mes-
+sage size of 1 Mb. In our protocol we use generalized division
+hashing in messaging stage. The repetition rate of the laser
+is 1 GHz. The distances between Alice-Bob and Alice-Charlie
+are assumed to be the same. The data size N is 1013 and the
+security bound is 10−10.
+To demonstrate the superiority of our proposal, we
+first build our protocol based on BB84-KGP, SNS-KGP,
+and TP-TFKGP, and compare them with decoy-state
+BB84-QDS [13] and SNS-QDS [21] which are single-bit
+QDS protocols based on BB84-KGP and SNS-KGP. We
+also compare SNS-QDS with random pairing [24], which
+shows improvement in the signature rate of SNS-QDS
+and can be applied to other QDS. More details of cal-
+culation are shown in Appendix C. In the simulations,
+we consider two common cases where each message to
+be signed is 103 bits (1 Kb) and 106 bits (1 Mb), re-
+spectively.
+The repetition rate of the laser is 1 GHz,
+and the distances between Alice-Bob and Alice-Charlie
+are assumed to be the same.
+It should be mentioned
+that all conventional QDS protocols sign only a one-bit
+message every time. When signing multi-bit message, an
+n-bit message must be encoded into a new sequence with
+length h by inserting ‘0’ and adding ‘1’ to the original
+sequence. The signing efficiency, i.e., ˆη = n/h, is obvi-
+ously less than 1. For simplicity we use the upper bound
+ˆη = 1, i.e., h = n, in our simulation. Detailed analysis
+is shown in Appendix A 2. Other simulation parameters
+are listed in Table I.
+Figures. 3 and 4 show the simulation results of all the
+protocols mentioned above. When the message size is 1
+Kb, our protocols show a superiority on signature rate
+of over five orders of magnitude compared with conven-
+tional QDS schemes, which is a quite larger improvement
+than SNS-QDS with random pairing.
+If the message
+size becomes 1 Mb, the signature rate of conventional
+BB84-QDS, SNS-QDS and SNS-QDS with random pair-
+
+10
+ing will decrease by three orders of magnitude, whereas
+that of our protocols decreases only slightly. Thus our
+QDS scheme has a superior signature rate of over eight
+orders of magnitude. It turns out that our protocol shows
+great robustness to message size. Furthermore, our TP-
+TFQDS scheme can reach a transmission distance of 650
+km as well as a signature rate of approximately 0.01 times
+per second (tps). It is a large breakthrough in terms of
+both distance and signature rate, indicating the great po-
+tential of our protocol in the practical implementation of
+QDS. In Fig. 5, we show the performance of our proto-
+col under different data sizes 109, 1011 and 1013. Even
+with a data size as small as 109, our protocol can reach a
+transmission distance of 350 km. Our protocol is robust
+to finite-size effects.
+Compared with OTUH-QDS, our protocol does not re-
+quire perfectly secret keys, and is thus requires no privacy
+amplification step.
+In other words, our protocol only
+consumes keys with limited information leakage, which
+is a cheap and practical resource compared with perfect
+quantum keys generated by quantum secure communi-
+cation. Error correction of quantum keys can be easily
+performed by classical Cascade protocol [63, 64] where
+the bit string is first blocked and then manipulated by
+blocks. Thus the complexity of error correction increases
+linearly with the data size N and can be performed by
+stream computing. Privacy amplification, however, re-
+quires a hash matrix multiplication step where the num-
+bers of columns and rows are all proportional to N. Thus
+the computational complexity of privacy amplification is
+O(N 2). The fast Fourier transform algorithm can reduce
+the complexity to O(N log N) [67], and one can also block
+the keys before performing privacy amplification. How-
+ever, to reduce the finite-size effect, the minimum blocks
+should be large, so actual computational cost and time
+delay of privacy amplification are still very large.
+Table II lists the time consumption of error correction,
+privacy amplification and data transmission, as well as
+total postprocessing time of both protocols, at a distance
+of 400 km with data sizes 1013 and 1011. Details of sim-
+ulation are introduced in Appendix D. If N = 1011, time
+consumption of postprocessing in OTUH-QDS is 6.6 sec-
+onds, while that of our protocol is 3.62 seconds. More-
+over, when N = 1013, time for privacy amplification is
+5.71 hours (2.057 × 104s), which will introduce a quite
+long delay in experiment. Our scheme can significantly
+save computational resources and time delays for post-
+processing.
+We also compare the signature rate of our protocol
+with that of OTUH-QDS [42]. Theoretically, the two sig-
+nature rates should be the same under ideal conditions.
+In practical cases, signature rate of our protocol will be
+slightly damaged by the statistical fluctuation of privacy
+leakage. In Fig. 6, we draw the ratio of signature rate of
+our protocol based on TP-TFKGP and that of OTUH-
+QDS [42] combined with TP-TFQKD, if not considering
+the postprocessing time, with data sizes of 1013 and 1011,
+and message size of 1Kb. The result shows that when the
+0
+100
+200
+300
+400
+500
+600
+Distance (km)
+10-2
+10-1
+100
+101
+102
+103
+104
+105
+Signature rate (tps)
+N=109
+N=1011
+N=1013
+FIG. 5. Signature rates of our protocols with TP-TFKGP un-
+der different data sizes N = 109, 1011 and 1013. The message
+size is assumed to be 1 Mb and the repetition rate of the laser
+is 1 GHz. The security bound is 10−10.
+0
+100
+200
+300
+400
+500
+600
+Distance (km)
+ 20%
+30%
+40%
+50%
+60%
+70%
+80%
+90%
+100%
+Ratio of signature rate
+N=1013
+N=1011
+FIG. 6. The ratio of signature rate of our TP-TFQDS proto-
+col and that of OTUH-QDS [42] combined with TP-TFQKD,
+if not considering the postprocessing time, with data size 1013
+and 1011. The message size is 1 Kb and the repetition rate of
+the laser is 1 GHz. The ratio is more than 0.8 with transmis-
+sion distance lower than 500 km.
+transmission distance is less than 500 km, the ratio is
+more than 80%. The signature rates of the two protocols
+are comparable. In addition, if assuming the repetition
+rate of the laser as 1 GHz, time consumption for postpro-
+cessing (2.057 × 104s) is even longer than time for data
+transmission (104s) when N = 1013. The signature rate
+of OTUH-QDS will be constrained by the efficiency of
+privacy amplification. That is, in practice the signature
+rate of OTUH-QDS is lower than the simulation result,
+while our scheme can compensate for this shortcoming.
+
+11
+TABLE II. Time consumption of error correction TEC and privacy amplification TP A under different data sizes N = 1013 and
+N = 1011 when the distance is 400 km. T1 = TEC and T2 = TEC + TPA represent the postprocessing time of our proposed
+scheme and OTUH-QDS, respectively. nZ is the number of raw bits generated in TP-TFKGP; l is the length of keys after
+privacy amplification. When N = 1013, postprocessing time of OTUH-QDS is 5.85 hours, and that of the proposed protocol is
+only 8.07 minutes.
+N
+nZ
+errors
+l
+TEC
+TPA
+T1
+T2
+1013
+1.695 × 108
+300
+4.87 × 107
+8.07 min
+5.71 h
+8.07 min
+5.85 h
+1011
+1.267 × 106
+39830
+2.51 × 105
+3.62 s
+2.98 s
+3.62 s
+6.6 s
+Considering the fact that our protocol can save postpro-
+cessing time by even one hundred times, our proposal
+shows significant superiority in the practical scenario es-
+pecially when the digital signature tasks are performed
+at high frequency and the data size is large .
+VI.
+CONCLUSION
+Overall, in this paper we prove that keys with limited
+secrecy leakage and full correctness can protect authen-
+ticity and integrity of messages if combined with AXU
+hash functions.
+Furthermore, we theoretically propose
+an efficient QDS protocol utilizing imperfect quantum
+keys without privacy amplification based on the frame-
+work of OTUH-QDS, reducing computational resources
+and time delays of postprocessing without compromising
+the security. The simulation results demonstrate that our
+protocol outperforms previous single-bit QDS protocols
+in terms of both signing efficiency and distance. For ex-
+ample, for the 1 MB message to be signed, the signature
+rate of our protocol is higher than that of single-bit QDS
+protocols by eight orders of magnitude. Specifically, for
+our protocol based on TP-TFKGP, the transmission dis-
+tance can reach 650 km and still holds a signature rate of
+0.01 tps. In addition, our protocol is robust against se-
+curity levels and finite-size effects. Moreover, compared
+with OUTH-QDS, our protocol notably saves the post-
+processing time into an endurable range and thus signifi-
+cantly improves the practicality. Our scheme is a general
+framework that can be applied to any existing QKD or
+QSS protocol, and is highly compatible with future quan-
+tum networks and feasible for many applications. Addi-
+tionally, our work, only requiring keys with imperfect
+secrecy, is a new approach of quantum communication
+that is different from other quantum secret communi-
+cation protocols. For the first time we propose crypto-
+graphic applications of imperfect quantum keys without
+full secrecy, indicating that this type of keys is a resource
+with huge potential. We hope that the proposed scheme
+and the idea of utilizing imperfect quantum keys provide
+a solution for the real implementation of practical and
+commercial QDS as well as other quantum cryptography
+tasks in future quantum networks.
+ACKNOWLEDGMENTS
+This study was supported by the National Natu-
+ral Science Foundation of China (No.
+12274223), the
+Natural Science Foundation of Jiangsu Province (No.
+BK20211145), the Fundamental Research Funds for the
+Central Universities (No. 020414380182), the Key Re-
+search and Development Program of Nanjing Jiangbei
+New Aera (No. ZDYD20210101), the Program for In-
+novative Talents and Entrepreneurs in Jiangsu (JSS-
+CRC2021484), and the Program of Song Shan Lab-
+oratory (Included in the management of Major Sci-
+ence and Technology Program of Henan Province) (No.
+221100210800).
+Appendix A: single-bit QDS
+1.
+schematic of single-bit QDS
+Here, we first introduce orthogonal encoding QDS [13]
+as an example of single-bit QDS schemes. Commonly,
+all single-bit QDS protocols can be segmented into two
+stages: the distribution stage and messaging stage. The
+schematic of orthogonal encoding QDS is shown in Fig. 7.
+distribution stage:
+(i)For each possible future message m = 0 or 1, Alice
+uses the KGP to generate four different length L keys,
+A0
+B, A1
+B, A0
+C, A1
+C, where the subscript denotes the par-
+ticipant with whom she performed the KGP and the su-
+perscript denotes the future message, to be decided later
+by Alice. Bob holds the length L strings K0
+B, K1
+B and
+Charlie holds the length L strings K0
+C, K1
+C.
+(ii)For each future message, Bob and Charlie sym-
+metrize their keys by choosing half of the bit values in
+their Km
+B , Km
+C and sending them (as well as the cor-
+responding positions) to the other participant using the
+Bob-Charlie secret classical channel. They will only use
+the bits they did not forward and those received from
+the other participant. Their final symmetrized keys are
+denoted as Sm
+B and Sm
+C . Bob (and Charlie) will keep a
+record of whether an element in Sm
+B (Sm
+C ) came directly
+from Alice or whether it was forwarded to him by Charlie
+(or Bob).
+messaging stage:
+(i) To send a signed one-bit message m, Alice sends
+
+12
+Alice
+Bob
+Charlie
+KGP
+Alice
+Bob
+Charlie
+Distribution stage
+Messaging stage
+classic channel
+quantum channel
+KGP
+FIG. 7. Orthogonal encoding QDS. In the distribution stage,
+Alice–Bob and Alice–Charlie independently perform KGP to
+generate correlated bit strings with limited mismatches. Then
+Bob and Charlie symmetrize their keys by exchanging half of
+their keys. In the messaging stage Alice generates the sig-
+nature depending on the message bit, and sends the message
+and signature to Bob, who will transfer it to Charlie. Bob
+and Charlie examine their mismatch and compare it with the
+threshold to verify the signed message..
+(m, Sigm) to the desired recipient (say Bob), where
+sigm = (Am
+B , Am
+C ).
+(ii) Bob checks whether (m, Sigm) matches his Sm
+B and
+records the number of mismatches he finds. He separately
+checks the part of his key received directly from Alice and
+the part of the key received from Charlie. If there are
+fewer than sa(L/2) mismatches in both halves of the key,
+where sa < 1/2 is a small threshold determined by the
+parameters and the desired security level of the protocol,
+then Bob accepts the message.
+(iii) To forward the message to Charlie, Bob forwards
+the pair (m, Sigm) that he received from Alice.
+(iv) Charlie tests for mismatches in the same way, but
+in order to protect against repudiation by Alice he uses a
+different threshold. Charlie accepts the forwarded mes-
+sage if the number of mismatches in both halves of his
+key is below sv(L/2) where sv is another threshold, with
+0 < sa < sv < 1/2.
+KGP is actually part of QKD protocol except for the
+error correction and privacy amplification steps. In dis-
+tribution stage, Am
+X and Km
+X generated through KGP are
+correlated with limited mismatch, and Am
+X contains fewer
+mismatches with Km
+X than does any string produced by
+an eavesdropper, where X ∈ {B, C} represents Bob and
+Charlie, m is the message. After Bob and Charlie’s sym-
+metrization step, Bob holds Sm
+B and Charlie holds Sm
+C ,
+each containing half of Km
+B and Km
+C . From the perspec-
+tive of Alice, Sm
+B and Sm
+C are symmetric. Alice has no in-
+formation on whether it is Bob’s Sm
+B or Charlie’s Sm
+C that
+contains a particular element of the string (Km
+B , Km
+C ).
+This protects against repudiation. From the perspective
+of Bob, Sm
+B and Sm
+C are asymmetric. Bob has access to
+all of Km
+B and only half of Km
+C , but, even if he is dishon-
+est, he does not know the half of Km
+C that Charlie chose
+to keep. This protects against forging.
+The framework of non-orthogonal encoding QDS is
+analogous to that of orthogonal encoding. The difference
+is that it does not require the symmetrization step. How-
+ever the signer needs to send the same quantum states
+to two receivers and only detection events where the two
+receivers both have clicks are valid.
+2.
+Signing a multi-bit message using single-bit QDS
+The framework above only offer a way for signing a
+one-bit message. To sign multi-bit messages with these
+protocols, it is not sufficient to directly iterate the proto-
+col on each bits of the message, which will give a chance
+for an outside or inside attacker to perform forgery at-
+tacks [37]. In order to offer information-theoretical secu-
+rity, one must reconstruct the multi-bit message and then
+sign it bit by bit. This step will make the new message
+become longer and thus damage the efficiency. To date,
+the most efficient coding rule is given in Ref. [39], which
+can be summarized as follows.
+Suppose the signer Alice needs to sign an n-bit message
+M = m1||m2||...||mn, mi ∈ {0, 1}, i = 1, 2, ..., n. She will
+encode M into
+ˆ
+M = 11||12||...||1x+1||0||m1||m2||...||mx||0||
+||mx+1||mx+2||...||m2x||0||
+...
+||m[ n
+x ]x+1||m[ n
+x ]x+2||...||mn||0||11||12||...||1x+1,
+(A1)
+where x refers to the coding interval, [x] is the round
+down function. To conclude, the coding rule is that the
+encoder replenishes a ‘0’ in the head of M, and another
+in the tail. Then, the encoder inserts a ‘0’ every x bits
+and adds ‘1’ with a number of x+1 to both the start and
+the end.
+Denote the length of
+ˆ
+M as h.
+An iteration of
+conventional QDS protocols with h rounds on
+ˆ
+M is
+an information-theoretically secure protocol to sign the
+multi-bit message M. According to the encoding rule,
+h = n + [ n
+x] + 2x + 4. For a given n we can optimize x to
+obtain the minimal h and the maximum efficiency η = n
+h.
+It is clear that if n is large, the maximum efficiency will
+be close to 1, but will definitely be less than 1. Thus
+in our simulation in Sec. V we use the upper bound of
+deficiency, i.e., assume h = n.
+Appendix B: Generating an irreducible polynomial
+In this section we introduce ways to generate an irre-
+ducible polynomial over Gloise fields GF(2) in random,
+which is the first step in the messaging stage of our pro-
+tocol.
+Suppose p(x) is a polynomial of order n in GF(2). p(x)
+is irreducible means that no polynomials can divide it
+except the identity element ’1’ and p(x) itself. The nec-
+essary and sufficient condition for p(x) being irreducible
+
+13
+can be expressed as:
+�
+�
+�
+x2n ≡ x mod p(x)
+gcf(x2
+n
+d − x, p(x)) = 1
+(B1)
+where d is any prime factor of n, gcf(f(x), g(x)) represents
+the greatest common factor (GCF) of f(x) and g(x).
+In order to randomly generate an irreducible polyno-
+mial, one way is to generate polynomials at random and
+test for irreducibility through the condition above. How-
+ever, this is quite time consuming and requires a lot of
+random bits.
+A better solution is proposed in Ref. [65]. We can first
+have an irreducible polynomial of order n, defining the
+extension field GF(2n). Given this, we generate a random
+element in GF(2n) and then compute the minimal poly-
+nomial of this element, which will be irreducible. This
+procedure only needs n random bits and consumes less
+time. The concrete procedure is as follows.
+Denote the initial irreducible polynomial as f(x) and
+the polynomial generated by random element as g(x).
+We will calculate the sequence a0 = g0(0), a1 = g1(0), ...,
+a2n−1 = g2n−1(0), where gi(x) = gi(x) mod f(x). This
+sequence of 2n elements can fully determine the minimal
+polynomial of g(x), which can be efficiently computed
+by Berlekamp-Massey algorithm [68].
+The result, i.e.,
+the minimal polynomial of g(x), will be the irreducible
+polynomial we generate.
+If choosing generalized division hashing, we need to
+generate an irreducible polynomial over GF(2k).
+The
+procedure is the same as that described above. The only
+difference is that all the calculations need to be done
+under GF(2k).
+Appendix C: Calculation details
+1.
+TP-TFQKD and TP-TFKGP in this work
+The calculation of TP-TFKGP in this work is analo-
+gous to that in TP-TFQKD [43].
+When Alice and Bob send intensities ka and kb with
+phase difference θ, the gain corresponding to only one
+detector (L or R) clicking is
+QLθ
+kakb =ykakb
+�
+eωkakb cos θ − ykakb
+�
+,
+QRθ
+kakb =ykakb
+�
+e−ωkakb cos θ − ykakb
+�
+.
+(C1)
+where ykakb = e
+−(ηaka+ηbkb)
+2
+(1 − pd), ωkakb = √ηakaηbkb.
+The
+overall
+gain
+can
+be
+expressed
+as
+Qkakb
+=
+1/2π
+� 2π
+0 (QLθ
+kakb + QRθ
+kakb)dθ = 2ykakb[I0(ωkakb) − ykakb],
+where I0(x) refers to the zero-order modified Bessel func-
+tions of the first kind.
+The total number for {ka, kb} is
+xkakb = NpkapkbQkakb.
+(C2)
+The
+valid
+post-matching
+events
+on
+the
+basis
+of
+Z
+can
+be
+divided
+into
+two
+types:
+correct
+events {µaoa, obµb}, {oaµa, µbob}, and incorrect events
+{µaoa, µbob}, {oaµa, obµb}. The corresponding numbers
+are denoted as nz
+C and nz
+E, respectively, which can be
+written as
+nz
+C = xmin
+xoaµb
+x0
+xµaob
+x1
+= xoaµbxµaob
+xmax
+,
+and
+nz
+E = xmin
+xoaob
+x0
+xµaµb
+x1
+= xoaobxµaµb
+xmax
+,
+where x0 = xoaµb + xoaob, x1 = xµaob + xµaµb, xmin =
+min{x0, x1}, and xmax = max{x0, x1}. sz
+11 corresponds
+to the number of successful detection events, where Alice
+and Bob emit a single photon in different time bins in
+the Z basis. The overall number of events in the Z basis
+is
+nz =nz
+C + nz
+E.
+(C3)
+Considering the misalignment error ez
+d, the number of bit
+errors in the Z basis is mz = (1 − ez
+d)nz
+E + ez
+dnz
+C. Thus,
+the bit error rate in the Z basis is
+Ez =mz
+nz .
+(C4)
+The overall number of “effective” events in the X basis
+is
+nx = 1
+π
+� δ
+0
+xθ
+νaνbdθ
+= Npνapνb
+π
+� σ+δ
+σ
+yνaνb(eωνaνb cos θ
++ e−ωνaνb cos θ − 2yνaνb)dθ.
+(C5)
+For simplicity, we only consider the case in which all
+matched events satisfy θi − θj = 0. In this case, when
+ri
+a ⊕ rj
+a ⊕ ri
+b ⊕ rj
+b = 0 (1), the {νi
+aνj
+a, νi
+bνj
+b} event is con-
+sidered to be an error event when different detectors (the
+same detector) click at time bins i and j.
+The overall error count in the X basis can be given as
+mx = 1
+π
+� σ+δ
+σ
+xθ
+νaνbpEdθ
+= 2Npνapνb
+π
+� σ+δ
+σ
+yνaνb×
+�
+(1 − yνaνb)2
+eωνaνb cos θ + e−ωνaνb cos θ − 2yνaνb
+− 1
+�
+dθ,
+(C6)
+where pE =
+2qLθ
+νaνbqRθ
+νaνb
+qθ
+νaνbqθ
+νaνb .
+We can then calculate the parameters in Eq. (1) to
+estimate the key rate and the information leaked after
+the distribution stage. In the following description, let
+x∗ denote the expected value of x. We denote the number
+
+14
+of {ka, kb} as xkakb. We denote the number and error
+number of events {ki
+akj
+a, ki
+bkj
+b} after post-matching as
+nkiakj
+a, ki
+bkj
+b and mkiakj
+a, ki
+bkj
+b, respectively. For simplicity,
+we abbreviate ki
+akj
+a, ki
+akj
+a as 2ka, 2kb when ki
+a = kj
+a and
+ki
+b = kj
+b.
+(1) sz
+11.
+sz
+11 corresponds to the number of successful detec-
+tion events, where Alice and Bob emit a single photon
+in different time bins in the Z basis.
+Define z10 (z01)
+as the number of events in which Alice (Bob) emits a
+single photon and Bob (Alice) emits a vacuum state
+in an {µa, ob} ({oa, µb}) event.
+The lower bounds of
+their expected values are z∗
+10 = Npµapobµae−µay∗
+10 and
+z∗
+01 = Npoapµbµbe−µby∗
+01, respectively, where y∗
+10 and y∗
+01
+are the corresponding yields. These can be estimated us-
+ing the decoy-state method
+y∗
+01 ≥
+µb
+N(µbνb − ν2
+b )
+�eνbx∗
+oaνb
+poapνb
+−ν2
+b
+µ2
+b
+eµbx∗
+ˆoaµb
+pˆoapµb
+− µ2
+b − ν2
+b
+µ2
+b
+xd∗
+oo
+pdoaob
+�
+,
+(C7)
+y∗
+10 ≥
+µa
+N(µaνa − ν2a)
+�eνax∗
+νaob
+pνapob
+− ν2
+a
+µ2a
+eµax∗
+µaˆob
+pµapˆob
+− µ2
+a − ν2
+a
+µ2a
+xd∗
+oo
+pdoaob
+�
+,
+(C8)
+where xd
+oo = xˆoaˆob +xˆoaob +xoaˆob represents the number
+of events where at least one user chooses the declare-
+vacuum state and pd
+oo = pˆoapˆob + pˆoapob + poapˆob refers
+to the corresponding probability. Thus, the lower bound
+of sz∗
+11 is given by
+sz∗
+11 = z∗
+10z∗
+01
+xmax
+.
+(C9)
+(2) sz
+0µb. sz
+0µb represents the number of events in the Z
+basis, Alice emits a zero-photon state in the two matched
+time bins, and the total intensity of Bob’s pulses is µb.
+We define z00 (z0µb) as the number of detection events
+where the state sent by Alice collapses to the vacuum
+state in the {µa, ob} ({µa, µb}) event. The lower bound
+of the expected values is z∗
+00 = pµapobe−µaxd∗
+oo/pd
+oaob
+and z∗
+0µb = pµapµbe−µax∗
+oaµb/poapµb, respectively. Here,
+we employ the relationship between the expected value
+x∗
+oaµb = poax∗
+ˆoaµb/pˆoa, and x∗
+oaob = poapobxd∗
+oo/pd
+oo. The
+lower bound of sz∗
+0µb can be written as
+sz∗
+0µb = x∗
+oaµbz∗
+00
+xmax
++ x∗
+oaobz∗
+0µb
+xmax
+(C10)
+(3) sx
+11. We define the phase difference between Al-
+ice and Bob as θ = θa − θb + φab. All valid events in the
+X basis can be grouped according to the phase difference
+θ (∈ {−δ, δ}∪{π−δ, π+δ}), and the corresponding num-
+ber in the {ka, kb} event is denoted as xθ
+kakb. In the post-
+matching step, two time bins are matched if they have the
+same phase difference θ. Suppose the global phase dif-
+ference θ is a randomly and uniformly distributed value,
+and considering the angle of misalignment in the X ba-
+sis σ, the expected number of single-photon pairs can be
+given by
+sx∗
+11 = 1
+π
+� σ+δ
+σ
+xθ
+νaνb × 2
+νbe−(νa+νb)y∗
+01
+qθνaνb
+νae−(νa+νb)y∗
+10
+qθνaνb
+dθ
+= Npνapνb
+π
+� σ+δ
+σ
+2νaνbe−2(νa+νb)y∗
+01y∗
+10
+qθνaνb
+,
+(C11)
+where qθ
+νaνb is the gain when Alice chooses intensity νa,
+and Bob chooses the intensity νb with phase difference θ
+and xθ
+νaνb = Npνapνbqθ
+νaνb.
+(4) ex
+11. For single-photon pairs, the expected value of
+the phase error rate in the Z basis is equal to the expected
+value of the bit error rate in the X basis. Therefore, we
+first calculate the number of errors of the single-photon
+pairs in the X basis tx
+11. The upper bound of tx
+11 can be
+expressed as
+t
+x
+11 ≤mx − (mνa0,νb0 + m0νa,0νb) + m00,00,
+(C12)
+where (mνa0,νb0 (m0νa,0νb) is the error count when the
+states sent by Alice and Bob in time bin i (j) both col-
+lapse to the vacuum state in events {2νa, 2νb}, and m00,00
+corresponds to the event where the states sent by Al-
+ice and Bob both collapse to vacuum states in events
+{2νa, 2νb}.
+The expected counts (nνa0,νb0 + n0νa,0νb)∗
+and n∗
+00,00 can be expressed as
+(nνa0,νb0 + n0νa,0νb)∗ = 2
+π
+� σ+δ
+σ
+xθ
+νaνb
+e−(νa+νb)q∗
+00
+qθνaνb
+dθ
+=
+δNpνapνbe−(νa+νb)q∗
+00
+π
+(C13)
+and
+n∗
+00,00 = 1
+π
+� σ+δ
+σ
+xθ
+νaνb
+�e−(νa+νb)q∗
+00
+qθνaνb
+�2
+dθ
+=Npνapνb
+π
+� σ+δ
+σ
+e−2(νa+νb)(q∗
+00)2
+qθνaνb
+dθ
+(C14)
+respectively. Here q∗
+00 = xd∗
+oaob/(Npd
+oo). Using the fact
+that the error rate of the vacuum state is always 1/2,
+we have (mνa0,νb0 + m0νa,0νb)∗ = 1
+2(nνa0,νb0 + n0νa,0νb)∗
+and m∗
+00,00 = 1
+2n∗
+00,00. Hence the upper bound of the bit
+error rate in the X basis can be given by
+ex
+11 = tx
+11/sx
+11.
+(C15)
+(5) φ
+z
+11. For a failure probability ε, the upper bound
+of the phase error rate φz
+11 can be obtained by using the
+random sampling without replacement [69]
+φ
+z
+11 ≤ex
+11 + γU (sz
+11, sx
+11, ex
+11, ϵ) ,
+(C16)
+
+15
+where
+γU(n, k, λ, ϵ) =
+(1−2λ)AG
+n+k
++
+�
+A2G2
+(n+k)2 + 4λ(1 − λ)G
+2 + 2
+A2G
+(n+k)2
+,
+(C17)
+with A = max{n, k} and G = n+k
+nk ln
+n+k
+2πnkλ(1−λ)ϵ2 .
+(6) szn
+11 , szn
+0µb and φ
+zn
+11 .
+Finally we can estimate the
+parameters in Eq. (1), i.e., the lower bound of vacuum
+events and single-photon pairs in a selected key group
+sz
+11L and sz
+0µbL, and the upper bound of the phase er-
+ror rate of the n-bit group φ
+z
+11U. They can be obtained
+from the parameters above by using the random sampling
+without replacement.
+szn
+11 ≥n
+�
+sz
+11/nz − γU (n, nz − n, sz
+11/nz, ϵ)
+�
+,
+szn
+0µb ≥n
+�
+sz
+0µb/nz − γU �
+n, nz − n, sz
+0µb/nz, ϵ
+��
+,
+φ
+zn
+11 ≤φ
+z
+11 + γU �
+szn
+11 , sz
+11 − szn
+11 , φ
+z
+11, ϵ
+�
+.
+(C18)
+(7)lkey. We can also get the length of final keys of TP-
+TFQKD, which can be used to simulate the performance
+of OTUH-QDS in Fig. 4.
+lkey =sz
+0µb + sz
+11[1 − H(φ
+z
+11)] − nzfH(Ez)
+− log2
+2
+ϵcor
+− 2 log2
+1
+2ϵP A
+,
+(C19)
+where ϵP A is the failure probability of privacy amplifica-
+tion.
+2.
+BB84-KGP in BB84-QDS and this work
+Both BB84-QDS and this work utilize decoy-state
+BB84-KGP to generate correlated bit strings. Accord-
+ing to Ref. [70] we can estimate the number of vacuum
+events and single-photon events under X basis,
+sX,0 ≥ τ0
+µ2n−
+X,µ3 − µ3n+
+X,µ2
+µ2 − µ3
+,
+(C20)
+sX,1 ≥
+τ1µ1
+�
+n−
+X,µ2 − n+
+X,µ3 − µ2
+2−µ2
+3
+µ2
+1
+(n+
+X,µ1 − sX,0
+τ0 )
+�
+µ1(µ2 − µ3) − µ2
+2 + µ2
+3
+.
+(C21)
+where τn := �
+k∈K e−kknpk/n! is the probability that
+Alice sends a n-photon state, and
+n±
+X,k := ek
+pk
+�
+nX,k ±
+�
+nX
+2 log 21
+εsec
+�
+, ∀ k ∈ K.
+We can also calculate the number of vacuum events,
+sZ,0, and the number of single-photon events, sZ,1, for
+Z = ∪k∈KZk, i.e., by using Eqs. (C20) and (C21) with
+statistics from the basis Z. Then we can obtain the phase
+error rate of the single-photon events in the X basis by
+φX,1 := cX,1
+sX,1
+≤ vZ,1
+sZ,1
++ γU
+�
+sZ,1, sX,1, vZ,1
+sZ,1
+, εsec
+�
+,
+(C22)
+where
+vZ,1 ≤ τ1
+m+
+Z,µ2 − m−
+Z,µ3
+µ2 − µ3
+,
+m±
+Z,k := ek
+pk
+�
+mZ,k ±
+�
+mZ
+2 log 21
+εsec
+�
+, ∀ k ∈ K,
+γU(n, k, λ, ϵ) =
+(1−2λ)AG
+n+k
++
+�
+A2G2
+(n+k)2 + 4λ(1 − λ)G
+2 + 2
+A2G
+(n+k)2
+The total number of events under Z basis is nZ =
+�
+k∈K nZ,k and the number of error events is mZ =
+�
+k∈K mZ,k.
+In BB84-QDS, the unknown information to the at-
+tacker is given by
+H = sX,0 + sX,1(1 − h(φX,1)).
+(C23)
+In our protocol based on BB84-KGP, we need to esti-
+mate parameters in a selected n-bit group, i.e., the lower
+bound of number of vacuum events and single-photon
+events under X basis sn
+X,0 and sn
+X,1, and the upper bound
+of the phase error rate of the single-photon events in the
+X basis φ
+n
+X,1.
+sn
+X,0 ≥n
+�
+sX,0/nZ − γU �
+n, nZ − n, sX,0/nZ, ϵ
+��
+, (C24)
+sn
+X,1 ≥n
+�
+sX,1/nZ − γU �
+n, nZ − n, sX,1/nZ, ϵ
+��
+, (C25)
+φn
+X,1 ≤φX,1 + γU �
+sn
+X,1, sX,1 − sn
+X,1, φX,1, ϵ
+�
+.
+(C26)
+Finally we can obtain
+H = sn
+X,0 + sn
+X,1(1 − h(φ
+n
+X,1)) − λEC,
+(C27)
+where λEC = nh(mZ/nZ).
+3.
+SNS-KGP and SNS-QDS with random pairing
+We first follow the calculation in Ref. [71].
+Alice and Bob obtain Njk(jk = {00, 01, 02, 10, 20}) in-
+stances when Alice sends intensity j and Bob sends state
+k. Here ’1’ and ’2’ represent the two intensities used in
+the KGP. After the sifted step, Alice and Bob obtain
+
+16
+njk one-detector heralded events. We denote the count-
+ing rate of source jk as Sjk = njk/Njk. With all these
+definitions, we have
+N00 =[(1 − pz)2p2
+0 + 2(1 − pz)pzp0pz0]N,
+N01 =N10 = [(1 − pz)2p0p1 + (1 − pz)pzpz0p1]N,
+N02 =N20 = [(1 − pz)2(1 − p0 − p1)p0
++ (1 − pz)pzpz0(1 − p0 − p1)]N.
+(C28)
+In addition, we need to define two new subsets of X1
+windows, C∆+ and C∆−, to estimate the upper bound of
+eph
+1 . The number of instances in C∆± is
+N∆± = ∆
+2π (1 − pz)2p2
+1N.
+(C29)
+We denote the number of effective events of right de-
+tectors responding from C∆+ as nR
+∆+, and the number
+of effective events of left detectors responding from C∆−
+as nL
+∆−. And we obtain the counting error rate of C∆±,
+T∆ =
+nR
+∆++nL
+∆−
+2N∆±
+.
+If we denote the expected value of the counting rate of
+untagged photons as sZ∗
+1 , the lower bound of sZ∗
+1
+is
+sZ∗
+1
+≥sZ∗
+1
+=
+1
+2µ1µ2(µ2 − µ1)[µ2
+2eµ1(S∗
+01 + S∗
+10)
+− µ2
+1eµ2(S
+∗
+02 + S
+∗
+20) − 2(µ2
+2 − µ2
+1)S
+∗
+00],
+(C30)
+where S∗
+jk is the expected value of Sjk, and S
+∗
+jk and S∗
+jk
+are the upper bound and lower bound of S∗
+jk when we
+estimate the expected value from its observed value.
+The expected value of the phase-flip error rate of the
+untagged photons satisfies
+eph∗
+1
+≤ eph∗
+1
+= T
+∗
+∆ − 1
+2e−2µ1S∗
+00
+2µ1e−2µ1sZ∗
+1
+.
+(C31)
+Here we use the fact that the error rate of vacuum state
+is always 1
+2.
+If the total transmittance of the experimental setups
+is η, then we have
+n00 = 2pd(1 − pd)N00,
+n01 = n10 = 2[(1 − pd)eηµ1/2 − (1 − pd)2e−ηµ1]N01,
+n02 = n20 = 2[(1 − pd)eηµ2/2 − (1 − pd)2e−ηµ2]N02,
+nt = nsignal + nerror,
+Ez = nerror
+nt
+,
+nR
+∆+ = nL
+∆− = [TX(1 − 2ed) + edSX]N∆±,
+where N00, N01, N10, N02, N20, N∆±
+are defined in
+Eqs. (C28) and (C29), and
+nsignal =4Np2
+zpz0(1 − pz0)[(1 − pd)e−ηµz/2
+− (1 − pd)2e−2ηµz],
+nerror =2Np2
+z(1 − pz0)2[(1 − pd)e−ηµzI0(ηµz)
+− (1 − pd)2e−2ηµz] + 2Np2
+zp2
+z0pd(1 − pd),
+TX = 1
+∆
+�
+∆
+2
+− ∆
+2
+(1 − pd)e−2ηµ1 cos2 δ
+2 dδ
+− (1 − pd)2e−2ηµ1,
+SX = 1
+∆
+�
+∆
+2
+− ∆
+2
+(1 − pd)e−2ηµ1 sin2 δ
+2 dδ
+− (1 − pd)2e−2ηµ1 + TX,
+where I0(x) is the 0-order hyperbolic Bessel functions of
+the first kind.
+In SNS-QDS, the unknown information to the attacker
+is given by
+H = sZ∗
+1 (1 − h(eph∗
+1
+)).
+(C32)
+In our protocol based on SNS-KGP, there is
+sZ∗
+1n =n
+�
+sZ∗
+1
+− γU �
+n, nZ − n, sZ∗
+1 /nZ, ϵ
+��
+,
+eph∗
+1n =n
+�
+eph∗
+1
++ γU �
+sZ∗
+1n , sZ∗
+1
+− sZ∗
+1n eph∗
+1
+, ϵ
+��
+.
+(C33)
+and
+H = sZ∗
+1n (1 − h(eph∗
+1n )) − λEC,
+(C34)
+where λEC = nh(Ez).
+In SNS-QDS with random pairing, we follow the cal-
+culation in Ref. [24]. After random pairing there are two
+different phase error rates
+˜e′ph
+1
+=
+(eph
+1 )2
+(eph
+1 )2 + (1 − eph
+1 )2 ,
+(C35)
+˜e′ph
+2
+= 1
+2,
+(C36)
+and a new bit flip error rate
+E′ = 2Ez(1 − Ez).
+(C37)
+The proportion of untagged bits after random pairing is
+∆′
+un = ∆2
+un + 2∆un(1 − ∆un),
+(C38)
+where ∆un = Np2
+zpz0(1 − pz0)sZ
+1 /nt is the proportion
+of untagged bits before random pairing. The unknown
+information to the attacker is given by
+H =∆′
+un − ∆2
+un[p1H(˜e′ph
+1 ) + (1 − p1)H(˜e′ph
+2 )]
+− 2∆un(1 − ∆un)H(eph
+1 ),
+(C39)
+where p1 = (eph
+1 )2 + (1 − eph
+1 )2.
+
+17
+Appendix D: Error correction and privacy
+amplification
+In this section we introduce our simulation of error
+correction and privacy amplification in Table II. We use
+the simulated data of TP-TFKGP at the distance of 400
+km, which can be calculated by Eqs. C3, C4, C19 in
+Appendix C. We implement our simulation on a desktop
+computer with an Intel i5-10400 CPU (with RAM of 8
+GB).
+We use improved Cascade protocol to perform er-
+ror correction to correct 300 (or 39830) errors among
+1.267 × 106 (or 1.695 × 108) bits. The detailed procedure
+of improved Cascade protocol can be seen in Ref. [64],
+where users first block their keys, and then perform bi-
+nary process on each block to correct the errors and start
+trace-back section to check the error, until there are no
+errors in each block.
+The results show that time con-
+sumption is 3.62 and 930.98 seconds with data sizes 1011
+and 1013, respectively.
+In privacy amplification step, Alice chooses a random
+universal2 hash function and performs it on the nZ-bit
+keys after error correction to obtain l-bit final keys. The
+choice of function is communicated to Bob, who also uses
+it to get his l-bit final keys. In the simulation we utilize
+a random Toeplitz matrix as the universal2 hash func-
+tion. When the data size is 1013, the matrix is too large
+that it exceeds the storage of our computer. Thus in the
+algorithm we block the matrix into 10 × 10 (100) subma-
+trices with the same size to accomplish the calculation.
+For hash manipulations of every submatrix we follow the
+procedure in Ref. [72], where fast Fourier transform is
+used to speed up calculation time. In the simulation it
+takes 2.98 and 2.057 × 104 seconds with data sizes 1011
+and 1013, respectively.
+[1] W. Diffie and M. Hellman, New directions in cryptogra-
+phy, IEEE trans. Inf. Theory 22, 644 (1976).
+[2] R. A. DeMillo, Foundations of secure computation, Tech.
+Rep. (Georgia Institute of Technology, 1978).
+[3] R. L. Rivest, A. Shamir, and L. Adleman, A method
+for obtaining digital signatures and public-key cryptosys-
+tems, Commun. ACM 21 (1978).
+[4] T. Elgamal, A public key cryptosystem and a signature
+scheme based on discrete logarithms, IEEE trans. Inf.
+Theory 31, 469 (1985).
+[5] M. Stevens, E. Bursztein, P. Karpman, A. Albertini,
+and Y. Markov, The first collision for full sha-1, in Ad-
+vances in Cryptology – CRYPTO 2017, LNCS, Vol. 10401
+(Springer, 2017) pp. 570–596.
+[6] F. Boudot,
+P. Gaudry,
+A. Guillevic,
+N. Heninger,
+E. Thom´e, and P. Zimmermann, Comparing the diffi-
+culty of factorization and discrete logarithm: a 240-digit
+experiment, in Annual International Cryptology Confer-
+ence (Springer, 2020) pp. 62–91.
+[7] P. W. Shor, Algorithms for quantum computation: dis-
+crete logarithms and factoring, in Proceedings 35th an-
+nual symposium on foundations of computer science
+(IEEE, 1994) pp. 124–134.
+[8] D. Gottesman and I. Chuang, Quantum digital signa-
+tures, arXiv preprint quant-ph/0105032 (2001).
+[9] P. J. Clarke, R. J. Collins, V. Dunjko, E. Andersson,
+J. Jeffers, and G. S. Buller, Experimental demonstration
+of quantum digital signatures using phase-encoded coher-
+ent states of light, Nat. Commun. 3, 1174 (2012).
+[10] R. J. Collins, R. J. Donaldson, V. Dunjko, P. Wallden,
+P. J. Clarke, E. Andersson, J. Jeffers, and G. S. Buller,
+Realization of quantum digital signatures without the re-
+quirement of quantum memory, Phys. Rev. Lett. 113,
+040502 (2014).
+[11] V. Dunjko, P. Wallden, and E. Andersson, Quantum
+digital signatures without quantum memory, Phys. Rev.
+Lett. 112, 040502 (2014).
+[12] H.-L. Yin, Y. Fu, and Z.-B. Chen, Practical quantum
+digital signature, Phys. Rev. A 93, 032316 (2016).
+[13] R. Amiri, P. Wallden, A. Kent, and E. Andersson, Se-
+cure quantum signatures using insecure quantum chan-
+nels, Phys. Rev. A 93, 032325 (2016).
+[14] I. V. Puthoor, R. Amiri, P. Wallden, M. Curty, and
+E. Andersson, Measurement-device-independent quan-
+tum digital signatures, Phys. Rev. A 94, 022328 (2016).
+[15] T. Shang, Q. Lei, and J. Liu, Quantum random oracle
+model for quantum digital signature, Phys. Rev. A 94,
+042314 (2016).
+[16] Y.-G. Yang, Z.-C. Liu, J. Li, X.-B. Chen, H.-J. Zuo, Y.-H.
+Zhou, and W.-M. Shi, Theoretically extensible quantum
+digital signature with starlike cluster states, Quantum
+Inf. Process. 16, 12 (2017).
+[17] M. Thornton, H. Scott, C. Croal, and N. Korolkova,
+Continuous-variable quantum digital signatures over in-
+secure channels, Phys. Rev. A 99, 032341 (2019).
+[18] W. Qu, Y. Zhang, H. Liu, T. Dou, J. Wang, Z. Li,
+S. Yang, and H. Ma, Multi-party ring quantum digital
+signatures, J. Opt. Soc. Am. B 36, 1335 (2019).
+[19] C.-M. Zhang, Y. Zhu, J.-J. Chen, and Q. Wang, Practical
+quantum digital signature with configurable decoy states,
+Quantum Inf. Process. 19, 151 (2020).
+[20] Y.-S. Lu, X.-Y. Cao, C.-X. Weng, J. Gu, Y.-M. Xie, M.-
+G. Zhou, H.-L. Yin, and Z.-B. Chen, Efficient quantum
+digital signatures without symmetrization step, Opt. Ex-
+press 29, 10162 (2021).
+[21] C.-H. Zhang, X. Zhou, C.-M. Zhang, J. Li, and Q. Wang,
+Twin-field quantum digital signatures, Opt. Lett. 46,
+3757 (2021).
+[22] W. Zhao, R. Shi, J. Shi, P. Huang, Y. Guo, and D. Huang,
+Multibit quantum digital signature with continuous vari-
+ables using basis encoding over insecure channels, Phys.
+Rev. A 103, 012410 (2021).
+[23] C.-X. Weng, Y.-S. Lu, R.-Q. Gao, Y.-M. Xie, J. Gu, C.-L.
+Li, B.-H. Li, H.-L. Yin, and Z.-B. Chen, Secure and prac-
+tical multiparty quantum digital signatures, Opt. Express
+29, 27661 (2021).
+[24] J.-Q. Qin, C. Jiang, Y.-L. Yu, and X.-B. Wang, Quan-
+tum digital signatures with random pairing, Phys. Rev.
+Applied 17, 044047 (2022).
+[25] M.-H. Zhang, J.-H. Xie, J.-Y. Wu, L.-Y. Yue, C. He,
+
+18
+Z.-W. Cao, and J.-Y. Peng, Practical long-distance twin-
+field quantum digital signatures, Quantum Inf. Process.
+21, 150 (2022).
+[26] H.-L. Yin, Y. Fu, H. Liu, Q.-J. Tang, J. Wang, L.-X.
+You, W.-J. Zhang, S.-J. Chen, Z. Wang, Q. Zhang, T.-Y.
+Chen, Z.-B. Chen, and J.-W. Pan, Experimental quan-
+tum digital signature over 102 km, Phys. Rev. A 95,
+032334 (2017).
+[27] R. J. Collins,
+R. Amiri,
+M. Fujiwara,
+T. Honjo,
+K. Shimizu, K. Tamaki, M. Takeoka, M. Sasaki, E. An-
+dersson, and G. S. Buller, Experimental demonstration
+of quantum digital signatures over 43 db channel loss us-
+ing differential phase shift quantum key distribution, Sci.
+Rep. 7, 3235 (2017).
+[28] H.-L. Yin, W.-L. Wang, Y.-L. Tang, Q. Zhao, H. Liu,
+X.-X. Sun, W.-J. Zhang, H. Li, I. V. Puthoor, L.-X.
+You, E. Andersson, Z. Wang, Y. Liu, X. Jiang, X. Ma,
+Q. Zhang, M. Curty, T.-Y. Chen, and J.-W. Pan, Exper-
+imental measurement-device-independent quantum digi-
+tal signatures over a metropolitan network, Phys. Rev.
+A 95, 042338 (2017).
+[29] G. Roberts, M. Lucamarini, Z. Yuan, J. Dynes, L. Co-
+mandar, A. Sharpe, A. Shields, M. Curty, I. Puthoor,
+and E. Andersson, Experimental measurement-device-
+independent quantum digital signatures, Nat. Commun.
+8, 1098 (2017).
+[30] C.-H. Zhang, X.-Y. Zhou, H.-J. Ding, C.-M. Zhang, G.-C.
+Guo, and Q. Wang, Proof-of-principle demonstration of
+passive decoy-state quantum digital signatures over 200
+km, Phys. Rev. Applied 10, 034033 (2018).
+[31] X.-B. An, H. Zhang, C.-M. Zhang, W. Chen, S. Wang,
+Z.-Q. Yin, Q. Wang, D.-Y. He, P.-L. Hao, S.-F. Liu, X.-
+Y. Zhou, G.-C. Guo, and Z.-F. Han, Practical quantum
+digital signature with a gigahertz bb84 quantum key dis-
+tribution system, Opt. Lett. 44, 139 (2019).
+[32] H.-J. Ding, J.-J. Chen, L. Ji, X.-Y. Zhou, C.-H. Zhang,
+C.-M. Zhang, and Q. Wang, 280-km experimental demon-
+stration of a quantum digital signature with one decoy
+state, Opt. Lett. 45, 1711 (2020).
+[33] S. Richter, M. Thornton, I. Khan, H. Scott, K. Jaksch,
+U. Vogl, B. Stiller, G. Leuchs, C. Marquardt, and N. Ko-
+rolkova, Agile and versatile quantum communication:
+Signatures and secrets, Phys. Rev. X 11, 011038 (2021).
+[34] M.-C. Roehsner, J. A. Kettlewell, J. Fitzsimons, and
+P. Walther, Probabilistic one-time programs using quan-
+tum entanglement, npj Quantum Inf. 7, 98 (2021).
+[35] M.-C. Roehsner, J. A. Kettlewell, T. B. Batalh˜ao, J. F.
+Fitzsimons, and P. Walther, Quantum advantage for
+probabilistic one-time programs, Nat. Commun. 9, 5225
+(2018).
+[36] Y.
+Pelet,
+I.
+V.
+Puthoor,
+N.
+Venkatachalam,
+S. Wengerovsky, M. Loncaric, S. P. Neumann, B. Liu,
+Z. Samec, M. Stipˇcevi´c, R. Ursin, et al., Unconditionally
+secure digital signatures implemented in an 8-user
+quantum network, New J. Phys. (2022).
+[37] T.-Y. Wang, X.-Q. Cai, Y.-L. Ren, and R.-L. Zhang, Se-
+curity of quantum digital signatures for classical mes-
+sages, Sci. Rep. 5, 9321 (2015).
+[38] T.-Y. Wang, J.-F. Ma, and X.-Q. Cai, The postprocessing
+of quantum digital signatures, Quantum Inf. Process. 16,
+19 (2017).
+[39] H. Zhang, X.-B. An, C.-H. Zhang, C.-M. Zhang, and
+Q. Wang, High-efficiency quantum digital signature
+scheme for signing long messages, Quantum Inf. Process.
+18, 3 (2019).
+[40] X.-Q. Cai, T.-Y. Wang, C.-Y. Wei, and F. Gao, Crypt-
+analysis of multiparty quantum digital signatures, Quan-
+tum Inf. Process. 18, 252 (2019).
+[41] X. Yao, X. Liu, R. Xue, H. Wang, H. Li, Z. Wang, L. You,
+Y. Huang, and W. Zhang, Multi-bit quantum digital sig-
+nature based on quantum temporal ghost imaging, arXiv
+preprint arXiv:1901.03004 (2019).
+[42] H.-L. Yin, Y. Fu, C.-L. Li, C.-X. Weng, B.-H. Li,
+J. Gu, Y.-S. Lu, S. Huang, and Z.-B. Chen, Experimen-
+tal quantum secure network with digital signatures and
+encryption, Natl. Sci. Rev. 10.1093/nsr/nwac228 (2022),
+nwac228.
+[43] Y.-M. Xie, C.-X. Weng, Y.-S. Lu, Y. Fu, Y. Wang,
+H.-L. Yin, and Z.-B. Chen, Scalable high-rate twin-
+field quantum key distribution networks without con-
+straint of probability and intensity, arXiv preprint
+arXiv:2112.11165 (2021).
+[44] C. H. Bennett and G. Brassard, Quantum cryptography:
+Public key distribution and coin tossing, Theor. Comput.
+Sci. 560, 7 (2014).
+[45] H.-K. Lo, M. Curty, and B. Qi, Measurement-device-
+independent quantum key distribution, Phy. Rev. Lett.
+108, 130503 (2012).
+[46] S. L. Braunstein and S. Pirandola, Side-channel-free
+quantum key distribution, Phy. Rev. Lett. 108, 130502
+(2012).
+[47] Y.-H. Zhou, Z.-W. Yu, and X.-B. Wang, Making the
+decoy-state measurement-device-independent quantum
+key distribution practically useful, Phys. Rev. A 93,
+042324 (2016).
+[48] M. Lucamarini, Z. L. Yuan, J. F. Dynes, and A. J.
+Shields, Overcoming the rate–distance limit of quantum
+key distribution without quantum repeaters, Nature 557,
+400 (2018).
+[49] X. Ma, P. Zeng, and H. Zhou, Phase-matching quantum
+key distribution, Phys. Rev. X 8, 031043 (2018).
+[50] X.-B. Wang, Z.-W. Yu, and X.-L. Hu, Twin-field quan-
+tum key distribution with large misalignment error, Phy.
+Rev. A 98, 062323 (2018).
+[51] W.-B. Liu, C.-L. Li, Y.-M. Xie, C.-X. Weng, J. Gu,
+X.-Y. Cao, Y.-S. Lu, B.-H. Li, H.-L. Yin, and Z.-B.
+Chen, Homodyne detection quadrature phase shift keying
+continuous-variable quantum key distribution with high
+excess noise tolerance, PRX Quantum 2, 040334 (2021).
+[52] Y.-M. Xie, Y.-S. Lu, C.-X. Weng, X.-Y. Cao, Z.-Y. Jia,
+Y. Bao, Y. Wang, Y. Fu, H.-L. Yin, and Z.-B. Chen,
+Breaking the rate-loss bound of quantum key distribution
+with asynchronous two-photon interference, PRX Quan-
+tum 3, 020315 (2022).
+[53] P. Zeng, H. Zhou, W. Wu, and X. Ma, Mode-pairing
+quantum key distribution, Nat. Commun. 13, 3903
+(2022).
+[54] J. Gu, X.-Y. Cao, Y. Fu, Z.-W. He, Z.-J. Yin, H.-L.
+Yin, and Z.-B. Chen, Experimental measurement-device-
+independent type quantum key distribution with flawed
+and correlated sources, Sci. Bull. 67, 2167 (2022).
+[55] Y.
+Fu,
+H.-L.
+Yin,
+T.-Y.
+Chen,
+and
+Z.-B.
+Chen,
+Long-distance measurement-device-independent multi-
+party quantum communication, Physical review letters
+114, 090501 (2015).
+[56] J. Gu, X.-Y. Cao, H.-L. Yin, and Z.-B. Chen, Differential
+phase shift quantum secret sharing using a twin field,
+Opt. Express 29, 9165 (2021).
+
+19
+[57] Z. Li, X.-Y. Cao, C.-L. Li, C.-X. Weng, J. Gu, H.-L. Yin,
+and Z.-B. Chen, Finite-key analysis for quantum confer-
+ence key agreement with asymmetric channels, Quantum
+Science and Technology 6, 045019 (2021).
+[58] A. J. Menezes, P. C. Van Oorschot, and S. A. Vanstone,
+Handbook of applied cryptography (CRC press, 2018).
+[59] H. Krawczyk, Lfsr-based hashing and authentication, in
+Annual International Cryptology Conference (1994) pp.
+129–139.
+[60] J. L. Carter and M. N. Wegman, Universal classes of
+hash functions, in Proceedings of the ninth annual ACM
+symposium on Theory of computing (1977) pp. 106–112.
+[61] X.-B. Wang, Beating the photon-number-splitting attack
+in practical quantum cryptography, Phys. Rev. Lett. 94,
+230503 (2005).
+[62] H.-K. Lo, X. Ma, and K. Chen, Decoy state quantum key
+distribution, Phys. Rev. Lett. 94, 230504 (2005).
+[63] G. Brassard and L. Salvail, Secret-key reconciliation by
+public discussion, in Workshop on the Theory and Appli-
+cation of of Cryptographic Techniques (Springer, 1993)
+pp. 410–423.
+[64] H. Yan, T. Ren, X. Peng, X. Lin, W. Jiang, T. Liu, and
+H. Guo, Information reconciliation protocol in quantum
+key distribution system, in 2008 Fourth International
+Conference on Natural Computation, Vol. 3 (IEEE, 2008)
+pp. 637–641.
+[65] V. Shoup, On fast and provably secure message authen-
+tication based on universal hashing, in Annual Interna-
+tional Cryptology Conference (Springer, 1996) pp. 313–
+328.
+[66] R. Konig, R. Renner, and C. Schaffner, The operational
+meaning of min-and max-entropy, IEEE trans. Inform.
+Theory 55, 4337 (2009).
+[67] M. Hayashi, Exponential decreasing rate of leaked infor-
+mation in universal random privacy amplification, IEEE
+trans. Inform. Theory 57, 3989 (2011).
+[68] J. Massey, Shift-register synthesis and bch decoding,
+IEEE trans. Inform. Theory 15, 122 (1969).
+[69] H.-L. Yin, M.-G. Zhou, J. Gu, Y.-M. Xie, Y.-S. Lu, and
+Z.-B. Chen, Tight security bounds for decoy-state quan-
+tum key distribution, Sci. Rep. 10, 14312 (2020).
+[70] C. C. W. Lim, M. Curty, N. Walenta, F. Xu, and
+H. Zbinden, Concise security bounds for practical decoy-
+state quantum key distribution, Phys. Rev. A 89, 022307
+(2014).
+[71] C. Jiang, Z.-W. Yu, X.-L. Hu, and X.-B. Wang, Uncondi-
+tional security of sending or not sending twin-field quan-
+tum key distribution with finite pulses, Phys. Rev. Ap-
+plied 12, 024061 (2019).
+[72] B.-Y. Tang, B. Liu, Y.-P. Zhai, C.-Q. Wu, and W.-R. Yu,
+High-speed and large-scale privacy amplification scheme
+for quantum key distribution, Sci. Rep. 9, 15733 (2019).
+
diff --git a/gNAzT4oBgHgl3EQfMftT/content/tmp_files/load_file.txt b/gNAzT4oBgHgl3EQfMftT/content/tmp_files/load_file.txt
new file mode 100644
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf,len=1533
+page_content='One-Time Universal Hashing Quantum Digital Signatures without Perfect Keys  Bing-Hong Li,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='1 Yuan-Mei Xie,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='1 Xiao-Yu Cao,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='1 Chen-Long Li,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='1 Yao Fu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' ∗ Hua-Lei Yin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' † and Zeng-Bing Chen1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' ‡  1National Laboratory of Solid State Microstructures and School of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Collaborative Innovation Center of Advanced Microstructures,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Nanjing University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Nanjing 210093,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' China  2Beijing National Laboratory for Condensed Matter Physics and Institute of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Beijing 100190,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' China  (Dated: January 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' 2023)  Quantum digital signatures (QDS),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' generating correlated bit strings among three remote parties  for signatures through quantum law,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' can guarantee non-repudiation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' authenticity and integrity of  messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Recently, one-time universal hashing QDS framework, exploiting the quantum asymmetric  encryption and universal hash functions, was proposed to significantly improve the signature rate  and ensure unconditional security by directly signing the hash value of long messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' However,  similar to quantum key distribution, this framework utilizes keys with perfect secrecy via performing  privacy amplification that introduces huge matrix operations, thus consuming large computational  resources, causing time delays and increasing failure probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Here, we prove that, different from  private communication, imperfect quantum keys with limited information leakage can be used for  digital signatures and authentication without compromising the security while having eight orders of  magnitude improvement on signature rate for signing a megabit message compared with conventional  single-bit schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Our work significantly reduces the time delay for data postprocessing and is  compatible with any quantum key generation protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  In our simulation, taking two-photon  twin-field key generation protocol as an example, QDS can be practically implemented over a fiber  distance of 600 km between the signer and receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Our work for the first time offers a cryptographic  application of quantum keys with imperfect secrecy and paves a way for the practical and agile  implementation of digital signatures in a future quantum network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  INTRODUCTION  Digital signatures are cryptographic primitives promis-  ing data authenticity and integrity [1], especially for the  non-repudiation of sensitive information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' It has become  an indispensable and essential technique in the global in-  ternet due to its wide application especially in digital  financial transactions, e-mail and digital currency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' How-  ever, the security of classical digital signatures, guaran-  teed by public-key infrastructure [2–4], is threatened by  rapidly developing algorithms [5, 6] and quantum com-  puting [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Different from classical digital signatures,  quantum digital signatures (QDSs) can provide a higher  level of security, information-theoretical security, by em-  ploying the fundamental principles of quantum mechan-  ics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' That is, QDS can protect data integrity, authentic-  ity and non-repudiation even if the attacker has unlim-  ited computational power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' The rudiment of the single-bit  QDS scheme was introduced in 2001 [8], but it could not  be implemented due to some impractical requirements  such as high-dimensional single-photon states and quan-  tum memories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Subsequently, there have been many de-  velopments to remove these impractical requirements [9–  11], making QDS closer to real implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Further-  more, based on non-orthogonal encoding [12] and orthog-  onal encoding [13], respectively, two independent single-  bit QDS protocols without secure quantum channels were  ∗ yfu@iphy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='cn  † hlyin@nju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='cn  ‡ zbchen@nju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='cn  proposed and proved to be secure for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  These two protocols have triggered numerous achieve-  ments of single-bit QDS theoretically [14–25] and exper-  imentally [26–36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  However, all these schemes still have several limita-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Protocols utilizing orthogonal encoding require  additional symmetrization steps which results in extra  secure channels [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' That is, to guarantee information-  theoretical security, quantum key distribution and one-  time pad encryption are still required between two re-  ceivers in orthogonal-type protocols [28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Single-bit  QDS schemes based on non-orthogonal encoding [12, 20,  23] are independent of additional quantum key distribu-  tion channels, but the signature rate is sensitive to the  misalignment error of the quantum channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  In addi-  tion, all these schemes can only sign a one-bit message in  each round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' If one wants to sign a multi-bit message by  single-bit QDS schemes, he needs to encode it into a new  message string and sign the new string bit by bit [22, 37–  41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' However, these solutions have not been completely  proved information-theoretically secure with the quanti-  fied failure probability, and the signature rate is very low  and far from implementation for long messages with a lot  of bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Recently, an efficient QDS scheme has been proposed  based on secret sharing, one-time pad, and one-time uni-  versal hashing (OTUH) [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Different from single-bit  QDS protocols that require a long key string to sign  a one-bit message, this OUTH-QDS protocol offers a  method to directly sign the hash value of multi-bit mes-  sages through one key string with information-theoretic  security, and thus drastically improves the QDS effi-  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='01132v1  [quant-ph]  3 Jan 2023  2  ciency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  However, this framework requires perfect keys  with full secrecy that is an expensive resource guaran-  teed by complete procedure of quantum key distribution  (QKD) or quantum secret sharing (QSS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' That is, pri-  vacy amplification steps are required, thereby adding to  the complexity of the algorithm and causing unendurable  time delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Here, we point out that quantum keys with imperfect  secrecy are sufficient for protecting authenticity and in-  tegrity of messages in such a digital signature scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Based on this idea, we propose a new OTUH-QDS pro-  tocol with imperfectly secret keys, utilizing only asym-  metric quantum keys without full secrecy to sign multi-  bit messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' We prove that our new scheme provides  information-theoretic security for digital signature tasks  and simulate the performance of our protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' The result  shows that our protocol outperforms other QDS schemes  in terms of signature rate and transmission distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' For  a practical case in which a megabit message is to be  signed, the proposed scheme has a superior signature  rate of nearly eight orders of magnitude, compared with  single-bit QDS schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' In addition, we show that our  scheme can significantly reduce the computational costs  and time delays of postprocessing due to the removal of  privacy amplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Our protocol is also robust against  message size, security bound and finite-size effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' More-  over, the proposed scheme is a general framework and  can be applied to all existing QKD protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' When uti-  lizing the idea of two-photon twin-field QKD [43], one of  the most efficient QKD protocols, to execute our work, a  transmission distance of 650 km can be achieved with a  signature rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='01 times per second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  To date, almost all quantum communication protocols  such as QKD [44–54], QSS [55, 56] and quantum confer-  ence key agreement [55, 57] aim at generating quantum  keys with perfect secrecy and correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' These keys are  then used to finish the corresponding cryptographic tasks  such as private communication, secret sharing and group  encryption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' In contrast, our proposed protocol offers a  new approach to digital signature tasks that only requires  keys with full correctness and imperfect secrecy through  quantum optical communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' It is the first time that  this kind of keys is applied to cryptographic tasks with  information-theoretic security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' We believe that our pro-  posed solution can provide a feasible approach to the  practical application of QDS and enlighten other applica-  tions of quantum keys with imperfect secrecy in a future  quantum network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' II  we review OTUH-QDS scheme and introduce the moti-  vation of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' III we propose our protocol  with two approaches of universal hashing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' IV we  give the security proof of authentication based on quan-  tum keys with imperfect secrecy and then, the security  analysis of our proposed QDS protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' V we dis-  cuss the performance of our scheme and compare it with  both single-bit QDS and OTUH-QDS schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Finally  we conclude the paper in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  PRELIMINARIES  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  OTUH-QDS protocol  In this section we review the schematic of OTUH-  QDS [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' The protocol can be segmented into the distri-  bution stage and messaging stage, consistent with that  in single-bit QDS that is introduced in Appendix A 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Denote the length of message as m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' The schematic of  OTUH-QDS is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' 1 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  distribution stage  Alice, Bob, and Charlie each have two key bit strings  {Xa, Xb, Xc} with n bits and {Ya, Yb, Yc} with 2n  bits, where the key bit strings meet the perfect correla-  tion Xa = Xb ⊕ Xc and Ya = Yb ⊕ Yc, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' The  distribution stage can be realized using quantum com-  munication protocols, such as QKD and QSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' It need  to be mentioned that single-bit QDS only requires the  quantum part of QKD protocols, also called key genera-  tion protocol (KGP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' In OTUH-QDS, the users need to  perform extra error correction and privacy amplification  steps after KGP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  messaging stage  (i) Signing of Alice—First, Alice uses a local quantum  random number, characterized by an n-bit string pa, to  randomly generate an irreducible polynomial p(x) of de-  gree n [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Second, she uses the initial vector (key bit  string Xa) and irreducible polynomial (quantum random  number pa) to generate a random linear feedback shift  register-based (LFSR-based) Toeplitz matrix [59] Hnm,  with n rows and m columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Third, she uses a hash op-  eration with Hash= Hnm · Doc to acquire an n-bit hash  value of the m-bit document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Fourth, she exploits the  hash value and the irreducible polynomial to constitute  the 2n-bit digest Dig = (Hash||pa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Fifth, she encrypts  the digest with her key bit string Ya to obtain the 2n-bit  signature Sig = Dig ⊕ Ya using OTP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Finally, she uses  the public channel to send the signature and document  {Sig, Doc} to Bob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  (ii) Verification of Bob—Bob uses the authentication  classical channel to transmit the received {Sig, Doc}, as  well as his key bit strings {Xb, Yb}, to Charlie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Then,  Charlie uses the same authentication channel to forward  his key bit strings {Xc, Yc} to Bob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Bob obtains two  new key bit strings {KXb = Xb ⊕ Xc, KYb = Yb ⊕ Yc} by  the XOR operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Bob exploits KYb to obtain an ex-  pected digest and bit string pb via XOR decryption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Bob  utilizes the initial vector KXb and irreducible polynomial  pb to establish an LFSR-based Toeplitz matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' He uses  a hash operation to acquire an n-bit hash value and then  constitutes a 2n-bit actual digest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Bob will accept the  signature if the actual digest is equal to the expected  3  Alice  Bob  Charlie  error correction  privacy amplification  Alice  Bob  Charlie  (a)  (b)  Distribution stage  Messaging stage  Alice  Bob  Charlie  Alice  Bob  Charlie  Distribution stage  Messaging stage  KGP  KGP  error correction  privacy amplification  KGP  KGP  error correction  error correction  classic channel  quantum channel  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  (a) OTUH-QDS [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' In the distribution stage, Alice, Bob and Charlie share key bit strings with perfect secret sharing  relationship through key generation protocol (KGP), error correction and privacy amplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' In the messaging stage Alice  generates the signature through AXU hashing, and sends the message and signature to Bob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Bob then sends his keys and  received information to Charlie, who will later send his keys to Bob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Bob and Charlie use their own and received keys to infer  Alice keys and then perform AXU hashing to verify the signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' (b) Schematic of our proposed protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' In the distribution  stage, the users only perform KGP and error correction to share keys with full correctness but some secrecy leakage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Their  keys still hold secret sharing relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' In the messaging stage the manipulation of classic information is analogous to that  in OUTH-QDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  digest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Then, he informs Charlie of the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Other-  wise, Bob rejects the signature and announces to abort  the protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  (iii) Verification of Charlie—If Bob announces that he  accepts the signature, Charlie then uses his original key  and the key sent to Bob to create two new key bit strings  {KXc = Xb ⊕ Xc, KYc = Yb ⊕ Yc}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Charlie employs  KYc to acquire an expected digest and bit string pc via  XOR decryption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Charlie uses a hash operation to ob-  tain an n-bit hash value and then constitutes a 2n-bit  actual digest, where the hash function is an LFSR-based  Toeplitz matrix generated by initial vector KXc and ir-  reducible polynomial pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Charlie accepts the signature if  the two digests are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Otherwise, Charlie rejects  the signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  The core point of this protocol is to realize the perfect  bits correlation of the three parties, building a completely  asymmetric key relationship for them, and perform one-  time almost XOR universal2 (AXU) hashing, specifically,  LFSR-based Toeplitz hashing, to generate the signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  AXU hash functions is a special class of hash functions  that can map the input value of arbitrary length into an  almost random hash value with a fixed length [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' The  signature generated in OTUH-QDS is just the AXU hash  value of the long message to be signed, where the AXU  hash function is determined by just one string of Alice  keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' After distribution stage, Alice, Bob and Charlie’s  keys are completely secret and correct with the relation-  ship of secret sharing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Bob (Charlie) can only obtain Al-  ice’s keys after he receives keys of Charlie (Bob).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Thus  Bob can obtain no information of Alice’s keys which de-  cides the AXU hash function before he transfer the mes-  sage and signature to Charlie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Then Bob’s forging attack  under this protocol is equivalent to Bob’s attack against  an authentication scenario where Alice sends an authenti-  cated message to Charlie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' It has been proved that such a  message authentication scheme based on AXU hashing  is information-theoretically secure [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Consequently,  forging attack is protected by one-time AXU hash func-  tions and key relationship among three parties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' From the  perspective of Alice, Bob and Charlie’s keys are totally  symmetric when they verify the signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Thus, Alice’s  repudiation attack is also prevented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Motivation of this work  Different from all single-bit QDS protocols that require  a long key string to sign a one-bit message, OUTH-QDS  offers a method to sign multi-bit messages through one  4  key string with information-theoretic security, and thus  drastically improves the QDS efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' This advantage  is essentially introduced by AXU hash functions, which is  only proved to be information-theoretically secure under  perfectly secret keys in previous studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Thus, OTUH-  QDS requires extra error correction and privacy amplifi-  cation steps to realize the perfect bits correlation in the  distribution stage, compared with single-bit QDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' These  postprocessing steps especially privacy amplification re-  quires calculations of a matrix with a comparable length  of data size, which introduces heavy computational costs  and unpleasant delays in practical scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' When the  data size is large, the delays will become unendurable  and constrain the practicality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  The process of AXU hashing is equivalent to the sce-  nario where the input value decides the function, map-  ping the initial input keys into almost random output  hash values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' We notice that limited secrecy leakage of in-  put value (keys) will be concealed in AXU hash value be-  cause of its randomness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Thus, such imperfect keys with  limited secrecy leakage will not undermine the authentic-  ity of messages in a QDS scheme like OTUH-QDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' More-  over, the integrity of messages is also not compromised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Based on this idea, in this paper we propose a solution  for OTUH-QDS protocols with imperfectly secret keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  In other words, we implement QDS with quantum keys  without privacy amplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Because the additional  computational cost and time delays of OTUH-QDS are  mainly introduced by privacy amplification, this concept  can effectively reduce the weaknesses of OTUH-QDS and  lay a ground for the future implementation of QDS in a  quantum internet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  The schematic of our proposed protocol is shown in  fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' 1 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' In the distribution stage users only perform  the error correction step after KGP, ensuring that their  keys have no mismatches, and build a secret sharing re-  lationship by Alice’s XOR operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' The final keys will  be randomly divided into several n-bit groups for AXU  hashing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Each of these groups of keys contains full cor-  rectness and some secrecy leakage with an upper bound  which can be estimated through finite-size analysis using  experimental data in KGP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' In the messaging stage, the  rules of information exchange are consistent with that in  OTUH-QDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' We will prove that the bit stings generated  in our distribution stage are sufficient for AXU hashing  and quantify the security bound in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' In addition,  we give two solutions based on different types of AXU  hash functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  QDS PROTOCOL  A schematic of setups of our QDS protocol is illus-  trated below and shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Distribution stage  Our proposal is a general framework in which KGP can  be derived from any type of QKD protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' As an ex-  ample, we demonstrate our scheme based on two-photon  twin-field (TP-TF) QKD [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' In the distribution stage  Alice–Bob and Alice–Charlie independently implement  TP-TF KGP (TP-TFKGP for simplicity) to share key  bit strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' At each time bin i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' , N}, Al-  ice and Bob (Alice and Charlie) each independently pre-  pare a weak coherent pulse |ei(θi  x+ri  xπ)�  kix⟩ with prob-  ability pkx, where the subscript x ∈ {a, b, c} repre-  sents the user Alice, Bob or Charlie, the phase θi  x  ∈ [0, 2π), classical bit ri  x ∈ {0, 1}, intensity ki  a(b) ∈  {µa(b),  νa(b),  oa(b),  ˆoa(b)} (represent signal,  decoy,  preserve-vacuum and declare-vacuum intensity, µa(b) >  νa(b) > oa(b) = ˆoa(b) = 0) are chosen randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Then  Alice and Bob (Alice and Charlie) send the correspond-  ing pulses to the untrusted relay Eve through insecure  quantum channels, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' In addition, they send a  bright reference light to Eve to measure the phase noise  difference φi  ab (φi  ac).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Eve performs interference measure-  ments on every received pulse pair with a beam splitter  and two detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' If one and only one detector clicks,  Eve publicly announces that she obtained a successful  detection event and which detector clicked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Sifting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' For successful detection events of Alice-Bob,  those when neither Alice nor Bob chooses the declare-  signal or decoy or declare-vacuum intensity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=', {µa, ob},  {µa, µb}, {oa, µb} and {oa, ob} will be used for generat-  ing data in the Z basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' For these events, Alice randomly  matches a time bin i of intensity µa with another time  bin j of intensity oa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Then she sets her bit value as 0 (1)  if i < j (i > j) and informs the serial numbers i and j  to Bob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' In the corresponding time bins, if Bob chooses  intensities kmin{i,j}  b  = µb (ob) and kmax{i,j}  b  = ob (µb), he  sets his bit value as 0 (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Bob announces to abort the  event where ki  b = kj  b = ob or µb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' To conclude, the pre-  served events in the Z basis are sifted as {µaoa, obµb},  {µaoa, µbob}, {oaµa, obµb} and {oaµa, µbob}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  For  other successful detection events, Alice and Bob commu-  nicate their intensities and phase information with each  other via an authenticated channel and later use them  for decoy analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Define the global phase difference at time bin i as  θi := θi  a − θi  b + φi  ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Alice and Bob keep detection events  {νi  a, νi  b} only if θi ∈ [−δ, δ] ∪ [π − δ, π + δ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' They ran-  domly select two retained detection events that satisfy  ��θi − θj�� = 0 or π, and then match these two events , de-  noted as {νi  aνj  a, νi  bνj  b}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' By calculating classical bits ri  a⊕rj  a  and ri  b ⊕ rj  b, Alice and Bob extract a bit value in the X  basis, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Subsequently, in the Z basis, Bob al-  ways flips his bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' In the X basis, Bob flips part of his  bits to correctly correlate them with those of Alice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' To  be specific, when the global phase difference between two  5  Eve  Charlie  Alice  Laser  SPD  BS  PM  AWG  VOA  PM  AWG  VOA  Laser  Bob  PM  AWG  VOA  Laser  IM  IM  IM  SPD  Optical  switch  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Schematic of the setup of our QDS protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' The red line represents quantum optical channel in the distribution stage,  and the black arrow line represents the information exchange through classic authenticated channel in the messaging stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' (i)  In the distribution stage, Alice, Bob and Charlie utilize a narrow-linewidth continuous-wave laser, intensity modulator (IM),  phase modulator (PM), arbitrary wave generator (AWG) and variable optical attenuator (VOA) to prepare a phase-randomized  weak coherent source with different intensities and phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' The signals from Bob and Charlie will both go through an optical  switch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' An untrusted relay Eve performs interference measurement on the signals from Alice and an optical switch with a beam  splitter (BS) and a single-photon detector (SPD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' After sifting, parameter estimation and error correction, Alice can share bit  strings with Bob and Charlie, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' (ii) In the messaging stage, Alice sends the desired message to Bob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Bob sends the  message as well as his keys to Charlie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Charlie will then send his keys to Bob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Then Bob verifies the signature by his own and  received keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' If he accepts the signature, he will inform Charlie and then Charlie will also verify the signature by his own and  received keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' The signature is successful if both Bob and Charlie accept it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  matching time bins is 0 (π) and the two clicking detectors  announced by Eve are different (same), Bob will flip his  bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Otherwise, Bob will directly save his bits for later  use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Similarly, Alice and Charlie will sift their successful  detection events in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Parameter estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Alice and Bob (Alice and  Charlie) form the nZ-length raw key bit from the random  bits under the Z basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' The remaining bits in the Z basis  are used to estimate the bit error rate Ez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' They also  communicate all bit values in the X basis to obtain the  total number of errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' The decoy-state method [61, 62]  is used to estimate the number of vacuum events in the  Z basis sz  0µb, the count of single-photon pairs sz  11, and  the phase error rate of single-photon pairs φz  11 on the Z  basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Error correction and examination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Alice and Bob  (Alice and Charlie) distill final keys by utilizing an error  correction algorithm with εcor-correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' [63, 64] The  size of the distilled key is still nZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' The unknown infor-  mation of a possible attacker can be expressed as H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Al-  ice then randomly disturbs the orders of the distilled key  and publicizes the new order to Bob (Charlie) through  the authenticated channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Subsequently, Alice and Bob  (Alice and Charlie) divide the final keys into several n-  bit strings, each of which is used to perform a task in the  messaging stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' This ensures that the attacker cannot  know the position of each bit of one concrete group be-  forehand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Thus the grouping process can be considered  as a random sampling in our finite-size analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  The  maximum unknown information of an n-bit group to the  attacker after error correction Hn can be estimated by  Hn ≤ szn  0µb + szn  11  �  1 − H(φzn  11 )  �  − nfH(Ez),  (1)  where f is the error correction efficiency, szn  0µb and szn  11 are  the lower bounds of vacuum events and single-photon  pairs, respectively, and φzn  11 is the upper bound of the  phase error rate of single-photon pairs in the n-bit string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Messaging stage  Various AXU hash functions can be used in the mes-  saging stage of our protocol by following the framework  of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' 1 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  To demonstrate the detailed procedure,  we here present two specific approaches to the messag-  ing stage utilizing LFSR based Toeplitz hashing and  generalized division hashing, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' LFSR-based  Toeplitz hashing is highly compatible with in hardware  systems and generalized division hashing is more suit-  able for achieving software systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' In a practical case  6  we choose either method of hashing depending on the  different application environments of users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  The mes-  sage to be signed is denoted as M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  For each M if  using LFSR-based Toeplitz hashing Alice generates six  bit strings XB, XC, YB, YC, ZB, ZC, each of length  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' If choosing generalized division hashing in the mes-  saging stage, Alice will only generate four bit strings  XB, XC, YB, YC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' The subscripts represent the partic-  ipants performing KGP with Alice, where B represents  Bob and C represents Charlie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Then Alice will generate  Xa = Xb⊕Xc, Ya = Yb⊕Yc and Za = Zb⊕Zc as her own  key strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' For the scheme with LFSR-based Toeplitz  hashing the signature rate is R = nZ/3n, whereas for  generalized division hashing there is R = nZ/2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Utilizing LFSR-based Toeplitz hashing  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' LFSR-based Toeplitz hash functions:  LFSR-based Toeplitz hash functions can be expressed as  hp,s(M) = Hnm · M, where p, s determines the function  and M is the message in the form of an m-bit vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  The detailed process of generating LFSR-based Toeplitz  hash function is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  A randomly selected irreducible polynomial of order  n in the field GF(2), p(x), determines the construction  of LFSR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' p(x) = xn + pn−1xn−1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' + p1x + p0 can  be characterized by its coefficients of order from 0 to  n − 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=', pa = (pn−1, pn−2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=', p1, p0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  For the ini-  tial state s which is also represented as an n-bit vector  s = (an, an−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=', a2, a1)T , the LFSR will be performed  n times to generate n vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Specifically, it will shift  down every element in the previous column, and add a  new element to the top of the column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  For example,  the LFSR transforms s into s1 = (an+1, an, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=', a3, a2)T ,  where an+1 = pa · s, and likewise, transforms s1 to s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Then the m vectors s, s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=', sm−1 will together construct  the Toeplitz matrix Hnm = (s, s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=', sm−1), and the hash  value of the message is Hnm · M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  (i) Alice obtains a string of random numbers through  a quantum random number generator and uses it to ran-  domly generate a monic irreducible polynomial in GF(2)  of order n, denoted as p(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' p(x) can be characterized  by its coefficients of order from 0 to n − 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=', an n-bit  string, denoted by pa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Details of generating p(x) can be  found in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  (ii) Alice uses her key bit string Ya and p(x) to per-  form LFSR-based Toeplitz hashing and generates an n-  bit hash value Dig = HY,pa(M), and encrypts it by Za to  obtain the final signature Sig = Dig ⊕ Za.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' In addition,  Alice encrypts pa by the key set Xa to get the encrypted  string p = pa ⊕ Xa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  She then sends {Sig, p, M} to Bob through an au-  thenticated classical channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  (iii) Bob transmits {Sig, p, M} as well as his key  bit strings {Xb, Yb, Zb} to Charlie so as to inform  Charlie that he has received the signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Then Char-  lie forwards his key bit strings {Xc, Yc, Zc} to Bob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  These data are all transmitted through an authenti-  cated channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Bob obtains three new key bit strings  KXb = Xb ⊕ Xc, KYb = Yb ⊕ Yc and KZb = Zb ⊕ Zc  using the XOR operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' He exploits KXb and KZb to  obtain the expected digest and string pb via XOR decryp-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' He utilizes KYb and pb to establish an LFSR-based  Toeplitz matrix and acquires an actual digest via a hash  operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Bob will accept the signature if the actual  digest is equal to the expected digest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Then he informs  Charlie of the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  (iv) If Bob announces that he accepts the signature,  Charlie creates three new key bit strings KXc = Xb ⊕  Xc, , KYc = Yb ⊕ Yc and KZc = Zb ⊕ Zc using his orig-  inal key and the key sent by Bob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' He employs KXc and  KZc to acquire the expected digest and variable pc via  XOR decryption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Charlie obtains an actual digest via  hash operation, where the hash function is an LFSR-  based Toeplitz matrix generated by KYc and pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Charlie  accepts the signature if the two digests are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Utilizing generalized division hashing  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Generalized division hash functions: The  generalized division hash functions can be expressed as  hP (M) = m(x)·xn/k mod P(x), where P(x) is a monic ir-  reducible polynomial of order n/k in the field GF(2k), M  is the message in the form of a polynomial of order m/k  in GF(2k) with every k bits corresponding to a coefficient  of the polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' The calculation is also performed in  GF(2k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' The final result is a polynomial of order n/k  in field GF(2k), and can be transformed into an n-bit  strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Commonly, k is set as k = 2x for simplicity, where x is  a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' In our scheme, we select k = 23 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  (i) In this case, Alice chooses Xa = Xb ⊕ Xc and  Ya = Yb ⊕ Yc as her own key sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Alice first obtains  a string of random numbers through a quantum random  number generator and uses it to randomly generate a  monic irreducible polynomial in GF(28) of order n/8, de-  noted by P(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Details of generating p(x) can be found  in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' P(x) can be characterized by its coef-  ficients of order from 0 to n/8 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' By encoding each  coefficient into an 8-bit string, we can use an n-bit string  to express P(x), denoted as Pa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Alice encrypts Pa by the  key set Xa to obtain the encrypted string P = Pa ⊕ Xa  (ii) Alice uses P(x) to perform the generalized divi-  sion hashing [65] to obtain an n-bit hash value Dig =  hPa(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' She encrypts Dig by Ya to obtain the signa-  ture Sig = Dig⊕Ya and sends the message and signature  {Sig, p, M} to Bob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  step (iii) and (iv) are similar to those utilizing LFSR-  based Toeplitz hashing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Bob and Charlie will exchange  their key strings in turn through an authenticated chan-  nel and examine their expected and received digests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Above is the whole procedure of our protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Note  that the TP-TFKGP can be replaced by any other KGP  7  such as BB84-KGP or sending-or-not-sending (SNS)-  KGP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Actually, in the distribution stage Alice shares bit  strings with Bob and Charlie in the relationship of secret  sharing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Thus, the distribution stage can also be per-  formed based on QSS without the privacy amplification  step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  SECURITY ANALYSIS  Similar to OTUH-QDS, the core point of our protocol  is the security of authentication based on AXU hash-  ing, which directly protects the security of QDS against  fogery [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' However, the security of our protocol differs  because of the information leakage during the distribu-  tion stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' In this section we provide a more detailed  security analysis of AXU hashing under imperfect keys  with limited secrecy leakage, and then demonstrate the  security of our protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Security of authentication based on hashing  In digital signatures, hashing is used to perform the  authentication task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Thus we first analyze the security  of authentication based on ϵ-AXU hash functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' ϵ-AXU hash functions: The class of func-  tions H = {h|h : M → T} is ϵ-AXU (ϵ-almost XOR  universal) if and only if, for a randomly selected hash  function h ∈ H and any m1, m2 ∈ M, t ∈ T, it holds  that the probability of h(m1) ⊕ h(m2) = t is at most ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  The hash functions used in our scheme (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=', LFSR-  based Toeplitz hash functions and generalized division  hash functions) are all linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' There is the relationship  that h(m1) ⊕ h(m2) = h(m1 ⊕ m2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Thus, the definition  of ϵ-AXU hash function is equivalent to:  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' ϵ-AXU hash functions: The class of func-  tions H = {h|h : M → T} is ϵ-AXU if and only if, for a  randomly selected hash function h ∈ H and any m ∈ M,  t ∈ T, the probability of h(m) = t is at most ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Suppose the authentication scenario that an attacker  captures a message and signature {M, Sig = h(M) ⊕ r}  from the sender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' h is randomly selected from the set of  ϵ-AXU hash functions H, and r is secret keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' He can  manage to forge if and only if he chooses a combination  {m, t} with the relationship h(m) = t, and sends {M ⊕  m, Sig ⊕ t} to the recipient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' In this case the recipient  will accept the message because of the relationship h(M⊕  m) = h(M)⊕h(m) = Sig⊕t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' It should be mentioned that  m ̸= 0 due to the requirement for a valid forge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Through  Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' 4, for message authentication schemes based on ϵ-  AXU hashing, the failure probability is no more than  ϵ [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Previous proof only applies for perfect keys with full  secrecy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Here we analyze the scenario with imperfect  keys without perfect secrecy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' The attacker is also to gen-  erate a combination m, t, as analyzed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  He can  perform various kinds of attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' The worst one is to  randomly generate m, t whose success probability is only  2−n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' We will leave out this attack in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Other  attacks include guessing keys and recovering keys from  signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  First we quantify the guessing probability of the at-  tacker when he is to guess an n-bit key string with limited  secrecy leakage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='Suppose an n-bit key string as X and the  attacker’s system is B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' We consider the general attack  where attackers can perform any operations on the sys-  tem of all quantum states, get a system ρx  B and perform  any positive operator-valued measure {Ex  B}x on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' The  probability that the attacker correctly guesses X when  using an optimal strategy is denoted as Pguess(X|B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Ac-  cording to the definition of min-entropy in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' [66],  Pguess(X|B) = max  {Ex  B}x  �  x  Px tr(Ex  Bρx  B) = 2−Hmin(X|B)ρ  (2)  where Hmin(X|B)ρ = Hn is the min-entropy of X and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  If X is generated in the distribution stage of our protocol,  Hmin(X|B)ρ can be estimated by  Hmin(X|B)ρ = Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  (3)  Before analyzing the attack strategy we first introduce  a proposition of LFSR-based Toeplitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' For the LFSR-based Toeplitz hash func-  tion hp,s(M)= Hnm ·M, if p(x)|m(x), then hp,s(M) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  hp,s(M) can be transformed into hp,M(s) =  �m−1  i=0 Mi+1W i · s, where W is the n × n circulant ma-  trix with the first row being p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Thus, p(x) is just the  characteristic polynomial of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Denote the characteris-  tic value of W by λi, where λi is an element in GF(2n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Then there is the relationship p(λi) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  We can di-  agonalize �m−1  i=0 Mi+1W i, and the diagonal elements are  m(λi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' hp,s(M) = 0 if and only if there exists a zero diag-  onal element, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=', there exists a λi satisfying m(λi) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  If p(x)|m(x), because p(λi) = 0, then the condition  m(λi) = 0 will be absolutely satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Attack of guessing keys  The LFSR-based Toeplitz hash function is hp,s(M)=  Hnm·M, where Hnm is determined by the two bit strings  p and s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Here we follow the terminology in the messaging  stage of our protocol where p is actually pa encrypted by  Xa, s is Ya and the hash value Dig is encrypted by Za.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  We show that only guessing Xa or in other words, only  guessing pa is enough to perform an attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Suppose the attacker obtain a string Xg as his estima-  tion of Xa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' He can decrypt it to obtain pg as his guessing  of pa and transform pg into a polynomial pg(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Then  the attacker can easily generate a bit string m satisfy-  ing pg(x)|m(x), and there is the relationship h(m) = 0 if  8  pg = pa (or equivalently Xg = Xa) according to Propo-  sitions 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Since m(x) is m-order and the polynomial  is n-order, the attacker can select no more than m/n  polynomials and multiply them to consist his choice of  m(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' In other words, he can guess the string Xa for no  more than m/n times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' It must be considered that the  attacker knows pa is irreducible, so he will only choose  those guesses that satisfy pa is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' The success  probability of this optimized strategy can be expressed  as  P1 = m  n · P(Xa = Xg|pg ∈ I),  (4)  where P(A|B) represents the probability of event A un-  der the condition that event B occurs, and I is the set  of all irreducible polynomials of order n in GF(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' The  cardinal number of I, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=', the number of all irreducible  polynomials of order n in GF(2), is more than 2n−1/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Thus P(pg ∈ I) ≤ (2n−1/n)/2n = 1/2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  It is obvi-  ous that P(Xa = Xg, pg ∈ I) = P(Xa = Xg) because if  Xa = Xg then pg = pa ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Then we can get the upper  bound of the success probability of this kind of attack,  denoted as ϵ1  LFSR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  P1 =m  n · P(Xa = Xg)  P(pg ∈ I)  (5)  ≤m  n · 2−Hn  1  2n  (6)  =m · 2−1−Hn = ϵ1  LFSR  (7)  The attacker can also guess the strings Xa and Ya to  obtain p and s so that he can guess a hash function and  make a successful attack for certainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Under this cir-  cumstance his success probability is no more than ϵ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  P2 =P(Xa = Xg, Ya = Yg|pg ∈ I)  ≤P(Xa = Xg|pg ∈ I)  ≤ϵ1  LFSR  The generalized division hash function hP (M)= m(x)·  xn/8 mod P(x) is determined only by P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' We also follow  the terminology in our protocol that P is actually Pa  encrypted by Xa and the hash value Dig is encrypted by  Ya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' The attacker’s strategy is to guess a string Xg so that  he can obtain Pg and then forge a message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Analogous  to the analysis above, the upper bound of the success  probability is  ϵ1  GDH = m  n · 2−Hn  4  n  = m · 22−Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  (8)  The only difference in calculation is that there are at  least 2n−1/(n/8) irreducible polynomials of order n/8 in  GF(28), so P(Pg ∈ I) ≤ (2n−1/(n/8))/2n = 4/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Attack of recovering keys from signature  The attacker can try to recover the desired keys from  the captured signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' In both kinds of hashing the hash  value is encrypted to generate the signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Thus the  attacker must first guess the corresponding key strings  (Za in LFSR-based Toeplitz hashing or Ya in general-  ized division hashing) and then perform the recovering  algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' The success probability of this strategy is no  more than that only guessing the bit string (P(Za = Zg)  or P(Ya = Yg)) and is obviously no more than ϵ1  LFSR or  ϵ1  GDH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  In conclusion, the optimal strategy on LFSR-based  Toeplitz hashing and generalized division hashing is to  guess the key string that encrypts the polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' We  can quantify the upper bound of failure probability of  authentication based on both types of hashing with im-  perfect keys of secrecy leakage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  ϵLFSR =m · 2−1−Hn  (9)  ϵGDH =m · 22−Hn  (10)  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Security of the QDS scheme  In a QDS scheme, the security analysis contains three  parts, robustness, repudiation, and forgery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  The honest run abortion occurs only when Alice and  Bob (or Charlie) share different key bits after the distri-  bution stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' In our protocol Alice and Bob (Charlie)  perform error correction in the distribution stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Thus,  they share the identical final key, and the robustness  bound is ϵrob = 2ϵcor, where ϵcor is the failure probability  of the error correction protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Repudiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Alice successfully repudiates when Bob accepts the  message while Charlie rejects it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  For Alice’s repudia-  tion attacks, Bob and Charlie are both honest and sym-  metric and have the same new key strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  They will  make the same decision for the same message and sig-  nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' In other words, when Bob rejects (accepts) the  message, Charlie also rejects (accepts) it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Therefore, our  QDS protocol is naturally immune to repudiation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=',  the repudiation bound ϵrep = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Forgery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Bob forges successfully when Charlie accepts the forged  message forwarded by Bob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' According to our protocol,  Charlie accepts the message if and only if Charlie ob-  tains the same result through one-time pad decryption  and one-time AXU hash functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Actually this is the  9  0  100  200  300  400  500  600  700  Distance (km)  10-6  10-4  10-2  100  102  104  Signature rate (tps)  This work with TP-TFKGP  This work with BB84-KGP  This work with SNS-KGP  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' [13]  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' [21]  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' [24] with SNS-KGP  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Signature rates of our protocol with TP-TFKGP,  BB84-KGP, SNS-KGP, and decoy state BB84-QDS [13], SNS-  QDS [21], SNS-QDS with random pairing [24] with the mes-  sage size of 1 Kb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' In our protocol we use generalized division  hashing in messaging stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' The repetition rate of the laser  is 1 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' The distances between Alice-Bob and Alice-Charlie  are assumed to be the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' The data size N is 1013 and the  security bound is 10−10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  same as an authentication scenario where Bob is the at-  tacker attempting to forge the information sent from Al-  ice to Charlie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Therefore, the probability of a successful  forgery can be determined by the failure probability of  hashing, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=', one chooses two distinct messages with iden-  tical hash values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' For the scheme utilizing LFSR-based  Toeplitz hash ϵfor = m · 2−1−Hn, and for generalized di-  vision hashing ϵfor = m · 22−Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  The total security bound of QDS, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=', the max-  imum failure probability of the protocol,  is ϵ  =  max{ϵrobb, ϵrep, ϵfor}  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  DISCUSSION  For two types of hashing, there is just differences in a  constant 2/3 between two signature rates and a constant  8 between two security parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' For simplicity in this  section we only analyze our protocol with generalized di-  vision hashing hashing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Simulation parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' ηd and pd are the detector  efficiency and dark count rate, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  ed is the mis-  alignment error rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' N is the data size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' α is the attenuation  coefficient of the fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' f is the error correction efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' ϵ  is the failure probability of QDS schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  ηd  pd  ed  N  α  f  ϵ  70%  10−8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='02  1013  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='165  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='1  10−10  0  100  200  300  400  500  600  700  Distance (km)  10-10  10-5  100  105  Signature rate (tps)  This work with TP-TFKGP  This work with BB84-KGP  This work with SNS-KGP  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' [13]  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' [21]  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' [24] with SNS-KGP  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Signature rates of our protocol with TP-TFKGP,  BB84-KGP, SNS-KGP, and decoy state BB84-QDS [13], SNS-  QDS [21], SNS-QDS with random pairing [24] with the mes-  sage size of 1 Mb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' In our protocol we use generalized division  hashing in messaging stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' The repetition rate of the laser  is 1 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' The distances between Alice-Bob and Alice-Charlie  are assumed to be the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' The data size N is 1013 and the  security bound is 10−10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  To demonstrate the superiority of our proposal, we  first build our protocol based on BB84-KGP, SNS-KGP,  and TP-TFKGP, and compare them with decoy-state  BB84-QDS [13] and SNS-QDS [21] which are single-bit  QDS protocols based on BB84-KGP and SNS-KGP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' We  also compare SNS-QDS with random pairing [24], which  shows improvement in the signature rate of SNS-QDS  and can be applied to other QDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' More details of cal-  culation are shown in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' In the simulations,  we consider two common cases where each message to  be signed is 103 bits (1 Kb) and 106 bits (1 Mb), re-  spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  The repetition rate of the laser is 1 GHz,  and the distances between Alice-Bob and Alice-Charlie  are assumed to be the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  It should be mentioned  that all conventional QDS protocols sign only a one-bit  message every time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' When signing multi-bit message, an  n-bit message must be encoded into a new sequence with  length h by inserting ‘0’ and adding ‘1’ to the original  sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' The signing efficiency, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=', ˆη = n/h, is obvi-  ously less than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' For simplicity we use the upper bound  ˆη = 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=', h = n, in our simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Detailed analysis  is shown in Appendix A 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Other simulation parameters  are listed in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' 3 and 4 show the simulation results of all the  protocols mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' When the message size is 1  Kb, our protocols show a superiority on signature rate  of over five orders of magnitude compared with conven-  tional QDS schemes, which is a quite larger improvement  than SNS-QDS with random pairing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  If the message  size becomes 1 Mb, the signature rate of conventional  BB84-QDS, SNS-QDS and SNS-QDS with random pair-  10  ing will decrease by three orders of magnitude, whereas  that of our protocols decreases only slightly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Thus our  QDS scheme has a superior signature rate of over eight  orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' It turns out that our protocol shows  great robustness to message size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Furthermore, our TP-  TFQDS scheme can reach a transmission distance of 650  km as well as a signature rate of approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='01 times  per second (tps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' It is a large breakthrough in terms of  both distance and signature rate, indicating the great po-  tential of our protocol in the practical implementation of  QDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' 5, we show the performance of our proto-  col under different data sizes 109, 1011 and 1013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Even  with a data size as small as 109, our protocol can reach a  transmission distance of 350 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Our protocol is robust  to finite-size effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Compared with OTUH-QDS, our protocol does not re-  quire perfectly secret keys, and is thus requires no privacy  amplification step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  In other words, our protocol only  consumes keys with limited information leakage, which  is a cheap and practical resource compared with perfect  quantum keys generated by quantum secure communi-  cation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Error correction of quantum keys can be easily  performed by classical Cascade protocol [63, 64] where  the bit string is first blocked and then manipulated by  blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Thus the complexity of error correction increases  linearly with the data size N and can be performed by  stream computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Privacy amplification, however, re-  quires a hash matrix multiplication step where the num-  bers of columns and rows are all proportional to N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Thus  the computational complexity of privacy amplification is  O(N 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' The fast Fourier transform algorithm can reduce  the complexity to O(N log N) [67], and one can also block  the keys before performing privacy amplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' How-  ever, to reduce the finite-size effect, the minimum blocks  should be large, so actual computational cost and time  delay of privacy amplification are still very large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Table II lists the time consumption of error correction,  privacy amplification and data transmission, as well as  total postprocessing time of both protocols, at a distance  of 400 km with data sizes 1013 and 1011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Details of sim-  ulation are introduced in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' If N = 1011, time  consumption of postprocessing in OTUH-QDS is 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='6 sec-  onds, while that of our protocol is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='62 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' More-  over, when N = 1013, time for privacy amplification is  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='71 hours (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='057 × 104s), which will introduce a quite  long delay in experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Our scheme can significantly  save computational resources and time delays for post-  processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  We also compare the signature rate of our protocol  with that of OTUH-QDS [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Theoretically, the two sig-  nature rates should be the same under ideal conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  In practical cases, signature rate of our protocol will be  slightly damaged by the statistical fluctuation of privacy  leakage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' 6, we draw the ratio of signature rate of  our protocol based on TP-TFKGP and that of OTUH-  QDS [42] combined with TP-TFQKD, if not considering  the postprocessing time, with data sizes of 1013 and 1011,  and message size of 1Kb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' The result shows that when the  0  100  200  300  400  500  600  Distance (km)  10-2  10-1  100  101  102  103  104  105  Signature rate (tps)  N=109  N=1011  N=1013  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Signature rates of our protocols with TP-TFKGP un-  der different data sizes N = 109, 1011 and 1013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' The message  size is assumed to be 1 Mb and the repetition rate of the laser  is 1 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' The security bound is 10−10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  0  100  200  300  400  500  600  Distance (km)  20%  30%  40%  50%  60%  70%  80%  90%  100%  Ratio of signature rate  N=1013  N=1011  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' The ratio of signature rate of our TP-TFQDS proto-  col and that of OTUH-QDS [42] combined with TP-TFQKD,  if not considering the postprocessing time, with data size 1013  and 1011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' The message size is 1 Kb and the repetition rate of  the laser is 1 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' The ratio is more than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='8 with transmis-  sion distance lower than 500 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  transmission distance is less than 500 km, the ratio is  more than 80%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' The signature rates of the two protocols  are comparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' In addition, if assuming the repetition  rate of the laser as 1 GHz, time consumption for postpro-  cessing (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='057 × 104s) is even longer than time for data  transmission (104s) when N = 1013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' The signature rate  of OTUH-QDS will be constrained by the efficiency of  privacy amplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' That is, in practice the signature  rate of OTUH-QDS is lower than the simulation result,  while our scheme can compensate for this shortcoming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  11  TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Time consumption of error correction TEC and privacy amplification TP A under different data sizes N = 1013 and  N = 1011 when the distance is 400 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' T1 = TEC and T2 = TEC + TPA represent the postprocessing time of our proposed  scheme and OTUH-QDS, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' nZ is the number of raw bits generated in TP-TFKGP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' l is the length of keys after  privacy amplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' When N = 1013, postprocessing time of OTUH-QDS is 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='85 hours, and that of the proposed protocol is  only 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='07 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  N  nZ  errors  l  TEC  TPA  T1  T2  1013  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='695 × 108  300  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='87 × 107  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='07 min  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='71 h  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='07 min  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='85 h  1011  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='267 × 106  39830  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='51 × 105  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='62 s  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='98 s  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='62 s  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='6 s  Considering the fact that our protocol can save postpro-  cessing time by even one hundred times, our proposal  shows significant superiority in the practical scenario es-  pecially when the digital signature tasks are performed  at high frequency and the data size is large .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  CONCLUSION  Overall, in this paper we prove that keys with limited  secrecy leakage and full correctness can protect authen-  ticity and integrity of messages if combined with AXU  hash functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Furthermore, we theoretically propose  an efficient QDS protocol utilizing imperfect quantum  keys without privacy amplification based on the frame-  work of OTUH-QDS, reducing computational resources  and time delays of postprocessing without compromising  the security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' The simulation results demonstrate that our  protocol outperforms previous single-bit QDS protocols  in terms of both signing efficiency and distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' For ex-  ample, for the 1 MB message to be signed, the signature  rate of our protocol is higher than that of single-bit QDS  protocols by eight orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Specifically, for  our protocol based on TP-TFKGP, the transmission dis-  tance can reach 650 km and still holds a signature rate of  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='01 tps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' In addition, our protocol is robust against se-  curity levels and finite-size effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Moreover, compared  with OUTH-QDS, our protocol notably saves the post-  processing time into an endurable range and thus signifi-  cantly improves the practicality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Our scheme is a general  framework that can be applied to any existing QKD or  QSS protocol, and is highly compatible with future quan-  tum networks and feasible for many applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Addi-  tionally, our work, only requiring keys with imperfect  secrecy, is a new approach of quantum communication  that is different from other quantum secret communi-  cation protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' For the first time we propose crypto-  graphic applications of imperfect quantum keys without  full secrecy, indicating that this type of keys is a resource  with huge potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' We hope that the proposed scheme  and the idea of utilizing imperfect quantum keys provide  a solution for the real implementation of practical and  commercial QDS as well as other quantum cryptography  tasks in future quantum networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  This study was supported by the National Natu-  ral Science Foundation of China (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  12274223), the  Natural Science Foundation of Jiangsu Province (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  BK20211145), the Fundamental Research Funds for the  Central Universities (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' 020414380182), the Key Re-  search and Development Program of Nanjing Jiangbei  New Aera (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' ZDYD20210101), the Program for In-  novative Talents and Entrepreneurs in Jiangsu (JSS-  CRC2021484), and the Program of Song Shan Lab-  oratory (Included in the management of Major Sci-  ence and Technology Program of Henan Province) (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  221100210800).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Appendix A: single-bit QDS  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  schematic of single-bit QDS  Here, we first introduce orthogonal encoding QDS [13]  as an example of single-bit QDS schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Commonly,  all single-bit QDS protocols can be segmented into two  stages: the distribution stage and messaging stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' The  schematic of orthogonal encoding QDS is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  distribution stage:  (i)For each possible future message m = 0 or 1, Alice  uses the KGP to generate four different length L keys,  A0  B, A1  B, A0  C, A1  C, where the subscript denotes the par-  ticipant with whom she performed the KGP and the su-  perscript denotes the future message, to be decided later  by Alice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Bob holds the length L strings K0  B, K1  B and  Charlie holds the length L strings K0  C, K1  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  (ii)For each future message, Bob and Charlie sym-  metrize their keys by choosing half of the bit values in  their Km  B , Km  C and sending them (as well as the cor-  responding positions) to the other participant using the  Bob-Charlie secret classical channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' They will only use  the bits they did not forward and those received from  the other participant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Their final symmetrized keys are  denoted as Sm  B and Sm  C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Bob (and Charlie) will keep a  record of whether an element in Sm  B (Sm  C ) came directly  from Alice or whether it was forwarded to him by Charlie  (or Bob).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  messaging stage:  (i) To send a signed one-bit message m, Alice sends  12  Alice  Bob  Charlie  KGP  Alice  Bob  Charlie  Distribution stage  Messaging stage  classic channel  quantum channel  KGP  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Orthogonal encoding QDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' In the distribution stage,  Alice–Bob and Alice–Charlie independently perform KGP to  generate correlated bit strings with limited mismatches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Then  Bob and Charlie symmetrize their keys by exchanging half of  their keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' In the messaging stage Alice generates the sig-  nature depending on the message bit, and sends the message  and signature to Bob, who will transfer it to Charlie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Bob  and Charlie examine their mismatch and compare it with the  threshold to verify the signed message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='.  (m, Sigm) to the desired recipient (say Bob), where  sigm = (Am  B , Am  C ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  (ii) Bob checks whether (m, Sigm) matches his Sm  B and  records the number of mismatches he finds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' He separately  checks the part of his key received directly from Alice and  the part of the key received from Charlie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' If there are  fewer than sa(L/2) mismatches in both halves of the key,  where sa < 1/2 is a small threshold determined by the  parameters and the desired security level of the protocol,  then Bob accepts the message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  (iii) To forward the message to Charlie, Bob forwards  the pair (m, Sigm) that he received from Alice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  (iv) Charlie tests for mismatches in the same way, but  in order to protect against repudiation by Alice he uses a  different threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Charlie accepts the forwarded mes-  sage if the number of mismatches in both halves of his  key is below sv(L/2) where sv is another threshold, with  0 < sa < sv < 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  KGP is actually part of QKD protocol except for the  error correction and privacy amplification steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' In dis-  tribution stage, Am  X and Km  X generated through KGP are  correlated with limited mismatch, and Am  X contains fewer  mismatches with Km  X than does any string produced by  an eavesdropper, where X ∈ {B, C} represents Bob and  Charlie, m is the message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' After Bob and Charlie’s sym-  metrization step, Bob holds Sm  B and Charlie holds Sm  C ,  each containing half of Km  B and Km  C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' From the perspec-  tive of Alice, Sm  B and Sm  C are symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Alice has no in-  formation on whether it is Bob’s Sm  B or Charlie’s Sm  C that  contains a particular element of the string (Km  B , Km  C ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  This protects against repudiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' From the perspective  of Bob, Sm  B and Sm  C are asymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Bob has access to  all of Km  B and only half of Km  C , but, even if he is dishon-  est, he does not know the half of Km  C that Charlie chose  to keep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' This protects against forging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  The framework of non-orthogonal encoding QDS is  analogous to that of orthogonal encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' The difference  is that it does not require the symmetrization step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' How-  ever the signer needs to send the same quantum states  to two receivers and only detection events where the two  receivers both have clicks are valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Signing a multi-bit message using single-bit QDS  The framework above only offer a way for signing a  one-bit message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' To sign multi-bit messages with these  protocols, it is not sufficient to directly iterate the proto-  col on each bits of the message, which will give a chance  for an outside or inside attacker to perform forgery at-  tacks [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' In order to offer information-theoretical secu-  rity, one must reconstruct the multi-bit message and then  sign it bit by bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' This step will make the new message  become longer and thus damage the efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' To date,  the most efficient coding rule is given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' [39], which  can be summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Suppose the signer Alice needs to sign an n-bit message  M = m1||m2||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='||mn, mi ∈ {0, 1}, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=', n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' She will  encode M into  ˆ  M = 11||12||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='||1x+1||0||m1||m2||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='||mx||0||  ||mx+1||mx+2||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='||m2x||0||  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  ||m[ n  x ]x+1||m[ n  x ]x+2||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='||mn||0||11||12||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='||1x+1,  (A1)  where x refers to the coding interval, [x] is the round  down function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' To conclude, the coding rule is that the  encoder replenishes a ‘0’ in the head of M, and another  in the tail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Then, the encoder inserts a ‘0’ every x bits  and adds ‘1’ with a number of x+1 to both the start and  the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Denote the length of  ˆ  M as h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  An iteration of  conventional QDS protocols with h rounds on  ˆ  M is  an information-theoretically secure protocol to sign the  multi-bit message M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' According to the encoding rule,  h = n + [ n  x] + 2x + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' For a given n we can optimize x to  obtain the minimal h and the maximum efficiency η = n  h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  It is clear that if n is large, the maximum efficiency will  be close to 1, but will definitely be less than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Thus  in our simulation in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' V we use the upper bound of  deficiency, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=', assume h = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Appendix B: Generating an irreducible polynomial  In this section we introduce ways to generate an irre-  ducible polynomial over Gloise fields GF(2) in random,  which is the first step in the messaging stage of our pro-  tocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Suppose p(x) is a polynomial of order n in GF(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' p(x)  is irreducible means that no polynomials can divide it  except the identity element ’1’ and p(x) itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' The nec-  essary and sufficient condition for p(x) being irreducible  13  can be expressed as:  �  �  �  x2n ≡ x mod p(x)  gcf(x2  n  d − x, p(x)) = 1  (B1)  where d is any prime factor of n, gcf(f(x), g(x)) represents  the greatest common factor (GCF) of f(x) and g(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  In order to randomly generate an irreducible polyno-  mial, one way is to generate polynomials at random and  test for irreducibility through the condition above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' How-  ever, this is quite time consuming and requires a lot of  random bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  A better solution is proposed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' We can first  have an irreducible polynomial of order n, defining the  extension field GF(2n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Given this, we generate a random  element in GF(2n) and then compute the minimal poly-  nomial of this element, which will be irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' This  procedure only needs n random bits and consumes less  time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' The concrete procedure is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Denote the initial irreducible polynomial as f(x) and  the polynomial generated by random element as g(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  We will calculate the sequence a0 = g0(0), a1 = g1(0), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=',  a2n−1 = g2n−1(0), where gi(x) = gi(x) mod f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' This  sequence of 2n elements can fully determine the minimal  polynomial of g(x), which can be efficiently computed  by Berlekamp-Massey algorithm [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  The result, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=',  the minimal polynomial of g(x), will be the irreducible  polynomial we generate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  If choosing generalized division hashing, we need to  generate an irreducible polynomial over GF(2k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  The  procedure is the same as that described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' The only  difference is that all the calculations need to be done  under GF(2k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Appendix C: Calculation details  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  TP-TFQKD and TP-TFKGP in this work  The calculation of TP-TFKGP in this work is analo-  gous to that in TP-TFQKD [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  When Alice and Bob send intensities ka and kb with  phase difference θ, the gain corresponding to only one  detector (L or R) clicking is  QLθ  kakb =ykakb  �  eωkakb cos θ − ykakb  �  ,  QRθ  kakb =ykakb  �  e−ωkakb cos θ − ykakb  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  (C1)  where ykakb = e  −(ηaka+ηbkb)  2  (1 − pd), ωkakb = √ηakaηbkb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  The  overall  gain  can  be  expressed  as  Qkakb  =  1/2π  � 2π  0 (QLθ  kakb + QRθ  kakb)dθ = 2ykakb[I0(ωkakb) − ykakb],  where I0(x) refers to the zero-order modified Bessel func-  tions of the first kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  The total number for {ka, kb} is  xkakb = NpkapkbQkakb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  (C2)  The  valid  post-matching  events  on  the  basis  of  Z  can  be  divided  into  two  types:  correct  events {µaoa, obµb}, {oaµa, µbob}, and incorrect events  {µaoa, µbob}, {oaµa, obµb}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' The corresponding numbers  are denoted as nz  C and nz  E, respectively, which can be  written as  nz  C = xmin  xoaµb  x0  xµaob  x1  = xoaµbxµaob  xmax  ,  and  nz  E = xmin  xoaob  x0  xµaµb  x1  = xoaobxµaµb  xmax  ,  where x0 = xoaµb + xoaob, x1 = xµaob + xµaµb, xmin =  min{x0, x1}, and xmax = max{x0, x1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' sz  11 corresponds  to the number of successful detection events, where Alice  and Bob emit a single photon in different time bins in  the Z basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' The overall number of events in the Z basis  is  nz =nz  C + nz  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  (C3)  Considering the misalignment error ez  d, the number of bit  errors in the Z basis is mz = (1 − ez  d)nz  E + ez  dnz  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Thus,  the bit error rate in the Z basis is  Ez =mz  nz .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  (C4)  The overall number of “effective” events in the X basis  is  nx = 1  π  � δ  0  xθ  νaνbdθ  = Npνapνb  π  � σ+δ  σ  yνaνb(eωνaνb cos θ  + e−ωνaνb cos θ − 2yνaνb)dθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  (C5)  For simplicity, we only consider the case in which all  matched events satisfy θi − θj = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' In this case, when  ri  a ⊕ rj  a ⊕ ri  b ⊕ rj  b = 0 (1), the {νi  aνj  a, νi  bνj  b} event is con-  sidered to be an error event when different detectors (the  same detector) click at time bins i and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  The overall error count in the X basis can be given as  mx = 1  π  � σ+δ  σ  xθ  νaνbpEdθ  = 2Npνapνb  π  � σ+δ  σ  yνaνb×  �  (1 − yνaνb)2  eωνaνb cos θ + e−ωνaνb cos θ − 2yνaνb  − 1  �  dθ,  (C6)  where pE =  2qLθ  νaνbqRθ  νaνb  qθ  νaνbqθ  νaνb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  We can then calculate the parameters in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' (1) to  estimate the key rate and the information leaked after  the distribution stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' In the following description, let  x∗ denote the expected value of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' We denote the number  14  of {ka, kb} as xkakb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' We denote the number and error  number of events {ki  akj  a, ki  bkj  b} after post-matching as  nkiakj  a, ki  bkj  b and mkiakj  a, ki  bkj  b, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' For simplicity,  we abbreviate ki  akj  a, ki  akj  a as 2ka, 2kb when ki  a = kj  a and  ki  b = kj  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  (1) sz  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  sz  11 corresponds to the number of successful detec-  tion events, where Alice and Bob emit a single photon  in different time bins in the Z basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Define z10 (z01)  as the number of events in which Alice (Bob) emits a  single photon and Bob (Alice) emits a vacuum state  in an {µa, ob} ({oa, µb}) event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  The lower bounds of  their expected values are z∗  10 = Npµapobµae−µay∗  10 and  z∗  01 = Npoapµbµbe−µby∗  01, respectively, where y∗  10 and y∗  01  are the corresponding yields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' These can be estimated us-  ing the decoy-state method  y∗  01 ≥  µb  N(µbνb − ν2  b )  �eνbx∗  oaνb  poapνb  −ν2  b  µ2  b  eµbx∗  ˆoaµb  pˆoapµb  − µ2  b − ν2  b  µ2  b  xd∗  oo  pdoaob  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  (C7)  y∗  10 ≥  µa  N(µaνa − ν2a)  �eνax∗  νaob  pνapob  − ν2  a  µ2a  eµax∗  µaˆob  pµapˆob  − µ2  a − ν2  a  µ2a  xd∗  oo  pdoaob  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  (C8)  where xd  oo = xˆoaˆob +xˆoaob +xoaˆob represents the number  of events where at least one user chooses the declare-  vacuum state and pd  oo = pˆoapˆob + pˆoapob + poapˆob refers  to the corresponding probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Thus, the lower bound  of sz∗  11 is given by  sz∗  11 = z∗  10z∗  01  xmax  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  (C9)  (2) sz  0µb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' sz  0µb represents the number of events in the Z  basis, Alice emits a zero-photon state in the two matched  time bins, and the total intensity of Bob’s pulses is µb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  We define z00 (z0µb) as the number of detection events  where the state sent by Alice collapses to the vacuum  state in the {µa, ob} ({µa, µb}) event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' The lower bound  of the expected values is z∗  00 = pµapobe−µaxd∗  oo/pd  oaob  and z∗  0µb = pµapµbe−µax∗  oaµb/poapµb, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Here,  we employ the relationship between the expected value  x∗  oaµb = poax∗  ˆoaµb/pˆoa, and x∗  oaob = poapobxd∗  oo/pd  oo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' The  lower bound of sz∗  0µb can be written as  sz∗  0µb = x∗  oaµbz∗  00  xmax  + x∗  oaobz∗  0µb  xmax  (C10)  (3) sx  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' We define the phase difference between Al-  ice and Bob as θ = θa − θb + φab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' All valid events in the  X basis can be grouped according to the phase difference  θ (∈ {−δ, δ}∪{π−δ, π+δ}), and the corresponding num-  ber in the {ka, kb} event is denoted as xθ  kakb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' In the post-  matching step, two time bins are matched if they have the  same phase difference θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Suppose the global phase dif-  ference θ is a randomly and uniformly distributed value,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  and considering the angle of misalignment in the X ba-  sis σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' the expected number of single-photon pairs can be  given by  sx∗  11 = 1  π  � σ+δ  σ  xθ  νaνb × 2  νbe−(νa+νb)y∗  01  qθνaνb  νae−(νa+νb)y∗  10  qθνaνb  dθ  = Npνapνb  π  � σ+δ  σ  2νaνbe−2(νa+νb)y∗  01y∗  10  qθνaνb  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  (C11)  where qθ  νaνb is the gain when Alice chooses intensity νa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  and Bob chooses the intensity νb with phase difference θ  and xθ  νaνb = Npνapνbqθ  νaνb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  (4) ex  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' For single-photon pairs, the expected value of  the phase error rate in the Z basis is equal to the expected  value of the bit error rate in the X basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Therefore, we  first calculate the number of errors of the single-photon  pairs in the X basis tx  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' The upper bound of tx  11 can be  expressed as  t  x  11 ≤mx − (mνa0,νb0 + m0νa,0νb) + m00,00,  (C12)  where (mνa0,νb0 (m0νa,0νb) is the error count when the  states sent by Alice and Bob in time bin i (j) both col-  lapse to the vacuum state in events {2νa, 2νb}, and m00,00  corresponds to the event where the states sent by Al-  ice and Bob both collapse to vacuum states in events  {2νa, 2νb}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  The expected counts (nνa0,νb0 + n0νa,0νb)∗  and n∗  00,00 can be expressed as  (nνa0,νb0 + n0νa,0νb)∗ = 2  π  � σ+δ  σ  xθ  νaνb  e−(νa+νb)q∗  00  qθνaνb  dθ  =  δNpνapνbe−(νa+νb)q∗  00  π  (C13)  and  n∗  00,00 = 1  π  � σ+δ  σ  xθ  νaνb  �e−(νa+νb)q∗  00  qθνaνb  �2  dθ  =Npνapνb  π  � σ+δ  σ  e−2(νa+νb)(q∗  00)2  qθνaνb  dθ  (C14)  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Here q∗  00 = xd∗  oaob/(Npd  oo).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Using the fact  that the error rate of the vacuum state is always 1/2,  we have (mνa0,νb0 + m0νa,0νb)∗ = 1  2(nνa0,νb0 + n0νa,0νb)∗  and m∗  00,00 = 1  2n∗  00,00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Hence the upper bound of the bit  error rate in the X basis can be given by  ex  11 = tx  11/sx  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  (C15)  (5) φ  z  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' For a failure probability ε, the upper bound  of the phase error rate φz  11 can be obtained by using the  random sampling without replacement [69]  φ  z  11 ≤ex  11 + γU (sz  11, sx  11, ex  11, ϵ) ,  (C16)  15  where  γU(n, k, λ, ϵ) =  (1−2λ)AG  n+k  +  �  A2G2  (n+k)2 + 4λ(1 − λ)G  2 + 2  A2G  (n+k)2  ,  (C17)  with A = max{n, k} and G = n+k  nk ln  n+k  2πnkλ(1−λ)ϵ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  (6) szn  11 , szn  0µb and φ  zn  11 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Finally we can estimate the  parameters in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' (1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=', the lower bound of vacuum  events and single-photon pairs in a selected key group  sz  11L and sz  0µbL, and the upper bound of the phase er-  ror rate of the n-bit group φ  z  11U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' They can be obtained  from the parameters above by using the random sampling  without replacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  szn  11 ≥n  �  sz  11/nz − γU (n, nz − n, sz  11/nz, ϵ)  �  ,  szn  0µb ≥n  �  sz  0µb/nz − γU �  n, nz − n, sz  0µb/nz, ϵ  ��  ,  φ  zn  11 ≤φ  z  11 + γU �  szn  11 , sz  11 − szn  11 , φ  z  11, ϵ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  (C18)  (7)lkey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' We can also get the length of final keys of TP-  TFQKD, which can be used to simulate the performance  of OTUH-QDS in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  lkey =sz  0µb + sz  11[1 − H(φ  z  11)] − nzfH(Ez)  − log2  2  ϵcor  − 2 log2  1  2ϵP A  ,  (C19)  where ϵP A is the failure probability of privacy amplifica-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  BB84-KGP in BB84-QDS and this work  Both BB84-QDS and this work utilize decoy-state  BB84-KGP to generate correlated bit strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Accord-  ing to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' [70] we can estimate the number of vacuum  events and single-photon events under X basis,  sX,0 ≥ τ0  µ2n−  X,µ3 − µ3n+  X,µ2  µ2 − µ3  ,  (C20)  sX,1 ≥  τ1µ1  �  n−  X,µ2 − n+  X,µ3 − µ2  2−µ2  3  µ2  1  (n+  X,µ1 − sX,0  τ0 )  �  µ1(µ2 − µ3) − µ2  2 + µ2  3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  (C21)  where τn := �  k∈K e−kknpk/n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' is the probability that  Alice sends a n-photon state, and  n±  X,k := ek  pk  �  nX,k ±  �  nX  2 log 21  εsec  �  , ∀ k ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  We can also calculate the number of vacuum events,  sZ,0, and the number of single-photon events, sZ,1, for  Z = ∪k∈KZk, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=', by using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' (C20) and (C21) with  statistics from the basis Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Then we can obtain the phase  error rate of the single-photon events in the X basis by  φX,1 := cX,1  sX,1  ≤ vZ,1  sZ,1  + γU  �  sZ,1, sX,1, vZ,1  sZ,1  , εsec  �  ,  (C22)  where  vZ,1 ≤ τ1  m+  Z,µ2 − m−  Z,µ3  µ2 − µ3  ,  m±  Z,k := ek  pk  �  mZ,k ±  �  mZ  2 log 21  εsec  �  , ∀ k ∈ K,  γU(n, k, λ, ϵ) =  (1−2λ)AG  n+k  +  �  A2G2  (n+k)2 + 4λ(1 − λ)G  2 + 2  A2G  (n+k)2  The total number of events under Z basis is nZ =  �  k∈K nZ,k and the number of error events is mZ =  �  k∈K mZ,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  In BB84-QDS, the unknown information to the at-  tacker is given by  H = sX,0 + sX,1(1 − h(φX,1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  (C23)  In our protocol based on BB84-KGP, we need to esti-  mate parameters in a selected n-bit group, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=', the lower  bound of number of vacuum events and single-photon  events under X basis sn  X,0 and sn  X,1, and the upper bound  of the phase error rate of the single-photon events in the  X basis φ  n  X,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  sn  X,0 ≥n  �  sX,0/nZ − γU �  n, nZ − n, sX,0/nZ, ϵ  ��  , (C24)  sn  X,1 ≥n  �  sX,1/nZ − γU �  n, nZ − n, sX,1/nZ, ϵ  ��  , (C25)  φn  X,1 ≤φX,1 + γU �  sn  X,1, sX,1 − sn  X,1, φX,1, ϵ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  (C26)  Finally we can obtain  H = sn  X,0 + sn  X,1(1 − h(φ  n  X,1)) − λEC,  (C27)  where λEC = nh(mZ/nZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  SNS-KGP and SNS-QDS with random pairing  We first follow the calculation in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Alice and Bob obtain Njk(jk = {00, 01, 02, 10, 20}) in-  stances when Alice sends intensity j and Bob sends state  k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Here ’1’ and ’2’ represent the two intensities used in  the KGP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' After the sifted step, Alice and Bob obtain  16  njk one-detector heralded events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' We denote the count-  ing rate of source jk as Sjk = njk/Njk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' With all these  definitions, we have  N00 =[(1 − pz)2p2  0 + 2(1 − pz)pzp0pz0]N,  N01 =N10 = [(1 − pz)2p0p1 + (1 − pz)pzpz0p1]N,  N02 =N20 = [(1 − pz)2(1 − p0 − p1)p0  + (1 − pz)pzpz0(1 − p0 − p1)]N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  (C28)  In addition, we need to define two new subsets of X1  windows, C∆+ and C∆−, to estimate the upper bound of  eph  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' The number of instances in C∆± is  N∆± = ∆  2π (1 − pz)2p2  1N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  (C29)  We denote the number of effective events of right de-  tectors responding from C∆+ as nR  ∆+, and the number  of effective events of left detectors responding from C∆−  as nL  ∆−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' And we obtain the counting error rate of C∆±,  T∆ =  nR  ∆++nL  ∆−  2N∆±  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  If we denote the expected value of the counting rate of  untagged photons as sZ∗  1 , the lower bound of sZ∗  1  is  sZ∗  1  ≥sZ∗  1  =  1  2µ1µ2(µ2 − µ1)[µ2  2eµ1(S∗  01 + S∗  10)  − µ2  1eµ2(S  ∗  02 + S  ∗  20) − 2(µ2  2 − µ2  1)S  ∗  00],  (C30)  where S∗  jk is the expected value of Sjk, and S  ∗  jk and S∗  jk  are the upper bound and lower bound of S∗  jk when we  estimate the expected value from its observed value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  The expected value of the phase-flip error rate of the  untagged photons satisfies  eph∗  1  ≤ eph∗  1  = T  ∗  ∆ − 1  2e−2µ1S∗  00  2µ1e−2µ1sZ∗  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  (C31)  Here we use the fact that the error rate of vacuum state  is always 1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  If the total transmittance of the experimental setups  is η, then we have  n00 = 2pd(1 − pd)N00,  n01 = n10 = 2[(1 − pd)eηµ1/2 − (1 − pd)2e−ηµ1]N01,  n02 = n20 = 2[(1 − pd)eηµ2/2 − (1 − pd)2e−ηµ2]N02,  nt = nsignal + nerror,  Ez = nerror  nt  ,  nR  ∆+ = nL  ∆− = [TX(1 − 2ed) + edSX]N∆±,  where N00, N01, N10, N02, N20, N∆±  are defined in  Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' (C28) and (C29), and  nsignal =4Np2  zpz0(1 − pz0)[(1 − pd)e−ηµz/2  − (1 − pd)2e−2ηµz],  nerror =2Np2  z(1 − pz0)2[(1 − pd)e−ηµzI0(ηµz)  − (1 − pd)2e−2ηµz] + 2Np2  zp2  z0pd(1 − pd),  TX = 1  ∆  �  ∆  2  − ∆  2  (1 − pd)e−2ηµ1 cos2 δ  2 dδ  − (1 − pd)2e−2ηµ1,  SX = 1  ∆  �  ∆  2  − ∆  2  (1 − pd)e−2ηµ1 sin2 δ  2 dδ  − (1 − pd)2e−2ηµ1 + TX,  where I0(x) is the 0-order hyperbolic Bessel functions of  the first kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  In SNS-QDS, the unknown information to the attacker  is given by  H = sZ∗  1 (1 − h(eph∗  1  )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  (C32)  In our protocol based on SNS-KGP, there is  sZ∗  1n =n  �  sZ∗  1  − γU �  n, nZ − n, sZ∗  1 /nZ, ϵ  ��  ,  eph∗  1n =n  �  eph∗  1  + γU �  sZ∗  1n , sZ∗  1  − sZ∗  1n eph∗  1  , ϵ  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  (C33)  and  H = sZ∗  1n (1 − h(eph∗  1n )) − λEC,  (C34)  where λEC = nh(Ez).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  In SNS-QDS with random pairing, we follow the cal-  culation in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' After random pairing there are two  different phase error rates  ˜e′ph  1  =  (eph  1 )2  (eph  1 )2 + (1 − eph  1 )2 ,  (C35)  ˜e′ph  2  = 1  2,  (C36)  and a new bit flip error rate  E′ = 2Ez(1 − Ez).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  (C37)  The proportion of untagged bits after random pairing is  ∆′  un = ∆2  un + 2∆un(1 − ∆un),  (C38)  where ∆un = Np2  zpz0(1 − pz0)sZ  1 /nt is the proportion  of untagged bits before random pairing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' The unknown  information to the attacker is given by  H =∆′  un − ∆2  un[p1H(˜e′ph  1 ) + (1 − p1)H(˜e′ph  2 )]  − 2∆un(1 − ∆un)H(eph  1 ),  (C39)  where p1 = (eph  1 )2 + (1 − eph  1 )2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  17  Appendix D: Error correction and privacy  amplification  In this section we introduce our simulation of error  correction and privacy amplification in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' We use  the simulated data of TP-TFKGP at the distance of 400  km, which can be calculated by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' C3, C4, C19 in  Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' We implement our simulation on a desktop  computer with an Intel i5-10400 CPU (with RAM of 8  GB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  We use improved Cascade protocol to perform er-  ror correction to correct 300 (or 39830) errors among  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='267 × 106 (or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='695 × 108) bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' The detailed procedure  of improved Cascade protocol can be seen in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' [64],  where users first block their keys, and then perform bi-  nary process on each block to correct the errors and start  trace-back section to check the error, until there are no  errors in each block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  The results show that time con-  sumption is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='62 and 930.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='98 seconds with data sizes 1011  and 1013, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  In privacy amplification step, Alice chooses a random  universal2 hash function and performs it on the nZ-bit  keys after error correction to obtain l-bit final keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' The  choice of function is communicated to Bob, who also uses  it to get his l-bit final keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' In the simulation we utilize  a random Toeplitz matrix as the universal2 hash func-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' When the data size is 1013, the matrix is too large  that it exceeds the storage of our computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Thus in the  algorithm we block the matrix into 10 × 10 (100) subma-  trices with the same size to accomplish the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  For hash manipulations of every submatrix we follow the  procedure in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' [72], where fast Fourier transform is  used to speed up calculation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' In the simulation it  takes 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='98 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='057 × 104 seconds with data sizes 1011  and 1013, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [1] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Diffie and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Hellman, New directions in cryptogra-  phy, IEEE trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Theory 22, 644 (1976).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [2] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' DeMillo, Foundations of secure computation, Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' (Georgia Institute of Technology, 1978).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [3] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Rivest, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Shamir, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Adleman, A method  for obtaining digital signatures and public-key cryptosys-  tems, Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' ACM 21 (1978).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [4] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Elgamal, A public key cryptosystem and a signature  scheme based on discrete logarithms, IEEE trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Theory 31, 469 (1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [5] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Stevens, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Bursztein, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Karpman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Albertini,  and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Markov, The first collision for full sha-1, in Ad-  vances in Cryptology – CRYPTO 2017, LNCS, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' 10401  (Springer, 2017) pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' 570–596.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [6] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Boudot,  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Gaudry,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Guillevic,  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Heninger,  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Thom´e, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Zimmermann, Comparing the diffi-  culty of factorization and discrete logarithm: a 240-digit  experiment, in Annual International Cryptology Confer-  ence (Springer, 2020) pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' 62–91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [7] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Shor, Algorithms for quantum computation: dis-  crete logarithms and factoring, in Proceedings 35th an-  nual symposium on foundations of computer science  (IEEE, 1994) pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' 124–134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [8] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Gottesman and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Chuang, Quantum digital signa-  tures, arXiv preprint quant-ph/0105032 (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [9] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Clarke, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Collins, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Dunjko, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Andersson,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Jeffers, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Buller, Experimental demonstration  of quantum digital signatures using phase-encoded coher-  ent states of light, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' 3, 1174 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [10] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Collins, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Donaldson, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Dunjko, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Wallden,  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Clarke, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Andersson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Jeffers, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Buller,  Realization of quantum digital signatures without the re-  quirement of quantum memory, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' 113,  040502 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [11] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Dunjko, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Wallden, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Andersson, Quantum  digital signatures without quantum memory, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' 112, 040502 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [12] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Yin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Fu, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Chen, Practical quantum  digital signature, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' A 93, 032316 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [13] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Amiri, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Wallden, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Kent, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Andersson, Se-  cure quantum signatures using insecure quantum chan-  nels, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' A 93, 032325 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [14] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Puthoor, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Amiri, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Wallden, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Curty, and  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Andersson, Measurement-device-independent quan-  tum digital signatures, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' A 94, 022328 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [15] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Shang, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Lei, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Liu, Quantum random oracle  model for quantum digital signature, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' A 94,  042314 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [16] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Yang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Liu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Chen, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Zuo, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Zhou, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Shi, Theoretically extensible quantum  digital signature with starlike cluster states, Quantum  Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' 16, 12 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [17] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Thornton, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Scott, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Croal, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Korolkova,  Continuous-variable quantum digital signatures over in-  secure channels, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' A 99, 032341 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [18] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Qu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Liu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Dou, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Wang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Li,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Yang, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Ma, Multi-party ring quantum digital  signatures, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' B 36, 1335 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [19] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Zhu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Chen, and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Wang, Practical  quantum digital signature with configurable decoy states,  Quantum Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' 19, 151 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [20] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Lu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Cao, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Weng, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Gu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Xie, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Zhou, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Yin, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Chen, Efficient quantum  digital signatures without symmetrization step, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Ex-  press 29, 10162 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [21] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Zhou, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Li, and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Wang,  Twin-field quantum digital signatures, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' 46,  3757 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [22] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Zhao, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Shi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Shi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Huang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Guo, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Huang,  Multibit quantum digital signature with continuous vari-  ables using basis encoding over insecure channels, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' A 103, 012410 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [23] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Weng, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Lu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Gao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Xie, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Gu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Li, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Yin, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Chen, Secure and prac-  tical multiparty quantum digital signatures, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Express  29, 27661 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [24] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Qin, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Jiang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Yu, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Wang, Quan-  tum digital signatures with random pairing, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Applied 17, 044047 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [25] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Xie, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Wu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Yue, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' He,  18  Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Cao, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Peng, Practical long-distance twin-  field quantum digital signatures, Quantum Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  21, 150 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [26] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Yin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Fu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Liu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Tang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Wang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  You, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Zhang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Chen, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Wang, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Zhang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Chen, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Chen, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Pan, Experimental quan-  tum digital signature over 102 km, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' A 95,  032334 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [27] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Collins,  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Amiri,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Fujiwara,  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Honjo,  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Shimizu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Tamaki, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Takeoka, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Sasaki, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' An-  dersson, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Buller, Experimental demonstration  of quantum digital signatures over 43 db channel loss us-  ing differential phase shift quantum key distribution, Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' 7, 3235 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [28] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Yin, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Tang, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Zhao, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Liu,  X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Sun, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Li, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Puthoor, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  You, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Andersson, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Liu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Jiang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Ma,  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Zhang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Curty, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Chen, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Pan, Exper-  imental measurement-device-independent quantum digi-  tal signatures over a metropolitan network, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  A 95, 042338 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [29] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Roberts, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Lucamarini, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Yuan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Dynes, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Co-  mandar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Sharpe, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Shields, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Curty, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Puthoor,  and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Andersson, Experimental measurement-device-  independent quantum digital signatures, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  8, 1098 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [30] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Zhou, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Ding, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Zhang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Guo, and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Wang, Proof-of-principle demonstration of  passive decoy-state quantum digital signatures over 200  km, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Applied 10, 034033 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [31] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' An, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Zhang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Zhang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Chen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Wang,  Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Yin, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Wang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' He, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Hao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Liu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Zhou, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Guo, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Han, Practical quantum  digital signature with a gigahertz bb84 quantum key dis-  tribution system, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' 44, 139 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [32] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Ding, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Chen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Ji, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Zhou, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Zhang,  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Zhang, and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Wang, 280-km experimental demon-  stration of a quantum digital signature with one decoy  state, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' 45, 1711 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [33] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Richter, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Thornton, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Khan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Scott, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Jaksch,  U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Vogl, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Stiller, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Leuchs, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Marquardt, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Ko-  rolkova, Agile and versatile quantum communication:  Signatures and secrets, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' X 11, 011038 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [34] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Roehsner, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Kettlewell, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Fitzsimons, and  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Walther, Probabilistic one-time programs using quan-  tum entanglement, npj Quantum Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' 7, 98 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [35] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Roehsner, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Kettlewell, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Batalh˜ao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Fitzsimons, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Walther, Quantum advantage for  probabilistic one-time programs, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' 9, 5225  (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [36] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Pelet,  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Puthoor,  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Venkatachalam,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Wengerovsky, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Loncaric, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Neumann, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Liu,  Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Samec, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Stipˇcevi´c, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Ursin, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=', Unconditionally  secure digital signatures implemented in an 8-user  quantum network, New J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [37] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Cai, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Ren, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Zhang, Se-  curity of quantum digital signatures for classical mes-  sages, Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' 5, 9321 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [38] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Ma, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Cai, The postprocessing  of quantum digital signatures, Quantum Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' 16,  19 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [39] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' An, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Zhang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Zhang, and  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Wang, High-efficiency quantum digital signature  scheme for signing long messages, Quantum Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  18, 3 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [40] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Cai, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Wang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Wei, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Gao, Crypt-  analysis of multiparty quantum digital signatures, Quan-  tum Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' 18, 252 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [41] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Yao, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Liu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Xue, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Wang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Li, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Wang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' You,  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Huang, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Zhang, Multi-bit quantum digital sig-  nature based on quantum temporal ghost imaging, arXiv  preprint arXiv:1901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='03004 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [42] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Yin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Fu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Li, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Weng, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Li,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Gu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Lu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Huang, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
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+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='1093/nsr/nwac228 (2022),  nwac228.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [43] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Xie, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Weng, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Lu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Fu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Wang,  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Yin, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Chen, Scalable high-rate twin-  field quantum key distribution networks without con-  straint of probability and intensity, arXiv preprint  arXiv:2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='11165 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [44] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Bennett and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Brassard, Quantum cryptography:  Public key distribution and coin tossing, Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' 560, 7 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [45] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Lo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Curty, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Qi, Measurement-device-  independent quantum key distribution, Phy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  108, 130503 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [46] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Braunstein and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Pirandola, Side-channel-free  quantum key distribution, Phy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' 108, 130502  (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [47] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Zhou, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Yu, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Wang, Making the  decoy-state measurement-device-independent quantum  key distribution practically useful, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' A 93,  042324 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [48] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Lucamarini, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Yuan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Dynes, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Shields, Overcoming the rate–distance limit of quantum  key distribution without quantum repeaters, Nature 557,  400 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [49] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Ma, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Zeng, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Zhou, Phase-matching quantum  key distribution, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' X 8, 031043 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [50] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Wang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Yu, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Hu, Twin-field quan-  tum key distribution with large misalignment error, Phy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' A 98, 062323 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [51] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Liu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Xie, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Weng, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Gu,  X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Cao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Lu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Yin, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Chen, Homodyne detection quadrature phase shift keying  continuous-variable quantum key distribution with high  excess noise tolerance, PRX Quantum 2, 040334 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [52] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Xie, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Lu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Weng, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Cao, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Jia,  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Bao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Fu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Yin, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Chen,  Breaking the rate-loss bound of quantum key distribution  with asynchronous two-photon interference, PRX Quan-  tum 3, 020315 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [53] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Zeng, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Zhou, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Wu, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Ma, Mode-pairing  quantum key distribution, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' 13, 3903  (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [54] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Gu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Cao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Fu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' He, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Yin, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Yin, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Chen, Experimental measurement-device-  independent type quantum key distribution with flawed  and correlated sources, Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' 67, 2167 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [55] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Fu,  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Yin,  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Chen,  and  Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Chen,  Long-distance measurement-device-independent multi-  party quantum communication, Physical review letters  114, 090501 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [56] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Gu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Cao, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Yin, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Chen, Differential  phase shift quantum secret sharing using a twin field,  Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Express 29, 9165 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  19  [57] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Cao, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Li, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Weng, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Gu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Yin,  and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Chen, Finite-key analysis for quantum confer-  ence key agreement with asymmetric channels, Quantum  Science and Technology 6, 045019 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [58] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Menezes, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Van Oorschot, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Vanstone,  Handbook of applied cryptography (CRC press, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [59] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Krawczyk, Lfsr-based hashing and authentication, in  Annual International Cryptology Conference (1994) pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  129–139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [60] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Carter and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Wegman, Universal classes of  hash functions, in Proceedings of the ninth annual ACM  symposium on Theory of computing (1977) pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' 106–112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [61] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Wang, Beating the photon-number-splitting attack  in practical quantum cryptography, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' 94,  230503 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [62] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Lo, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Ma, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Chen, Decoy state quantum key  distribution, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' 94, 230504 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [63] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Brassard and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Salvail, Secret-key reconciliation by  public discussion, in Workshop on the Theory and Appli-  cation of of Cryptographic Techniques (Springer, 1993)  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' 410–423.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [64] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Yan, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Ren, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Peng, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Lin, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Jiang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Liu, and  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Guo, Information reconciliation protocol in quantum  key distribution system, in 2008 Fourth International  Conference on Natural Computation, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' 3 (IEEE, 2008)  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' 637–641.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [65] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Shoup, On fast and provably secure message authen-  tication based on universal hashing, in Annual Interna-  tional Cryptology Conference (Springer, 1996) pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' 313–  328.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [66] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Konig, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Renner, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Schaffner, The operational  meaning of min-and max-entropy, IEEE trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Inform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  Theory 55, 4337 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [67] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Hayashi, Exponential decreasing rate of leaked infor-  mation in universal random privacy amplification, IEEE  trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Inform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Theory 57, 3989 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [68] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Massey, Shift-register synthesis and bch decoding,  IEEE trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Inform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Theory 15, 122 (1969).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [69] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Yin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Zhou, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Gu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Xie, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Lu, and  Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Chen, Tight security bounds for decoy-state quan-  tum key distribution, Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' 10, 14312 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [70] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Lim, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Curty, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Walenta, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Xu, and  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Zbinden, Concise security bounds for practical decoy-  state quantum key distribution, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' A 89, 022307  (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [71] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Jiang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Yu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Hu, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Wang, Uncondi-  tional security of sending or not sending twin-field quan-  tum key distribution with finite pulses, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Ap-  plied 12, 024061 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='  [72] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Tang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Zhai, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Wu, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Yu,  High-speed and large-scale privacy amplification scheme  for quantum key distribution, Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
+page_content=' 9, 15733 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfMftT/content/2301.01132v1.pdf'}
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+arXiv:2301.04789v1  [cs.CC]  12 Jan 2023
+Isomorphisms Between Impossible and Hard
+Tasks
+Hunter Monroe
+January 13, 2023
+Abstract
+If no efficient proof shows that an unprovable arithmetic sentence
+‘x is Kolmogorov random’ (‘x∈R’) lacks a length t proof, an isomor-
+phism associates for each x impossible and hard tasks: ruling out any
+proof and length t proofs respectively. This resembles Pudl´ak’s feasi-
+ble incompleteness. This possible isomorphism implies widely-believed
+complexity theoretic conjectures hold—in effect, translating theorems
+from noncomputability about proof speedup and average-case hard-
+ness directly to complexity.
+Formally, we conjecture: sentence “Peano arithmetic (PA) lacks
+any length t proof of ‘x∈R’” lacks tO(1) length proofs in any consis-
+tent extension T of PA if and only if T cannot prove ‘x∈R’. If so,
+tautologies encoding the sentence lack tO(1) length proofs in any proof
+system P for x∈R sufficiently long (relative to the description of a
+program enumerating theorems of a theory T proving ‘P is sound’).
+R’s density implies: TAUT/∈AvgP, Feige’s hypothesis holds, and, a
+new conjecture, P’s nonoptimality has dense witnesses. If the isomor-
+phism holds for any Π0 sentence, PH does not collapse, because the
+arithmetic hierarchy does not collapse.
+1
+Introduction
+It seems implausible that theories such as PA or ZFC set theory could
+efficiently show that their unprovable (independent) sentences lack short
+proofs—they would seem to have too much information about their own
+1
+
+blind spots.1 Even a theory T stronger than PA might be unable to show
+efficiently that PA lacks short proofs of a sentence unprovable in both PA
+and T . We formalize this intuition focusing on true but unprovable sentences
+regarding Kolmogorov random strings (those strings incompressible by half).
+Define R = {x|∀p: if |p|≤|x|/2, then U(p)̸=x or pր}, U a deterministic
+universal Turing machine (TM) with no limit on its running time, x and p
+binary strings with |x| denoting x’s length, pր and p↓ signifying program
+p does or does not halt, and ‘x∈R’ is an arithmetic sentence encoding x∈R
+(single and double quotes signify the sentence encoding a mathematical state-
+ment). Let PA be any arithmetically sound theory that can encode p↓. Proof
+length is the length of the string representing the proof.2
+By analogy with Chaitin’s Incompleteness Theorem—which states that
+any consistent theory T with a computably enumerable (c.e.) set of proofs
+cannot prove or refute ‘x∈R’ for x∈R almost always (e.g., with x∈R suffi-
+ciently long)—we conjecture:
+Conjecture 1.1 For each x∈R, the sentence “PA lacks any length t proof
+of ‘x∈R’” lacks tO(1) length proofs in any consistent extension T of PA if and
+only if (iff) T cannot prove ‘x∈R’ (e.g., T
+̸
+tO(1) “PA̸
+t ‘x∈R’” iff T ̸
+‘x∈R’).3
+In the notation above, write T
+φ or T ̸
+φ respectively if T does or does
+not have a proof of φ of any length. Write T
+t φ for “theory T has a length
+t (or shorter) proof of sentence φ” and T ̸
+t φ if not. Write P
+t ψ (e.g. ψ is
+“PA̸
+t ‘x∈R’”) if proof system P proves tautologies encoding ψ in t steps,
+where the encoding indicates “no t step proof is a valid proof” as described
+in Section 4.
+1The ideas in this paper and earlier versions have benefited from discussions with
+the following: Scott Aaronson, Eric Allender, Olaf Beyersdorff, Ilario Bonacino, Cristian
+Calude, Yuval Filmus, Bill Gasarch, Valentina Harizonov, Pavel Hrubeˇs, Russell Impagli-
+azzo, Valentine Kabanets, Mehmet Kayaalp, Yanyi Liu, Ian Mertz, Daniel Monroe, Igor
+Oliveira, Pavel Pudl´ak, Rahul Santhanam, Till Tantau, Neil Thapen, Luca Trevisan, Avi
+Wigderson, Ryan Williams, Marius Zimand, and participants in seminars at George Wash-
+ington University and Davidson College, the Computational Complexity Conference 2022
+rump session, the Workshop on Proof Complexity 2022, and the 2022 Conference on Com-
+plexity with a Human Face. Remaining errors are my own.
+2Pudl´ak[27] surveys the literature on the lengths of proofs. Note that the conjecture
+implies PA is consistent—otherwise, all sentences have short proofs by contradiction.
+3The second clause refers to T ̸
+‘x∈R’ rather than T ̸
+“PA̸
+‘x∈R’”. These are equiv-
+alent since PA
+‘x∈R’ implies T
+‘x∈R’ by the assumption that T extends PA.
+2
+
+Conjecture 1.1 has immediate implications for the length of proofs of
+tautologies (under the encoding in Section 4) in proof systems:4
+Corollary 1.2 Conjecture 1.1 implies tautologies encoding “PA lacks any
+length t proof of ‘x∈R’” lack tO(1) length proofs in any proof system P, for x
+sufficiently long (e.g. P ̸
+tO(1) “PA̸
+t ‘x∈R’”).
+To give context for Conjecture 1.1, a theory T like PA might intuitively be
+expected to have limited information about sentences φ that are unprovable
+in T , that is, φ is independent of T with T ̸ φ and T ̸ ¬φ. In particular, we
+might expect that T lacks efficient proofs that its unprovable sentences lack
+short proofs (T
+̸
+tO(1) “T ̸
+t φ”). Contrary to this intuition, Pudl´ak[25] showed
+that T , if consistent, can in fact efficiently show there is no proof of contradic-
+tion derivable from the axioms of T (no T -proof of ‘0=1’) within t steps, e.g.,
+he showed T
+tO(1) “T ̸
+t ‘0=1’”.5 Thus, a finite analog of G¨odel’s Second In-
+completeness Theorem fails to hold. However, Kraj´ıˇcek and Pudl´ak[18] later
+showed that if for any consistent theory S, there exists a (stronger) con-
+sistent theory T such that S ̸
+tO(1) “T ̸
+t ‘0=1’”, then no optimal propositional
+proof system exists, a key open problem in proof complexity. It is further
+conjectured that this holds if T proves the consistency of S. Our conjecture
+differs by choosing φ nonconstructively after S and T that is unprovable in
+both theories.
+This paper shows that if φ is ‘x∈R’, other rich results follow given that
+R is infinite, dense, immune, noncomputable, and nonconstructive and given
+that choosing x after T provides a second-mover advantage:6
+• Existing conjectures hold—TAUT/∈AvgP, Feige’s hypothesis holds, no
+complete disjoint coNP pair exists, and (a new conjecture) P’s nonop-
+timality has dense witnesses.
+4A
+similar
+conjecture
+in
+an
+earlier
+version
+of
+this
+paper
+fails
+to
+hold:
+T
+̸tO(1)
+“PA̸ t ‘x/∈R’”. There are a fixed number of p such that |p|≤|x|/2, so there is a
+linear-size refutation that simulates each p for t steps. I appreciate this observation by
+Pavel Pudl´ak.
+5Pudl´ak and Friedman independently formulated this finite version of G¨odel’s second
+incompleteness theorem (a theory cannot prove its own consistency). Pudl´ak[26] improved
+his result from tO(1) to O(t) with additional assumptions.
+6For an overview of Kolmogorov randomness, see Li and Vitanyi[20] and Calude[5]. An
+extensive literature has examined the computational power of random strings. The halting
+problem can be computed using R as an oracle (Li and Vitanyi[20]), and R is truth-table
+complete (Kummer[19]. See also Allender et al[3][4] and Hirihara[15].
+3
+
+• An isomorphism maps impossible tasks, proving sentences ‘x∈R’, to
+hard tasks, proving tautology families encoding “PA̸
+t ‘x∈R’”, pre-
+serving average-case unprovability/hardness and theory/proof system
+nonoptimality.
+• The isomorphism gives a procedure to find hard tautologies for a given
+proof system P with high probability: randomly choose x twice as long
+as a program enumerating theorems of a theory T proving ‘P is sound’,
+and construct tautologies encoding ‘x∈R’.
+• The isomorphism translates classic computability theory results im-
+plied by Chaitin’s Theorem and their proofs to complexity theory. The
+computability and complexity results and their proofs are so closely
+identified, they are essentially equivalent but with different terminol-
+ogy.
+• A stronger conjectured isomorphism, which holds for any true φ ∈ Π0,
+translates the stronger result that PH does not collapse because the
+arithmetic hierarchy does not collapse.
+The paper is organized as follows. Section 2 provides preliminaries. Sec-
+tion 3 presents Calude and J¨urgensen’s[6] proof that the share of length n
+arithmetic sentences that are true and unprovable is bounded above zero.
+Section 4 discusses implications of Conjecture 1.1 for tautologies and proof
+systems.
+Section 5 discusses the isomorphism implied by Conjecture 1.1.
+Section 6 concludes.
+2
+Preliminaries
+Strings: With a binary alphabet {0, 1}, let Sn be the set of length n strings,
+which are ordered n-tuples. Let |x| be the length of a string and |S| be the
+cardinality of set S. A language L is a subset of ∪n≥0Sn.
+Density: Say the share of length n strings in L is bounded above zero
+if there exists c > 0 such that |L ∩ Sn|/n ≥ c for for sufficiently large n.
+This implies the weaker condition that L has positive upper density, i.e.,
+that lim sup
+n→∞
+|L ∩ {1, 2, . . . , n}|
+n
+> 0. If an event depending on n occurs with
+probability that tends to one as n tends to infinity, such as x∈R where |x|=n,
+say that it occurs with high probability.
+4
+
+Kolmogorov Random Strings: Fix a universal TM U (not necessarily pre-
+fix free).
+Define the set of Kolmogorov random strings as R = {x|∀p: if
+|p|≤|x|/2, then U(p)̸=x ∨ pր}.
+Theories: Theories are assumed to be PA or an extension of PA—any
+arithmetically sound theory that can encode and (if true) prove p↓.7 The
+standard definition of PA is modified so binary strings are encoded in arith-
+metic sentences as natural numbers, in binary not unary, adding a leading
+“1” to avoid losing leading zeros.
+PA under a standard approach writes
+numbers in unary, for instance, “SSS0”, the third successor of 0, encodes the
+number 3.
+Proof Systems: A propositional proof system is a polynomial time func-
+tion h ∈ FP with range TAUT (Cook and Reckhow[9]). For tautology τ, any
+string w such that h(w) = τ is a proof of τ. The proof system h is p-optimal
+if for any other proof system f, there exists g ∈ FP such that h(g(x)) = f(x).
+The proof system h is optimal if there exists c ≥ 1 such that the length of
+minimal f proofs of x are polynomially bounded in |x| with exponent c by
+minimal h proofs (Kraj´ıˇcek and Pudl´ak[18]).
+3
+Density of Unprovable Sentences
+To help understand Conjecture 1.1, we first describe an unconditional result
+by Calude and J¨urgensen[6] that the share of length n arithmetic sentences
+that are true and unprovable is bounded above zero—informally, proving
+theorems is impossible in the average case. The result relies on two facts:
+‘x∈R’ is almost always unprovable, by Chaitin’s Incompleteness Theorem,
+and length n strings are in R with high probability.8
+With that context,
+Conjecture 1.1 asserts that analogous statements hold for coTHEOREMS≤t=
+{⟨φ, 1t⟩|PA̸
+t φ} and, as shown in Section 4, TAUT.
+Chaitin’s Incompleteness Theorem states:9
+Theorem 3.1 For every consistent, arithmetically sound theory T with a
+c.e. set of theorems, ∃k∀x:|x|>k, T cannot prove ‘x∈R’.
+7This does not require induction; see Papadimitriou[24] Section 6.2.
+8See the proof of Theorem 5.2 in [6]. We note: the share of true, provable sentences is
+also bounded above zero; consider ‘|x|≥0’, which trivially provable for all x. These results
+imply positive upper density in either case, as the share at position 2n−1 is also bounded
+above zero.
+9See Li and Vitanyi[20] Corollary 2.7.2.
+5
+
+Proof:
+In brief, the proof is: otherwise, a string x could be concisely
+described as “the first string x of length n such that T proves ‘x∈R”’, con-
+trary to the definition of R. To spell this out, suppose T can prove ‘x∈R’
+infinitely often. Construct a program E that on binary input n enumerates
+all proofs in T and outputs the first string x with |x|≥n for which a proof
+of ‘x∈R’ appears. The input has log n bits, so for n sufficiently large, E(n)
+is a short description of the string x printed by E: “the first length n string
+x for which a proof of ‘x∈R’ appears in an enumeration of all proofs in T ”.
+For n sufficiently large, n/2≥|E|+ log n. For such an n, let p=E(n) and let
+k = n. This contradicts the definition of R and proves the theorem.
+Theorem 3.1 implies that R is immune: infinite with no infinite c.e. subset.
+R’s pathology—no program or theory can identify more than a finite number
+of its members—works to our favor. Furthermore, Lemma 3.2 shows strings
+are in R with high probability.
+This gives us a second-mover advantage
+against any theory T —we can (noncomputably) choose x∈R after seeing T
+to highlight T ’s inefficiency.
+Lemma 3.2 x∈R with high probability.
+Proof:
+Compare the number of length n strings, 2n, with the number of
+programs p with |p|≤n/2, which is 2n/2+1−1. The number of length n strings
+not in R is therefore at most 2n/2+1 − 1. Therefore, R’s share of length n
+strings is at least 1 − 2−n/2, so x∈R with high probability.
+Calude and J¨urgensen’s result implies:
+Theorem 3.3 For every theory T extending PA, the set of sentences {‘x∈R’|
+x∈R} which are true and unprovable has a share in length n arithmetic sen-
+tences that is bounded above zero, for n sufficiently large.10
+Proof:
+Theory T can prove sentences ‘x∈R’ where x∈R only in finite
+cases, by Theorem 3.1. The sentences ‘x∈R’ satisfy |‘x∈R’| = |x| + c, where
+c is a constant not depending on |x|, giving the overhead of encoding ‘x∈R’
+beyond x itself. Therefore, the share length n sentences of form ‘x∈R’ is
+10The share of true and provable sentences is also bounded above zero: consider the
+sentences |x|≥0. In the terminology of Downey et al[10], the set of arithmetic sentences
+is “partially computable at density r”. Section 6 provides three new proofs that the share
+of length n arithmetic sentences that are unprovable is bounded above zero.
+6
+
+exactly 2−c and the share of form ‘x∈R’ for which in addition x∈R is at least
+1 − 2−n/2 for each n per Lemma 3.2 and is asymptotically one.
+Therefore, length n arithmetic sentences of the form ‘x∈R’ with x∈R have
+share bounded above zero by 2−c(1 − 2−n/2) − ǫ for ǫ>0, for n sufficiently
+large.
+4
+Tautologies and Proof Systems
+This section specifies how a tautology will encode the sentence “PA̸
+t ‘x∈R’”
+in Corollary 1.2 and shows that Conjecture 1.1 therefore implies some widely
+believed conjectures about tautologies and proof systems, implies one new
+conjecture, and gives a procedure to find hard tautologies for a proof system
+with high probability.
+To specify the encoding: for a given x, “PA̸
+t ‘x∈R’” is equivalent to
+⟨‘x∈R’, 1t⟩∈coTHEOREMS≤t. coTHEOREMS≤t and TAUT are both coNP-complete
+languages, so some polynomial-time reduction r from coTHEOREMS≤t to TAUT
+maps ⟨φ, 1t⟩ to tautology r(⟨φ, 1t⟩).11 Conjecture 1.1 implies that for any
+proof system P, for any sufficiently long x∈R, the family of tautologies
+r(⟨‘x∈R’, 1t⟩) lacks tO(1) length proofs. Equivalently, r(⟨‘x∈R’, 1t⟩) has tO(1)
+length proofs in P for only a finite set of x∈R.
+4.1
+Implications
+Three interesting implications immediately follow: dense witnesses to the
+nonoptimality of proof systems; TAUT≤t /∈AvgP; and Feige’s hypothesis.
+4.1.1
+Dense Witnesses to Nonoptimality
+Chen et al[8] and Kraj´ıˇcek[17] establish that no optimal propositional proof
+system exists iff, for each proof system, there is an easily described (P-
+uniform) family of tautologies requiring proofs of superpolynomial length,
+11Tautologies produced by r confirm that every possible proof is not a valid proof, not
+that every set of bits satisfies a sentence with a single universal bounded quantifier. These
+tautologies do not result from a propositional translation, which takes a sentence with a
+single universal bounded quantifier, i.e. applying to numbers up to a ceiling, and replaces
+each bit of a number with a logical variable in a tautology, and so on.
+7
+
+called a hard sequence. Conjecture 1.1 has a stronger implication not con-
+sidered in the literature: for each proof system, there is a dense set of hard
+sequences r(⟨‘x∈R’, 1t⟩) letting x range over all x∈R sufficiently long. To be
+precise:
+Theorem 4.1 Under Conjecture 1.1, there is no optimal proof system P.
+Proof:
+This follows immediately from Theorem 3.1.
+Theorem 4.2 Under Conjecture 1.1, there are a dense set of hard sequences
+for each P witnessing its nonoptimality. That is, in an enumeration over all
+φ of tautology families r(⟨φ, 1t⟩), the subset of the form r(⟨‘x∈R’, 1t⟩) with
+x∈R has positive upper density.
+Proof:
+This follows immediately from Theorem 3.3.
+4.1.2
+TAUT≤t /∈AvgP
+R’s density implies TAUT≤t /∈AvgP by Conjecture 1.1 and Theorem 3.3. Re-
+call that a distributional problem (L, D) (where D is an ensemble of distri-
+butions for length n inputs) is in AvgP if there is a TM M accepting L and
+constants C and ǫ > 0 such for that every n:
+E
+y∈RDn
+�TM(y)ǫ
+n
+�
+≤ C, where
+TM is the function that maps a string x to how many steps M(x) takes, and
+E is the expectation operator.
+Theorem 4.3 Conjecture 1.1 implies: (i) coTHEOREMS≤t /∈AvgP and (ii)
+TAUT≤t /∈AvgP, for any distribution applying non-negligible weight to ele-
+ments of the form ⟨‘x∈R’, 1t⟩ and r(⟨‘x∈R’, 1t⟩).
+Proof:
+(i) Conjecture 1.1 implies that the share of length n elements in
+the language coTHEOREMS≤t of the form ⟨‘x∈R’, 1t⟩ is bounded above zero,
+which is sufficient to show coTHEOREMS≤t /∈AvgP, by the following argu-
+ment similar to that of Calude and J¨urgensen[6].
+That proof notes that
+|‘x∈R’|=|x|+c. Therefore, |⟨‘x∈R’, 1t⟩|=|x|+c+t for a slightly larger c. Fix
+n, let |x| range from 0 to n−c, and let t = n−|x|−c. All but x is fixed,
+so among length n elements, and the number of possibilities for x is the
+sum 20+ . . . +2n−c=2n−c+1−1. Any string x is in R with high probability by
+8
+
+Lemma 3.2. Therefore, the share of length n elements of the form ⟨‘x∈R’, 1t⟩
+with x∈R is roughly 2n−c+1/2n, so the share exceeds 2−c+1−ǫ for ǫ>0 for
+n sufficiently large. This implies coTHEOREMS≤t /∈AvgP for any distribution
+applying non-negligible weight to ⟨‘x∈R’, 1t⟩ where x∈R.
+(ii) The same argument holds for tautologies r(⟨‘x∈R’, 1t⟩), if r maps all
+elements in its domain of length n to tautologies all of the same length, say,
+64n, implying the share of length n tautologies of the form r(⟨‘x∈R’, 1t⟩) with
+x∈R is bounded above zero by 2−c+1−64−ǫ with ǫ>0 for n sufficiently large.
+Then TAUT≤t /∈AvgP for any distribution applying non-negligible weight to
+the tautologies r(⟨‘x∈R’, 1t⟩).
+4.1.3
+Feige’s Hypothesis
+A similar argument shows that Feige’s hypothesis follows from Conjecture
+1.1. Recall that in the study of random k-CNF, with n Boolean variables and
+k variables per clause, there are
+�n
+k
+�
+choices for variables in a clause, each
+of which may or may not be negated. Suppose a random k-CNF consists of
+a fraction ∆n from these possibilities for a constant ∆ > 0. For ∆ above
+a threshold, the formula is unsatisfiable with high probability, with a sharp
+threshold by Friedgut[12], estimated empirically at ∆ ≈ 4.26 for 3-CNFs.
+Feige[11] conjectures that k-CNF refutation is hard above this threshold and
+below nk/2.
+The set UNSATk of unsatisfiable k-CNFs is coNP-complete, so there is a
+reduction u to this language from coTHEOREMS≤t. Again choose u to maps
+all elements in its domain of length n to all of the same length elements of
+UNSATk of the same length.
+Theorem 4.4 Conjecture 1.1 implies that Feige’s hypothesis holds for a suf-
+ficiently high clause density ∆ determined by u.
+Proof:
+The share of length n inputs ⟨‘x∈R’, 1t⟩ in coTHEOREMS≤t is bounded
+above zero by Theorem 4.3(i).
+Therefore, the share of length n inputs
+u(⟨‘x∈R’, 1t⟩)∈UNSATk will also be bounded above zero. The reduction u will
+imply that these k-CNFs have a certain clause density ∆. Feige’s Hypothesis
+holds at this clause density or higher.
+9
+
+4.2
+Procedures for Creating Hard Tautologies
+Under Conjecture 1.1, for a given proof system P, each of the following
+procedures yield a hard sequence of tautologies. All but the last procedure
+are noncomputable (each step is labeled computable or noncomputable):
+Theorem 4.5 Each of the following procedures yields a hard sequence of
+tautologies, given a proof system P:
+• (i) make a (finite) list of x∈R for which P has proofs of length tO(1)
+for tautologies r(⟨‘x∈R’, 1t⟩) (possibly noncomputable); (ii) choose any
+x∈R not on this list (noncomputable); and (iii) create tautologies
+r(⟨‘x∈R’, 1t⟩) (computable).
+• An alternative procedure is: (i) given P, choose theory T extend-
+ing PA that proves ‘P is sound’ (computable); (ii) calculate the k in
+Theorem 3.1 for T (computable); (iii) choose any ‘x∈R’ of length k
+or longer (noncomputable); and (iv) create tautologies r(⟨‘x∈R’, 1t⟩)
+(computable).
+• A probabilistic, polynomial-time computable procedure exists, which
+is the same as the previous procedure, but choosing a random string x
+of length k or longer. Then, x∈R with high probability by Lemma 3.2,
+so tautologies r(⟨‘x∈R’, 1t⟩) are hard for P with high probability.12
+Proof:
+These follow from Chaitin’s Theorem and Conjecture 1.1.
+The Cook-Reckhow program aims to show NP̸=coNP by proving super-
+polynomial lower bounds on proof length for specified tautologies for par-
+ticular proof systems P of increasing power.13 Conjecture 1.1 is so strong,
+it would if true complete the Cook-Reckhow program, by identifying hard
+tautologies for any proof system.
+4.3
+Relationship to the Literature
+Identifying hard families of tautologies has been a key objective in the fields
+of proof and computational complexity. As noted above, Conjecture 1.1 pulls
+12Pseudorandom bits with a seed less than |x|/2 are not sufficient for this procedure,
+which therefore cannot be derandomized.
+13The Proof Complexity Zoo (Vinyals[32]) lists 35 proof systems and 23 candidates for
+hard tautologies, which provide lower bounds. The literature includes many other proof
+systems and candidates.
+10
+
+together the literature’s three primary proposals for hard tautologies: ran-
+dom tautologies, tautologies encoding unprovable sentences, and tautologies
+created from strings outside a PRG’s range.14
+4.3.1
+Random Tautologies
+Conjecture 1.1 non-trivially derandomizes the long-standing conjecture that
+random tautologies are hard any proof system,15 and in particular random
+k-CNF refutation is hard (Feige’s Hypothesis). Under the conjecture, each
+x∈R is mapped to a family of tautologies, using the random bits frugally
+relying on the fact that R is immune. A trivial derandomization might use
+x∈R less frugally to create a single tautology rather than a family.16
+4.3.2
+Independent Sentences
+Conjecture 1.1 is related to Pudl´ak’s Feasible Incompleteness Thesis: “The
+phenomenon of incompleteness manifests itself at the level of polynomial time
+computations”. A related conjecture is the nonexistence of an optimal proof
+system for TAUT (an NTM accepting TAUT).17 If there is no optimal proof
+system for TAUT, there is no optimal NTM accepting any paddable coNP-
+complete language (Kraj´ıˇcek and Pudl´ak[18]), and there are hard P-uniform
+families of inputs (hard sequences) for each such language (Chen et al[8]).
+Monroe[22] shows that certain hard sequences have a link to noncomputabil-
+ity:
+i. If hard sequences exist, one exists referring to non-halting TMs: for
+any M accepting coBHP = {⟨N, x, 1t⟩| there is no accepting path of
+nondeterministic TM N on input x with t or fewer steps}, there exists
+14This by no means a complete list of proposals for hard tautologies, as we have focused
+on those some how related to our conjecture below. See the survey in Kraj´ıˇcek[17] Chapter
+19.
+15Kraj´ıˇcek[17] states: “Most researchers in proof complexity agree, I suppose, with the
+folklore informal conjecture that for any proof system most formulas are hard.”
+16A standard technique, the “incompressibility method”, analyzes objects randomly
+selected according to a probability distribution by studying instead Kolmogorov random
+objects which have the same properties—in our case, families of tautologies. See Li and
+Vitanyi[20] Chapter 6. Allender et al[3] notes that significant work in derandomization
+studies the interplay between strings chosen at random and Kolmogorov random strings.
+17For broader overviews of that rich literature, see Pudl´ak[28] Section 6.4 and [29] and
+Kraj´ıˇcek[17] Chapter 21.
+11
+
+some ⟨N′, x′⟩ where N′ does not halt on x′ such that ⟨N′, x′, 1t⟩ is a
+hard sequence.
+ii. Equivalently,18 for any M accepting coTHEOREMS≤t, there exists some
+sentence φ′ with no proof in PA of any length, such that ⟨φ′, 1t⟩ is a
+hard sequence.
+Conjecture 1.1 specifies φ′ =‘x∈R’ where x∈R.
+4.3.3
+Proof Complexity Generators
+The literature on proof complexity generators suggests that tautologies en-
+coding “string x lies outside the range of a PRG” are hard.19 In Conjecture
+1.1, tautologies encode that there is no PA proof within t steps that a string x
+lies outside the range of every program p with |p|≤|x|/2 run for an unlimited
+number of steps. Thus, these conjectures are thematically similar.
+5
+The Isomorphism
+Conjecture 1.1 asserts that for each x, T
+̸
+tO(1) “PA̸
+t ‘x∈R’” iff T ̸ ‘x∈R’. This
+is a monumental claim—a fact about noncomputability (T ̸ ‘x∈R’) provides
+a complete understanding of a question in complexity (T
+̸
+tO(1) “PA̸
+t ‘x∈R’”).
+Theorems 3.1 and 3.3 and other classic results from computability theory
+translate to complexity theory under the isomorphism.
+Conjecture 1.1 implies there is an isomorphism I between impossible-to-
+prove sentences of the form ‘x∈R’ and hard-to-prove sentences of the form
+“PA̸
+t ‘x∈R’” (the family indexed by t), so I(‘x∈R’) =“PA̸
+t ‘x∈R’”. The
+isomorphism preserves the following properties:
+• Hardness: For each T , T ̸ ‘x∈R’ iff T
+̸
+tO(1) “PA̸
+t ‘x∈R’”, that is, I
+maps the property “impossible-to-prove” to “hard-to-prove”.
+• Density: For each T , the share of “impossible-to-prove”sentences in
+length n sentences is bounded above zero, just as the share of “hard-
+to-prove” sentences is bounded above zero.
+18Chen and Flum[7] Theorem 31 shows that coBHP and coTHEOREMS≤t are fixed param-
+eter tractable reducible to each other.
+19See Alekhnovich et al[2] and Kraj´ıˇcek[16] for more information on proof complexity
+generators.
+12
+
+• Theory/Proof System Nonoptimality: Just as there is no best theory
+for proving ‘x∈R’, there is not best proof system for proving tautologies
+r(⟨‘x∈R’, 1t⟩).20 In both cases, adding a new axiom ‘x∈R’ leads to a
+stronger theory or proof system.
+• Blind Spots: Theorem 3.1 establishes that T proves x∈R in at most
+finite cases. Under Conjecture 1.1, T efficiently proves “PA̸
+t ‘x∈R’”
+in exactly the same cases. T ’s limited information about R constrains
+its capacity to prove in an identical manner.
+5.1
+Another Implication
+This section presents another implication of Conjecture 1.1: there does not
+exist a complete disjoint coNP pair with respect to polynomial reductions.21
+A disjoint coNP pair is two sets (A,B) with A,B⊂coNP and A∩B=∅. Say
+disjoint coNP pair (A,B) is polynomially reducible to disjoint coNP pair
+(C,D) if there if a polynomial time function that maps A into C and B into
+D. Say that two sets are p-inseparable, if there does not exist S∈P such that
+S contains one set and is disjoint with the other.
+Razborov[30] defines the canonical NP pair of a proof system. We de-
+fine the canonical coNP pair of a theory T consisting of the set of φ not
+provable within t steps and the set not refutable within t steps, i.e., as
+({⟨φ, 1t⟩|T ̸
+t φ}, {⟨φ, 1t⟩|T ̸
+t ¬φ}).
+Theorem 5.1 Under Conjecture 1.1, there is no complete disjoint coNP
+pair among canonical coNP pairs for theories. Canonical coNP pairs for
+theories are universal among disjoint coNP pairs, so there are no complete
+disjoint coNP pairs. There exists a p-inseparable disjoint coNP pair.
+Proof:
+Suppose the canonical coNP pair for T is a complete disjoint
+coNP pair. Choose x∈R such that T ̸
+‘x∈R’. Let T ′ be T plus the new
+axiom ‘x∈R’. Then, the canonical coNP pair for T can be reduced to the
+the canonical coNP pair for T ′, but the reverse is not true. Otherwise, the
+reduction could be used to create efficient proofs in T for “PA̸
+t ‘x∈R’”, using
+the efficient proofs in T ′. Therefore, the canonical pair for T is not complete.
+20Pudl´ak[29] states “the fact that the logical strength of theories cannot be bounded is
+likely to be projected into these proof systems”.
+21Pudl´ak[29] explores these conjectures that are stronger than the nonexistence of an
+optimal proof system.
+13
+
+Canonical coNP pairs for theories are universal among disjoint coNP
+pairs: for any such pair, there is a theory that proves each member of the
+pair is in coNP and proves that the pair is disjoint.
+Then this pair is
+reducible to the canonical coNP pairs for that theory.
+The canonical coNP pair for PA is p-inseparable, as Conjecture 1.1 im-
+plies that NP̸=coNP.
+5.2
+Translating Theorems from Computability Theory
+to Complexity Theory
+The isomorphism in Conjecture 1.1 implies that the above results and their
+proofs can be interpreted in two ways. First, they can be seen as conditional
+results in complexity theory, proved by complexity theoretic arguments, as-
+suming the conjecture holds. Second, they can be seen as classic results and
+their proofs in computability theory, which immediately hold in complexity
+theory by referring to the isomorphism. Under this view, the two approaches
+are equally valid.
+Consider the second view for each of the results above. The following are
+alternative proofs, citing a result in computability theory and referring to
+the isomorphism.
+Theorem 5.2 Theorem 3.1 (Chaitin) implies that there is no optimal theory
+T , as a new axiom ‘x∈R’ can always be added. This implies there is no
+optimal proof system P, by the isomorphism in Conjecture 1.1.
+Theorem 3.3 (Calude and J¨urgensen) implies Theorems 4.2 (dense hard
+sequences), 4.3 (TAUT≤t /∈AvgP), and 4.4 (Feige’s Hypothesis), by the iso-
+morphism in Conjecture 1.1.
+Theorem 3.1 implies that the procedures in Theorem 4.5 yield x∈R that
+is hard for theory T . Therefore, they yield x∈R that is hard for proof system
+P, by the isomorphism in Conjecture 1.1.
+For a consistent axiomatizable theory in which all recursive sets are de-
+finable, there is no complete pair ({φ|T ̸
+φ}, {φ|T ̸
+¬φ}) under computable
+reductions, and the two sets are computably inseparable (Smullyan[31]). This
+implies Theorem 5.1 (disjoint coNP pairs), by the isomorphism in Conjec-
+ture 1.1.
+The isomorphism translates unprovable sentences to hard-to-prove tau-
+14
+
+tologies. This contrasts with propositional translation, which translates prov-
+able sentences with bounded quantifiers to easy-to-prove tautologies.
+An important question is how to distinguish which results translate from
+computability theory.
+We conjecture that any result that can be proven
+relying only on Chaitin’s Theorem translates under Conjecture 1.1. This
+raises the question of whether a stronger isomorphism could translate more
+results. The next section considers such isomorphisms.
+5.3
+Other Isomorphisms
+A more granular version of Conjecture 1.1 can replace the universal quantifier
+in the definition of R with a separate statement for each p. Given x, with
+|p|≤|x|/2, there are 2|x|/2−1 such p. For each p, the definition requires either
+U(p)̸=x or pր.
+If U(p) halts with U(p)̸=x, then any T (including PA)
+can prove this by simulating the computation of U on p to verify U(p)̸=x.
+The only interesting case is when T ̸
+t ‘pր’. This yields the more granular
+conjecture.
+Conjecture 5.3 For each p, T
+̸
+tO(1) “PA̸
+t ‘pր’” iff T ̸
+‘pր’.22
+Note that in an unbounded setting, we know unconditionally that proving
+or accepting ‘pր’ is impossible on average, as a corollary of Theorem 3.3:
+Corollary 5.4 (i) For any theory T , the share of p of length n is bounded
+above zero for which pր but T ̸
+‘pր’, for n sufficiently large.
+(ii) For any NTM M which accepts only elements of {p|pր} (the set of
+nonhalting programs), the share of length n inputs for which M fails to accept
+correctly is bounded above zero, for n sufficiently large.23
+Proof:
+(i) Consider the set of p which have the following form: for a given
+x∈R, p enumerates all proofs in T and halts if one proof shows x∈R (the
+input x is hard coded). By Theorem 3.3, the share of p of length n of this
+form for which pր but T ̸
+‘pր’ is bounded above zero.
+22A weaker conjecture could assert that the equivalence holds on average: The share
+of p of length n is bounded above zero for which: T
+̸tO(1)
+“PA̸ t ‘pր’” iff T ̸
+‘pր’, for n
+sufficiently large.
+23See however Hamkins and Miasnikov[14], which is sensitive to the computational
+model.
+15
+
+(ii) follows immediately from (i); otherwise M could serve as a theory
+contradicting part (i).
+Consider what kind of isomorphism would be sufficient to imply that
+the polynomial hierarchy (PH) does not collapse. Conjecture 1.1 implies
+NP ̸= coNP24 and therefore, Πp
+0 ̸= Πp
+1, but it does not seem to imply
+Πp
+i−1 ̸= Πp
+i for i > 1. The sentences ‘x∈R’ and ‘pր’ each employ a single
+universal quantifier.25
+This suggests considering an isomorphism allowing
+sentences φ with more quantifiers:
+Conjecture 5.5 For each true26 φ ∈ Π0, T
+̸
+tO(1) “PA̸
+t φ” iff T ̸
+φ.
+This very strong (and possibly false) conjectured isomorphism is sufficient to
+imply PH does not collapse:
+Theorem 5.6 (i) If Conjecture 5.5 holds, PH does not collapse. (ii) Be-
+cause the arithmetic hierarchy does not collapse, PH does not collapse by the
+isomorphism in Conjecture 5.5.
+Proof:
+(i) Consider the theory T +Π0
+1, which is T with every true Π0
+1 sen-
+tence added as an axiom. Choose any true φ in Π0
+2/Π0
+1 such that (T +Π0
+1)̸
+φ,
+which exists because the arithmetic hierarchy does not collapse.
+Then,
+(T +Π0
+1) ̸
+tO(1) “PA̸
+t φ” by Conjecture 5.5. Let coTHEOREMS2
+≤t= {⟨φ, 1t⟩|PA̸
+t φ
+and φ/∈Π0
+1}, which is Πp
+2-complete. Then, coTHEOREMS2
+≤t has hard sequences,
+even for an NTM accepting proofs for T with an oracle for coTHEOREMS≤t.
+Therefore, Πp
+1̸=Πp
+2, and likewise Πp
+i−1̸=Πp
+i , so PH does not collapse.
+(ii) See (i).
+Theorem 5.6(ii) confirms that the stronger isomorphism Conjecture 5.5,
+if correct, allows stronger results from computability theory to be translated
+to complexity theory.
+24See Kraj´ıˇcek and Pudl´ak[18]; it also implies NEXP ̸= coNEXP.
+25The sentence ‘pր’, spelled out, states that for any computation, it does not describe
+a halting computation for p. The sentence ‘x∈R’ employs a second universal quantifier:
+for any p with |p|≤|x|/2, pր or U(p)̸=x.
+26Examples of true φ ∈ Π0 are ‘pր’, which is true if p does not have a halting com-
+putation, and ‘x∈R’, which is true if it is not falsified by some p with |p|≤|x|/2. The
+requirement that φ be true rules out the counterexamples in footnotes 4 and 5, which
+involve the false statements ‘x/∈R’ and ‘0=1’.
+16
+
+5.4
+Plausibility Arguments
+Informally, Conjectures 1.1, 5.3, and 5.5 assert that there is no “free lunch”
+obtained by trying to prove hard (e.g., “PA̸
+t ‘x∈R’”) rather than impossible
+(‘x∈R’) sentences.
+Consider these plausibility arguments for the conjectures. First, suppose
+T ̸ ‘x∈R’ but T
+tO(1) “PA̸
+t ‘x∈R’”, contrary to Conjecture 1.1. The patho-
+logical nature of R might overwhelm T ’s ability to prove, so as to rule out
+tO(1) length proofs, necessitating a brute force approach. R’s pathology might
+imply that T cannot distinguish among possible proofs in PA of ‘x∈R’. This
+is a vague argument.
+Second, for t sufficiently large, T might be expected to lose the ability
+to prove statements about halting.
+To be more precise, define BB(c) =
+max{k:|p|≤c and p halts in k steps})—this is a busy beaver function defined
+with respect to the length of program description. Given a theory T , there
+exists cT such that T cannot prove the value of BB(c) for c≥cT ; otherwise, T
+could solve the halting problem.27 Given T , choose x∈R such that cT <|x|/2.
+For t>BB(|x|/2), there are programs p with p≤|x|/2 which have not halted
+by t; none of these will ever halt, but T cannot prove pր. For such x, p,
+and t, plausibly, T
+̸
+tO(1) “PA̸
+t ‘pր’” and therefore T
+̸
+tO(1) “PA̸
+t ‘x∈R’”.
+An informal argument in favor of Conjecture 5.3 is: if T ̸
+‘pր’, then T
+lacks any information that would allow efficient proofs as in T
+̸
+tO(1) “PA̸
+t ‘pր’”.
+This seems like a weaker argument than the one for Conjecture 1.1, which can
+be seen as an average of Conjecture 5.3 across all programs p with |p|≤|x|/2.
+The possibility that Conjecture 1.1 is independent of ZFC is heightened by
+its reliance on independent sentences such as ‘x/∈R’. The following argument
+suggests that Conjecture 1.1 is not independent of ZFC. Consider the weaker
+conjecture: for all T , there exists x∈R such that T
+̸
+tO(1) “PA̸
+t ‘x∈R’”. For
+any x that is predicted by this weaker conjecture, add the “false axiom”
+‘x/∈R’ to ZFC to create ZFC’ (which is not arithmetically sound and has
+an non-c.e. infinite set of axioms). It may be possible to do this such that
+ZFC’ is consistent, indicating that the weaker conjecture is false in ZFC’ and
+therefore independent of ZFC. However, this technique does not work for
+Conjecture 1.1, as it would require adding ‘x/∈R’ for x of length n with high
+probability. This implies a contradiction, as x∈R with high probability, by
+27See Aaronson[1].
+17
+
+Lemma 3.2.28
+6
+Conclusion
+This paper proposes that tautologies r(⟨‘x∈R’, 1t⟩) are hard by analogy to
+Chaitin’s Theorem. It conjectures that exactly the same x leads to hard-
+to-prove tautologies and impossible-to-prove arithmetic sentences. This sets
+up an isomorphism which carries over results from computability theory to
+complexity theory.
+This paper notes that the share of length n arithmetic sentences which
+are unprovable is bounded above zero, pointing Calude and J¨urgensen’s re-
+sult (Theorem 3.3) relying on properties of R. Here are three other proofs
+of the same result which also involve stuffing in any string into a slot to
+create density not using R. First, a trivial argument uses “dense padding”:
+choose any φ unprovable in theory T ; then the sentences φ ∧ (|x| > 0) are
+also unprovable. A preprocessor could remove this trivial type of unprovabil-
+ity bounded above zero. The next two approaches extends the encoding of
+arithmetic sentences to allow a function enc(x) to encode the conjunction of
+a series of unprovable sentences or their negations based on each bit in string
+x. Mostowski’s flexible formula µT(n) yields an infinite set of mutually in-
+dependent sentences indexed by n.29 Let encM(x) represent the conjunction
+of |x| clauses, including µT(i) or its negation as the ith clause depending on
+whether the ith bit of x is 1 or 0. As an alternative to Mostowski’s formula,
+given a binary string x and a theory T , let φ0 be the formula ‘0=0’, and let
+φi be “theory ‘T +φi−1’ is consistent” if the ith digit of x is 1 and otherwise
+“theory ‘T +φi−1’ is inconsistent” if the ith digit of x is 0. Extend the en-
+coding of arithmetic sentences such that encG(x) encodes φ|x|.30 With this
+encoding, encG(x) is unprovable in T for any x, so the share of unprovable
+sentences is again bounded above zero.31
+28The fact that this leads to contradiction means that adding false axioms ‘x/∈R’ makes
+it easy to prove ‘x∈R’, and adding a set of false axioms with measure zero to ZFC allows
+it always prove ‘x∈R’ if it is true.
+29See Pudl´ak[28], p. 286.
+30The last idea is similar to Hamkins[13], but for finite strings.
+31The four cases give different infinite sets of dense unprovable sentences, one for each
+string x. An interesting question is whether the infinite set of sentences can all be added
+as new axioms, or only a subset due to interdependencies between the axioms.
+For
+Mostowski’s flexible formula, any set of strings can be added.
+For the last proposal,
+18
+
+Similar conjectures link many open complexity theory problems to com-
+putability theory, potentially offering a unified view. Monroe[23] exhibits
+variants implying that the polynomial hierarchy does not collapse, P=BPP,
+and SAT has exponential circuits. Liu and Pass[21] have shown an equiva-
+lence between the existence of one-way functions and the mild average case
+hardness of time-bounded Kolmogorov complexity. It may be possible to dis-
+cuss coherently hardness not only asymptotically but for small inputs, since
+x is chosen after T , defeating any lookup table; an argument would also be
+needed on why inputs with small t are hard.
+References
+[1] Scott Aaronson, The busy beaver frontier, SIGACT News 51 (2020),
+no. 3, 32–54.
+[2] Michael Alekhnovich, Eli Ben-Sasson, Alexander A. Razborov, and Avi
+Wigderson, Pseudorandom generators in propositional proof complexity,
+SIAM Journal on Computing 34 (2004), no. 1, 67–88.
+[3] Eric Allender, Harry Buhrman, Michal Kouck´y, Dieter van Melkebeek,
+and Detlef Ronneburger, Power from random strings, SIAM J. Comput.
+35 (2006), no. 6, 1467–1493.
+[4] Eric Allender, Luke Friedman, and William I. Gasarch, Limits on the
+computational power of random strings, Inf. Comput. 222 (2013), 80–92.
+[5] Cristian S. Calude, Information and randomness - an algorithmic per-
+spective, Texts in Theoretical Computer Science. An EATCS Series,
+Springer, 2002.
+[6] Cristian S. Calude and Helmut J¨urgensen, Is complexity a source of
+incompleteness?, Advances in Applied Mathematics 35 (2005), no. 1,
+1–15.
+only strings that have another string as prefix can be added–the ith position cannot have
+conflicting values. A positive density of false axioms ‘x/∈R’ cannot be added, as that would
+contradict the theorem that x∈R with high probability. These dense sets of unprovable
+sentences resemble random variables that may or may not be correlated.
+19
+
+[7] Yijia Chen and J¨org Flum, A parameterized halting problem, Lecture
+Notes in Computer Science, vol. 7370, Springer, 2012, pp. 364–397.
+[8] Yijia Chen, J¨org Flum, and Moritz M¨uller, Hard instances of algo-
+rithms and proof systems, How the World Computes (Berlin, Heidel-
+berg) (S. Barry Cooper, Anuj Dawar, and Benedikt L¨owe, eds.), Springer
+Berlin Heidelberg, 2012, pp. 118–128.
+[9] Stephen A. Cook and Robert A. Reckhow, The relative efficiency of
+propositional proof systems, J. Symb. Log. 44 (1979), 36–50.
+[10] Rodney G. Downey, Carl G. Jockusch Jr., and Paul E. Schupp, Asymp-
+totic density and computably enumerable sets, J. Math. Log. 13 (2013),
+no. 2, 1350005.
+[11] Uriel Feige, Relations between average case complexity and approxima-
+tion complexity, Proceedings on 34th Annual ACM Symposium on The-
+ory of Computing (John H. Reif, ed.), ACM, 2002, pp. 534–543.
+[12] Ehud Friedgut, Sharp thresholds of graph properties, and the k-SAT
+problem, J. Amer. Math. Soc. 12 (1999), no. 4, 1017–1054.
+[13] Joel David Hamkins, Every function can be computable!, Blog post on
+http://jdh.hamkins.org/, Mar 2016.
+[14] Joel David Hamkins and Alexei Miasnikov, The halting problem is de-
+cidable on a set of asymptotic probability one, Notre Dame J. Formal
+Log. 47 (2006), no. 4, 515–524.
+[15] Shuichi Hirahara, Unexpected hardness results for Kolmogorov complex-
+ity under uniform reductions, Symposium on Theory of Computing,
+ACM, 2020, pp. 1038–1051.
+[16] Jan Kraj´ıcek, On the existence of strong proof complexity genera-
+tors, Electronic Colloquium on Computational Complexity TR22-120
+(2022).
+[17] Jan Kraj´ıˇcek, Proof complexity, Cambridge University Press, New York,
+NY, 2019.
+20
+
+[18] Jan Kraj´ıˇcek and Pavel Pudl´ak, Propositional proof systems, the con-
+sistency of first order theories and the complexity of computations, J.
+Symb. Log. 54 (1989), 1063–79.
+[19] Martin Kummer, On the complexity of random strings, Lecture Notes
+in Computer Science, vol. 1046, Springer, 1996, pp. 25–36.
+[20] Ming Li and Paul M. B. Vit´anyi, An introduction to Kolmogorov com-
+plexity and its applications, Texts in Computer Science, Springer, 2008.
+[21] Yanyi Liu and Rafael Pass, On one-way functions and Kolmogorov com-
+plexity, 2020 IEEE 61st Annual Symposium on Foundations of Computer
+Science (FOCS), 2020, pp. 1243–1254.
+[22] Hunter Monroe, Speedup for natural problems and noncomputability.,
+Theoretical Computer Science 412 (2011), no. 4-5, 478–481.
+[23]
+, Yes Virginia, there is a Santa Claus for complexity theorists, if
+you only believe, Computational Complexity Blog (2021).
+[24] Christos H. Papadimitriou, Computational complexity, Addison-Wesley,
+Reading, MA, 1994.
+[25] Pavel Pudl´ak, On the length of proofs of finitistic consistency statements
+in first order theories, Studies in Logic and the Foundations of Mathe-
+matics, vol. 120, Elsevier, 1986, pp. 165–196.
+[26]
+, Improved bounds to the length of proofs of finitistic consistency
+statements, Logic and combinatorics (Arcata, Calif., 1985) 65 (1987),
+309–331.
+[27]
+, The lengths of proofs, Handbook of Proof Theory (Samuel R.
+Buss, ed.), Elsevier, 1998.
+[28]
+, Logical foundations of mathematics and computational complex-
+ity: A gentle introduction, Springer, 2013.
+[29]
+, Incompleteness in the finite domain, Bull. Symb. Log. 23
+(2017), no. 4, 405–441.
+[30] Alexander A. Razborov, On provably disjoint NP-pairs, Electronic Col-
+loquium on Computational Complexity TR94-006 (1994).
+21
+
+[31] Raymond M Smullyan, Undecidability and Recursive Inseparability, Re-
+cursion Theory for Metamathematics, Oxford University Press, 1993.
+[32] Marc
+Vinyals,
+Proof
+complexity
+zoo,
+https://proofcomplexityzoo.gitlab.io/zoo/, Jul 2022.
+22
+
diff --git a/mNE3T4oBgHgl3EQf6QvR/content/tmp_files/load_file.txt b/mNE3T4oBgHgl3EQf6QvR/content/tmp_files/load_file.txt
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@@ -0,0 +1,608 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf,len=607
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='04789v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='CC]  12 Jan 2023  Isomorphisms Between Impossible and Hard  Tasks  Hunter Monroe  January 13, 2023  Abstract  If no efficient proof shows that an unprovable arithmetic sentence  ‘x is Kolmogorov random’ (‘x∈R’) lacks a length t proof, an isomor-  phism associates for each x impossible and hard tasks: ruling out any  proof and length t proofs respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' This resembles Pudl´ak’s feasi-  ble incompleteness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' This possible isomorphism implies widely-believed  complexity theoretic conjectures hold—in effect, translating theorems  from noncomputability about proof speedup and average-case hard-  ness directly to complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  Formally, we conjecture: sentence “Peano arithmetic (PA) lacks  any length t proof of ‘x∈R’” lacks tO(1) length proofs in any consis-  tent extension T of PA if and only if T cannot prove ‘x∈R’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' If so,  tautologies encoding the sentence lack tO(1) length proofs in any proof  system P for x∈R sufficiently long (relative to the description of a  program enumerating theorems of a theory T proving ‘P is sound’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  R’s density implies: TAUT/∈AvgP, Feige’s hypothesis holds, and, a  new conjecture, P’s nonoptimality has dense witnesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' If the isomor-  phism holds for any Π0 sentence, PH does not collapse, because the  arithmetic hierarchy does not collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  1  Introduction  It seems implausible that theories such as PA or ZFC set theory could  efficiently show that their unprovable (independent) sentences lack short  proofs—they would seem to have too much information about their own  1  blind spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1 Even a theory T stronger than PA might be unable to show  efficiently that PA lacks short proofs of a sentence unprovable in both PA  and T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' We formalize this intuition focusing on true but unprovable sentences  regarding Kolmogorov random strings (those strings incompressible by half).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  Define R = {x|∀p: if |p|≤|x|/2, then U(p)̸=x or pր}, U a deterministic  universal Turing machine (TM) with no limit on its running time, x and p  binary strings with |x| denoting x’s length, pր and p↓ signifying program  p does or does not halt, and ‘x∈R’ is an arithmetic sentence encoding x∈R  (single and double quotes signify the sentence encoding a mathematical state-  ment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Let PA be any arithmetically sound theory that can encode p↓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Proof  length is the length of the string representing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='2  By analogy with Chaitin’s Incompleteness Theorem—which states that  any consistent theory T with a computably enumerable (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=') set of proofs  cannot prove or refute ‘x∈R’ for x∈R almost always (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=', with x∈R suffi-  ciently long)—we conjecture:  Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1 For each x∈R, the sentence “PA lacks any length t proof  of ‘x∈R’” lacks tO(1) length proofs in any consistent extension T of PA if and  only if (iff) T cannot prove ‘x∈R’ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=', T  ̸  tO(1) “PA̸  t ‘x∈R’” iff T ̸  ‘x∈R’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='3  In the notation above, write T  φ or T ̸  φ respectively if T does or does  not have a proof of φ of any length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Write T  t φ for “theory T has a length  t (or shorter) proof of sentence φ” and T ̸  t φ if not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Write P  t ψ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' ψ is  “PA̸  t ‘x∈R’”) if proof system P proves tautologies encoding ψ in t steps,  where the encoding indicates “no t step proof is a valid proof” as described  in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  1The ideas in this paper and earlier versions have benefited from discussions with  the following: Scott Aaronson,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Eric Allender,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Olaf Beyersdorff,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Ilario Bonacino,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Cristian  Calude,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Yuval Filmus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Bill Gasarch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Valentina Harizonov,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Pavel Hrubeˇs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Russell Impagli-  azzo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Valentine Kabanets,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Mehmet Kayaalp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Yanyi Liu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Ian Mertz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Daniel Monroe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Igor  Oliveira,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Pavel Pudl´ak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Rahul Santhanam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Till Tantau,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Neil Thapen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Luca Trevisan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Avi  Wigderson,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Ryan Williams,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Marius Zimand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' and participants in seminars at George Wash-  ington University and Davidson College,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' the Computational Complexity Conference 2022  rump session,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' the Workshop on Proof Complexity 2022,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' and the 2022 Conference on Com-  plexity with a Human Face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Remaining errors are my own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  2Pudl´ak[27] surveys the literature on the lengths of proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Note that the conjecture  implies PA is consistent—otherwise, all sentences have short proofs by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  3The second clause refers to T ̸  ‘x∈R’ rather than T ̸  “PA̸  ‘x∈R’”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' These are equiv-  alent since PA  ‘x∈R’ implies T  ‘x∈R’ by the assumption that T extends PA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  2  Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1 has immediate implications for the length of proofs of  tautologies (under the encoding in Section 4) in proof systems:4  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='2 Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1 implies tautologies encoding “PA lacks any  length t proof of ‘x∈R’” lack tO(1) length proofs in any proof system P, for x  sufficiently long (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' P ̸  tO(1) “PA̸  t ‘x∈R’”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  To give context for Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1, a theory T like PA might intuitively be  expected to have limited information about sentences φ that are unprovable  in T , that is, φ is independent of T with T ̸ φ and T ̸ ¬φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' In particular, we  might expect that T lacks efficient proofs that its unprovable sentences lack  short proofs (T  ̸  tO(1) “T ̸  t φ”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Contrary to this intuition, Pudl´ak[25] showed  that T , if consistent, can in fact efficiently show there is no proof of contradic-  tion derivable from the axioms of T (no T -proof of ‘0=1’) within t steps, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=',  he showed T  tO(1) “T ̸  t ‘0=1’”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='5 Thus, a finite analog of G¨odel’s Second In-  completeness Theorem fails to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' However, Kraj´ıˇcek and Pudl´ak[18] later  showed that if for any consistent theory S, there exists a (stronger) con-  sistent theory T such that S ̸  tO(1) “T ̸  t ‘0=1’”, then no optimal propositional  proof system exists, a key open problem in proof complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' It is further  conjectured that this holds if T proves the consistency of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Our conjecture  differs by choosing φ nonconstructively after S and T that is unprovable in  both theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  This paper shows that if φ is ‘x∈R’, other rich results follow given that  R is infinite, dense, immune, noncomputable, and nonconstructive and given  that choosing x after T provides a second-mover advantage:6  Existing conjectures hold—TAUT/∈AvgP, Feige’s hypothesis holds, no  complete disjoint coNP pair exists, and (a new conjecture) P’s nonop-  timality has dense witnesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  4A  similar  conjecture  in  an  earlier  version  of  this  paper  fails  to  hold:  T  ̸tO(1)  “PA̸ t ‘x/∈R’”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' There are a fixed number of p such that |p|≤|x|/2, so there is a  linear-size refutation that simulates each p for t steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' I appreciate this observation by  Pavel Pudl´ak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  5Pudl´ak and Friedman independently formulated this finite version of G¨odel’s second  incompleteness theorem (a theory cannot prove its own consistency).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Pudl´ak[26] improved  his result from tO(1) to O(t) with additional assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  6For an overview of Kolmogorov randomness, see Li and Vitanyi[20] and Calude[5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' An  extensive literature has examined the computational power of random strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' The halting  problem can be computed using R as an oracle (Li and Vitanyi[20]), and R is truth-table  complete (Kummer[19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' See also Allender et al[3][4] and Hirihara[15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  3  An isomorphism maps impossible tasks, proving sentences ‘x∈R’, to  hard tasks, proving tautology families encoding “PA̸  t ‘x∈R’”, pre-  serving average-case unprovability/hardness and theory/proof system  nonoptimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  The isomorphism gives a procedure to find hard tautologies for a given  proof system P with high probability: randomly choose x twice as long  as a program enumerating theorems of a theory T proving ‘P is sound’,  and construct tautologies encoding ‘x∈R’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  The isomorphism translates classic computability theory results im-  plied by Chaitin’s Theorem and their proofs to complexity theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' The  computability and complexity results and their proofs are so closely  identified, they are essentially equivalent but with different terminol-  ogy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  A stronger conjectured isomorphism, which holds for any true φ ∈ Π0,  translates the stronger result that PH does not collapse because the  arithmetic hierarchy does not collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Section 2 provides preliminaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Sec-  tion 3 presents Calude and J¨urgensen’s[6] proof that the share of length n  arithmetic sentences that are true and unprovable is bounded above zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  Section 4 discusses implications of Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1 for tautologies and proof  systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  Section 5 discusses the isomorphism implied by Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  Section 6 concludes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  2  Preliminaries  Strings: With a binary alphabet {0, 1}, let Sn be the set of length n strings,  which are ordered n-tuples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Let |x| be the length of a string and |S| be the  cardinality of set S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' A language L is a subset of ∪n≥0Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  Density: Say the share of length n strings in L is bounded above zero  if there exists c > 0 such that |L ∩ Sn|/n ≥ c for for sufficiently large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  This implies the weaker condition that L has positive upper density, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=',  that lim sup  n→∞  |L ∩ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' , n}|  n  > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' If an event depending on n occurs with  probability that tends to one as n tends to infinity, such as x∈R where |x|=n,  say that it occurs with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  4  Kolmogorov Random Strings: Fix a universal TM U (not necessarily pre-  fix free).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  Define the set of Kolmogorov random strings as R = {x|∀p: if  |p|≤|x|/2, then U(p)̸=x ∨ pր}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  Theories: Theories are assumed to be PA or an extension of PA—any  arithmetically sound theory that can encode and (if true) prove p↓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='7 The  standard definition of PA is modified so binary strings are encoded in arith-  metic sentences as natural numbers, in binary not unary, adding a leading  “1” to avoid losing leading zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  PA under a standard approach writes  numbers in unary, for instance, “SSS0”, the third successor of 0, encodes the  number 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  Proof Systems: A propositional proof system is a polynomial time func-  tion h ∈ FP with range TAUT (Cook and Reckhow[9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' For tautology τ, any  string w such that h(w) = τ is a proof of τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' The proof system h is p-optimal  if for any other proof system f, there exists g ∈ FP such that h(g(x)) = f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  The proof system h is optimal if there exists c ≥ 1 such that the length of  minimal f proofs of x are polynomially bounded in |x| with exponent c by  minimal h proofs (Kraj´ıˇcek and Pudl´ak[18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  3  Density of Unprovable Sentences  To help understand Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1, we first describe an unconditional result  by Calude and J¨urgensen[6] that the share of length n arithmetic sentences  that are true and unprovable is bounded above zero—informally, proving  theorems is impossible in the average case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' The result relies on two facts:  ‘x∈R’ is almost always unprovable, by Chaitin’s Incompleteness Theorem,  and length n strings are in R with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='8  With that context,  Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1 asserts that analogous statements hold for coTHEOREMS≤t=  {⟨φ, 1t⟩|PA̸  t φ} and, as shown in Section 4, TAUT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  Chaitin’s Incompleteness Theorem states:9  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1 For every consistent, arithmetically sound theory T with a  c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' set of theorems, ∃k∀x:|x|>k, T cannot prove ‘x∈R’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  7This does not require induction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' see Papadimitriou[24] Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  8See the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='2 in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' We note: the share of true, provable sentences is  also bounded above zero;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' consider ‘|x|≥0’, which trivially provable for all x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' These results  imply positive upper density in either case, as the share at position 2n−1 is also bounded  above zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  9See Li and Vitanyi[20] Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  5  Proof:  In brief, the proof is: otherwise, a string x could be concisely  described as “the first string x of length n such that T proves ‘x∈R”’, con-  trary to the definition of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' To spell this out, suppose T can prove ‘x∈R’  infinitely often.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Construct a program E that on binary input n enumerates  all proofs in T and outputs the first string x with |x|≥n for which a proof  of ‘x∈R’ appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' The input has log n bits, so for n sufficiently large, E(n)  is a short description of the string x printed by E: “the first length n string  x for which a proof of ‘x∈R’ appears in an enumeration of all proofs in T ”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  For n sufficiently large, n/2≥|E|+ log n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' For such an n, let p=E(n) and let  k = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' This contradicts the definition of R and proves the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1 implies that R is immune: infinite with no infinite c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  R’s pathology—no program or theory can identify more than a finite number  of its members—works to our favor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Furthermore, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='2 shows strings  are in R with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  This gives us a second-mover advantage  against any theory T —we can (noncomputably) choose x∈R after seeing T  to highlight T ’s inefficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='2 x∈R with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  Proof:  Compare the number of length n strings, 2n, with the number of  programs p with |p|≤n/2, which is 2n/2+1−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' The number of length n strings  not in R is therefore at most 2n/2+1 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Therefore, R’s share of length n  strings is at least 1 − 2−n/2, so x∈R with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  Calude and J¨urgensen’s result implies:  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='3 For every theory T extending PA, the set of sentences {‘x∈R’|  x∈R} which are true and unprovable has a share in length n arithmetic sen-  tences that is bounded above zero, for n sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='10  Proof:  Theory T can prove sentences ‘x∈R’ where x∈R only in finite  cases, by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' The sentences ‘x∈R’ satisfy |‘x∈R’| = |x| + c, where  c is a constant not depending on |x|, giving the overhead of encoding ‘x∈R’  beyond x itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Therefore, the share length n sentences of form ‘x∈R’ is  10The share of true and provable sentences is also bounded above zero: consider the  sentences |x|≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' In the terminology of Downey et al[10], the set of arithmetic sentences  is “partially computable at density r”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Section 6 provides three new proofs that the share  of length n arithmetic sentences that are unprovable is bounded above zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  6  exactly 2−c and the share of form ‘x∈R’ for which in addition x∈R is at least  1 − 2−n/2 for each n per Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='2 and is asymptotically one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  Therefore, length n arithmetic sentences of the form ‘x∈R’ with x∈R have  share bounded above zero by 2−c(1 − 2−n/2) − ǫ for ǫ>0, for n sufficiently  large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  4  Tautologies and Proof Systems  This section specifies how a tautology will encode the sentence “PA̸  t ‘x∈R’”  in Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='2 and shows that Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1 therefore implies some widely  believed conjectures about tautologies and proof systems, implies one new  conjecture, and gives a procedure to find hard tautologies for a proof system  with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  To specify the encoding: for a given x, “PA̸  t ‘x∈R’” is equivalent to  ⟨‘x∈R’, 1t⟩∈coTHEOREMS≤t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' coTHEOREMS≤t and TAUT are both coNP-complete  languages, so some polynomial-time reduction r from coTHEOREMS≤t to TAUT  maps ⟨φ, 1t⟩ to tautology r(⟨φ, 1t⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='11 Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1 implies that for any  proof system P, for any sufficiently long x∈R, the family of tautologies  r(⟨‘x∈R’, 1t⟩) lacks tO(1) length proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Equivalently, r(⟨‘x∈R’, 1t⟩) has tO(1)  length proofs in P for only a finite set of x∈R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1  Implications  Three interesting implications immediately follow: dense witnesses to the  nonoptimality of proof systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' TAUT≤t /∈AvgP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' and Feige’s hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1  Dense Witnesses to Nonoptimality  Chen et al[8] and Kraj´ıˇcek[17] establish that no optimal propositional proof  system exists iff, for each proof system, there is an easily described (P-  uniform) family of tautologies requiring proofs of superpolynomial length,  11Tautologies produced by r confirm that every possible proof is not a valid proof, not  that every set of bits satisfies a sentence with a single universal bounded quantifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' These  tautologies do not result from a propositional translation, which takes a sentence with a  single universal bounded quantifier, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' applying to numbers up to a ceiling, and replaces  each bit of a number with a logical variable in a tautology, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  7  called a hard sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1 has a stronger implication not con-  sidered in the literature: for each proof system, there is a dense set of hard  sequences r(⟨‘x∈R’, 1t⟩) letting x range over all x∈R sufficiently long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' To be  precise:  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1 Under Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1, there is no optimal proof system P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  Proof:  This follows immediately from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='2 Under Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1, there are a dense set of hard sequences  for each P witnessing its nonoptimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' That is, in an enumeration over all  φ of tautology families r(⟨φ, 1t⟩), the subset of the form r(⟨‘x∈R’, 1t⟩) with  x∈R has positive upper density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  Proof:  This follows immediately from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='2  TAUT≤t /∈AvgP  R’s density implies TAUT≤t /∈AvgP by Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Re-  call that a distributional problem (L, D) (where D is an ensemble of distri-  butions for length n inputs) is in AvgP if there is a TM M accepting L and  constants C and ǫ > 0 such for that every n:  E  y∈RDn  �TM(y)ǫ  n  �  ≤ C, where  TM is the function that maps a string x to how many steps M(x) takes, and  E is the expectation operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='3 Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1 implies: (i) coTHEOREMS≤t /∈AvgP and (ii)  TAUT≤t /∈AvgP, for any distribution applying non-negligible weight to ele-  ments of the form ⟨‘x∈R’, 1t⟩ and r(⟨‘x∈R’, 1t⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  Proof:  (i) Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1 implies that the share of length n elements in  the language coTHEOREMS≤t of the form ⟨‘x∈R’, 1t⟩ is bounded above zero,  which is sufficient to show coTHEOREMS≤t /∈AvgP, by the following argu-  ment similar to that of Calude and J¨urgensen[6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  That proof notes that  |‘x∈R’|=|x|+c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Therefore, |⟨‘x∈R’, 1t⟩|=|x|+c+t for a slightly larger c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Fix  n, let |x| range from 0 to n−c, and let t = n−|x|−c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' All but x is fixed,  so among length n elements, and the number of possibilities for x is the  sum 20+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' +2n−c=2n−c+1−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Any string x is in R with high probability by  8  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Therefore, the share of length n elements of the form ⟨‘x∈R’, 1t⟩  with x∈R is roughly 2n−c+1/2n, so the share exceeds 2−c+1−ǫ for ǫ>0 for  n sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' This implies coTHEOREMS≤t /∈AvgP for any distribution  applying non-negligible weight to ⟨‘x∈R’, 1t⟩ where x∈R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  (ii) The same argument holds for tautologies r(⟨‘x∈R’, 1t⟩), if r maps all  elements in its domain of length n to tautologies all of the same length, say,  64n, implying the share of length n tautologies of the form r(⟨‘x∈R’, 1t⟩) with  x∈R is bounded above zero by 2−c+1−64−ǫ with ǫ>0 for n sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  Then TAUT≤t /∈AvgP for any distribution applying non-negligible weight to  the tautologies r(⟨‘x∈R’, 1t⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='3  Feige’s Hypothesis  A similar argument shows that Feige’s hypothesis follows from Conjecture  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Recall that in the study of random k-CNF, with n Boolean variables and  k variables per clause, there are  �n  k  �  choices for variables in a clause, each  of which may or may not be negated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Suppose a random k-CNF consists of  a fraction ∆n from these possibilities for a constant ∆ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' For ∆ above  a threshold, the formula is unsatisfiable with high probability, with a sharp  threshold by Friedgut[12], estimated empirically at ∆ ≈ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='26 for 3-CNFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  Feige[11] conjectures that k-CNF refutation is hard above this threshold and  below nk/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  The set UNSATk of unsatisfiable k-CNFs is coNP-complete, so there is a  reduction u to this language from coTHEOREMS≤t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Again choose u to maps  all elements in its domain of length n to all of the same length elements of  UNSATk of the same length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='4 Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1 implies that Feige’s hypothesis holds for a suf-  ficiently high clause density ∆ determined by u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  Proof:  The share of length n inputs ⟨‘x∈R’, 1t⟩ in coTHEOREMS≤t is bounded  above zero by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='3(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  Therefore, the share of length n inputs  u(⟨‘x∈R’, 1t⟩)∈UNSATk will also be bounded above zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' The reduction u will  imply that these k-CNFs have a certain clause density ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Feige’s Hypothesis  holds at this clause density or higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  9  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='2  Procedures for Creating Hard Tautologies  Under Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1, for a given proof system P, each of the following  procedures yield a hard sequence of tautologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' All but the last procedure  are noncomputable (each step is labeled computable or noncomputable):  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='5 Each of the following procedures yields a hard sequence of  tautologies, given a proof system P:  (i) make a (finite) list of x∈R for which P has proofs of length tO(1)  for tautologies r(⟨‘x∈R’, 1t⟩) (possibly noncomputable);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' (ii) choose any  x∈R not on this list (noncomputable);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' and (iii) create tautologies  r(⟨‘x∈R’, 1t⟩) (computable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  An alternative procedure is: (i) given P, choose theory T extend-  ing PA that proves ‘P is sound’ (computable);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' (ii) calculate the k in  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1 for T (computable);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' (iii) choose any ‘x∈R’ of length k  or longer (noncomputable);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' and (iv) create tautologies r(⟨‘x∈R’, 1t⟩)  (computable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  A probabilistic, polynomial-time computable procedure exists, which  is the same as the previous procedure, but choosing a random string x  of length k or longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Then, x∈R with high probability by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='2,  so tautologies r(⟨‘x∈R’, 1t⟩) are hard for P with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='12  Proof:  These follow from Chaitin’s Theorem and Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  The Cook-Reckhow program aims to show NP̸=coNP by proving super-  polynomial lower bounds on proof length for specified tautologies for par-  ticular proof systems P of increasing power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='13 Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1 is so strong,  it would if true complete the Cook-Reckhow program, by identifying hard  tautologies for any proof system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='3  Relationship to the Literature  Identifying hard families of tautologies has been a key objective in the fields  of proof and computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' As noted above, Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1 pulls  12Pseudorandom bits with a seed less than |x|/2 are not sufficient for this procedure,  which therefore cannot be derandomized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  13The Proof Complexity Zoo (Vinyals[32]) lists 35 proof systems and 23 candidates for  hard tautologies, which provide lower bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' The literature includes many other proof  systems and candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  10  together the literature’s three primary proposals for hard tautologies: ran-  dom tautologies, tautologies encoding unprovable sentences, and tautologies  created from strings outside a PRG’s range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='14  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1  Random Tautologies  Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1 non-trivially derandomizes the long-standing conjecture that  random tautologies are hard any proof system,15 and in particular random  k-CNF refutation is hard (Feige’s Hypothesis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Under the conjecture, each  x∈R is mapped to a family of tautologies, using the random bits frugally  relying on the fact that R is immune.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' A trivial derandomization might use  x∈R less frugally to create a single tautology rather than a family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='16  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='2  Independent Sentences  Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1 is related to Pudl´ak’s Feasible Incompleteness Thesis: “The  phenomenon of incompleteness manifests itself at the level of polynomial time  computations”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' A related conjecture is the nonexistence of an optimal proof  system for TAUT (an NTM accepting TAUT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='17 If there is no optimal proof  system for TAUT, there is no optimal NTM accepting any paddable coNP-  complete language (Kraj´ıˇcek and Pudl´ak[18]), and there are hard P-uniform  families of inputs (hard sequences) for each such language (Chen et al[8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  Monroe[22] shows that certain hard sequences have a link to noncomputabil-  ity:  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' If hard sequences exist, one exists referring to non-halting TMs: for  any M accepting coBHP = {⟨N, x, 1t⟩| there is no accepting path of  nondeterministic TM N on input x with t or fewer steps}, there exists  14This by no means a complete list of proposals for hard tautologies, as we have focused  on those some how related to our conjecture below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' See the survey in Kraj´ıˇcek[17] Chapter  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  15Kraj´ıˇcek[17] states: “Most researchers in proof complexity agree, I suppose, with the  folklore informal conjecture that for any proof system most formulas are hard.”  16A standard technique, the “incompressibility method”, analyzes objects randomly  selected according to a probability distribution by studying instead Kolmogorov random  objects which have the same properties—in our case, families of tautologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' See Li and  Vitanyi[20] Chapter 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Allender et al[3] notes that significant work in derandomization  studies the interplay between strings chosen at random and Kolmogorov random strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  17For broader overviews of that rich literature, see Pudl´ak[28] Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='4 and [29] and  Kraj´ıˇcek[17] Chapter 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  11  some ⟨N′, x′⟩ where N′ does not halt on x′ such that ⟨N′, x′, 1t⟩ is a  hard sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Equivalently,18 for any M accepting coTHEOREMS≤t, there exists some  sentence φ′ with no proof in PA of any length, such that ⟨φ′, 1t⟩ is a  hard sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1 specifies φ′ =‘x∈R’ where x∈R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='3  Proof Complexity Generators  The literature on proof complexity generators suggests that tautologies en-  coding “string x lies outside the range of a PRG” are hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='19 In Conjecture  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1, tautologies encode that there is no PA proof within t steps that a string x  lies outside the range of every program p with |p|≤|x|/2 run for an unlimited  number of steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Thus, these conjectures are thematically similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  5  The Isomorphism  Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1 asserts that for each x, T  ̸  tO(1) “PA̸  t ‘x∈R’” iff T ̸ ‘x∈R’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' This  is a monumental claim—a fact about noncomputability (T ̸ ‘x∈R’) provides  a complete understanding of a question in complexity (T  ̸  tO(1) “PA̸  t ‘x∈R’”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='3 and other classic results from computability theory  translate to complexity theory under the isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1 implies there is an isomorphism I between impossible-to-  prove sentences of the form ‘x∈R’ and hard-to-prove sentences of the form  “PA̸  t ‘x∈R’” (the family indexed by t), so I(‘x∈R’) =“PA̸  t ‘x∈R’”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' The  isomorphism preserves the following properties:  Hardness: For each T , T ̸ ‘x∈R’ iff T  ̸  tO(1) “PA̸  t ‘x∈R’”, that is, I  maps the property “impossible-to-prove” to “hard-to-prove”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  Density: For each T , the share of “impossible-to-prove”sentences in  length n sentences is bounded above zero, just as the share of “hard-  to-prove” sentences is bounded above zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  18Chen and Flum[7] Theorem 31 shows that coBHP and coTHEOREMS≤t are fixed param-  eter tractable reducible to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  19See Alekhnovich et al[2] and Kraj´ıˇcek[16] for more information on proof complexity  generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  12  Theory/Proof System Nonoptimality: Just as there is no best theory  for proving ‘x∈R’, there is not best proof system for proving tautologies  r(⟨‘x∈R’, 1t⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='20 In both cases, adding a new axiom ‘x∈R’ leads to a  stronger theory or proof system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  Blind Spots: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1 establishes that T proves x∈R in at most  finite cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Under Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1, T efficiently proves “PA̸  t ‘x∈R’”  in exactly the same cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' T ’s limited information about R constrains  its capacity to prove in an identical manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1  Another Implication  This section presents another implication of Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1: there does not  exist a complete disjoint coNP pair with respect to polynomial reductions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='21  A disjoint coNP pair is two sets (A,B) with A,B⊂coNP and A∩B=∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Say  disjoint coNP pair (A,B) is polynomially reducible to disjoint coNP pair  (C,D) if there if a polynomial time function that maps A into C and B into  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Say that two sets are p-inseparable, if there does not exist S∈P such that  S contains one set and is disjoint with the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  Razborov[30] defines the canonical NP pair of a proof system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' We de-  fine the canonical coNP pair of a theory T consisting of the set of φ not  provable within t steps and the set not refutable within t steps, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=', as  ({⟨φ, 1t⟩|T ̸  t φ}, {⟨φ, 1t⟩|T ̸  t ¬φ}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1 Under Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1, there is no complete disjoint coNP  pair among canonical coNP pairs for theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Canonical coNP pairs for  theories are universal among disjoint coNP pairs, so there are no complete  disjoint coNP pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' There exists a p-inseparable disjoint coNP pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  Proof:  Suppose the canonical coNP pair for T is a complete disjoint  coNP pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Choose x∈R such that T ̸  ‘x∈R’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Let T ′ be T plus the new  axiom ‘x∈R’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Then, the canonical coNP pair for T can be reduced to the  the canonical coNP pair for T ′, but the reverse is not true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Otherwise, the  reduction could be used to create efficient proofs in T for “PA̸  t ‘x∈R’”, using  the efficient proofs in T ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Therefore, the canonical pair for T is not complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  20Pudl´ak[29] states “the fact that the logical strength of theories cannot be bounded is  likely to be projected into these proof systems”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  21Pudl´ak[29] explores these conjectures that are stronger than the nonexistence of an  optimal proof system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  13  Canonical coNP pairs for theories are universal among disjoint coNP  pairs: for any such pair, there is a theory that proves each member of the  pair is in coNP and proves that the pair is disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  Then this pair is  reducible to the canonical coNP pairs for that theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  The canonical coNP pair for PA is p-inseparable, as Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1 im-  plies that NP̸=coNP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='2  Translating Theorems from Computability Theory  to Complexity Theory  The isomorphism in Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1 implies that the above results and their  proofs can be interpreted in two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' First, they can be seen as conditional  results in complexity theory, proved by complexity theoretic arguments, as-  suming the conjecture holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Second, they can be seen as classic results and  their proofs in computability theory, which immediately hold in complexity  theory by referring to the isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Under this view, the two approaches  are equally valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  Consider the second view for each of the results above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' The following are  alternative proofs, citing a result in computability theory and referring to  the isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='2 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1 (Chaitin) implies that there is no optimal theory  T , as a new axiom ‘x∈R’ can always be added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' This implies there is no  optimal proof system P, by the isomorphism in Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='3 (Calude and J¨urgensen) implies Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='2 (dense hard  sequences), 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='3 (TAUT≤t /∈AvgP), and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='4 (Feige’s Hypothesis), by the iso-  morphism in Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1 implies that the procedures in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='5 yield x∈R that  is hard for theory T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Therefore, they yield x∈R that is hard for proof system  P, by the isomorphism in Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  For a consistent axiomatizable theory in which all recursive sets are de-  finable, there is no complete pair ({φ|T ̸  φ}, {φ|T ̸  ¬φ}) under computable  reductions, and the two sets are computably inseparable (Smullyan[31]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' This  implies Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1 (disjoint coNP pairs), by the isomorphism in Conjec-  ture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  The isomorphism translates unprovable sentences to hard-to-prove tau-  14  tologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' This contrasts with propositional translation, which translates prov-  able sentences with bounded quantifiers to easy-to-prove tautologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  An important question is how to distinguish which results translate from  computability theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  We conjecture that any result that can be proven  relying only on Chaitin’s Theorem translates under Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' This  raises the question of whether a stronger isomorphism could translate more  results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' The next section considers such isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='3  Other Isomorphisms  A more granular version of Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1 can replace the universal quantifier  in the definition of R with a separate statement for each p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Given x, with  |p|≤|x|/2, there are 2|x|/2−1 such p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' For each p, the definition requires either  U(p)̸=x or pր.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  If U(p) halts with U(p)̸=x, then any T (including PA)  can prove this by simulating the computation of U on p to verify U(p)̸=x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  The only interesting case is when T ̸  t ‘pր’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' This yields the more granular  conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  Conjecture 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='3 For each p, T  ̸  tO(1) “PA̸  t ‘pր’” iff T ̸  ‘pր’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='22  Note that in an unbounded setting, we know unconditionally that proving  or accepting ‘pր’ is impossible on average, as a corollary of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='3:  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='4 (i) For any theory T , the share of p of length n is bounded  above zero for which pր but T ̸  ‘pր’, for n sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  (ii) For any NTM M which accepts only elements of {p|pր} (the set of  nonhalting programs), the share of length n inputs for which M fails to accept  correctly is bounded above zero, for n sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='23  Proof:  (i) Consider the set of p which have the following form: for a given  x∈R, p enumerates all proofs in T and halts if one proof shows x∈R (the  input x is hard coded).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='3, the share of p of length n of this  form for which pր but T ̸  ‘pր’ is bounded above zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  22A weaker conjecture could assert that the equivalence holds on average: The share  of p of length n is bounded above zero for which: T  ̸tO(1)  “PA̸ t ‘pր’” iff T ̸  ‘pր’, for n  sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  23See however Hamkins and Miasnikov[14], which is sensitive to the computational  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  15  (ii) follows immediately from (i);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' otherwise M could serve as a theory  contradicting part (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  Consider what kind of isomorphism would be sufficient to imply that  the polynomial hierarchy (PH) does not collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1 implies  NP ̸= coNP24 and therefore, Πp  0 ̸= Πp  1, but it does not seem to imply  Πp  i−1 ̸= Πp  i for i > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' The sentences ‘x∈R’ and ‘pր’ each employ a single  universal quantifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='25  This suggests considering an isomorphism allowing  sentences φ with more quantifiers:  Conjecture 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='5 For each true26 φ ∈ Π0, T  ̸  tO(1) “PA̸  t φ” iff T ̸  φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  This very strong (and possibly false) conjectured isomorphism is sufficient to  imply PH does not collapse:  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='6 (i) If Conjecture 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='5 holds, PH does not collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' (ii) Be-  cause the arithmetic hierarchy does not collapse, PH does not collapse by the  isomorphism in Conjecture 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  Proof:  (i) Consider the theory T +Π0  1, which is T with every true Π0  1 sen-  tence added as an axiom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Choose any true φ in Π0  2/Π0  1 such that (T +Π0  1)̸  φ,  which exists because the arithmetic hierarchy does not collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  Then,  (T +Π0  1) ̸  tO(1) “PA̸  t φ” by Conjecture 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Let coTHEOREMS2  ≤t= {⟨φ, 1t⟩|PA̸  t φ  and φ/∈Π0  1}, which is Πp  2-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Then, coTHEOREMS2  ≤t has hard sequences,  even for an NTM accepting proofs for T with an oracle for coTHEOREMS≤t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  Therefore, Πp  1̸=Πp  2, and likewise Πp  i−1̸=Πp  i , so PH does not collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  (ii) See (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='6(ii) confirms that the stronger isomorphism Conjecture 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='5,  if correct, allows stronger results from computability theory to be translated  to complexity theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  24See Kraj´ıˇcek and Pudl´ak[18];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' it also implies NEXP ̸= coNEXP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  25The sentence ‘pր’, spelled out, states that for any computation, it does not describe  a halting computation for p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' The sentence ‘x∈R’ employs a second universal quantifier:  for any p with |p|≤|x|/2, pր or U(p)̸=x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  26Examples of true φ ∈ Π0 are ‘pր’, which is true if p does not have a halting com-  putation, and ‘x∈R’, which is true if it is not falsified by some p with |p|≤|x|/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' The  requirement that φ be true rules out the counterexamples in footnotes 4 and 5, which  involve the false statements ‘x/∈R’ and ‘0=1’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  16  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='4  Plausibility Arguments  Informally, Conjectures 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='3, and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='5 assert that there is no “free lunch”  obtained by trying to prove hard (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=', “PA̸  t ‘x∈R’”) rather than impossible  (‘x∈R’) sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  Consider these plausibility arguments for the conjectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' First, suppose  T ̸ ‘x∈R’ but T  tO(1) “PA̸  t ‘x∈R’”, contrary to Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' The patho-  logical nature of R might overwhelm T ’s ability to prove, so as to rule out  tO(1) length proofs, necessitating a brute force approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' R’s pathology might  imply that T cannot distinguish among possible proofs in PA of ‘x∈R’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' This  is a vague argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  Second, for t sufficiently large, T might be expected to lose the ability  to prove statements about halting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  To be more precise, define BB(c) =  max{k:|p|≤c and p halts in k steps})—this is a busy beaver function defined  with respect to the length of program description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Given a theory T , there  exists cT such that T cannot prove the value of BB(c) for c≥cT ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' otherwise, T  could solve the halting problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='27 Given T , choose x∈R such that cT <|x|/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  For t>BB(|x|/2), there are programs p with p≤|x|/2 which have not halted  by t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' none of these will ever halt, but T cannot prove pր.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' For such x, p,  and t, plausibly, T  ̸  tO(1) “PA̸  t ‘pր’” and therefore T  ̸  tO(1) “PA̸  t ‘x∈R’”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  An informal argument in favor of Conjecture 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='3 is: if T ̸  ‘pր’, then T  lacks any information that would allow efficient proofs as in T  ̸  tO(1) “PA̸  t ‘pր’”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  This seems like a weaker argument than the one for Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1, which can  be seen as an average of Conjecture 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='3 across all programs p with |p|≤|x|/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  The possibility that Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1 is independent of ZFC is heightened by  its reliance on independent sentences such as ‘x/∈R’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' The following argument  suggests that Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1 is not independent of ZFC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Consider the weaker  conjecture: for all T , there exists x∈R such that T  ̸  tO(1) “PA̸  t ‘x∈R’”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' For  any x that is predicted by this weaker conjecture, add the “false axiom”  ‘x/∈R’ to ZFC to create ZFC’ (which is not arithmetically sound and has  an non-c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' infinite set of axioms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' It may be possible to do this such that  ZFC’ is consistent, indicating that the weaker conjecture is false in ZFC’ and  therefore independent of ZFC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' However, this technique does not work for  Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='1, as it would require adding ‘x/∈R’ for x of length n with high  probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' This implies a contradiction, as x∈R with high probability, by  27See Aaronson[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  17  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='28  6  Conclusion  This paper proposes that tautologies r(⟨‘x∈R’, 1t⟩) are hard by analogy to  Chaitin’s Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' It conjectures that exactly the same x leads to hard-  to-prove tautologies and impossible-to-prove arithmetic sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' This sets  up an isomorphism which carries over results from computability theory to  complexity theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  This paper notes that the share of length n arithmetic sentences which  are unprovable is bounded above zero, pointing Calude and J¨urgensen’s re-  sult (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='3) relying on properties of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Here are three other proofs  of the same result which also involve stuffing in any string into a slot to  create density not using R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' First, a trivial argument uses “dense padding”:  choose any φ unprovable in theory T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' then the sentences φ ∧ (|x| > 0) are  also unprovable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' A preprocessor could remove this trivial type of unprovabil-  ity bounded above zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' The next two approaches extends the encoding of  arithmetic sentences to allow a function enc(x) to encode the conjunction of  a series of unprovable sentences or their negations based on each bit in string  x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Mostowski’s flexible formula µT(n) yields an infinite set of mutually in-  dependent sentences indexed by n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='29 Let encM(x) represent the conjunction  of |x| clauses, including µT(i) or its negation as the ith clause depending on  whether the ith bit of x is 1 or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' As an alternative to Mostowski’s formula,  given a binary string x and a theory T , let φ0 be the formula ‘0=0’, and let  φi be “theory ‘T +φi−1’ is consistent” if the ith digit of x is 1 and otherwise  “theory ‘T +φi−1’ is inconsistent” if the ith digit of x is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Extend the en-  coding of arithmetic sentences such that encG(x) encodes φ|x|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='30 With this  encoding, encG(x) is unprovable in T for any x, so the share of unprovable  sentences is again bounded above zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='31  28The fact that this leads to contradiction means that adding false axioms ‘x/∈R’ makes  it easy to prove ‘x∈R’, and adding a set of false axioms with measure zero to ZFC allows  it always prove ‘x∈R’ if it is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  29See Pudl´ak[28], p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' 286.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  30The last idea is similar to Hamkins[13], but for finite strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  31The four cases give different infinite sets of dense unprovable sentences, one for each  string x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' An interesting question is whether the infinite set of sentences can all be added  as new axioms, or only a subset due to interdependencies between the axioms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  For  Mostowski’s flexible formula, any set of strings can be added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  For the last proposal,  18  Similar conjectures link many open complexity theory problems to com-  putability theory, potentially offering a unified view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Monroe[23] exhibits  variants implying that the polynomial hierarchy does not collapse, P=BPP,  and SAT has exponential circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Liu and Pass[21] have shown an equiva-  lence between the existence of one-way functions and the mild average case  hardness of time-bounded Kolmogorov complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' It may be possible to dis-  cuss coherently hardness not only asymptotically but for small inputs, since  x is chosen after T , defeating any lookup table;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' an argument would also be  needed on why inputs with small t are hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  References  [1] Scott Aaronson, The busy beaver frontier, SIGACT News 51 (2020),  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' 3, 32–54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  [2] Michael Alekhnovich, Eli Ben-Sasson, Alexander A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Razborov, and Avi  Wigderson, Pseudorandom generators in propositional proof complexity,  SIAM Journal on Computing 34 (2004), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' 1, 67–88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  [3] Eric Allender, Harry Buhrman, Michal Kouck´y, Dieter van Melkebeek,  and Detlef Ronneburger, Power from random strings, SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  35 (2006), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' 6, 1467–1493.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  [4] Eric Allender, Luke Friedman, and William I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Gasarch, Limits on the  computational power of random strings, Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' 222 (2013), 80–92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  [5] Cristian S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Calude, Information and randomness - an algorithmic per-  spective, Texts in Theoretical Computer Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' An EATCS Series,  Springer, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  [6] Cristian S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Calude and Helmut J¨urgensen, Is complexity a source of  incompleteness?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=', Advances in Applied Mathematics 35 (2005), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' 1,  1–15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  only strings that have another string as prefix can be added–the ith position cannot have  conflicting values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' A positive density of false axioms ‘x/∈R’ cannot be added, as that would  contradict the theorem that x∈R with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' These dense sets of unprovable  sentences resemble random variables that may or may not be correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  19  [7] Yijia Chen and J¨org Flum, A parameterized halting problem, Lecture  Notes in Computer Science, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' 7370, Springer, 2012, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' 364–397.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  [8] Yijia Chen, J¨org Flum, and Moritz M¨uller, Hard instances of algo-  rithms and proof systems, How the World Computes (Berlin, Heidel-  berg) (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Barry Cooper, Anuj Dawar, and Benedikt L¨owe, eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' ), Springer  Berlin Heidelberg, 2012, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' 118–128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  [9] Stephen A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Cook and Robert A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Reckhow, The relative efficiency of  propositional proof systems, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Symb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' 44 (1979), 36–50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  [10] Rodney G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Downey, Carl G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Jockusch Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=', and Paul E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Schupp, Asymp-  totic density and computably enumerable sets, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' 13 (2013),  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' 2, 1350005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  [11] Uriel Feige, Relations between average case complexity and approxima-  tion complexity, Proceedings on 34th Annual ACM Symposium on The-  ory of Computing (John H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Reif, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' ), ACM, 2002, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' 534–543.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  [12] Ehud Friedgut, Sharp thresholds of graph properties, and the k-SAT  problem, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' 12 (1999), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' 4, 1017–1054.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  [13] Joel David Hamkins, Every function can be computable!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=', Blog post on  http://jdh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='hamkins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='org/, Mar 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  [14] Joel David Hamkins and Alexei Miasnikov, The halting problem is de-  cidable on a set of asymptotic probability one, Notre Dame J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Formal  Log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' 47 (2006), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' 4, 515–524.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  [15] Shuichi Hirahara, Unexpected hardness results for Kolmogorov complex-  ity under uniform reductions, Symposium on Theory of Computing,  ACM, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' 1038–1051.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  [16] Jan Kraj´ıcek, On the existence of strong proof complexity genera-  tors, Electronic Colloquium on Computational Complexity TR22-120  (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  [17] Jan Kraj´ıˇcek, Proof complexity, Cambridge University Press, New York,  NY, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  20  [18] Jan Kraj´ıˇcek and Pavel Pudl´ak, Propositional proof systems, the con-  sistency of first order theories and the complexity of computations, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  Symb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' 54 (1989), 1063–79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  [19] Martin Kummer, On the complexity of random strings, Lecture Notes  in Computer Science, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' 1046, Springer, 1996, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' 25–36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  [20] Ming Li and Paul M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Vit´anyi, An introduction to Kolmogorov com-  plexity and its applications, Texts in Computer Science, Springer, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  [21] Yanyi Liu and Rafael Pass, On one-way functions and Kolmogorov com-  plexity, 2020 IEEE 61st Annual Symposium on Foundations of Computer  Science (FOCS), 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' 1243–1254.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  [22] Hunter Monroe, Speedup for natural problems and noncomputability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=',  Theoretical Computer Science 412 (2011), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' 4-5, 478–481.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  [23]  , Yes Virginia, there is a Santa Claus for complexity theorists, if  you only believe, Computational Complexity Blog (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  [24] Christos H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Papadimitriou, Computational complexity, Addison-Wesley,  Reading, MA, 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  [25] Pavel Pudl´ak, On the length of proofs of finitistic consistency statements  in first order theories, Studies in Logic and the Foundations of Mathe-  matics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' 120, Elsevier, 1986, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' 165–196.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  [26]  , Improved bounds to the length of proofs of finitistic consistency  statements, Logic and combinatorics (Arcata, Calif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=', 1985) 65 (1987),  309–331.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  [27]  , The lengths of proofs, Handbook of Proof Theory (Samuel R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  Buss, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' ), Elsevier, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  [28]  , Logical foundations of mathematics and computational complex-  ity: A gentle introduction, Springer, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  [29]  , Incompleteness in the finite domain, Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Symb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' 23  (2017), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' 4, 405–441.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  [30] Alexander A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content=' Razborov, On provably disjoint NP-pairs, Electronic Col-  loquium on Computational Complexity TR94-006 (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  21  [31] Raymond M Smullyan, Undecidability and Recursive Inseparability, Re-  cursion Theory for Metamathematics, Oxford University Press, 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  [32] Marc  Vinyals,  Proof  complexity  zoo,  https://proofcomplexityzoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='gitlab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='io/zoo/, Jul 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
+page_content='  22' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQf6QvR/content/2301.04789v1.pdf'}
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+arXiv:2301.04800v1  [math.PR]  12 Jan 2023
+Minimum Weight Random Graphs with Edge Constraints
+Ghurumuruhan Ganesan1
+1IISER, Bhopal
+E-mail: gganesan82@gmail.com
+Abstract: In this paper, we study two examples of minimum weight random graphs
+with edge constraints. First we consider the complete graph on n vertices equipped with
+uniformly heavy edge weights and use iteration methods to obtain deviation estimates
+for the minimum weight of subtrees with a given number of edges. Next we analyze
+edge constrained minimum weight paths in the integer lattice Zd and employ martingale
+difference techniques to describe the behaviour of the scaled minimum weight in terms
+of the edge constraint.
+Keywords: Minimum spanning tree, heavy weights, minimum passage time paths, edge
+constraints.
+2010 Mathematics Subject Classification: Primary: 60J10, 60K35; Secondary:
+60C05, 62E10, 90B15, 91D30.
+1
+Introduction
+Trees of complete graphs with random edge weights are important from both theoreti-
+cal and practical perspectives. For independent and identically distributed (i.i.d.) edge
+weights with a common cumulative distribution function (cdf) F(.) that varies linearly
+close to zero, [5] studied convergence of the weight of the minimum spanning tree (MST)
+of the complete graph Kn on n vertices. Later [2] studied asymptotics for the expected
+value of the MST weight, when the edge weight distributions follow a power law distri-
+bution. The paper [6] studied central limit theorems for a scaled and centred version
+of the MST weight and more recently [1] studied bounds on the diameter of the MST.
+The methods involve a combination of graph evolution via Kruskal’s agorithm along with
+a component analysis of random graphs. For MSTs with nonidentical edge weight dis-
+tributions, [10] use the Tutte polynomial approach [11] to compute expressions for the
+expected value of MSTn.
+In Section 2 of our paper, we study “approximate” MSTs containing O(n) edges,
+obtained by placing random heavy weights in each edge of Kn that are not necessarily
+identically distributed but are uniformly heavy. We use stochastic domination to obtain
+1
+
+Minimum Weight Random Graphs
+2
+deviation type estimates for the minimum weight and use the martingale method to
+bound the variance (see Theorem 2.1).
+Next we study constrained paths in the integer lattice. Consider the following sce-
+nario where each edge in the square lattice Zd is associated with a random passage
+time and it is of interest to determine the minimum passage time Tn between the ori-
+gin and (n, 0, . . . , 0). The case of independent and identically distributed (i.i.d.) passage
+times has been well-studied and detailed results are known regarding the almost sure
+convergence and convergence in mean of the scaled passage time Tn
+n (see [8]). Later [4]
+studied central limit theorems for first passage across thin cylinders and recently [7] have
+studied critical first passage percolation in the triangular lattice.
+In many applications, the passage times may not be i.i.d. For example, if we model
+vertices of Zd as mobiles stations and the edge passage times as the delay in sending a
+packet between two adjacent stations, then depending on external conditions, the edges
+may have different passage time distributions. In such cases, it is of interest to study
+convergence properties of the minimum passage time Tn, with appropriate centering and
+scaling. In Section 3 our paper, we state and prove our result (Theorem 3.1) regarding the
+behaviour of the constrained minimum passage times as a function of the edge constraint.
+The paper is organized as follows. In Section 2, we state and prove our result regarding
+the asymptotic behaviour of weighted trees of the complete graph, with edge constraints.
+Finally, in Section 3, we describe the behaviour of constrained minimum passage time
+paths in the integer lattice Zd.
+2
+Edge Constrained Minimum Weight Trees
+For n ≥ 1, let Kn be the complete graph with vertex set {1, 2, . . . , n}. Let {w(i, j)}1≤i<j≤n
+be independent random variables with corresponding cumulative distribution functions
+(cdfs) {Fi,j}1≤i<j≤n and for 1 ≤ j < i ≤ n, set w(i, j) := w(j, i). We define w(e) := w(i, j)
+to be the weight of the edge e = (i, j) ∈ Kn and assume throughout that w(e) ≤ 1, for
+simplicity.
+A tree in Kn is a connected acyclic subgraph. For a tree T with vertex set {v1, . . . , vt},
+the weight of T is the sum of the weights of the edges in T ; i.e., W(T ) := �
+e∈T w(e).
+For 1 ≤ τ ≤ n − 1 we define
+Mn = Mn(τ) := min
+T
+W(T ),
+(2.1)
+where the minimum is taken over all trees T ⊂ Kn having at least τ edges.
+The following is the main result of this section. Constants throughout do not depend
+on n.
+Theorem 2.1. Suppose τ ≥ ρn for some 0 < ρ ≤ 1 and also suppose there are positive
+constants D1 ≤ D2 and 0 < α < 1 such that
+D1x
+1
+α ≤ Fi,j(x) ≤ D2x
+1
+α
+(2.2)
+2
+
+Minimum Weight Random Graphs
+3
+for 0 ≤ x ≤ 1. There are positive constants Ci, 1 ≤ i ≤ 3 such that
+P
+�
+C1n1−α ≤ Mn ≤ C2n1−α�
+≥ 1 − e−C3n1−α
+(2.3)
+and C1n1−α ≤ EMn ≤ C2n1−α. Moreover, var(Mn) ≤ 2n.
+To prove Theorem 2.1, we use the following preliminary Lemma regarding the be-
+haviour of the exponential moments of small edge weights. For 1 ≤ j ≤ n and distinct
+deterministic integers 1 ≤ a1, . . . , aj ≤ n let
+Yj = Yj(a1, . . . , aj) :=
+min
+a/∈{a1,...,aj−1} w(aj, a).
+(2.4)
+Lemma 2.2. There are positive constants C1 and C2 not depending on the choice of {ai}
+or j such that for any 1 ≤ j ≤ n − 1,
+C1
+(n − j)α ≤ EYj ≤
+C2
+(n − j)α .
+(2.5)
+Moreover for every s > 1 there are constants K, C ≥ 1 not depending on the choice
+of {ai} or j such that for all 1 ≤ j ≤ n − K
+EesYj ≤ exp
+�
+C
+(n − j)α
+�
+.
+(2.6)
+Proof of Lemma 2.2: In what follows, we use the following standard deviation es-
+timate.
+Suppose Wi, 1 ≤ i ≤ m are independent Bernoulli random variables satisfy-
+ing P(W1 = 1) = 1 − P(W1 = 0) ≤ µ2. For any 0 < ǫ < 1
+2,
+P
+� m
+�
+i=1
+Wi > mµ2(1 + ǫ)
+�
+≤ exp
+�
+−ǫ2
+4 mµ2
+�
+.
+(2.7)
+For a proof of (2.7, we refer to Corollary A.1.14, pp. 312 of Alon and Spencer (2008).
+We first find the lower bound for EYj and then upper bound EYj and EesYj for
+constant s > 1 in that order.
+The term Yj is the minimum of n − j edge weights
+and so for 0 < x < (n − j)α we use the upper bound for the cdfs in (2.2) to get
+that P
+�
+Yj >
+x
+(n−j)α
+�
+≥
+�
+1 − D2 x
+1
+α
+n−j
+�n−j
+, where D2 ≥ 1 is as in (2.2). Thus
+(n − j)αEYj =
+� (n−j)α
+0
+P
+�
+Yj >
+x
+(n − j)α
+�
+dx ≥
+� (n−j)α
+0
+�
+1 − D2
+x
+1
+α
+n − j
+�n−j
+dx. (2.8)
+To evaluate the integral in (2.8), we use 1 − y ≥ e−2y for all 0 < y < 1
+2. Letting y =
+D2x
+1
+α
+n−j
+for 0 < x <
+�
+n−j
+2D2
+�α
+we then have (1 − y)n−j ≥ e−2D2x
+1
+α and substituting this
+3
+
+Minimum Weight Random Graphs
+4
+in (2.8) and using D2 ≥ 1, we get
+(n − j)αEYj ≥
+� �
+n−j
+2D2
+�α
+0
+e−2D2x
+1
+α dx ≥
+� �
+1
+2D2
+� 1
+α
+0
+e−2D2x
+1
+α dx =: C1
+for all 1 ≤ j ≤ n − 1.
+For upper bounding EYj, we again use the fact that the term Yj is the minimum
+of n − j edge weights and so for 0 < x < (n − j)α we use the lower bound for the cdfs
+in (2.2) to get P
+�
+Yj >
+x
+(n−j)α
+�
+≤
+�
+1 − D1 x
+1
+α
+n−j
+�n−j
+≤ e−D1x
+1
+α . Thus
+(n − j)αEYj =
+� (n−j)α
+0
+P
+�
+Yj >
+x
+(n − j)α
+�
+dx ≤
+� ∞
+0
+e−D1x
+1
+α dx =: C2,
+(2.9)
+a finite positive constant not depending on the choice of {ai}.
+To compute EesYj we split EesYj = I1 + I2, where I1 = EesYj11
+�
+Yj ≤ 1
+2s
+�
+and I2 =
+EesYj11
+�
+Yj > 1
+2s
+�
+and estimate each term separately. To evaluate I1, we bound ex ≤ 1+2x
+for x ≤ 1
+2 and set x = sYj ≤ 1
+2 to get that esYj ≤ 1 + 2sYj. Thus
+I1 ≤ 1 + 2sEYj ≤ 1 +
+2sC2
+(n − j)α ,
+(2.10)
+using (2.9).
+To evaluate I2, we recall that Yj = mina/∈{a1,...,aj−1} w(aj, a) ≤ 1 is the minimum
+of n − j independent edge weights.
+Using the upper bound for the cdfs in (2.2) we
+have P
+�
+w(aj, a) > 1
+2s
+�
+≤ 1−
+D1
+(2s)
+1
+α < 1, since D1 ≤ 1 and s > 1. Setting e−θ = 1−
+D1
+(2s)
+1
+α ,
+we therefore have θ > 0 and that P
+�
+Yj > 1
+2s
+�
+≤ e−θ(n−j). Finally using Yj ≤ 1 (since all
+edge weights are at most one) we get
+I2 = EesYj11
+�
+Yj > 1
+2s
+�
+≤ esP
+�
+Yj > 1
+2s
+�
+≤ ese−θ(n−j) ≤
+esC2
+(n − j)α ,
+(2.11)
+where C2 > 0 is as in (2.10), provided n − j ≥ K + 1 and K = K(s, C2) is large. The
+final estimate in (2.11) is obtained using xαe−θx −→ 0 as x → ∞.
+From (2.10) and (2.11), we therefore get for 1 ≤ j ≤ n − K that
+EesYj ≤ 1 +
+2sC2
+(n − j)α +
+esC2
+(n − j)α ≤ exp
+�2sC2 + esC2
+(n − j)α
+�
+,
+proving (2.6).
+Proof of Theorem 2.1: We obtain the lower deviation bound by counting the number of
+edges with large enough weight and the upper deviation bound by constructing a spanning
+4
+
+Minimum Weight Random Graphs
+5
+path with low weight analogous to Aldous (1990). The expectation bounds then follow
+from (2.3). To compute the variance, we use the martingale difference method. In fact,
+from the variance bound, we get that var
+� Mn
+n1−α
+�
+≤
+C
+n1−2α and so if α < 1
+2, then Mn−EMn
+n1−α
+converges to zero in probability. Details follow.
+We begin with the proof of the lower deviation bound. For γ > 0 a small constant,
+let Rtot := �
+e∈Kn 11
+�
+w(e) <
+� γ
+n
+�α�
+be the number of edges of weight at most
+� γ
+n
+�α .
+We estimate Rtot using the standard deviation bound (2.7). First, we have from the
+bounds for the cdfs in (2.2) that P
+�
+w(e) <
+� γ
+n
+�α�
+< D2γ
+n
+and since the edge weights are
+independent, we use (2.7) with m =
+�n
+2
+�
+, µ2 = D2γ
+n
+and ǫ = 1
+4 to get that
+P
+�
+Rtot > 5mD2γ
+4n
+�
+≤ exp
+�
+−mD2γ
+64n
+�
+≤ exp
+�
+−nD2γ
+256
+�
+,
+since m = n(n−1)
+2
+> n2
+4 . Let Tn be any tree with weight Mn and containing at least τ ≥ ρn
+edges. Using m < n2
+2 , we get that with probability at least
+1 − e− nD2γ
+256 , the weight of Tn is at least
+�
+ρn − 5mD2γ
+4n
+�
+·
+�γ
+n
+�α
+≥
+�
+ρn − 5nD2γ
+8
+�
+·
+�γ
+n
+�α
+≥ Cn1−α
+for some constant C > 0, provided γ > 0 is small. This completes the proof of the lower
+deviation bound in (2.3).
+For the upper deviation bound, we consider the spanning path obtained by an in-
+cremental approach similar to Aldous (1990).
+Let i1 = 1 and among all edges with
+endvertex i1, let i2 be the index such that w(i1, i2) has the least weight.
+Similarly,
+among all edges with endvertex in i2 \ {i1}, let i3 be such that w(i2, i3) has the least
+weight. Continuing this way, the path Piter := (i1, . . . , in) is a spanning path containing
+all the nodes and so letting Zj = w(Xij−1, Xij) be the weight of the jth edge in Piter, we
+have Mn ≤ W(Piter) = �n
+j=1 Zj. For s > 0 we therefore have
+EesMn ≤ Ee
+�n−1
+j=1 sZj
+(2.12)
+and in what follows, we find an upper bound for the right hand side of (2.12).
+Let a1 := 1 and al, 2 ≤ l ≤ j−1 be deterministic numbers and suppose the event {i1 =
+a1, . . . , ij−1 = aj−1} occurs so that Zl = w(al, al+1) for 1 ≤ l ≤ j − 1 and Zj = Yj =
+Yj(a1, . . . , aj) = mina/∈{a1,...,aj−1} w(aj, a) is as in (2.4). The event {i1 = a1, . . . , ij−1 =
+aj−1} and the random variables w(al, al+1), 1 ≤ l ≤ j − 1 depend only the state of
+edges having at least one endvertex in {a1, . . . , aj−1}. On the other hand, the random
+variable Yj depends only on the state of edges having both endvertices in {1, . . . , n} \
+5
+
+Minimum Weight Random Graphs
+6
+{a1, . . . , aj−1}. Thus
+Ees �j
+l=1 Zl11(i1 = a1, . . . , ij−1 = aj−1)
+= EesYje
+�j−1
+l=1 sZl11(i1 = a1, . . . , ij−1 = aj−1)
+= EesYjEe
+�j−1
+l=1 sZl11(i1 = a1, . . . , ij−1 = aj−1).
+(2.13)
+Using (2.5) we have EesYj ≤ exp
+�
+C
+(n−j)α
+�
+for all 1 ≤ j ≤ n − K, where K and C do
+not depend on the choice of {ai}. Thus summing (2.13) over all possible a1, . . . , aj−1, we
+get Ees �j
+l=1 Zl ≤ exp
+�
+C
+(n−j)α
+�
+Ees �j−1
+l=1 Zl and continuing iteratively, we get Ees �j
+l=1 Zl ≤
+exp
+�
+C �j
+l=1
+1
+(n−l)α
+�
+for n − j ≥ K. For n − j < K, we use the bound EesYj ≤ es
+since Yj ≤ 1 (all the edge weights are at most one) and argue as before to get that
+Ees �n−1
+l=1 Zl ≤ exp
+�
+C
+n−K
+�
+l=1
+1
+(n − l)α
+�
+esK.
+(2.14)
+Comparing with integrals, the term
+n−K
+�
+l=1
+1
+(n − l)α =
+n−1
+�
+j=K+1
+1
+jα ≤ C3
+� n−1
+K
+1
+xα dx ≤ C4n1−α
+for some positive constants C3, C4. We therefore get from (2.14) and (2.12) that
+EesMn ≤ eC5n1−α for some constant C5 = C5(s). Therefore by Chernoff estimate, we
+have P(Mn ≥ C6n1−α) ≤ e−C7n1−α for some positive constants C6, C7. This completes
+the proof of the lower deviation bound in (2.3).
+Finally, the lower bound on the expectation EMn follows directly from the lower
+deviation bound (2.3). For the expectation upper bound, we use the fact that the edge
+weights are at most one and so total weight of any tree containing at least ρn edges is
+at most n. Consequently from the upper deviation bound in (2.3), we get that EMn ≤
+C2n1−α + n · e−Cn1−α ≤ 2C2n1−α. The proof of the variance bound is analogous to the
+pivotal edge argument in Kesten (1993) together with the fact that the number of edges
+in a spanning tree is at most n. This completes the proof of the Theorem.
+3
+Edge Constrained Minimum Passage Time Paths
+Consider the square lattice Zd, where two vertices w1 = (w1,1, . . . , w1,d) and w2 =
+(w2,1, . . . , w2,d) are adjacent if �d
+i=1 |w1,i − w2,i| = 1 and adjacent vertices are joined
+together by an edge. Let {qi}i≥1 denote the set of edges. Each edge qi is equipped with
+6
+
+Minimum Weight Random Graphs
+7
+a random passage time t(qi) and we define the random sequence (t(q1), t(q2), . . .) on the
+probability space (Ω, F, P).
+A path π is a sequence of distinct adjacent vertices (w1, . . . , wr+1). If ei, 1 ≤ i ≤ r is
+the edge with endvertices wi and wi+1, then we denote π = (e1, ..., er). By definition, π
+is self-avoiding and w1 and wr+1 are said to be the endvertices of π. The length of π is
+the number of edges in π and the passage time of π is defined as T(π) := �r
+i=1 t(ei).
+Definition 3.1. For k ≥ 1 we define the k−constrained minimum passage time between
+the origin and the vertex (n, 0) as Tn(k) := minπ T(π), where the minimum is over
+all paths π of length at most k and with endvertices (0, 0) and (n, 0). We define the
+unconstrained minimum passage time as Tn := infk≥1 Tn(k).
+By definition Tn(k) ↓ Tn a.s. as k → ∞. In this section, we are primarily interested in
+studying how Tn(k) varies as the constraint parameter k increases and also how fast Tn(k)
+converges to Tn.
+The following are the main results of this section.
+Throughout constants do not
+depend on n.
+Theorem 3.1. Suppose
+sup
+i≥1
+P(t(qi) ≤ ǫ) −→ 0 as ǫ ↓ 0 and µ2 := sup
+i≥1
+Et2(qi) < ∞.
+(3.1)
+(a) There are constants C1, C2 > 0 such that for every k ≥ n :
+P (C1n ≤ Tn ≤ Tn(k) ≤ C2n) ≥ 1 − C2
+n ,
+var(Tn(k)) ≤ C2n
+(3.2)
+and C1n ≤ ETn ≤ ETn(k) ≤ C2n.
+(b) There exists a constant C3 > 0 such that if k ≥ C3n then
+P(Tn ̸= Tn(k)) ≤ C3
+k . If k ≥ n1+ǫ for some ǫ > 0, then both Tn(k)−ETn(k)
+n
+and Tn−ETn
+n
+converge to zero a.s. as n → ∞.
+(c) If the edge weights are uniformly square integrable in the sense that
+supi≥1 Et2(qi)11(t(qi) ≥ M) −→ 0 as M → ∞, then var
+� Tn
+n
+�
+−→ 0 as n → ∞.
+If supi≥1 Etp(qi) < ∞ for some p > 2, then var(Tn) ≤ Cn for some constant C > 0.
+Proof of Theorem 3.1(a): Let µ := supf Et(f) be the maximum expected passage
+time of an edge. We begin by showing that there exists a constant 0 < β ≤ µ such for
+any integer m ≥ 1 and any path π containing m edges,
+P (T(π) ≤ βm) ≤ e−dm
+(3.3)
+for some positive constant β = β(d) ≤ µ, not depending on m or π. Here d is the
+dimension of the integer lattice under consideration.
+7
+
+Minimum Weight Random Graphs
+8
+Indeed, let π = (e1, . . . , em) so that T(π) = �m
+i=1 t(ei). Using the Chernoff bound we
+obtain for δ, s > 0 that
+P(T(π) ≤ δm) = P
+� m
+�
+i=1
+t(ei) ≤ δm
+�
+≤ esδm
+m
+�
+i=1
+E
+�
+e−st(ei)�
+.
+(3.4)
+For a fixed η > 0, we write Ee−st(ei) =
+�
+t(ei)<η e−st(ei)dP +
+�
+t(ei)≥η e−st(ei)dP and use
+�
+t(ei)<η
+e−st(ei)dP ≤ P(t(ei) < η) and
+�
+t(ei)≥η
+e−st(ei)dP ≤ e−sη
+to get that
+Ee−st(ei) ≤ P(t(ei) < η) + e−sη.
+Since F(0) = 0, we choose η > 0 small so that P(t(ei) < η) ≤ e−6ǫ
+2 . Fixing such an η
+we choose s = s(η, ǫ) > 0 large so that the second term e−sη <
+e−6ǫ
+2 . This implies
+that Ee−st(ei) ≤ e−6ǫ and so from (3.4) we then get that P(W(P) ≤ δm) ≤ esδme−6ǫm ≤
+e−2ǫm for all m ≥ 1, provided δ = δ(s, ǫ) > 0 is small. This completes the proof of (3.3).
+Next, for integer m ≥ 1 define the event Em as
+Em :=
+�
+r≥ 3µ
+β m
+�
+π
+{T(π) ≥ βr}
+(3.5)
+where the second intersection is over all paths with origin as an endvertex and consisting
+of r edges. Thus Em is the event that every path π with origin as an endvertex and
+consisting of r ≥ 3µ
+β m edges has passage time T(π) ≥ βr. Since there are at most (2d)r
+paths of length r starting from the origin, the estimate (3.3) gives
+P(Ec
+m) ≤
+�
+r≥3µβ−1m
+(2d)re−dr ≤
+�
+r≥3µβ−1m
+(2e−1)r ≤
+e−δm
+1 − 2e−1
+(3.6)
+for all m ≥ 1 and some positive constant δ = δ(d, µ), not depending on m. Here, the
+second inequality in (3.6) is obtained using the fact that the function xe−x attains its
+maximum at x = 1 and so 2de−d ≤ 2e−1 < 1 for all d ≥ 2.
+Let Fm be the event that �m
+i=1 t(fi) ≤ 2µm where fi is the horizontal edge with
+endvertices (i − 1, 0) and (i, 0). Letting Xi := t(fi) − Et(fi) and using the fact that {Xi}
+are independent, we then get from Chebychev’s inequality that
+P(F c
+m) ≤ P
+� m
+�
+i=1
+Xi ≥ µm
+�
+≤
+�m
+i=1 var(Xi)
+µ2m2
+≤ C
+m
+(3.7)
+for some constant C > 0. Now set m = βn
+3µ < n and suppose Em∩Fn occurs. From (3.6), (3.7)
+and the union bound, we get that P(Em ∩Fn) ≥ 1− C
+n for some constant C > 0. Since Fn
+8
+
+Minimum Weight Random Graphs
+9
+occurs, we get that Tn(k) ≤ 2µn and since Em occurs, we get that any path starting
+from the origin and containing r ≥ 3µ
+β m = n edges has weight at least βr ≥ 3µm = βn.
+This obtains the first estimate (3.2).
+Next, using the bounded second moment assumption in (3.1) and arguing as in the
+variance estimate in Theorem 1, Kesten (1993) we get that var(Tn(k)) ≤ CENn(k),
+where Nn(k) is the number of edges in the path with passage time Tn(k). If Em ∩ Fn
+occurs, then from the discussion in the previous paragraph, we get that Nn(k) ≤ 3µ
+β n.
+For x ≥ 3µ
+β n we assume for simplicity that y = βx
+3µ is an integer and write
+P(Nn(k) ≥ x)
+≤
+P ({Nn(k) ≥ x} ∩ Ey) + P
+�
+Ec
+y
+�
+≤
+P ({Nn(k) ≥ x} ∩ Em) + D1e−D2x
+(3.8)
+for some constants D1, D2 > 0 by (3.6). If Nn(k) ≥ x and the event Ey occurs, then
+every path containing r ≥ 3µ
+β y = x edges has weight at least βr ≥ βx. Thus Tn(k) ≥ βx
+and so
+P ({Nn(k) ≥ x} ∩ Ey) ≤ P(Tn(k) ≥ βx) ≤ P
+� n
+�
+i=1
+t(fi) ≥ βx
+�
+.
+Since βx ≥ 3µn we have that βx − �n
+i=1 Et(fi) ≥ βx − µn ≥ 2βx
+3 . Consequently,
+recalling that Xi = t(fi)−Et(fi) and using the fact that var(Xi) ≤ Et2(fi) ≤ C for some
+constant C > 0, we get from Chebychev’s inequality that
+P ({Nn(k) ≥ x} ∩ Ey) ≤ P
+� n
+�
+i=1
+Xi ≥ 2βx
+3
+�
+≤ D3
+�n
+i=1 var(Xi)
+x2
+≤ D4n
+x2
+(3.9)
+for some constants D3, D4 > 0. Combining (3.8) and (3.9), we get that P(Nn(k) ≥ x) ≤
+D5
+x2 for x ≥ 3µ
+β n and so ENn(k) ≤ D6n for some constant D6 > 0. Plugging this into
+the variance estimate for Tn(k) obtained in the previous paragraph, we get the second
+estimate in (3.2).
+The lower expectation bound follows directly from the lower deviation bound in (3.2).
+The upper expectation bound follows from the fact that ETn ≤ �n
+i=1 Et(fi) ≤ µn. This
+completes the proof of part (a).
+Proof of Theorem 3.1(b): Let β, µ be as in part (a) and for k ≥ n suppose that the
+event Ek∩Fk occurs. From the discussion following (3.7), we know that P(Ek∩Fk) ≥ 1− C
+k
+for some constant C > 0. The minimum passage time between the origin and (n, 0) is
+at most 2µk and any path starting from the origin and containing r ≥ 3µ
+β k edges has
+passage time at least βr ≥ 3µk. Thus Tn
+�
+3µk
+β
+�
+= Tn and this obtains the probability
+estimate in (b) with C3 =
+3µ
+β .
+We prove the a.s. convergence in two steps. In the first step, we use a subsequence
+argument to show that Tn(k)−ETn(k)
+n
+converges to zero a.s. In the second step we show
+9
+
+Minimum Weight Random Graphs
+10
+that the difference
+Tn(k)−Tn
+n
+converges to zero a.s. and in L1, provided k is sufficiently
+large. This then obtains the a.s. convergence for Tn−ETn
+n
+.
+Step 1: We begin with a description of the sub-additivity property of the uncon-
+strained passage time Tn. If Tn,m is the minimum passage time between (n, 0) and (m, 0),
+then Tn ≤ Tm + Tn,m. This is because the concatenation of the minimum passage time
+path with endvertices (0, 0) and (m, 0) and the minimum passage time path with endver-
+tices (m, 0) and (n, 0) contains a path with endvertices (0, 0) and (n, 0). Switching the
+roles of m and n we therefore have that |Tn − Tm| ≤ Tn,m and we refer to this estimate
+as the sub-additive property of Tn.
+Letting k ≥ n1+ǫ with ǫ > 0, we now perform the subsequence argument.
+Set-
+ting Un := Tn(k) and Sn := Un − EUn, we first show that Sn
+n −→ 0 a.s. as n → ∞.
+Indeed, from the variance estimate for Un in (3.2), we know that ES2
+n ≤ Cn for some
+constant C > 0 and so for a fixed δ > 0, the sum
+�
+n≥1
+P(|Sn2| > n2δ) ≤
+�
+n≥1
+ES2
+n2
+δ2n4 ≤
+�
+n≥1
+C
+δ2n2 < ∞.
+Since this is true for all δ > 0, Borel-Cantelli Lemma implies that Sn2
+n2 converges to zero
+a.s. as n → ∞.
+To estimate the intermediate values of Sj, we let n2 ≤ j < (n + 1)2 and set Rn :=
+maxn2≤j<(n+1)2 |Sj − Sn2| and show below that Rn
+n2 −→ 0 a.s. as n → ∞. This would
+imply that for n2 ≤ j < (n + 1)2,
+|Sj|
+j
+≤ |Sj − Sn2|
+j
++ |Sn2|
+j
+≤ |Sj − Sn2|
+n2
++ |Sn2|
+n2
+≤ Dn
+n2 + |Sn2|
+n2
+and so Sj
+j converges to zero a.s. as j → ∞.
+To estimate Rn, use first the triangle inequality to get that
+|Sj − Sn2| ≤ |Uj − Un2| + E|Uj − Un2|
+(3.10)
+We know that P(Tj ̸= Uj) ≤
+D1
+k(j) ≤
+D2
+j1+ǫ for some constants D1, D2 > 0 and so set-
+ting Etot := �(n+1)2
+j=n2 {Tj = Uj}, we get from the union bound that
+P(Etot) ≥ 1 −
+(n+1)2
+�
+j=n2
+D2
+j1+ǫ ≥ 1 − D3n
+n2+2ǫ = 1 −
+D3
+n1+2ǫ
+(3.11)
+for some constant D3 > 0.
+We now write |Uj −Un2| = |Tj −Tn2|11(Etot)+|Uj −Un2|11(Ec
+tot) and evaluate each term
+separately. For n2 ≤ j < (n+1)2 we know by the subadditivity property that |Tj −Tn2| ≤
+Tj,n2 ≤ �(n+1)2
+i=n2
+t(fi) =: An and by definition, we have that Uj ≤ �(n+1)2
+i=1
+t(fi) =: Jn.
+10
+
+Minimum Weight Random Graphs
+11
+Thus |Uj − Un2| ≤ An + 2Jn11(Ec
+tot) for all n2 ≤ j ≤ (n + 1)2 and so from (3.10), we see
+that
+Rn ≤ An + 2Jn11(Ec
+tot) + EAn + 2EJn11(Ec
+tot).
+(3.12)
+Based on (3.12), it suffices to show that both the terms An
+n2 and Jn1
+1(Ec
+tot)
+n2
+converge
+to zero a.s. and in L1 as n → ∞. First, from the estimate EAn ≤ �(n+1)2
+i=n2
+Et(fi) ≤
+Cn for some constant C > 0, we get that
+EAn
+n2
+−→ 0 as n
+→
+∞. Next, let 0 <
+θ < 1 be any constant. Using Chebychev’s inequality and arguing as in (3.7), we get
+that P(An ≥ 2µn1+θ) ≤
+C
+n1+2θ for all n large and so by the Borel-Cantelli Lemma we
+get that P
+�
+An ≤ 2µn1+θ for all large n
+�
+= 1. Since θ < 1, we get that An
+n2 −→ 0 a.s.
+as n → ∞.
+To evaluate Jn11(Ec
+tot), we use (3.11) and the Borel-Cantelli Lemma to get 11(Ec
+tot) −→ 0
+a.s. as n → ∞. Thus Jn1
+1(Ec
+tot)
+n2
+−→ 0 a.s. as n → ∞. Using the Cauchy-Schwartz inequality,
+we also get that EJn11(Ec
+tot) ≤
+�
+EJ2
+n
+�1/2 (P(Ec
+tot))1/2 . By the AM-GM inequality and the
+bounded second moment assumption in (3.1), we get that
+EJ2
+n ≤ (n + 1)2
+(n+1)2
+�
+i=1
+Et2(fi) ≤ C2n4
+for some constant C2 > 0 and so using (3.11), we get that
+EJn11(Ec
+tot) ≤
+�
+C2n2 ·
+�
+D
+n1+2ǫ
+�1/2
+.
+Thus EJn1
+1(Ec
+tot)
+n2
+−→ 0 as n → ∞ as well and this completes the proof of Sn
+n −→ 0 a.s.
+as n → ∞.
+Step 2: We now set k = n1+ǫ and show that Tn(k)−Tn
+n
+converges to zero a.s. and
+in L1. First using P(Tn(k) ̸= Tn) ≤
+D1
+n1+ǫ for some constant D1 > 0, we get from the
+Borel-Cantelli Lemma that Tn(k)−Tn
+n
+−→ 0 a.s. as n → ∞. Next using the fact that Tn(k)
+and Tn are both bounded above by �n
+i=1 t(fi) we have that E|Tn(k) − Tn| = E|Tn(k) −
+Tn|11(Tn(k) ̸= Tn) is bounded above by
+E
+n
+�
+i=1
+t(fi)11(Tn(k) ̸= Tn)
+≤
+E1/2
+� n
+�
+i=1
+t(fi)
+�2
+P1/2(Tn(k) ̸= Tn)
+≤
+E1/2
+� n
+�
+i=1
+t(fi)
+�2 � D1
+n1+ǫ
+�1/2
+.
+Using (�l
+i=1 ai)2 ≤ l �
+i a2
+i we have E (�n
+i=1 t(fi))2 ≤ n �n
+i=1 Et2(fi) ≤ D2n2 for some
+constant D2 > 0. Thus E|Tn(k) − Tn| ≤ D3(n1−ǫ)1/2 = o(n) and so
+E|Tn(k)−Tn|
+n
+−→ 0 as n → ∞. This completes the proof of a.s. convergence in part (b).
+11
+
+Minimum Weight Random Graphs
+12
+Proof of Theorem 3.1(c): Using (a+b)2 ≤ 2(a2 +b2) for any two real numbers a and b
+we have that the variance of the sum of any two random variables X and Y satisfies
+var(X + Y ) = E ((X − EX) + (Y − EY ))2 ≤ 2(var(X) + var(Y )).
+(3.13)
+Setting T = Tn, U = Tn(k), X = T − U and Y = U we get that
+var(T) ≤ 2var(T − U) + 2var(U) ≤ 2var(T − U) + D1n
+(3.14)
+for all n large and some constant D1 > 0, by the variance estimate for U in part (a) of
+this Theorem.
+We estimate var(T − U) as follows. Using T ≤ U we write
+E(T − U)2 = E(T − U)211(T ̸= U) ≤ 2E(T 2 + U 2)11(T ̸= U) ≤ 4EU 211(T ̸= U).
+Since U ≤ �n
+i=1 t(fi), we have that U 2 ≤ n �n
+i=1 t2(fi) and so var(T − U) is bounded
+above by
+E(T − U)2 ≤ 4n
+n
+�
+i=1
+Et2(fi)11(T ̸= U) ≤ 4n2 sup
+i
+Et2(fi)11(T ̸= U).
+(3.15)
+Let θ > 0 be a constant and split
+Et2(fi)11(T ̸= U)
+=
+Et2(fi)11({T ̸= U} ∩ {t(fi) < nθ})
++
+Et2(fi)11({T ̸= U} ∩ {t(fi) ≥ nθ}).
+(3.16)
+From (3.2), we know that there are constants D2, D3 > 0 such that if k ≥ D2n then
+P(T ̸= U) ≤ D3
+k . With this choice of k, the first term in (3.16) is bounded above by
+Et2(fi)11({T ̸= U} ∩ {t(fi) < nθ}) ≤ n2θP(T ̸= U) ≤ D2n2θ
+k
+≤ 1
+n3
+(3.17)
+provided we choose k larger if necessary so that k ≥ n2θ log n.
+We now consider the case where the uniform square integrability condition holds. For
+any η > 0 and all n large, the final term in (3.16) is then at most Et2(fi)11(t(fi) ≥ nθ) ≤ η.
+Combining this estimate with (3.17), we get that Et2(fi)11(T ̸= U) ≤
+1
+n3 +η ≤ 2η for all n
+large and so from (3.15) we get that E(T − U)2 ≤ 8n2η. Plugging this into (3.14) we get
+that var(T) ≤ D1n + 8n2η and since η > 0 is arbitrary, this implies that var
+� T
+n
+�
+= o(1).
+Suppose now that bounded pth moment condition holds for some p > 2. Using H¨older’s
+inequality and Markov inequality in succession, the final term in (3.16) is at most
+Et2(fi)11(t(fi) ≥ nθ)
+≤
+(Etp(fi))2/p P
+�
+t(fi) ≥ nθ�1−2/p
+≤
+(Etp(fi))2/p
+�Etp(fi)
+nθp
+�1−2/p
+≤
+D4
+nθ(p−2)
+(3.18)
+12
+
+Minimum Weight Random Graphs
+13
+for some constant D4 > 0, by the bounded pth moment assumption. We choose θ > 0
+large so that the final term in (3.18) is at most
+1
+n3. With this choice of θ we get from (3.17)
+that Et2(fi)11(T ̸= U) ≤
+2
+n3 and plugging this into (3.15), we get that E(T −U)2 ≤ D5
+n for
+constant D5 > 0. From (3.14), we then get that var(T) ≤ D6n for some constant D6 > 0.
+This completes the proof of part (c).
+Remark: For p > 2, the bounded pth moment condition in Theorem 3.1(c) is stronger
+than the uniformly square integrable condition which in turn is stronger than the bounded
+second moment condition in (3.1). For the particular case of i.i.d. passage times, uniform
+square integrability is implied by the bounded second moment condition and the first
+condition in (3.1) above simply states that the passage times are a.s. positive.
+Acknowledgement
+I thank Professors Rahul Roy, C. R. Subramanian and the referee for crucial comments
+that led to an improvement of the paper. I also thank IMSc and IISER Bhopal for my
+fellowships.
+References
+[1] L. Addario-Berry, N. Broutin and B. Reed. (2006).
+The diameter of the minimum
+spanning tree of a complete graph. Proceedings DMTCS, AG, pp. 237—248.
+[2] D. Aldous. (1990). A random tree model associated with random graphs. Random
+Structures and Algorithms, 1, 383–202.
+[3] N. Alon and J. Spencer. (2008). The Probabilistic Method. Wiley.
+[4] S. Chatterjee and P. Dey. (2013). Central limit theorem for first-passage percolation
+time across thin cylinders. Probability Theory and Related Fields, 156, 613–663.
+[5] A. M. Frieze. (1985). On the value of a random minimum spanning tree problem.
+Discrete Applied Mathematics, 10, 47–56.
+[6] S. Janson. (1995). The minimal spanning tree in a complete graph and a functional
+limit theorem for trees in a random graph. Random Structres and Algorithms, 7,
+337–355.
+[7] J. Jiang and C-L. Yao. (2019).
+Critical first-passage percolation starting on the
+boundary. Stochastic Processes and their Applications, 129, 2049–2065.
+[8] H. Kesten. (1986). Aspects of First Passage Percolation. Springer.
+13
+
+Minimum Weight Random Graphs
+14
+[9] H. Kesten. (1993), On the Speed of convergence in first-passage percolation. Annals
+Appl. Prob., 3, 296–338.
+[10] W. V. Li and X. Zhang. (2010). Expected lengths of minimum spanning trees for
+non-identical edge distributions. Electronic Journal of Probability, 15, pp. 110–141.
+[11] J. Steele. (2002).
+Minimal spanning trees for graphs with random edge lengths.
+Mathematics and computer science II. Algorithms, trees, combinatorics and proba-
+bilities, pp. 223—245.
+14
+
diff --git a/o9E3T4oBgHgl3EQf7gve/content/tmp_files/load_file.txt b/o9E3T4oBgHgl3EQf7gve/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..8e5b41ca4913d17ccc259e084826feb5615089d6
--- /dev/null
+++ b/o9E3T4oBgHgl3EQf7gve/content/tmp_files/load_file.txt
@@ -0,0 +1,515 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf,len=514
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='04800v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='PR]  12 Jan 2023  Minimum Weight Random Graphs with Edge Constraints  Ghurumuruhan Ganesan1  1IISER, Bhopal  E-mail: gganesan82@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='com  Abstract: In this paper, we study two examples of minimum weight random graphs  with edge constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' First we consider the complete graph on n vertices equipped with  uniformly heavy edge weights and use iteration methods to obtain deviation estimates  for the minimum weight of subtrees with a given number of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Next we analyze  edge constrained minimum weight paths in the integer lattice Zd and employ martingale  difference techniques to describe the behaviour of the scaled minimum weight in terms  of the edge constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  Keywords: Minimum spanning tree, heavy weights, minimum passage time paths, edge  constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  2010 Mathematics Subject Classification: Primary: 60J10, 60K35;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Secondary:  60C05, 62E10, 90B15, 91D30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  1  Introduction  Trees of complete graphs with random edge weights are important from both theoreti-  cal and practical perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' For independent and identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=') edge  weights with a common cumulative distribution function (cdf) F(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=') that varies linearly  close to zero, [5] studied convergence of the weight of the minimum spanning tree (MST)  of the complete graph Kn on n vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Later [2] studied asymptotics for the expected  value of the MST weight, when the edge weight distributions follow a power law distri-  bution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' The paper [6] studied central limit theorems for a scaled and centred version  of the MST weight and more recently [1] studied bounds on the diameter of the MST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  The methods involve a combination of graph evolution via Kruskal’s agorithm along with  a component analysis of random graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' For MSTs with nonidentical edge weight dis-  tributions, [10] use the Tutte polynomial approach [11] to compute expressions for the  expected value of MSTn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  In Section 2 of our paper, we study “approximate” MSTs containing O(n) edges,  obtained by placing random heavy weights in each edge of Kn that are not necessarily  identically distributed but are uniformly heavy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' We use stochastic domination to obtain  1  Minimum Weight Random Graphs  2  deviation type estimates for the minimum weight and use the martingale method to  bound the variance (see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  Next we study constrained paths in the integer lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Consider the following sce-  nario where each edge in the square lattice Zd is associated with a random passage  time and it is of interest to determine the minimum passage time Tn between the ori-  gin and (n, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' , 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' The case of independent and identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=') passage  times has been well-studied and detailed results are known regarding the almost sure  convergence and convergence in mean of the scaled passage time Tn  n (see [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Later [4]  studied central limit theorems for first passage across thin cylinders and recently [7] have  studied critical first passage percolation in the triangular lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  In many applications, the passage times may not be i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' For example, if we model  vertices of Zd as mobiles stations and the edge passage times as the delay in sending a  packet between two adjacent stations, then depending on external conditions, the edges  may have different passage time distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' In such cases, it is of interest to study  convergence properties of the minimum passage time Tn, with appropriate centering and  scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' In Section 3 our paper, we state and prove our result (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='1) regarding the  behaviour of the constrained minimum passage times as a function of the edge constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' In Section 2, we state and prove our result regarding  the asymptotic behaviour of weighted trees of the complete graph, with edge constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  Finally, in Section 3, we describe the behaviour of constrained minimum passage time  paths in the integer lattice Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  2  Edge Constrained Minimum Weight Trees  For n ≥ 1, let Kn be the complete graph with vertex set {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Let {w(i, j)}1≤i<j≤n  be independent random variables with corresponding cumulative distribution functions  (cdfs) {Fi,j}1≤i<j≤n and for 1 ≤ j < i ≤ n, set w(i, j) := w(j, i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' We define w(e) := w(i, j)  to be the weight of the edge e = (i, j) ∈ Kn and assume throughout that w(e) ≤ 1, for  simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  A tree in Kn is a connected acyclic subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' For a tree T with vertex set {v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' , vt},  the weight of T is the sum of the weights of the edges in T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=', W(T ) := �  e∈T w(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  For 1 ≤ τ ≤ n − 1 we define  Mn = Mn(τ) := min  T  W(T ),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='1)  where the minimum is taken over all trees T ⊂ Kn having at least τ edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  The following is the main result of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Constants throughout do not depend  on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Suppose τ ≥ ρn for some 0 < ρ ≤ 1 and also suppose there are positive  constants D1 ≤ D2 and 0 < α < 1 such that  D1x  1  α ≤ Fi,j(x) ≤ D2x  1  α  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='2)  2  Minimum Weight Random Graphs  3  for 0 ≤ x ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' There are positive constants Ci, 1 ≤ i ≤ 3 such that  P  �  C1n1−α ≤ Mn ≤ C2n1−α�  ≥ 1 − e−C3n1−α  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='3)  and C1n1−α ≤ EMn ≤ C2n1−α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Moreover, var(Mn) ≤ 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  To prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='1, we use the following preliminary Lemma regarding the be-  haviour of the exponential moments of small edge weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' For 1 ≤ j ≤ n and distinct  deterministic integers 1 ≤ a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' , aj ≤ n let  Yj = Yj(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' , aj) :=  min  a/∈{a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=',aj−1} w(aj, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='4)  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' There are positive constants C1 and C2 not depending on the choice of {ai}  or j such that for any 1 ≤ j ≤ n − 1,  C1  (n − j)α ≤ EYj ≤  C2  (n − j)α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='5)  Moreover for every s > 1 there are constants K, C ≥ 1 not depending on the choice  of {ai} or j such that for all 1 ≤ j ≤ n − K  EesYj ≤ exp  �  C  (n − j)α  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='6)  Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='2: In what follows, we use the following standard deviation es-  timate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  Suppose Wi, 1 ≤ i ≤ m are independent Bernoulli random variables satisfy-  ing P(W1 = 1) = 1 − P(W1 = 0) ≤ µ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' For any 0 < ǫ < 1  2,  P  � m  �  i=1  Wi > mµ2(1 + ǫ)  �  ≤ exp  �  −ǫ2  4 mµ2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='7)  For a proof of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='7, we refer to Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='14, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' 312 of Alon and Spencer (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  We first find the lower bound for EYj and then upper bound EYj and EesYj for  constant s > 1 in that order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  The term Yj is the minimum of n − j edge weights  and so for 0 < x < (n − j)α we use the upper bound for the cdfs in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='2) to get  that P  �  Yj >  x  (n−j)α  �  ≥  �  1 − D2 x  1  α  n−j  �n−j  , where D2 ≥ 1 is as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Thus  (n − j)αEYj =  � (n−j)α  0  P  �  Yj >  x  (n − j)α  �  dx ≥  � (n−j)α  0  �  1 − D2  x  1  α  n − j  �n−j  dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='8)  To evaluate the integral in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='8), we use 1 − y ≥ e−2y for all 0 < y < 1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Letting y =  D2x  1  α  n−j  for 0 < x <  �  n−j  2D2  �α  we then have (1 − y)n−j ≥ e−2D2x  1  α and substituting this  3  Minimum Weight Random Graphs  4  in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='8) and using D2 ≥ 1, we get  (n − j)αEYj ≥  � �  n−j  2D2  �α  0  e−2D2x  1  α dx ≥  � �  1  2D2  � 1  α  0  e−2D2x  1  α dx =: C1  for all 1 ≤ j ≤ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  For upper bounding EYj, we again use the fact that the term Yj is the minimum  of n − j edge weights and so for 0 < x < (n − j)α we use the lower bound for the cdfs  in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='2) to get P  �  Yj >  x  (n−j)α  �  ≤  �  1 − D1 x  1  α  n−j  �n−j  ≤ e−D1x  1  α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Thus  (n − j)αEYj =  � (n−j)α  0  P  �  Yj >  x  (n − j)α  �  dx ≤  � ∞  0  e−D1x  1  α dx =: C2,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='9)  a finite positive constant not depending on the choice of {ai}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  To compute EesYj we split EesYj = I1 + I2, where I1 = EesYj11  �  Yj ≤ 1  2s  �  and I2 =  EesYj11  �  Yj > 1  2s  �  and estimate each term separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' To evaluate I1, we bound ex ≤ 1+2x  for x ≤ 1  2 and set x = sYj ≤ 1  2 to get that esYj ≤ 1 + 2sYj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Thus  I1 ≤ 1 + 2sEYj ≤ 1 +  2sC2  (n − j)α ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='10)  using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  To evaluate I2, we recall that Yj = mina/∈{a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=',aj−1} w(aj, a) ≤ 1 is the minimum  of n − j independent edge weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  Using the upper bound for the cdfs in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='2) we  have P  �  w(aj, a) > 1  2s  �  ≤ 1−  D1  (2s)  1  α < 1, since D1 ≤ 1 and s > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Setting e−θ = 1−  D1  (2s)  1  α ,  we therefore have θ > 0 and that P  �  Yj > 1  2s  �  ≤ e−θ(n−j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Finally using Yj ≤ 1 (since all  edge weights are at most one) we get  I2 = EesYj11  �  Yj > 1  2s  �  ≤ esP  �  Yj > 1  2s  �  ≤ ese−θ(n−j) ≤  esC2  (n − j)α ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='11)  where C2 > 0 is as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='10), provided n − j ≥ K + 1 and K = K(s, C2) is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' The  final estimate in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='11) is obtained using xαe−θx −→ 0 as x → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='10) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='11), we therefore get for 1 ≤ j ≤ n − K that  EesYj ≤ 1 +  2sC2  (n − j)α +  esC2  (n − j)α ≤ exp  �2sC2 + esC2  (n − j)α  �  ,  proving (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='1: We obtain the lower deviation bound by counting the number of  edges with large enough weight and the upper deviation bound by constructing a spanning  4  Minimum Weight Random Graphs  5  path with low weight analogous to Aldous (1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' The expectation bounds then follow  from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' To compute the variance, we use the martingale difference method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' In fact,  from the variance bound, we get that var  � Mn  n1−α  �  ≤  C  n1−2α and so if α < 1  2, then Mn−EMn  n1−α  converges to zero in probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Details follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  We begin with the proof of the lower deviation bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' For γ > 0 a small constant,  let Rtot := �  e∈Kn 11  �  w(e) <  � γ  n  �α�  be the number of edges of weight at most  � γ  n  �α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  We estimate Rtot using the standard deviation bound (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' First, we have from the  bounds for the cdfs in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='2) that P  �  w(e) <  � γ  n  �α�  < D2γ  n  and since the edge weights are  independent, we use (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='7) with m =  �n  2  �  , µ2 = D2γ  n  and ǫ = 1  4 to get that  P  �  Rtot > 5mD2γ  4n  �  ≤ exp  �  −mD2γ  64n  �  ≤ exp  �  −nD2γ  256  �  ,  since m = n(n−1)  2  > n2  4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Let Tn be any tree with weight Mn and containing at least τ ≥ ρn  edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Using m < n2  2 , we get that with probability at least  1 − e− nD2γ  256 , the weight of Tn is at least  �  ρn − 5mD2γ  4n  � �γ  n  �α  ≥  �  ρn − 5nD2γ  8  � �γ  n  �α  ≥ Cn1−α  for some constant C > 0, provided γ > 0 is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' This completes the proof of the lower  deviation bound in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  For the upper deviation bound, we consider the spanning path obtained by an in-  cremental approach similar to Aldous (1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  Let i1 = 1 and among all edges with  endvertex i1, let i2 be the index such that w(i1, i2) has the least weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  Similarly,  among all edges with endvertex in i2 \\ {i1}, let i3 be such that w(i2, i3) has the least  weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Continuing this way, the path Piter := (i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' , in) is a spanning path containing  all the nodes and so letting Zj = w(Xij−1, Xij) be the weight of the jth edge in Piter, we  have Mn ≤ W(Piter) = �n  j=1 Zj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' For s > 0 we therefore have  EesMn ≤ Ee  �n−1  j=1 sZj  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='12)  and in what follows, we find an upper bound for the right hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  Let a1 := 1 and al, 2 ≤ l ≤ j−1 be deterministic numbers and suppose the event {i1 =  a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' , ij−1 = aj−1} occurs so that Zl = w(al, al+1) for 1 ≤ l ≤ j − 1 and Zj = Yj =  Yj(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' , aj) = mina/∈{a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=',aj−1} w(aj, a) is as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' The event {i1 = a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' , ij−1 =  aj−1} and the random variables w(al, al+1), 1 ≤ l ≤ j − 1 depend only the state of  edges having at least one endvertex in {a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' , aj−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' On the other hand, the random  variable Yj depends only on the state of edges having both endvertices in {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' , n} \\  5  Minimum Weight Random Graphs  6  {a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' , aj−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Thus  Ees �j  l=1 Zl11(i1 = a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' , ij−1 = aj−1)  = EesYje  �j−1  l=1 sZl11(i1 = a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' , ij−1 = aj−1)  = EesYjEe  �j−1  l=1 sZl11(i1 = a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' , ij−1 = aj−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='13)  Using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='5) we have EesYj ≤ exp  �  C  (n−j)α  �  for all 1 ≤ j ≤ n − K, where K and C do  not depend on the choice of {ai}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Thus summing (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='13) over all possible a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' , aj−1, we  get Ees �j  l=1 Zl ≤ exp  �  C  (n−j)α  �  Ees �j−1  l=1 Zl and continuing iteratively, we get Ees �j  l=1 Zl ≤  exp  �  C �j  l=1  1  (n−l)α  �  for n − j ≥ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' For n − j < K, we use the bound EesYj ≤ es  since Yj ≤ 1 (all the edge weights are at most one) and argue as before to get that  Ees �n−1  l=1 Zl ≤ exp  �  C  n−K  �  l=1  1  (n − l)α  �  esK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='14)  Comparing with integrals, the term  n−K  �  l=1  1  (n − l)α =  n−1  �  j=K+1  1  jα ≤ C3  � n−1  K  1  xα dx ≤ C4n1−α  for some positive constants C3, C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' We therefore get from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='14) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='12) that  EesMn ≤ eC5n1−α for some constant C5 = C5(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Therefore by Chernoff estimate, we  have P(Mn ≥ C6n1−α) ≤ e−C7n1−α for some positive constants C6, C7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' This completes  the proof of the lower deviation bound in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  Finally, the lower bound on the expectation EMn follows directly from the lower  deviation bound (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' For the expectation upper bound, we use the fact that the edge  weights are at most one and so total weight of any tree containing at least ρn edges is  at most n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Consequently from the upper deviation bound in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='3), we get that EMn ≤  C2n1−α + n · e−Cn1−α ≤ 2C2n1−α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' The proof of the variance bound is analogous to the  pivotal edge argument in Kesten (1993) together with the fact that the number of edges  in a spanning tree is at most n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' This completes the proof of the Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  3  Edge Constrained Minimum Passage Time Paths  Consider the square lattice Zd, where two vertices w1 = (w1,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' , w1,d) and w2 =  (w2,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' , w2,d) are adjacent if �d  i=1 |w1,i − w2,i| = 1 and adjacent vertices are joined  together by an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Let {qi}i≥1 denote the set of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Each edge qi is equipped with  6  Minimum Weight Random Graphs  7  a random passage time t(qi) and we define the random sequence (t(q1), t(q2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=') on the  probability space (Ω, F, P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  A path π is a sequence of distinct adjacent vertices (w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' , wr+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' If ei, 1 ≤ i ≤ r is  the edge with endvertices wi and wi+1, then we denote π = (e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=', er).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' By definition, π  is self-avoiding and w1 and wr+1 are said to be the endvertices of π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' The length of π is  the number of edges in π and the passage time of π is defined as T(π) := �r  i=1 t(ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' For k ≥ 1 we define the k−constrained minimum passage time between  the origin and the vertex (n, 0) as Tn(k) := minπ T(π), where the minimum is over  all paths π of length at most k and with endvertices (0, 0) and (n, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' We define the  unconstrained minimum passage time as Tn := infk≥1 Tn(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  By definition Tn(k) ↓ Tn a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' In this section, we are primarily interested in  studying how Tn(k) varies as the constraint parameter k increases and also how fast Tn(k)  converges to Tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  The following are the main results of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  Throughout constants do not  depend on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Suppose  sup  i≥1  P(t(qi) ≤ ǫ) −→ 0 as ǫ ↓ 0 and µ2 := sup  i≥1  Et2(qi) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='1)  (a) There are constants C1, C2 > 0 such that for every k ≥ n :  P (C1n ≤ Tn ≤ Tn(k) ≤ C2n) ≥ 1 − C2  n ,  var(Tn(k)) ≤ C2n  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='2)  and C1n ≤ ETn ≤ ETn(k) ≤ C2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  (b) There exists a constant C3 > 0 such that if k ≥ C3n then  P(Tn ̸= Tn(k)) ≤ C3  k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' If k ≥ n1+ǫ for some ǫ > 0, then both Tn(k)−ETn(k)  n  and Tn−ETn  n  converge to zero a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  (c) If the edge weights are uniformly square integrable in the sense that  supi≥1 Et2(qi)11(t(qi) ≥ M) −→ 0 as M → ∞, then var  � Tn  n  �  −→ 0 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  If supi≥1 Etp(qi) < ∞ for some p > 2, then var(Tn) ≤ Cn for some constant C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='1(a): Let µ := supf Et(f) be the maximum expected passage  time of an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' We begin by showing that there exists a constant 0 < β ≤ µ such for  any integer m ≥ 1 and any path π containing m edges,  P (T(π) ≤ βm) ≤ e−dm  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='3)  for some positive constant β = β(d) ≤ µ, not depending on m or π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Here d is the  dimension of the integer lattice under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  7  Minimum Weight Random Graphs  8  Indeed, let π = (e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' , em) so that T(π) = �m  i=1 t(ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Using the Chernoff bound we  obtain for δ, s > 0 that  P(T(π) ≤ δm) = P  � m  �  i=1  t(ei) ≤ δm  �  ≤ esδm  m  �  i=1  E  �  e−st(ei)�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='4)  For a fixed η > 0, we write Ee−st(ei) =  �  t(ei)<η e−st(ei)dP +  �  t(ei)≥η e−st(ei)dP and use  �  t(ei)<η  e−st(ei)dP ≤ P(t(ei) < η) and  �  t(ei)≥η  e−st(ei)dP ≤ e−sη  to get that  Ee−st(ei) ≤ P(t(ei) < η) + e−sη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  Since F(0) = 0, we choose η > 0 small so that P(t(ei) < η) ≤ e−6ǫ  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Fixing such an η  we choose s = s(η, ǫ) > 0 large so that the second term e−sη <  e−6ǫ  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' This implies  that Ee−st(ei) ≤ e−6ǫ and so from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='4) we then get that P(W(P) ≤ δm) ≤ esδme−6ǫm ≤  e−2ǫm for all m ≥ 1, provided δ = δ(s, ǫ) > 0 is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' This completes the proof of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  Next, for integer m ≥ 1 define the event Em as  Em :=  �  r≥ 3µ  β m  �  π  {T(π) ≥ βr}  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='5)  where the second intersection is over all paths with origin as an endvertex and consisting  of r edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Thus Em is the event that every path π with origin as an endvertex and  consisting of r ≥ 3µ  β m edges has passage time T(π) ≥ βr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Since there are at most (2d)r  paths of length r starting from the origin, the estimate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='3) gives  P(Ec  m) ≤  �  r≥3µβ−1m  (2d)re−dr ≤  �  r≥3µβ−1m  (2e−1)r ≤  e−δm  1 − 2e−1  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='6)  for all m ≥ 1 and some positive constant δ = δ(d, µ), not depending on m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Here, the  second inequality in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='6) is obtained using the fact that the function xe−x attains its  maximum at x = 1 and so 2de−d ≤ 2e−1 < 1 for all d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  Let Fm be the event that �m  i=1 t(fi) ≤ 2µm where fi is the horizontal edge with  endvertices (i − 1, 0) and (i, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Letting Xi := t(fi) − Et(fi) and using the fact that {Xi}  are independent, we then get from Chebychev’s inequality that  P(F c  m) ≤ P  � m  �  i=1  Xi ≥ µm  �  ≤  �m  i=1 var(Xi)  µ2m2  ≤ C  m  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='7)  for some constant C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Now set m = βn  3µ < n and suppose Em∩Fn occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='6), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='7)  and the union bound, we get that P(Em ∩Fn) ≥ 1− C  n for some constant C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Since Fn  8  Minimum Weight Random Graphs  9  occurs, we get that Tn(k) ≤ 2µn and since Em occurs, we get that any path starting  from the origin and containing r ≥ 3µ  β m = n edges has weight at least βr ≥ 3µm = βn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  This obtains the first estimate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  Next, using the bounded second moment assumption in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='1) and arguing as in the  variance estimate in Theorem 1, Kesten (1993) we get that var(Tn(k)) ≤ CENn(k),  where Nn(k) is the number of edges in the path with passage time Tn(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' If Em ∩ Fn  occurs, then from the discussion in the previous paragraph, we get that Nn(k) ≤ 3µ  β n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  For x ≥ 3µ  β n we assume for simplicity that y = βx  3µ is an integer and write  P(Nn(k) ≥ x)  ≤  P ({Nn(k) ≥ x} ∩ Ey) + P  �  Ec  y  �  ≤  P ({Nn(k) ≥ x} ∩ Em) + D1e−D2x  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='8)  for some constants D1, D2 > 0 by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' If Nn(k) ≥ x and the event Ey occurs, then  every path containing r ≥ 3µ  β y = x edges has weight at least βr ≥ βx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Thus Tn(k) ≥ βx  and so  P ({Nn(k) ≥ x} ∩ Ey) ≤ P(Tn(k) ≥ βx) ≤ P  � n  �  i=1  t(fi) ≥ βx  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  Since βx ≥ 3µn we have that βx − �n  i=1 Et(fi) ≥ βx − µn ≥ 2βx  3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Consequently,  recalling that Xi = t(fi)−Et(fi) and using the fact that var(Xi) ≤ Et2(fi) ≤ C for some  constant C > 0, we get from Chebychev’s inequality that  P ({Nn(k) ≥ x} ∩ Ey) ≤ P  � n  �  i=1  Xi ≥ 2βx  3  �  ≤ D3  �n  i=1 var(Xi)  x2  ≤ D4n  x2  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='9)  for some constants D3, D4 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='8) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='9), we get that P(Nn(k) ≥ x) ≤  D5  x2 for x ≥ 3µ  β n and so ENn(k) ≤ D6n for some constant D6 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Plugging this into  the variance estimate for Tn(k) obtained in the previous paragraph, we get the second  estimate in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  The lower expectation bound follows directly from the lower deviation bound in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  The upper expectation bound follows from the fact that ETn ≤ �n  i=1 Et(fi) ≤ µn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' This  completes the proof of part (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='1(b): Let β, µ be as in part (a) and for k ≥ n suppose that the  event Ek∩Fk occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' From the discussion following (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='7), we know that P(Ek∩Fk) ≥ 1− C  k  for some constant C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' The minimum passage time between the origin and (n, 0) is  at most 2µk and any path starting from the origin and containing r ≥ 3µ  β k edges has  passage time at least βr ≥ 3µk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Thus Tn  �  3µk  β  �  = Tn and this obtains the probability  estimate in (b) with C3 =  3µ  β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  We prove the a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' convergence in two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' In the first step, we use a subsequence  argument to show that Tn(k)−ETn(k)  n  converges to zero a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' In the second step we show  9  Minimum Weight Random Graphs  10  that the difference  Tn(k)−Tn  n  converges to zero a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' and in L1, provided k is sufficiently  large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' This then obtains the a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' convergence for Tn−ETn  n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  Step 1: We begin with a description of the sub-additivity property of the uncon-  strained passage time Tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' If Tn,m is the minimum passage time between (n, 0) and (m, 0),  then Tn ≤ Tm + Tn,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' This is because the concatenation of the minimum passage time  path with endvertices (0, 0) and (m, 0) and the minimum passage time path with endver-  tices (m, 0) and (n, 0) contains a path with endvertices (0, 0) and (n, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Switching the  roles of m and n we therefore have that |Tn − Tm| ≤ Tn,m and we refer to this estimate  as the sub-additive property of Tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  Letting k ≥ n1+ǫ with ǫ > 0, we now perform the subsequence argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  Set-  ting Un := Tn(k) and Sn := Un − EUn, we first show that Sn  n −→ 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  Indeed, from the variance estimate for Un in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='2), we know that ES2  n ≤ Cn for some  constant C > 0 and so for a fixed δ > 0, the sum  �  n≥1  P(|Sn2| > n2δ) ≤  �  n≥1  ES2  n2  δ2n4 ≤  �  n≥1  C  δ2n2 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  Since this is true for all δ > 0, Borel-Cantelli Lemma implies that Sn2  n2 converges to zero  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  To estimate the intermediate values of Sj, we let n2 ≤ j < (n + 1)2 and set Rn :=  maxn2≤j<(n+1)2 |Sj − Sn2| and show below that Rn  n2 −→ 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' This would  imply that for n2 ≤ j < (n + 1)2,  |Sj|  j  ≤ |Sj − Sn2|  j  + |Sn2|  j  ≤ |Sj − Sn2|  n2  + |Sn2|  n2  ≤ Dn  n2 + |Sn2|  n2  and so Sj  j converges to zero a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' as j → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  To estimate Rn, use first the triangle inequality to get that  |Sj − Sn2| ≤ |Uj − Un2| + E|Uj − Un2|  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='10)  We know that P(Tj ̸= Uj) ≤  D1  k(j) ≤  D2  j1+ǫ for some constants D1, D2 > 0 and so set-  ting Etot := �(n+1)2  j=n2 {Tj = Uj}, we get from the union bound that  P(Etot) ≥ 1 −  (n+1)2  �  j=n2  D2  j1+ǫ ≥ 1 − D3n  n2+2ǫ = 1 −  D3  n1+2ǫ  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='11)  for some constant D3 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  We now write |Uj −Un2| = |Tj −Tn2|11(Etot)+|Uj −Un2|11(Ec  tot) and evaluate each term  separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' For n2 ≤ j < (n+1)2 we know by the subadditivity property that |Tj −Tn2| ≤  Tj,n2 ≤ �(n+1)2  i=n2  t(fi) =: An and by definition, we have that Uj ≤ �(n+1)2  i=1  t(fi) =: Jn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  10  Minimum Weight Random Graphs  11  Thus |Uj − Un2| ≤ An + 2Jn11(Ec  tot) for all n2 ≤ j ≤ (n + 1)2 and so from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='10), we see  that  Rn ≤ An + 2Jn11(Ec  tot) + EAn + 2EJn11(Ec  tot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='12)  Based on (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='12), it suffices to show that both the terms An  n2 and Jn1  1(Ec  tot)  n2  converge  to zero a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' and in L1 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' First, from the estimate EAn ≤ �(n+1)2  i=n2  Et(fi) ≤  Cn for some constant C > 0, we get that  EAn  n2  −→ 0 as n  →  ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Next, let 0 <  θ < 1 be any constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Using Chebychev’s inequality and arguing as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='7), we get  that P(An ≥ 2µn1+θ) ≤  C  n1+2θ for all n large and so by the Borel-Cantelli Lemma we  get that P  �  An ≤ 2µn1+θ for all large n  �  = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Since θ < 1, we get that An  n2 −→ 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  To evaluate Jn11(Ec  tot), we use (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='11) and the Borel-Cantelli Lemma to get 11(Ec  tot) −→ 0  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Thus Jn1  1(Ec  tot)  n2  −→ 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Using the Cauchy-Schwartz inequality,  we also get that EJn11(Ec  tot) ≤  �  EJ2  n  �1/2 (P(Ec  tot))1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' By the AM-GM inequality and the  bounded second moment assumption in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='1), we get that  EJ2  n ≤ (n + 1)2  (n+1)2  �  i=1  Et2(fi) ≤ C2n4  for some constant C2 > 0 and so using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='11), we get that  EJn11(Ec  tot) ≤  �  C2n2 ·  �  D  n1+2ǫ  �1/2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  Thus EJn1  1(Ec  tot)  n2  −→ 0 as n → ∞ as well and this completes the proof of Sn  n −→ 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  Step 2: We now set k = n1+ǫ and show that Tn(k)−Tn  n  converges to zero a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' and  in L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' First using P(Tn(k) ̸= Tn) ≤  D1  n1+ǫ for some constant D1 > 0, we get from the  Borel-Cantelli Lemma that Tn(k)−Tn  n  −→ 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Next using the fact that Tn(k)  and Tn are both bounded above by �n  i=1 t(fi) we have that E|Tn(k) − Tn| = E|Tn(k) −  Tn|11(Tn(k) ̸= Tn) is bounded above by  E  n  �  i=1  t(fi)11(Tn(k) ̸= Tn)  ≤  E1/2  � n  �  i=1  t(fi)  �2  P1/2(Tn(k) ̸= Tn)  ≤  E1/2  � n  �  i=1  t(fi)  �2 � D1  n1+ǫ  �1/2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  Using (�l  i=1 ai)2 ≤ l �  i a2  i we have E (�n  i=1 t(fi))2 ≤ n �n  i=1 Et2(fi) ≤ D2n2 for some  constant D2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Thus E|Tn(k) − Tn| ≤ D3(n1−ǫ)1/2 = o(n) and so  E|Tn(k)−Tn|  n  −→ 0 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' This completes the proof of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' convergence in part (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  11  Minimum Weight Random Graphs  12  Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='1(c): Using (a+b)2 ≤ 2(a2 +b2) for any two real numbers a and b  we have that the variance of the sum of any two random variables X and Y satisfies  var(X + Y ) = E ((X − EX) + (Y − EY ))2 ≤ 2(var(X) + var(Y )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='13)  Setting T = Tn, U = Tn(k), X = T − U and Y = U we get that  var(T) ≤ 2var(T − U) + 2var(U) ≤ 2var(T − U) + D1n  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='14)  for all n large and some constant D1 > 0, by the variance estimate for U in part (a) of  this Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  We estimate var(T − U) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Using T ≤ U we write  E(T − U)2 = E(T − U)211(T ̸= U) ≤ 2E(T 2 + U 2)11(T ̸= U) ≤ 4EU 211(T ̸= U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  Since U ≤ �n  i=1 t(fi), we have that U 2 ≤ n �n  i=1 t2(fi) and so var(T − U) is bounded  above by  E(T − U)2 ≤ 4n  n  �  i=1  Et2(fi)11(T ̸= U) ≤ 4n2 sup  i  Et2(fi)11(T ̸= U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='15)  Let θ > 0 be a constant and split  Et2(fi)11(T ̸= U)  =  Et2(fi)11({T ̸= U} ∩ {t(fi) < nθ})  +  Et2(fi)11({T ̸= U} ∩ {t(fi) ≥ nθ}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='16)  From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='2), we know that there are constants D2, D3 > 0 such that if k ≥ D2n then  P(T ̸= U) ≤ D3  k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' With this choice of k, the first term in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='16) is bounded above by  Et2(fi)11({T ̸= U} ∩ {t(fi) < nθ}) ≤ n2θP(T ̸= U) ≤ D2n2θ  k  ≤ 1  n3  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='17)  provided we choose k larger if necessary so that k ≥ n2θ log n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  We now consider the case where the uniform square integrability condition holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' For  any η > 0 and all n large, the final term in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='16) is then at most Et2(fi)11(t(fi) ≥ nθ) ≤ η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  Combining this estimate with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='17), we get that Et2(fi)11(T ̸= U) ≤  1  n3 +η ≤ 2η for all n  large and so from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='15) we get that E(T − U)2 ≤ 8n2η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Plugging this into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='14) we get  that var(T) ≤ D1n + 8n2η and since η > 0 is arbitrary, this implies that var  � T  n  �  = o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  Suppose now that bounded pth moment condition holds for some p > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Using H¨older’s  inequality and Markov inequality in succession, the final term in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='16) is at most  Et2(fi)11(t(fi) ≥ nθ)  ≤  (Etp(fi))2/p P  �  t(fi) ≥ nθ�1−2/p  ≤  (Etp(fi))2/p  �Etp(fi)  nθp  �1−2/p  ≤  D4  nθ(p−2)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='18)  12  Minimum Weight Random Graphs  13  for some constant D4 > 0, by the bounded pth moment assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' We choose θ > 0  large so that the final term in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='18) is at most  1  n3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' With this choice of θ we get from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='17)  that Et2(fi)11(T ̸= U) ≤  2  n3 and plugging this into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='15), we get that E(T −U)2 ≤ D5  n for  constant D5 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='14), we then get that var(T) ≤ D6n for some constant D6 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  This completes the proof of part (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  Remark: For p > 2, the bounded pth moment condition in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='1(c) is stronger  than the uniformly square integrable condition which in turn is stronger than the bounded  second moment condition in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' For the particular case of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' passage times, uniform  square integrability is implied by the bounded second moment condition and the first  condition in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='1) above simply states that the passage times are a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  Acknowledgement  I thank Professors Rahul Roy, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Subramanian and the referee for crucial comments  that led to an improvement of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' I also thank IMSc and IISER Bhopal for my  fellowships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  References  [1] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Addario-Berry, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Broutin and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Reed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  The diameter of the minimum  spanning tree of a complete graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Proceedings DMTCS, AG, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' 237—248.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  [2] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Aldous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' (1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' A random tree model associated with random graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Random  Structures and Algorithms, 1, 383–202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  [3] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Alon and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Spencer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' The Probabilistic Method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Wiley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  [4] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Chatterjee and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Dey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Central limit theorem for first-passage percolation  time across thin cylinders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Probability Theory and Related Fields, 156, 613–663.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  [5] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Frieze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' (1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' On the value of a random minimum spanning tree problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  Discrete Applied Mathematics, 10, 47–56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  [6] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Janson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' The minimal spanning tree in a complete graph and a functional  limit theorem for trees in a random graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Random Structres and Algorithms, 7,  337–355.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  [7] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Jiang and C-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Yao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  Critical first-passage percolation starting on the  boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Stochastic Processes and their Applications, 129, 2049–2065.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  [8] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Kesten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' (1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Aspects of First Passage Percolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  13  Minimum Weight Random Graphs  14  [9] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Kesten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' (1993), On the Speed of convergence in first-passage percolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Annals  Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=', 3, 296–338.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  [10] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Li and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Expected lengths of minimum spanning trees for  non-identical edge distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Electronic Journal of Probability, 15, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' 110–141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  [11] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Steele.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  Minimal spanning trees for graphs with random edge lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  Mathematics and computer science II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' Algorithms, trees, combinatorics and proba-  bilities, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content=' 223—245.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
+page_content='  14' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQf7gve/content/2301.04800v1.pdf'}
diff --git a/p9FAT4oBgHgl3EQfex0W/content/tmp_files/2301.08577v1.pdf.txt b/p9FAT4oBgHgl3EQfex0W/content/tmp_files/2301.08577v1.pdf.txt
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+Ontology Pre-training for Poison Prediction
+Martin Glauer1, Fabian Neuhaus1, Till Mossakowski1, and Janna Hastings2,3
+1Institute for Intelligent Cooperating Systems, Otto-von-G¨uricke University Magdeburg
+2Faculty of Medicine, Institute for Implementation Science in Health Care, University of
+Zurich
+3School of Medicine, University of St. Gallen
+January 23, 2023
+Abstract
+Integrating human knowledge into neural networks has the potential to improve their robustness and inter-
+pretability. We have developed a novel approach to integrate knowledge from ontologies into the structure of
+a Transformer network which we call ontology pre-training: we train the network to predict membership in
+ontology classes as a way to embed the structure of the ontology into the network, and subsequently fine-tune
+the network for the particular prediction task. We apply this approach to a case study in predicting the potential
+toxicity of a small molecule based on its molecular structure, a challenging task for machine learning in life
+sciences chemistry. Our approach improves on the state of the art, and moreover has several additional benefits.
+First, we are able to show that the model learns to focus attention on more meaningful chemical groups when
+making predictions with ontology pre-training than without, paving a path towards greater robustness and inter-
+pretability. Second, the training time is reduced after ontology pre-training, indicating that the model is better
+placed to learn what matters for toxicity prediction with the ontology pre-training than without. This strategy
+has general applicability as a neuro-symbolic approach to embed meaningful semantics into neural networks.
+1
+Introduction
+Deep neural networks have recently led to breakthrough performance for a wide range of tasks such as protein
+folding [15] and image generation [21]. However, they still suffer from challenges in generalisability, robustness,
+and interpretability. Approaches that incorporate human knowledge alongside learning from data, which have
+been called hybrid, knowledge-aware or informed [27], have the potential to improve the correspondence between
+what the model learns and the structure of the human world, which in turn allows the model to learn more
+generalisable representations from smaller datasets.
+Human knowledge is carefully curated into ontologies [9, 18], making them a prime candidate source of
+knowledge to incorporate into learning. Many different approaches have already been developed with the objec-
+tive of harnessing prior knowledge to improve machine learning. The most common approach is enrichment of
+the training data with additional information from ontologies (see Section 4.2). In this paper we present a novel
+methodology, which uses an ontology, namely the Chemical Entities of Biological Interest (ChEBI), to create a
+pre-training task for a Transformer model (see Section 2). This pre-training task consists of predicting super-
+classes in ChEBI’s taxonomic hierarchy for molecule classes represented by input chemical structures. Thus,
+during this pre-training the model learns to recognise categories of chemical entities that are chemically mean-
+ingful. After the ontology pre-training the model is fine-tuned for the task of toxicity prediction using the dataset
+from the well-known Tox21 challenge [12]. This dataset consists of 12 different toxicity endpoints, including 7
+nuclear receptor signals and 5 stress response indicators.
+As we show in Section 3, for the purpose of toxicity prediction the ontological pre-training step showed
+the following benefits: First, the model converges faster during fine-tuning. Second, an inspection of the atten-
+tion heads indicates that the model pays attention to chemical structures that correspond to structural chemical
+annotations that are associated with classes in ChEBI. Since ChEBI classes represent chemical categories that
+are meaningful to humans, this connection improves the interpretability of the model’s predictions. Third, the
+predictive performance is improved significantly compared to the performance without pre-training. Indeed, our
+ontology pre-trained model outperforms the state of the art for toxicity prediction on the Tox21 dataset from
+structures without additional input features (see Section 4.2).
+1
+arXiv:2301.08577v1  [cs.AI]  20 Jan 2023
+
+data:ChEBI
+data:Unlabeled molecules
+data:Tox21
+Train: Electra Pre-training
+Pre-trained Model
+Train: Ontology-based Task
+Ontology
+Pre-trained Model
+Toxicity Prediction
+Toxicity Prediction
+Figure 1: Training stack for standard training and ontology pre-training
+Hyperparameter
+Value
+Vocab. size
+1,400
+Hidden size
+256
+Num. of attention heads
+8
+Num. of hidden layers
+6
+Epochs
+100
+Learning Rate
+1e−4
+Optimizer
+Adamax
+Table 1: Hyperparameters shared by all models.
+These results seem to indicate that the ontological pre-training is enabling the model to learn some of the
+knowledge that is represented by the ontology. However, there are important limitations with respect to the
+knowledge that is learned by the model. Further, our proposed methodology is only applicable to ontologies
+that either contain rich structural annotations or which are associated with suitable datasets that link the input
+datatype intended for the learning task to the ontology classes. We will discuss these limitations in Section 4.3.
+2
+Methods
+The usual process used to train a transformer-based model consists of two steps: pre-training and fine-tuning.
+The intention behind the pre-training step is to give the model a solid foundation training in the kind of data
+that will be used in downstream tasks in order for it to gain a preliminary understanding of the target domain
+that can the transferred to more specific tasks later. Most transformer-based architectures are built for language
+problems, and the respective pre-training is often limited to masked-language tasks (BERT [6], RoBERTa) or
+token discrimination tasks (Electra). This kind of training enables the model to learn the syntactic relations
+between words, but it does not get any information about their semantics, aside from context similarity when
+words are interchangeably used in similar contexts.
+In this study, we introduced an additional ontology pre-training stage after the usual pre-training and be-
+fore the fine-tuning. We present a case-study for the use of ontology pre-training to improve chemical toxicity
+prediction. Figure 1 depicts the process using the boxology notation [25] for the novel approach as well as a
+comparison approach without ontology pre-training which serves as our baseline. In the remainder of this sec-
+tion, we will detail the setup for each individual step. All models are based on the Electra model as implemented
+in the ChEBai1 tool used in previous work [8]. All models share the hyperparameters detailed in Table 1 with
+different classification heads for different tasks.
+2.1
+Step 1: Standard Pre-Training
+The first step of this architecture is based on the pre-training mechanism for Electra [5]. This process extends the
+general masked-language model (MLM) used to pre-train transformers by a competing discriminator that aims to
+1https://github.com/MGlauer/ChEBai
+2
+
+predict which tokens have been replaced as part of the MLM task. For the molecular data that serves as the input
+for the prediction task, the language of the molecular representations is SMILES [28], a molecular line notation
+in which atoms and bonds are represented in a linear sequence, and the masking of tokens affects sub-elements
+within the molecule. The generator part of Electra uses a simple linear layer with as softmax to predict the token
+that was most likely masked, while the discriminator uses the same setup, but one additional dense layer to guess
+which token had been replaced by the generator.
+In our case-study, we use the same dataset for pre-training that has been used in a previous task on ontology
+extension [8]. This dataset consists of 365,512 SMILES representations for molecular structures that have been
+extracted from PubChem [23] and the ChEBI ontology [11]. Notably, 152.205 of these substances are known to
+be hazardous substances as they are associated with a hazard class annotation in PubChem.
+2.2
+Step 2: Ontology Pre-Training
+The standard pre-training teaches the network the foundations of how chemical structures are composed and
+represented in SMILES strings, but it does not give the network any insights into which parts of these molecules
+may be important and chemically active. These functional properties are semantic information that are used by
+experts to distinguish classes of molecules within the ChEBI ontology. They are therefore inherently encoded in
+the subsumption relations of the ontology. We used the subsumption hierarchy to create a dataset for an ontology-
+based classification task by extracting all subclasses of ‘molecular entity’ that had at least 100 subclasses with
+SMILES strings attached to them. This resulted in a collection of 856 classes. We then collected all classes in
+ChEBI that had a SMILES string attached to them and annotated them with their subsumption relation for each
+of the 856 classes. The resulting dataset is similar to the ones used in [10] and [8], but covers a wider range of
+classes as labels (856 instead of 500) and also a larger number of molecules (129,187 instead of 31,280). We then
+use the pre-trained model from Step 1 to predict the superclasses of a class of molecules based on its annotated
+SMILES string.
+2.3
+Step 3: Toxicity prediction
+In order to assess the impact of this additional ontology-based pre-training step, we compare the model that re-
+sulted from the ontology pre-training in step 2 with the one from step 1 that did not receive that training. This
+comparison is based on each model’s performance on toxicity prediction using the Tox21 dataset. This dataset
+was created by the National Center for Advanced Translational Sciences (NCATS) of the National Institutes of
+Health (NIH), and constitutes a widely used benchmark for research on machine learning for toxicity prediction
+from small molecules [12]. Notably, there are currently two versions of the Tox21 dataset available in bench-
+marks. The first one originates from the Tox21 Challenge that was conducted in 2014. This dataset consists of
+three different subsets, one for training and two for different steps of the challenge evaluation. In our study, we
+will use the “testing dataset” that was used for an initial ranking of models as a validation set and the dataset that
+was used for the final evaluation as our test set. This version of the Tox21 dataset suffers from several issues
+regarding the consistency and quality of different entries [13]. A more modern version of this dataset has been
+published as part of the MoleculeNet benchmark [29]. This version of Tox21 consists of only of 7,831 molecules.
+We split this dataset into a training (85%), validation (7.5%) and test set (7.5%). There are, however, still two
+major challenges that need to be considered when working with this dataset. First, the number of datapoints is
+rather low. Molecular structures are complex graphs, which makes it hard for any model to derive a sufficient
+amount of information from this dataset alone. Second, the information available in the dataset is not complete: a
+substantial amount of toxicity information is missing in this dataset. There are 4,752 molecules for which at least
+one of the 12 kinds of toxicity is not known. In the prior literature, this issue has been approached in different
+ways. Some approaches perform data cleaning which limits the number of available data points even further, e.g.
+[13] excluded all datapoints that contained any missing labels. We decided to keep these datapoints as part of our
+dataset, but exclude the missing labels from the calculation of all loss functions and metrics. Any outputs that
+the model generates for these missing labels is considered as correct and does not influence the training gradient.
+This approach allows the network to fill these gaps autonomously.
+Preliminary results showed that both models were prone to overfitting, when used with the same settings as
+the model from step 2. We found that this behaviour could be prevented by using strong regularisations. The final
+experiments used the Adamax optimizer with dropouts on input embeddings of 0.2, dropouts on hidden states of
+0.4 and L2-regularisation of 0.0001. All data and code used in this study is publicly available.2
+2https://doi.org/10.5281/zenodo.7548313
+3
+
+(a) ROC-AUC / MoleculeNet Tox21 dataset
+(b) F1 score (micro) / MoleculeNet Tox21 dataset
+(c) ROC-AUC / original TOX21 dataset
+(d) F1 score (micro) / original TOX21 dataset
+Figure 2: Development of ROC-AUC and F1 score (micro) during training on the validation sets of the Tox21 dataset available
+as part of MoleculeNet and the original TOX21 challenge
+3
+Results
+3.1
+Predictive performance
+The final result of our training stack are four models: with or without ontology pre-training, and fine-tuned
+on the original Tox21 competition dataset or on the smaller version of the Tox21 dataset published as part of
+MoleculeNet. The semantic pre-training already showed a clear impact during the training phase. Figures 2a-
+2d depict the curves for two metrics (F1 score and ROC-AUC) on our validation set as evaluated at the end
+of each epoch during training. It can be seen that models with ontology pre-training start with a better initial
+performance and also retain this margin throughout the training. This behaviour is also reflected in the predictive
+performance on both test sets. Table 2 shows the predictive behaviour for the dataset from MoleculeNet and the
+original challenge. The leading model (highlighted in bold) is predominantly the one that received additional
+ontology pre-training. This is particularly visible for the more noisy and sparse dataset used in the original
+Tox21 competition. The overall improved performance shows that pre-training with a more general ontology
+pre-training does support the network for the more specialised toxicity prediction. The drastic drop that can be
+seen around epoch 50 in Figure 2d but not for the pre-trained model in Figure 2b further indicates that ontology
+pre-training hedges the model against early overfitting. The reported results, and in particular the F1 scores,
+however, show that there is still a large margin of error for this task.
+3.2
+Interpretability
+Attention weights in Transformer networks can be visualised to enable a kind of salience-based visual interpre-
+tation for predictions directly connected with the input data [26]. Previous work [8] explored the link between
+attention weights to the prediction of ontology classes, and observed that the network learned to pay attention to
+relevant substructures when making predictions of ontology classes to which the molecule belongs.
+In the current work, our hypothesis was that the additional information from the ontology would both enhance
+the prediction and enhance the coherence of the attention weights for explanatory visualisations. To test this
+hypothesis we explored the attention visualisations for the predictions by the ontology pre-trained network as
+compared to the normal network.
+Figure 3 shows an individual example of the attention weight that the network uses when visiting specific
+4
+
+0.88
+0.86
+0.84
+0.82
+0.80
+0.78
+With ontology pre-training
+Without ontology pre-training
+0.76
+0
+20
+40
+60
+80
+1000.5
+0.4 -
+0.3 
+0.2
+0.1
+With ontology pre-training
+0.0
+Without ontology pre-training
+0
+20
+40
+60
+80
+1000.82
+0.80
+0.78
+0.76
+0.74
+0.72
+0.70
+With ontology pre-training
+Without ontology pre-training
+0
+20
+40
+60
+80
+1000.30
+0.25
+0.20
+0.15
+0.10
+0.05
+With ontology pre-training
+0.00
+Without ontology pre-training
+0
+20
+40
+60
+80
+100Dataset
+Tox 21 (MoleculeNet)
+Tox21 (Challenge)
+Metric
+F1
+ROC-AUC
+F1
+ROC-AUC
+Model
+Our Model
+SSL-GCN
+Our Model
+Ontology Pre-training
+yes
+no
+yes
+no
+-
+yes
+no
+yes
+no
+NR-AR
+0.41
+0.52
+0.82
+0.76
+0.80
+0.1
+0.14
+0.63
+0.62
+NR-AR-LBD
+0.51
+0.5
+0.85
+0.77
+0.76
+0.05
+0.1
+0.69
+0.67
+NR-AhR
+0.53
+0.45
+0.81
+0.82
+0.83
+0.23
+0.05
+0.8
+0.69
+NR-Aromatase
+0.33
+0.15
+0.84
+0.8
+0.73
+0.25
+0.04
+0.75
+0.69
+NR-ER
+0.44
+0.4
+0.74
+0.71
+0.72
+0.16
+0.09
+0.64
+0.62
+NR-ER-LBD
+0.37
+0.3
+0.84
+0.76
+0.69
+0.14
+0.12
+0.66
+0.63
+NR-PPAR-gamma
+0.29
+-
+0.84
+0.83
+0.76
+0.14
+-
+0.67
+0.66
+SR-ARE
+0.48
+0.53
+0.8
+0.84
+0.73
+0.37
+0.23
+0.71
+0.69
+SR-ATAD5
+0.14
+0.19
+0.75
+0.74
+0.72
+0.16
+-
+0.65
+0.65
+SR-HSE
+0.24
+0.22
+0.82
+0.82
+0.78
+0.13
+0.09
+0.76
+0.68
+SR-MMP
+0.62
+0.53
+0.9
+0.88
+0.81
+0.48
+0.21
+0.86
+0.82
+SR-p53
+0.39
+0.35
+0.83
+0.8
+0.75
+0.3
+-
+0.82
+0.78
+Table 2: Class-wise scores on the test set on both Tox21 datasets. Bold values denote the best value for a particular com-
+bination of dataset and metric. NR - nuclear receptor; AR - androgen receptor; LBD - luciferase; AhR - aryl hydrocarbon
+receptor; ER - estrogen receptor; PPAR - peroxisome proliferator-activated receptor; SR - stress response; ARE - nuclear
+factor antioxidant response; ATAD5 - genotoxicity; HSE - heat shock factor response; MMP - mitochondrial response; p53 -
+DNA damage response.
+Ontology Pre-training
+yes
+no
+Tox21 Challenge
+0.86
+0.90
+Tox21 MoleculeNet
+0.79
+0.85
+Table 3: Entropy values averaged over all tokens, attention heads, layers and atoms in the respective test set.
+input tokens. The molecule depicted is TOX25530, corresponding to the sulfonamide anti-glaucoma drug dorzo-
+lamide, which is not toxic. Dark green lines indicate strong, focused attention, while more opaque lines indicate
+broader, less focused attention. As this example illustrates, we observed that the ontology pre-trained network
+often shows more coherence and structure in its attention weights compared to the baseline network without
+ontology pre-training. This is reflected in the triangular clusters of darker attention weights in the top rows of
+Figure 3a and b. Clusters reflect that from a particular position in the input token sequence, strong attention
+is placed on a subset of the molecule, reflecting relevant substructures within the molecule. Figure 3c shows
+how the attention weights relate to the molecular graph for this molecule. Attention weight relationships may be
+short-range (nearby atoms or groups) or long-range (atoms or groups further away within the molecule).
+To test this visual intuition more systematically, we computed the entropy for each attention head in each
+layer. Attention is computed using softmax and can therefore be interpreted as a probability distribution. In order
+to evaluate our hypothesis that the ontology pre-training also impacts the way a model focuses its attention, we
+calculated the average entropy of these distributions. Entropy is a measure for the randomness of a distribution.
+A perfectly even distribution would result in a entropy value of 1 (complete uncertainty), while a distribution in
+which only one event can possibly occur will result in an entropy value of 0 (complete certainty). That means that
+an attention distribution with an entropy value of 1 is not paying attention to any particular part of the molecule,
+but is spreading attention evenly. An entropy value of 0 indicates that the model paid attention to only a single
+token of the molecule. Table 3 shows the aggregated entropy values for our models. It can be seen that the model
+that received additional ontology pre-training has a lower entropy value for both datasets. That means, that the
+attention is generally spent less evenly and is therefore more focused in comparison to the model that did not
+receive that additional training. This indicates that the ontology pre-trained model’s decisions are based on more
+concise substructures within the molecule.
+5
+
+(a) Attention plots for layers 2-3
+(b) Attention plots for layers 4-5
+(c) Attention clusters relate to the molecule structure
+Figure 3: Visualisation of the attention weights for the layers 2-3 in subfigure a) and layers 4-5 in subfigure b). We omit
+layers 1 and 6 as they had largely uniform attention weights. Each subfigure compares the ontology pre-trained network (first
+row) to the prediction network without pre-training (second row). c) The molecular structure processed in the attention plots
+is depicted with attention from layer 4 heads 1-2, showing how attention clusters relate to the molecular structure.
+6
+
+[CLS]
+[CLS]
+[CLS]
+[CLS]
+[CLS]
+[CLS]
+[CLS]
+[CLS]
+[CLS]
+(s12]
+[CLS]
+[CLS]
+[CLS]
+[CLS]
+[CLS]
+[CLS]
+[CLS]
+UUZ
+UUz
+[CGH)
+[CGH]
+[CGH]
+N
+[CGH]
+[CGH]
+[C@H]
+[CGH]
+[HO]
+[Ho]
+[CGH]
+[CgH]
+(CGH]
+[CGH]
+(CGH)
+[C@H]
+[CGH]
+[CGH]
+[CGH]
+AU
+[C@H]
+[CGH]
+[C@H]
+[CGH]
+[HO]
+(CGH)
+[CGH]
+(COH)
+[CGH]
+(C@H)
+[C@H]
+(HOD]
+(C@H)
+[CGH]
+U
+U
+IO
+Q
+IO-UN
+0
+Q
+Io.
+o.
+0
+OT
+UUNA
+UN
+[570]
+[CLS]
+[CLS]
+[CLS]
+UUZ
+UUZ
+U U Z
+[C@H]
+[(CoH)
+H
+[CGH]
+(CGH)
+[CGH]
+[CGH]
+(CGH)
+[CGH]
+ICGHI
+[CGH)
+[HBD]
+(C@H)
+[C@H]
+c
+c
+c
+[C@H]
+[CGH]
+[C@H]
+[CeH]
+(CH]
+(C@H)
+[CGH]
+[C@H]
+[C@H]
+[C@H]
+[(C@H]
+[(CeH]
+[C@H]
+[CeH)
+[HO)]
+[C@H]
+[C@H]
+U -M
+o-
+0
+O-
+UN
+UN
+U N
+U2
+2
+U N
+5
+N
+N
+N
+Z 
+HOMFOMUUNR
+10-
+OT
+IO-
+OUN
+IOUUN
+UNHICgHI
+:
+[CGH]
+(CGHI
+[CGH]
+(CoH)
+(CM)
+[CGH]
+(Hea]
+(CGH)
+c
+UW
+ro
+d
+[He)]
+(CGH)
+(CGM)
+z ~ WN
+N
+COH)
+[COH]
+[C
+C
+C
+CGH)
+[C@H]
+[CG
+HN
+H2N-
+UN
+U
+2
+N
+1
+1
+14
+Discussion
+4.1
+Significance
+Our approach introduces a new way to embed knowledge from an ontology into a neural network. The underlying
+assumption of this approach is the following: A well-designed ontology represents a classification of its domain
+that has proven useful for the various goals and tasks of experts in that area. Thus, it is possible (and even likely)
+that some of the classes of the ontology reflect features that are relevant for a prediction task in that domain. For
+example, the classification of chemical entities in ChEBI is based on empirical knowledge about the pertinent
+features of molecules and their chemical behaviour, and it is reasonable to expect that some of these pertinent
+features are, at least indirectly, correlated with toxicity. The goal of ontology pre-training is to enable a model to
+benefit from the knowledge represented in an ontology by training it to categorise its input according to the class
+hierarchy from the ontology.
+This approach is applicable in any case where a dataset is available that links the input datatype for the
+prediction task to the classification hierarchy from the ontology. In the case of our example, the SMILES input
+structures are directly associated with the leaf classes of the ChEBI ontology, thus we can prepare a training
+dataset directly from the ontology. However, for other ontologies, the dataset may be assembled from ontology
+annotations which are external to the ontology but serve the purpose of linking the ontology classes to examples
+of instances of the corresponding input datatype.
+The results in Table 2 show that we were able to improve the performance of our model for toxicity prediction
+with the help of ontology pre-training. The inspection of the attention head weights indicates that the system
+indeed learned meaningful aspects of life science chemistry from the pre-training task. Further, as we will
+discuss next, the performance of the ontology pre-trained model compares favourably with the state of the art.
+4.2
+Related work
+4.2.1
+Toxicity prediction
+The prediction, based on chemical structure, of whether a molecule has the potential to be harmful or poisonous
+to living systems, is a challenging task for life science chemistry [1, 30]. The Tox21 dataset has become a widely
+used benchmark for evaluating machine learning approaches to this task, thus there have been multiple previous
+publications using different strategies. Most approaches to toxicity prediction supplement the input data chemical
+structures with additional calculated features, such as known toxicophores, in order to enhance performance on
+the toxicity prediction task. This was the case for the winner of the original Tox21 challenge, who used a deep
+learning implementation together with three other types of classifier in an ensemble [17], and more recently [19],
+which augments the input molecular structure with physico-chemical properties. Another recent approach uses
+‘geometric graph learning’ [14] which augments the input molecular graphs with multi-scale weighted coloured
+graph descriptors. Some approaches additionally augment the training data with more examples in order to
+mitigate the fact that the Tox21 dataset is small. In [13], chemical descriptors were calculated and in addition, a
+strategy to remove class imbalance was applied including over-sampling from the minority classes in the dataset
+followed by cleaning of mislabelled instances. In these cases, which use a different input dataset whether through
+feature augmentation, data augmentation or data cleaning, we cannot directly compare their results to ours. We
+anticipate that feature and data augmentation approaches such as these would potentially improve our method’s
+performance as well, but we would need to develop a more complex attention visualisation mechanism to operate
+across augmented inputs. Since our objective is rather to describe a new approach to incorporating ontology
+knowledge into a neural network, we here focus our performance evaluation on those approaches that are more
+directly comparable to ours.
+[3] uses a graph neural network and a semi-supervised learning approach known as Mean Teacher which
+augments the training data with additional unlabelled examples. This network and training approach achieving
+a ROC-AUC score of 0.757 in the test set, which our approach outperforms without data augmentation. Table 2
+shows a comparison of the ROC-AUC values achieved by our model against those reported for the best model
+(SSL-GCN) reported in [3]. With the exception of one class, our model shows better performance for all target
+classes.
+ChemBERTa [4] is the most similar to our approach in that it also uses a Transformer-based network. Its
+original publication also contains an evaluation on the p53 stress-response pathway activation (SR-p53) target of
+the Tox21 dataset from MoleculeNet. Our model exceeds the reported ROC-AUC value (ChemBERTa: 0.728,
+our model: 0.83).
+7
+
+4.2.2
+Knowledge-aware pre-training with an ontology
+Approaches to add knowledge from an ontology into a machine learning model follow several different strategies.
+The most common is that the information from the ontology is used to supplement the input data in some
+form, such as by adding synonyms and classification parent labels to the input data. For example, in [22] an
+ontology is used to supplement the input data with an ontology-based ‘feature engineering’ strategy.
+A second approach is that the ontology content is itself provided as input that is embedded into the network,
+for example by using random walks through the ontology content to create sentences representing the structure
+of the ontology for input into a language model. Ontology embedding strategies include OWL2Vec∗ [2] and
+OPA2Vec [24]. These approaches are suitable for tasks such as knowledge graph completion or link predic-
+tion, but the additional information provided by such embeddings is not inherently connected in the internal
+representation space to the other information learned by the network, and this limits their potential benefit if the
+input datatype is complex and internally structured. For example, in the chemistry case, the information about
+the molecular structure of the chemical that is provided in the SMILES input strings would not be connected
+intrinsically to the information about the class hierarchy provided to the network by an ontology embedding
+strategy.
+There are some examples of the use of biological ontologies together with biomolecular input data types that
+are closer to our approach. OntoProtein [32] combines background knowledge from the Gene Ontology with
+protein sequences to improve the prediction of protein functions.OntoProtein uses the Gene Ontology in pre-
+training a protein embedding. Existing protein embeddings such as ProtBERT are enhanced by embedding the
+Gene Ontology as a knowledge graph (following approaches such as OWL2Vec∗) and then explicitly aligning
+the two embeddings. By contrast, our approach uses the ontology more directly. Namely, our pre-training
+exploits both ChEBI’s class structure as well its class hierarchy. The class structure leads to an aggregation
+of molecules with similar sub-structures and properties, which can enhance the learning process. The class
+hierarchy is indirectly influencing the learning process as well, because a subclass relation corresponds to a
+subset relation for the training samples. OntoProtein uses the subclass hierarchy only for defining depth of
+ontology terms, which influences learning only very indirectly. Hence, our model incorporates expert knowledge
+in a more direct way. In the future, we will try to incorporate OntoProtein’s approach of contrastive learning
+using knowledge-aware negative sampling into our approach.
+Other approaches have developed custom architectures for the incorporation of knowledge from an ontology
+into the network. For example, DeepPheno [16] predicts phenotypes from combinations of genetic mutations,
+where phenotypes are encoded into the network through a novel hierarchical classification layer that encodes
+almost 4,000 classes from the Human Phenotype Ontology together with their hierarchical dependencies as an
+ontology prediction layer that informs the remainder of the training of the network. A similar approach is used in
+[31], in which an ontology layer adds information from an ontology to a neural network to enhance the prediction
+of microbial biomes. Instead our approach uses the generic architecture of a standard Transformer network and
+learns the information from the ontology through an additional pre-training task.
+4.3
+Limitations
+Our current approach only uses a fraction of the information available in the ChEBI ontology, since we only
+consider the structural classification of chemical entities beneath the ‘molecular entity’ class. Hence, we currently
+consider neither classes of biological and chemical roles nor pharmaceutical applications. These classes have the
+potential to further enhance the knowledge that is provided to the network, and will be explored in future work.
+Another limitation is related to the way we create the ontology pre-training task: We use the leaf nodes of
+ChEBI as examples for training a model to predict the subsumption relation for more general classes. Or, to put
+it differently, while from an ontological perspective the leaf nodes of ChEBI are classes in ChEBI’s taxonomy,
+we are treating them as instances and, thus, turning subsumption prediction into a classification task from a
+machine learning perspective. Consequently, while we use the whole structural hierarchy of ChEBI for creating
+the pre-training task, the model learns to classify only 856 classes, those that have a sufficient number of example
+structures to learn from, which is a relatively small number compared to the number of classes in ChEBI. Further,
+this approach of creating a subsumption prediction pre-training task requires rich structural annotations linked to
+the learning input datatype (which is the SMILES in our example case), which many ontologies do not contain.
+As indicated in Section 4.1, both of these limitations may be addressed by using class membership prediction
+for ontology pre-training. All that is required for ontology pre-training is a dataset that (a) is of the same input
+datatype as the fine-tuning task, (b) is annotated with terms from the ontology, and (c) contains sufficient training
+examples to train the model to classify the input with the terms of the ontology. Because we treat subsumption
+prediction as a classification problem anyway, both approaches are functionally equivalent. However, using an
+external dataset for training (instead of generating it from the ontology itself), has the benefit that the ontology
+8
+
+pre-training might cover the whole ontology instead of just a subset of the ontology. Further, this approach does
+not rely on the existence of appropriate annotations in the associated input data type.
+A further limitation of the approach is that the interpretability offered by the attention weights is limited to
+visual inspection. In future work we aim to develop an algorithm that is able to systematically determine clusters
+of attention mapped to the relevant parts of the input molecular structure.
+5
+Conclusion
+This paper presents the results of training an Electra model for toxicity prediction using the Tox21 dataset as
+a benchmark. The model was able to achieve state-of-the-art performance on the task of toxicity prediction,
+outperforming comparable models that have been trained on the same dataset.
+While improving the state of the art of toxicity prediction is in itself an interesting result, the main contri-
+bution of the paper is the presentation of a novel approach to combining symbolic and connectionist AI. This
+is because our result was achieved with the help of an additional pre-training step, which trains the model with
+the help of background knowledge from an ontology. For the presented work the pre-training task consisted of
+predicting subclass relationships between classes in the ChEBI ontology, but other tasks (e.g., predicting class
+membership) are likely to be equally suitable to achieve the purpose of ontology pre-training, namely, to train the
+model to recognise the meaningful classification of the entities in a given domain, as represented by the ontology.
+As we have illustrated in this paper, ontology pre-training has the potential benefit of reducing the time
+needed for fine-tuning and improving performance. Further, as we have illustrated in Section 3.2, an inspection
+of attention heads indicates that some of the attention patterns of the model correlate to the substructures that are
+pertinent for chemical categories in ChEBI. Thus, since ontological categories are meaningful for humans, an-
+other potential benefit of ontology pre-training is an improved interpretability of the trained model. In the future,
+we are planning to further investigate this line of research by systematically analysing the attention patterns of
+the pre-trained model and automatically linking these patterns to the ontology.
+As we discussed in Section 4.3, currently we are only using some of the knowledge available in the ontology
+for pre-training purposes. In particular, we do not include classes from other parts of the ontology, nor do we
+include other axioms aside from the hierarchy. In the future we are planning to overcome these limitations
+by sampling a wider set of classes from ChEBI and by using a more complex architecture that combines a
+Transformer with a logical neural network (LNN) [20]. The LNN is able to represent logical axioms from the
+ontology as a first-order theory, translated from OWL [7]. This will enable us to use logical axioms from the
+ontology (and therefore also its binary relations) to influence both the ontology pre-training as well as the fine-
+tuning.
+References
+[1] Claudio N. Cavasotto and Valeria Scardino. Machine Learning Toxicity Prediction: Latest Advances by
+Toxicity End Point. ACS Omega, 7(51):47536–47546, December 2022. Publisher: American Chemical
+Society.
+[2] Jiaoyan Chen, Pan Hu, Ernesto Jim´enez-Ruiz, Ole Magnus Holter, Denvar Antonyrajah, and Ian Horrocks.
+Owl2vec*: embedding of OWL ontologies. Mach. Learn., 110(7):1813–1845, 2021.
+[3] Jiarui Chen, Yain-Whar Si, Chon-Wai Un, and Shirley W. I. Siu. Chemical toxicity prediction based on
+semi-supervised learning and graph convolutional neural network. Journal of Cheminformatics, 13(1):93,
+November 2021.
+[4] Seyone Chithrananda, Gabriel Grand, and Bharath Ramsundar. Chemberta: large-scale self-supervised
+pretraining for molecular property prediction. arXiv preprint arXiv:2010.09885, 2020.
+[5] Kevin Clark, Minh-Thang Luong, Quoc V Le, and Christopher D Manning. Electra: Pre-training text
+encoders as discriminators rather than generators. arXiv preprint arXiv:2003.10555, 2020.
+[6] Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. BERT: pre-training of deep bidi-
+rectional transformers for language understanding. In Jill Burstein, Christy Doran, and Thamar Solorio,
+editors, Proceedings of the 2019 Conference of the North American Chapter of the Association for Com-
+putational Linguistics: Human Language Technologies, NAACL-HLT 2019, Minneapolis, MN, USA, June
+2-7, 2019, Volume 1 (Long and Short Papers), pages 4171–4186. Association for Computational Linguis-
+tics, 2019.
+9
+
+[7] Simon Fl¨ugel, Martin Glauer, Fabian Neuhaus, and Janna Hastings. When one logic is not enough: Inte-
+grating first-order annotations in OWL ontologies. Semantic web journal, 2023.
+[8] Martin Glauer, Adel Memariani, Fabian Neuhaus, Till Mossakowski, and Janna Hastings. Interpretable
+Ontology Extension in Chemistry. Semantic Web Journal, February 2022.
+[9] Janna Hastings. Primer on Ontologies. In Christophe Dessimoz and Nives ˇSkunca, editors, The Gene
+Ontology Handbook, volume 1446, pages 3–13. Springer New York, New York, NY, 2017. Series Title:
+Methods in Molecular Biology.
+[10] Janna Hastings, Martin Glauer, Adel Memariani, Fabian Neuhaus, and Till Mossakowski. Learning Chem-
+istry: Exploring the Suitability of Machine Learning for the Task of Structure-based Chemical Ontology
+Classification. Journal of Cheminformatics, 13(23), 2021.
+[11] Janna Hastings, Gareth Owen, Adriano Dekker, Marcus Ennis, Namrata Kale, Venkatesh Muthukrishnan,
+Steve Turner, Neil Swainston, Pedro Mendes, and Christoph Steinbeck. ChEBI in 2016: Improved services
+and an expanding collection of metabolites. Nucleic Acids Research, 44(D1):D1214–D1219, January 2016.
+[12] Ruili Huang, Menghang Xia, Dac-Trung Nguyen, Tongan Zhao, Srilatha Sakamuru, Jinghua Zhao, Sam-
+pada A. Shahane, Anna Rossoshek, and Anton Simeonov. Tox21Challenge to Build Predictive Models of
+Nuclear Receptor and Stress Response Pathways as Mediated by Exposure to Environmental Chemicals
+and Drugs. Frontiers in Environmental Science, 3, 2016.
+[13] Gabriel Idakwo, Sundar Thangapandian, Joseph Luttrell, Yan Li, Nan Wang, Zhaoxian Zhou, Huixiao
+Hong, Bei Yang, Chaoyang Zhang, and Ping Gong. Structure–activity relationship-based chemical classi-
+fication of highly imbalanced tox21 datasets. Journal of cheminformatics, 12(1):1–19, 2020.
+[14] Jian Jiang, Rui Wang, and Guo-Wei Wei.
+GGL-Tox: Geometric Graph Learning for Toxicity Predic-
+tion. Journal of Chemical Information and Modeling, 61(4):1691–1700, April 2021. Publisher: American
+Chemical Society.
+[15] John Jumper, Richard Evans, Alexander Pritzel, Tim Green, Michael Figurnov, Olaf Ronneberger, Kathryn
+Tunyasuvunakool, Russ Bates, Augustin ˇZ´ıdek, Anna Potapenko, Alex Bridgland, Clemens Meyer, Simon
+A. A. Kohl, Andrew J. Ballard, Andrew Cowie, Bernardino Romera-Paredes, Stanislav Nikolov, Rishub
+Jain, Jonas Adler, Trevor Back, Stig Petersen, David Reiman, Ellen Clancy, Michal Zielinski, Martin
+Steinegger, Michalina Pacholska, Tamas Berghammer, Sebastian Bodenstein, David Silver, Oriol Vinyals,
+Andrew W. Senior, Koray Kavukcuoglu, Pushmeet Kohli, and Demis Hassabis. Highly accurate protein
+structure prediction with AlphaFold. Nature, pages 1–11, July 2021. Publisher: Nature Publishing Group.
+[16] Maxat Kulmanov and Robert Hoehndorf. DeepPheno: Predicting single gene loss-of-function phenotypes
+using an ontology-aware hierarchical classifier. PLoS Comput. Biol., 16(11), 2020.
+[17] Andreas Mayr, G¨unter Klambauer, Thomas Unterthiner, and Sepp Hochreiter. DeepTox: Toxicity Predic-
+tion using Deep Learning. Frontiers in Environmental Science, 3, 2016.
+[18] Fabian Neuhaus and Janna Hastings. Ontology development is consensus creation, not (merely) represen-
+tation. Appl. Ontology, 17(4):495–513, 2022.
+[19] Yuzhong Peng, Ziqiao Zhang, Qizhi Jiang, Jihong Guan, and Shuigeng Zhou. TOP: A deep mixture repre-
+sentation learning method for boosting molecular toxicity prediction. Methods, 179:55–64, July 2020.
+[20] Ryan Riegel, Alexander Gray, Francois Luus, Naweed Khan, Ndivhuwo Makondo, Ismail Yunus Akhal-
+waya, Haifeng Qian, Ronald Fagin, Francisco Barahona, Udit Sharma, et al. Logical neural networks. arXiv
+preprint arXiv:2006.13155, 2020.
+[21] Robin Rombach, Andreas Blattmann, Dominik Lorenz, Patrick Esser, and Bj¨orn Ommer. High-Resolution
+Image Synthesis with Latent Diffusion Models, April 2022. arXiv:2112.10752 [cs].
+[22] Satya S. Sahoo, Katja Kobow, Jianzhe Zhang, Jeffrey Buchhalter, Mojtaba Dayyani, Dipak P. Upadhyaya,
+Katrina Prantzalos, Meenakshi Bhattacharjee, Ingmar Blumcke, Samuel Wiebe, and Samden D. Lhatoo.
+Ontology-based feature engineering in machine learning workflows for heterogeneous epilepsy patient
+records. Scientific Reports, 12(1):19430, November 2022. Number: 1 Publisher: Nature Publishing Group.
+10
+
+[23] E. Sayers. PubChem: An Entrez Database of Small Molecules. NLM Tech Bull., 2005 Jan-Feb(342:e2),
+2005.
+[24] Fatima Zohra Smaili, Xin Gao, and Robert Hoehndorf. OPA2Vec: combining formal and informal content
+of biomedical ontologies to improve similarity-based prediction. Bioinformatics, 35(12):2133–2140, June
+2019.
+[25] Michael van Bekkum, Maaike de Boer, Frank van Harmelen, Andr´e Meyer-Vitali, and Annette ten Teije.
+Modular design patterns for hybrid learning and reasoning systems. Applied Intelligence, 51(9):6528–6546,
+2021.
+[26] Jesse Vig, Ali Madani, Lav R. Varshney, Caiming Xiong, Richard Socher, and Nazneen Fatema Rajani.
+BERTology Meets Biology: Interpreting Attention in Protein Language Models. arXiv:2006.15222 [cs,
+q-bio], March 2021. arXiv: 2006.15222.
+[27] Laura von Rueden, Sebastian Mayer, Katharina Beckh, Bogdan Georgiev, Sven Giesselbach, Raoul Heese,
+Birgit Kirsch, Michal Walczak, Julius Pfrommer, Annika Pick, Rajkumar Ramamurthy, Jochen Garcke,
+Christian Bauckhage, and Jannis Schuecker. Informed Machine Learning - A Taxonomy and Survey of
+Integrating Prior Knowledge into Learning Systems. IEEE Transactions on Knowledge and Data Engi-
+neering, pages 1–1, 2021. Conference Name: IEEE Transactions on Knowledge and Data Engineering.
+[28] David Weininger. SMILES, a chemical language and information system. 1. Introduction to methodology
+and encoding rules. Journal of Chemical Information and Computer Sciences, 28(1):31–36, February 1988.
+[29] Zhenqin Wu, Bharath Ramsundar, Evan N Feinberg, Joseph Gomes, Caleb Geniesse, Aneesh S Pappu, Karl
+Leswing, and Vijay Pande. Moleculenet: a benchmark for molecular machine learning. Chemical science,
+9(2):513–530, 2018.
+[30] Hongbin Yang, Lixia Sun, Weihua Li, Guixia Liu, and Yun Tang. In Silico Prediction of Chemical Toxicity
+for Drug Design Using Machine Learning Methods and Structural Alerts. Frontiers in Chemistry, 6, 2018.
+[31] Yuguo Zha, Hui Chong, Hao Qiu, Kai Kang, Yuzheng Dun, Zhixue Chen, Xuefeng Cui, and Kang Ning.
+Ontology-aware deep learning enables ultrafast and interpretable source tracking among sub-million mi-
+crobial community samples from hundreds of niches. Genome Medicine, 14(1):43, April 2022.
+[32] Ningyu Zhang, Zhen Bi, Xiaozhuan Liang, Siyuan Cheng, Haosen Hong, Shumin Deng, Qiang Zhang,
+Jiazhang Lian, and Huajun Chen. Ontoprotein: Protein pretraining with gene ontology embedding. In The
+Tenth International Conference on Learning Representations, ICLR 2022, Virtual Event, April 25-29, 2022.
+OpenReview.net, 2022.
+11
+
diff --git a/p9FAT4oBgHgl3EQfex0W/content/tmp_files/load_file.txt b/p9FAT4oBgHgl3EQfex0W/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf,len=626
+page_content='Ontology Pre-training for Poison Prediction  Martin Glauer1, Fabian Neuhaus1, Till Mossakowski1, and Janna Hastings2,3  1Institute for Intelligent Cooperating Systems, Otto-von-G¨uricke University Magdeburg  2Faculty of Medicine, Institute for Implementation Science in Health Care, University of  Zurich  3School of Medicine, University of St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Gallen  January 23, 2023  Abstract  Integrating human knowledge into neural networks has the potential to improve their robustness and inter-  pretability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' We have developed a novel approach to integrate knowledge from ontologies into the structure of  a Transformer network which we call ontology pre-training: we train the network to predict membership in  ontology classes as a way to embed the structure of the ontology into the network, and subsequently fine-tune  the network for the particular prediction task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' We apply this approach to a case study in predicting the potential  toxicity of a small molecule based on its molecular structure, a challenging task for machine learning in life  sciences chemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Our approach improves on the state of the art, and moreover has several additional benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  First, we are able to show that the model learns to focus attention on more meaningful chemical groups when  making predictions with ontology pre-training than without, paving a path towards greater robustness and inter-  pretability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Second, the training time is reduced after ontology pre-training, indicating that the model is better  placed to learn what matters for toxicity prediction with the ontology pre-training than without.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' This strategy  has general applicability as a neuro-symbolic approach to embed meaningful semantics into neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  1  Introduction  Deep neural networks have recently led to breakthrough performance for a wide range of tasks such as protein  folding [15] and image generation [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' However, they still suffer from challenges in generalisability, robustness,  and interpretability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Approaches that incorporate human knowledge alongside learning from data, which have  been called hybrid, knowledge-aware or informed [27], have the potential to improve the correspondence between  what the model learns and the structure of the human world, which in turn allows the model to learn more  generalisable representations from smaller datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  Human knowledge is carefully curated into ontologies [9, 18], making them a prime candidate source of  knowledge to incorporate into learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Many different approaches have already been developed with the objec-  tive of harnessing prior knowledge to improve machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' The most common approach is enrichment of  the training data with additional information from ontologies (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' In this paper we present a novel  methodology, which uses an ontology, namely the Chemical Entities of Biological Interest (ChEBI), to create a  pre-training task for a Transformer model (see Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' This pre-training task consists of predicting super-  classes in ChEBI’s taxonomic hierarchy for molecule classes represented by input chemical structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Thus,  during this pre-training the model learns to recognise categories of chemical entities that are chemically mean-  ingful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' After the ontology pre-training the model is fine-tuned for the task of toxicity prediction using the dataset  from the well-known Tox21 challenge [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' This dataset consists of 12 different toxicity endpoints, including 7  nuclear receptor signals and 5 stress response indicators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  As we show in Section 3, for the purpose of toxicity prediction the ontological pre-training step showed  the following benefits: First, the model converges faster during fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Second, an inspection of the atten-  tion heads indicates that the model pays attention to chemical structures that correspond to structural chemical  annotations that are associated with classes in ChEBI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Since ChEBI classes represent chemical categories that  are meaningful to humans, this connection improves the interpretability of the model’s predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Third, the  predictive performance is improved significantly compared to the performance without pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Indeed, our  ontology pre-trained model outperforms the state of the art for toxicity prediction on the Tox21 dataset from  structures without additional input features (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='08577v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='AI]  20 Jan 2023  data:ChEBI  data:Unlabeled molecules  data:Tox21  Train: Electra Pre-training  Pre-trained Model  Train: Ontology-based Task  Ontology  Pre-trained Model  Toxicity Prediction  Toxicity Prediction  Figure 1: Training stack for standard training and ontology pre-training  Hyperparameter  Value  Vocab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' size  1,400  Hidden size  256  Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' of attention heads  8  Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' of hidden layers  6  Epochs  100  Learning Rate  1e−4  Optimizer  Adamax  Table 1: Hyperparameters shared by all models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  These results seem to indicate that the ontological pre-training is enabling the model to learn some of the  knowledge that is represented by the ontology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' However, there are important limitations with respect to the  knowledge that is learned by the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Further, our proposed methodology is only applicable to ontologies  that either contain rich structural annotations or which are associated with suitable datasets that link the input  datatype intended for the learning task to the ontology classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' We will discuss these limitations in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  2  Methods  The usual process used to train a transformer-based model consists of two steps: pre-training and fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  The intention behind the pre-training step is to give the model a solid foundation training in the kind of data  that will be used in downstream tasks in order for it to gain a preliminary understanding of the target domain  that can the transferred to more specific tasks later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Most transformer-based architectures are built for language  problems, and the respective pre-training is often limited to masked-language tasks (BERT [6], RoBERTa) or  token discrimination tasks (Electra).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' This kind of training enables the model to learn the syntactic relations  between words, but it does not get any information about their semantics, aside from context similarity when  words are interchangeably used in similar contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  In this study, we introduced an additional ontology pre-training stage after the usual pre-training and be-  fore the fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' We present a case-study for the use of ontology pre-training to improve chemical toxicity  prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Figure 1 depicts the process using the boxology notation [25] for the novel approach as well as a  comparison approach without ontology pre-training which serves as our baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' In the remainder of this sec-  tion, we will detail the setup for each individual step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' All models are based on the Electra model as implemented  in the ChEBai1 tool used in previous work [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' All models share the hyperparameters detailed in Table 1 with  different classification heads for different tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='1  Step 1: Standard Pre-Training  The first step of this architecture is based on the pre-training mechanism for Electra [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' This process extends the  general masked-language model (MLM) used to pre-train transformers by a competing discriminator that aims to  1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='com/MGlauer/ChEBai  2  predict which tokens have been replaced as part of the MLM task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' For the molecular data that serves as the input  for the prediction task, the language of the molecular representations is SMILES [28], a molecular line notation  in which atoms and bonds are represented in a linear sequence, and the masking of tokens affects sub-elements  within the molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' The generator part of Electra uses a simple linear layer with as softmax to predict the token  that was most likely masked, while the discriminator uses the same setup, but one additional dense layer to guess  which token had been replaced by the generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  In our case-study, we use the same dataset for pre-training that has been used in a previous task on ontology  extension [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' This dataset consists of 365,512 SMILES representations for molecular structures that have been  extracted from PubChem [23] and the ChEBI ontology [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Notably, 152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='205 of these substances are known to  be hazardous substances as they are associated with a hazard class annotation in PubChem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='2  Step 2: Ontology Pre-Training  The standard pre-training teaches the network the foundations of how chemical structures are composed and  represented in SMILES strings, but it does not give the network any insights into which parts of these molecules  may be important and chemically active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' These functional properties are semantic information that are used by  experts to distinguish classes of molecules within the ChEBI ontology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' They are therefore inherently encoded in  the subsumption relations of the ontology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' We used the subsumption hierarchy to create a dataset for an ontology-  based classification task by extracting all subclasses of ‘molecular entity’ that had at least 100 subclasses with  SMILES strings attached to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' This resulted in a collection of 856 classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' We then collected all classes in  ChEBI that had a SMILES string attached to them and annotated them with their subsumption relation for each  of the 856 classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' The resulting dataset is similar to the ones used in [10] and [8], but covers a wider range of  classes as labels (856 instead of 500) and also a larger number of molecules (129,187 instead of 31,280).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' We then  use the pre-trained model from Step 1 to predict the superclasses of a class of molecules based on its annotated  SMILES string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='3  Step 3: Toxicity prediction  In order to assess the impact of this additional ontology-based pre-training step, we compare the model that re-  sulted from the ontology pre-training in step 2 with the one from step 1 that did not receive that training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' This  comparison is based on each model’s performance on toxicity prediction using the Tox21 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' This dataset  was created by the National Center for Advanced Translational Sciences (NCATS) of the National Institutes of  Health (NIH), and constitutes a widely used benchmark for research on machine learning for toxicity prediction  from small molecules [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Notably, there are currently two versions of the Tox21 dataset available in bench-  marks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' The first one originates from the Tox21 Challenge that was conducted in 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' This dataset consists of  three different subsets, one for training and two for different steps of the challenge evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' In our study, we  will use the “testing dataset” that was used for an initial ranking of models as a validation set and the dataset that  was used for the final evaluation as our test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' This version of the Tox21 dataset suffers from several issues  regarding the consistency and quality of different entries [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' A more modern version of this dataset has been  published as part of the MoleculeNet benchmark [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' This version of Tox21 consists of only of 7,831 molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  We split this dataset into a training (85%), validation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='5%) and test set (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='5%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' There are, however, still two  major challenges that need to be considered when working with this dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' First, the number of datapoints is  rather low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Molecular structures are complex graphs, which makes it hard for any model to derive a sufficient  amount of information from this dataset alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Second, the information available in the dataset is not complete: a  substantial amount of toxicity information is missing in this dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' There are 4,752 molecules for which at least  one of the 12 kinds of toxicity is not known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' In the prior literature, this issue has been approached in different  ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Some approaches perform data cleaning which limits the number of available data points even further, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  [13] excluded all datapoints that contained any missing labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' We decided to keep these datapoints as part of our  dataset, but exclude the missing labels from the calculation of all loss functions and metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Any outputs that  the model generates for these missing labels is considered as correct and does not influence the training gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  This approach allows the network to fill these gaps autonomously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  Preliminary results showed that both models were prone to overfitting, when used with the same settings as  the model from step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' We found that this behaviour could be prevented by using strong regularisations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' The final  experiments used the Adamax optimizer with dropouts on input embeddings of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='2, dropouts on hidden states of  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='4 and L2-regularisation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' All data and code used in this study is publicly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='2  2https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='7548313  3  (a) ROC-AUC / MoleculeNet Tox21 dataset  (b) F1 score (micro) / MoleculeNet Tox21 dataset  (c) ROC-AUC / original TOX21 dataset  (d) F1 score (micro) / original TOX21 dataset  Figure 2: Development of ROC-AUC and F1 score (micro) during training on the validation sets of the Tox21 dataset available  as part of MoleculeNet and the original TOX21 challenge  3  Results  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='1  Predictive performance  The final result of our training stack are four models: with or without ontology pre-training, and fine-tuned  on the original Tox21 competition dataset or on the smaller version of the Tox21 dataset published as part of  MoleculeNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' The semantic pre-training already showed a clear impact during the training phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Figures 2a-  2d depict the curves for two metrics (F1 score and ROC-AUC) on our validation set as evaluated at the end  of each epoch during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' It can be seen that models with ontology pre-training start with a better initial  performance and also retain this margin throughout the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' This behaviour is also reflected in the predictive  performance on both test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Table 2 shows the predictive behaviour for the dataset from MoleculeNet and the  original challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' The leading model (highlighted in bold) is predominantly the one that received additional  ontology pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' This is particularly visible for the more noisy and sparse dataset used in the original  Tox21 competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' The overall improved performance shows that pre-training with a more general ontology  pre-training does support the network for the more specialised toxicity prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' The drastic drop that can be  seen around epoch 50 in Figure 2d but not for the pre-trained model in Figure 2b further indicates that ontology  pre-training hedges the model against early overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' The reported results, and in particular the F1 scores,  however, show that there is still a large margin of error for this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='2  Interpretability  Attention weights in Transformer networks can be visualised to enable a kind of salience-based visual interpre-  tation for predictions directly connected with the input data [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Previous work [8] explored the link between  attention weights to the prediction of ontology classes, and observed that the network learned to pay attention to  relevant substructures when making predictions of ontology classes to which the molecule belongs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  In the current work, our hypothesis was that the additional information from the ontology would both enhance  the prediction and enhance the coherence of the attention weights for explanatory visualisations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' To test this  hypothesis we explored the attention visualisations for the predictions by the ontology pre-trained network as  compared to the normal network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  Figure 3 shows an individual example of the attention weight that the network uses when visiting specific  4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
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+page_content='78  Table 2: Class-wise scores on the test set on both Tox21 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Bold values denote the best value for a particular com-  bination of dataset and metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' NR - nuclear receptor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' AR - androgen receptor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' LBD - luciferase;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' AhR - aryl hydrocarbon  receptor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' ER - estrogen receptor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' PPAR - peroxisome proliferator-activated receptor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' SR - stress response;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' ARE - nuclear  factor antioxidant response;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' ATAD5 - genotoxicity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' HSE - heat shock factor response;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' MMP - mitochondrial response;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' p53 -  DNA damage response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  Ontology Pre-training  yes  no  Tox21 Challenge  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='86  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='90  Tox21 MoleculeNet  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='79  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='85  Table 3: Entropy values averaged over all tokens, attention heads, layers and atoms in the respective test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  input tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' The molecule depicted is TOX25530, corresponding to the sulfonamide anti-glaucoma drug dorzo-  lamide, which is not toxic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Dark green lines indicate strong, focused attention, while more opaque lines indicate  broader, less focused attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' As this example illustrates, we observed that the ontology pre-trained network  often shows more coherence and structure in its attention weights compared to the baseline network without  ontology pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' This is reflected in the triangular clusters of darker attention weights in the top rows of  Figure 3a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Clusters reflect that from a particular position in the input token sequence, strong attention  is placed on a subset of the molecule, reflecting relevant substructures within the molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Figure 3c shows  how the attention weights relate to the molecular graph for this molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Attention weight relationships may be  short-range (nearby atoms or groups) or long-range (atoms or groups further away within the molecule).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  To test this visual intuition more systematically, we computed the entropy for each attention head in each  layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Attention is computed using softmax and can therefore be interpreted as a probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' In order  to evaluate our hypothesis that the ontology pre-training also impacts the way a model focuses its attention, we  calculated the average entropy of these distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Entropy is a measure for the randomness of a distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  A perfectly even distribution would result in a entropy value of 1 (complete uncertainty), while a distribution in  which only one event can possibly occur will result in an entropy value of 0 (complete certainty).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' That means that  an attention distribution with an entropy value of 1 is not paying attention to any particular part of the molecule,  but is spreading attention evenly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' An entropy value of 0 indicates that the model paid attention to only a single  token of the molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Table 3 shows the aggregated entropy values for our models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' It can be seen that the model  that received additional ontology pre-training has a lower entropy value for both datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' That means, that the  attention is generally spent less evenly and is therefore more focused in comparison to the model that did not  receive that additional training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' This indicates that the ontology pre-trained model’s decisions are based on more  concise substructures within the molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  5  (a) Attention plots for layers 2-3  (b) Attention plots for layers 4-5  (c) Attention clusters relate to the molecule structure  Figure 3: Visualisation of the attention weights for the layers 2-3 in subfigure a) and layers 4-5 in subfigure b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
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+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='1  Significance  Our approach introduces a new way to embed knowledge from an ontology into a neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' The underlying  assumption of this approach is the following: A well-designed ontology represents a classification of its domain  that has proven useful for the various goals and tasks of experts in that area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Thus, it is possible (and even likely)  that some of the classes of the ontology reflect features that are relevant for a prediction task in that domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' For  example, the classification of chemical entities in ChEBI is based on empirical knowledge about the pertinent  features of molecules and their chemical behaviour, and it is reasonable to expect that some of these pertinent  features are, at least indirectly, correlated with toxicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' The goal of ontology pre-training is to enable a model to  benefit from the knowledge represented in an ontology by training it to categorise its input according to the class  hierarchy from the ontology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  This approach is applicable in any case where a dataset is available that links the input datatype for the  prediction task to the classification hierarchy from the ontology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' In the case of our example, the SMILES input  structures are directly associated with the leaf classes of the ChEBI ontology, thus we can prepare a training  dataset directly from the ontology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' However, for other ontologies, the dataset may be assembled from ontology  annotations which are external to the ontology but serve the purpose of linking the ontology classes to examples  of instances of the corresponding input datatype.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  The results in Table 2 show that we were able to improve the performance of our model for toxicity prediction  with the help of ontology pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' The inspection of the attention head weights indicates that the system  indeed learned meaningful aspects of life science chemistry from the pre-training task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Further, as we will  discuss next, the performance of the ontology pre-trained model compares favourably with the state of the art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='2  Related work  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='1  Toxicity prediction  The prediction, based on chemical structure, of whether a molecule has the potential to be harmful or poisonous  to living systems, is a challenging task for life science chemistry [1, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' The Tox21 dataset has become a widely  used benchmark for evaluating machine learning approaches to this task, thus there have been multiple previous  publications using different strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Most approaches to toxicity prediction supplement the input data chemical  structures with additional calculated features, such as known toxicophores, in order to enhance performance on  the toxicity prediction task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' This was the case for the winner of the original Tox21 challenge, who used a deep  learning implementation together with three other types of classifier in an ensemble [17], and more recently [19],  which augments the input molecular structure with physico-chemical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Another recent approach uses  ‘geometric graph learning’ [14] which augments the input molecular graphs with multi-scale weighted coloured  graph descriptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Some approaches additionally augment the training data with more examples in order to  mitigate the fact that the Tox21 dataset is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' In [13], chemical descriptors were calculated and in addition, a  strategy to remove class imbalance was applied including over-sampling from the minority classes in the dataset  followed by cleaning of mislabelled instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' In these cases, which use a different input dataset whether through  feature augmentation, data augmentation or data cleaning, we cannot directly compare their results to ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' We  anticipate that feature and data augmentation approaches such as these would potentially improve our method’s  performance as well, but we would need to develop a more complex attention visualisation mechanism to operate  across augmented inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Since our objective is rather to describe a new approach to incorporating ontology  knowledge into a neural network, we here focus our performance evaluation on those approaches that are more  directly comparable to ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  [3] uses a graph neural network and a semi-supervised learning approach known as Mean Teacher which  augments the training data with additional unlabelled examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' This network and training approach achieving  a ROC-AUC score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='757 in the test set, which our approach outperforms without data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Table 2  shows a comparison of the ROC-AUC values achieved by our model against those reported for the best model  (SSL-GCN) reported in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' With the exception of one class, our model shows better performance for all target  classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  ChemBERTa [4] is the most similar to our approach in that it also uses a Transformer-based network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Its  original publication also contains an evaluation on the p53 stress-response pathway activation (SR-p53) target of  the Tox21 dataset from MoleculeNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Our model exceeds the reported ROC-AUC value (ChemBERTa: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='728,  our model: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='83).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  7  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='2  Knowledge-aware pre-training with an ontology  Approaches to add knowledge from an ontology into a machine learning model follow several different strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  The most common is that the information from the ontology is used to supplement the input data in some  form, such as by adding synonyms and classification parent labels to the input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' For example, in [22] an  ontology is used to supplement the input data with an ontology-based ‘feature engineering’ strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  A second approach is that the ontology content is itself provided as input that is embedded into the network,  for example by using random walks through the ontology content to create sentences representing the structure  of the ontology for input into a language model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Ontology embedding strategies include OWL2Vec∗ [2] and  OPA2Vec [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' These approaches are suitable for tasks such as knowledge graph completion or link predic-  tion, but the additional information provided by such embeddings is not inherently connected in the internal  representation space to the other information learned by the network, and this limits their potential benefit if the  input datatype is complex and internally structured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' For example, in the chemistry case, the information about  the molecular structure of the chemical that is provided in the SMILES input strings would not be connected  intrinsically to the information about the class hierarchy provided to the network by an ontology embedding  strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  There are some examples of the use of biological ontologies together with biomolecular input data types that  are closer to our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' OntoProtein [32] combines background knowledge from the Gene Ontology with  protein sequences to improve the prediction of protein functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='OntoProtein uses the Gene Ontology in pre-  training a protein embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Existing protein embeddings such as ProtBERT are enhanced by embedding the  Gene Ontology as a knowledge graph (following approaches such as OWL2Vec∗) and then explicitly aligning  the two embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' By contrast, our approach uses the ontology more directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Namely, our pre-training  exploits both ChEBI’s class structure as well its class hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' The class structure leads to an aggregation  of molecules with similar sub-structures and properties, which can enhance the learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' The class  hierarchy is indirectly influencing the learning process as well, because a subclass relation corresponds to a  subset relation for the training samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' OntoProtein uses the subclass hierarchy only for defining depth of  ontology terms, which influences learning only very indirectly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Hence, our model incorporates expert knowledge  in a more direct way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' In the future, we will try to incorporate OntoProtein’s approach of contrastive learning  using knowledge-aware negative sampling into our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  Other approaches have developed custom architectures for the incorporation of knowledge from an ontology  into the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' For example, DeepPheno [16] predicts phenotypes from combinations of genetic mutations,  where phenotypes are encoded into the network through a novel hierarchical classification layer that encodes  almost 4,000 classes from the Human Phenotype Ontology together with their hierarchical dependencies as an  ontology prediction layer that informs the remainder of the training of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' A similar approach is used in  [31], in which an ontology layer adds information from an ontology to a neural network to enhance the prediction  of microbial biomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Instead our approach uses the generic architecture of a standard Transformer network and  learns the information from the ontology through an additional pre-training task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='3  Limitations  Our current approach only uses a fraction of the information available in the ChEBI ontology, since we only  consider the structural classification of chemical entities beneath the ‘molecular entity’ class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Hence, we currently  consider neither classes of biological and chemical roles nor pharmaceutical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' These classes have the  potential to further enhance the knowledge that is provided to the network, and will be explored in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  Another limitation is related to the way we create the ontology pre-training task: We use the leaf nodes of  ChEBI as examples for training a model to predict the subsumption relation for more general classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Or, to put  it differently, while from an ontological perspective the leaf nodes of ChEBI are classes in ChEBI’s taxonomy,  we are treating them as instances and, thus, turning subsumption prediction into a classification task from a  machine learning perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Consequently, while we use the whole structural hierarchy of ChEBI for creating  the pre-training task, the model learns to classify only 856 classes, those that have a sufficient number of example  structures to learn from, which is a relatively small number compared to the number of classes in ChEBI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Further,  this approach of creating a subsumption prediction pre-training task requires rich structural annotations linked to  the learning input datatype (which is the SMILES in our example case), which many ontologies do not contain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  As indicated in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='1, both of these limitations may be addressed by using class membership prediction  for ontology pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' All that is required for ontology pre-training is a dataset that (a) is of the same input  datatype as the fine-tuning task, (b) is annotated with terms from the ontology, and (c) contains sufficient training  examples to train the model to classify the input with the terms of the ontology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Because we treat subsumption  prediction as a classification problem anyway, both approaches are functionally equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' However, using an  external dataset for training (instead of generating it from the ontology itself), has the benefit that the ontology  8  pre-training might cover the whole ontology instead of just a subset of the ontology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Further, this approach does  not rely on the existence of appropriate annotations in the associated input data type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  A further limitation of the approach is that the interpretability offered by the attention weights is limited to  visual inspection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' In future work we aim to develop an algorithm that is able to systematically determine clusters  of attention mapped to the relevant parts of the input molecular structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  5  Conclusion  This paper presents the results of training an Electra model for toxicity prediction using the Tox21 dataset as  a benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' The model was able to achieve state-of-the-art performance on the task of toxicity prediction,  outperforming comparable models that have been trained on the same dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  While improving the state of the art of toxicity prediction is in itself an interesting result, the main contri-  bution of the paper is the presentation of a novel approach to combining symbolic and connectionist AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' This  is because our result was achieved with the help of an additional pre-training step, which trains the model with  the help of background knowledge from an ontology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' For the presented work the pre-training task consisted of  predicting subclass relationships between classes in the ChEBI ontology, but other tasks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=', predicting class  membership) are likely to be equally suitable to achieve the purpose of ontology pre-training, namely, to train the  model to recognise the meaningful classification of the entities in a given domain, as represented by the ontology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  As we have illustrated in this paper, ontology pre-training has the potential benefit of reducing the time  needed for fine-tuning and improving performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Further, as we have illustrated in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='2, an inspection  of attention heads indicates that some of the attention patterns of the model correlate to the substructures that are  pertinent for chemical categories in ChEBI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Thus, since ontological categories are meaningful for humans, an-  other potential benefit of ontology pre-training is an improved interpretability of the trained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' In the future,  we are planning to further investigate this line of research by systematically analysing the attention patterns of  the pre-trained model and automatically linking these patterns to the ontology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  As we discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='3, currently we are only using some of the knowledge available in the ontology  for pre-training purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' In particular, we do not include classes from other parts of the ontology, nor do we  include other axioms aside from the hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' In the future we are planning to overcome these limitations  by sampling a wider set of classes from ChEBI and by using a more complex architecture that combines a  Transformer with a logical neural network (LNN) [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' The LNN is able to represent logical axioms from the  ontology as a first-order theory, translated from OWL [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' This will enable us to use logical axioms from the  ontology (and therefore also its binary relations) to influence both the ontology pre-training as well as the fine-  tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  References  [1] Claudio N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Cavasotto and Valeria Scardino.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Machine Learning Toxicity Prediction: Latest Advances by  Toxicity End Point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' ACS Omega, 7(51):47536–47546, December 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Publisher: American Chemical  Society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  [2] Jiaoyan Chen, Pan Hu, Ernesto Jim´enez-Ruiz, Ole Magnus Holter, Denvar Antonyrajah, and Ian Horrocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  Owl2vec*: embedding of OWL ontologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Mach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=', 110(7):1813–1845, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  [3] Jiarui Chen, Yain-Whar Si, Chon-Wai Un, and Shirley W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Siu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Chemical toxicity prediction based on  semi-supervised learning and graph convolutional neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Journal of Cheminformatics, 13(1):93,  November 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  [4] Seyone Chithrananda, Gabriel Grand, and Bharath Ramsundar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Chemberta: large-scale self-supervised  pretraining for molecular property prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' arXiv preprint arXiv:2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='09885, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  [5] Kevin Clark, Minh-Thang Luong, Quoc V Le, and Christopher D Manning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Electra: Pre-training text  encoders as discriminators rather than generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' arXiv preprint arXiv:2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='10555, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  [6] Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' BERT: pre-training of deep bidi-  rectional transformers for language understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' In Jill Burstein, Christy Doran, and Thamar Solorio,  editors, Proceedings of the 2019 Conference of the North American Chapter of the Association for Com-  putational Linguistics: Human Language Technologies, NAACL-HLT 2019, Minneapolis, MN, USA, June  2-7, 2019, Volume 1 (Long and Short Papers), pages 4171–4186.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Association for Computational Linguis-  tics, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  9  [7] Simon Fl¨ugel, Martin Glauer, Fabian Neuhaus, and Janna Hastings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' When one logic is not enough: Inte-  grating first-order annotations in OWL ontologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Semantic web journal, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  [8] Martin Glauer, Adel Memariani, Fabian Neuhaus, Till Mossakowski, and Janna Hastings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Interpretable  Ontology Extension in Chemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Semantic Web Journal, February 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  [9] Janna Hastings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Primer on Ontologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' In Christophe Dessimoz and Nives ˇSkunca, editors, The Gene  Ontology Handbook, volume 1446, pages 3–13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Springer New York, New York, NY, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Series Title:  Methods in Molecular Biology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  [10] Janna Hastings, Martin Glauer, Adel Memariani, Fabian Neuhaus, and Till Mossakowski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Learning Chem-  istry: Exploring the Suitability of Machine Learning for the Task of Structure-based Chemical Ontology  Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Journal of Cheminformatics, 13(23), 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  [11] Janna Hastings, Gareth Owen, Adriano Dekker, Marcus Ennis, Namrata Kale, Venkatesh Muthukrishnan,  Steve Turner, Neil Swainston, Pedro Mendes, and Christoph Steinbeck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' ChEBI in 2016: Improved services  and an expanding collection of metabolites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Nucleic Acids Research, 44(D1):D1214–D1219, January 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  [12] Ruili Huang, Menghang Xia, Dac-Trung Nguyen, Tongan Zhao, Srilatha Sakamuru, Jinghua Zhao, Sam-  pada A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Shahane, Anna Rossoshek, and Anton Simeonov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Tox21Challenge to Build Predictive Models of  Nuclear Receptor and Stress Response Pathways as Mediated by Exposure to Environmental Chemicals  and Drugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Frontiers in Environmental Science, 3, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  [13] Gabriel Idakwo, Sundar Thangapandian, Joseph Luttrell, Yan Li, Nan Wang, Zhaoxian Zhou, Huixiao  Hong, Bei Yang, Chaoyang Zhang, and Ping Gong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Structure–activity relationship-based chemical classi-  fication of highly imbalanced tox21 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Journal of cheminformatics, 12(1):1–19, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  [14] Jian Jiang, Rui Wang, and Guo-Wei Wei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  GGL-Tox: Geometric Graph Learning for Toxicity Predic-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Journal of Chemical Information and Modeling, 61(4):1691–1700, April 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Publisher: American  Chemical Society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  [15] John Jumper, Richard Evans, Alexander Pritzel, Tim Green, Michael Figurnov, Olaf Ronneberger, Kathryn  Tunyasuvunakool, Russ Bates, Augustin ˇZ´ıdek, Anna Potapenko, Alex Bridgland, Clemens Meyer, Simon  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Kohl, Andrew J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Ballard, Andrew Cowie, Bernardino Romera-Paredes, Stanislav Nikolov, Rishub  Jain, Jonas Adler, Trevor Back, Stig Petersen, David Reiman, Ellen Clancy, Michal Zielinski, Martin  Steinegger, Michalina Pacholska, Tamas Berghammer, Sebastian Bodenstein, David Silver, Oriol Vinyals,  Andrew W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Senior, Koray Kavukcuoglu, Pushmeet Kohli, and Demis Hassabis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Highly accurate protein  structure prediction with AlphaFold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Nature, pages 1–11, July 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Publisher: Nature Publishing Group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  [16] Maxat Kulmanov and Robert Hoehndorf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' DeepPheno: Predicting single gene loss-of-function phenotypes  using an ontology-aware hierarchical classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' PLoS Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Biol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=', 16(11), 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  [17] Andreas Mayr, G¨unter Klambauer, Thomas Unterthiner, and Sepp Hochreiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' DeepTox: Toxicity Predic-  tion using Deep Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Frontiers in Environmental Science, 3, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  [18] Fabian Neuhaus and Janna Hastings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Ontology development is consensus creation, not (merely) represen-  tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Ontology, 17(4):495–513, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  [19] Yuzhong Peng, Ziqiao Zhang, Qizhi Jiang, Jihong Guan, and Shuigeng Zhou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' TOP: A deep mixture repre-  sentation learning method for boosting molecular toxicity prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Methods, 179:55–64, July 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  [20] Ryan Riegel, Alexander Gray, Francois Luus, Naweed Khan, Ndivhuwo Makondo, Ismail Yunus Akhal-  waya, Haifeng Qian, Ronald Fagin, Francisco Barahona, Udit Sharma, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Logical neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' arXiv  preprint arXiv:2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='13155, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  [21] Robin Rombach, Andreas Blattmann, Dominik Lorenz, Patrick Esser, and Bj¨orn Ommer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' High-Resolution  Image Synthesis with Latent Diffusion Models, April 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' arXiv:2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='10752 [cs].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  [22] Satya S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Sahoo, Katja Kobow, Jianzhe Zhang, Jeffrey Buchhalter, Mojtaba Dayyani, Dipak P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Upadhyaya,  Katrina Prantzalos, Meenakshi Bhattacharjee, Ingmar Blumcke, Samuel Wiebe, and Samden D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Lhatoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  Ontology-based feature engineering in machine learning workflows for heterogeneous epilepsy patient  records.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Scientific Reports, 12(1):19430, November 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Number: 1 Publisher: Nature Publishing Group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  10  [23] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Sayers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' PubChem: An Entrez Database of Small Molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' NLM Tech Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=', 2005 Jan-Feb(342:e2),  2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  [24] Fatima Zohra Smaili, Xin Gao, and Robert Hoehndorf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' OPA2Vec: combining formal and informal content  of biomedical ontologies to improve similarity-based prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Bioinformatics, 35(12):2133–2140, June  2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  [25] Michael van Bekkum, Maaike de Boer, Frank van Harmelen, Andr´e Meyer-Vitali, and Annette ten Teije.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  Modular design patterns for hybrid learning and reasoning systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Applied Intelligence, 51(9):6528–6546,  2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  [26] Jesse Vig, Ali Madani, Lav R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Varshney, Caiming Xiong, Richard Socher, and Nazneen Fatema Rajani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
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+page_content=' arXiv:2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='15222 [cs,  q-bio], March 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' arXiv: 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='15222.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  [27] Laura von Rueden, Sebastian Mayer, Katharina Beckh, Bogdan Georgiev, Sven Giesselbach, Raoul Heese,  Birgit Kirsch, Michal Walczak, Julius Pfrommer, Annika Pick, Rajkumar Ramamurthy, Jochen Garcke,  Christian Bauckhage, and Jannis Schuecker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Informed Machine Learning - A Taxonomy and Survey of  Integrating Prior Knowledge into Learning Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' IEEE Transactions on Knowledge and Data Engi-  neering, pages 1–1, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Conference Name: IEEE Transactions on Knowledge and Data Engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  [28] David Weininger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' SMILES, a chemical language and information system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Introduction to methodology  and encoding rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Journal of Chemical Information and Computer Sciences, 28(1):31–36, February 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  [29] Zhenqin Wu, Bharath Ramsundar, Evan N Feinberg, Joseph Gomes, Caleb Geniesse, Aneesh S Pappu, Karl  Leswing, and Vijay Pande.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Moleculenet: a benchmark for molecular machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Chemical science,  9(2):513–530, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  [30] Hongbin Yang, Lixia Sun, Weihua Li, Guixia Liu, and Yun Tang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
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+page_content=' Frontiers in Chemistry, 6, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  [31] Yuguo Zha, Hui Chong, Hao Qiu, Kai Kang, Yuzheng Dun, Zhixue Chen, Xuefeng Cui, and Kang Ning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
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+page_content='  [32] Ningyu Zhang, Zhen Bi, Xiaozhuan Liang, Siyuan Cheng, Haosen Hong, Shumin Deng, Qiang Zhang,  Jiazhang Lian, and Huajun Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' Ontoprotein: Protein pretraining with gene ontology embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content=' In The  Tenth International Conference on Learning Representations, ICLR 2022, Virtual Event, April 25-29, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  OpenReview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='net, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
+page_content='  11' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FAT4oBgHgl3EQfex0W/content/2301.08577v1.pdf'}
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+arXiv:2301.05243v1  [gr-qc]  12 Jan 2023
+Are slowly rotating Ellis-Bronnikov wormholes stable?
+Bahareh Azad1,∗ Jose Luis Bl´azquez-Salcedo2,† Fech Scen Khoo1,‡ and Jutta Kunz1§
+1Institut f¨ur Physik, Universit¨at Oldenburg, Postfach 2503, D-26111 Oldenburg, Germany
+2Departamento de F´ısica Te´orica and IPARCOS,
+Facultad de Ciencias F´ısicas, Universidad Complutense de Madrid, Spain
+(Dated: January 16, 2023)
+We investigate the radial perturbations of Ellis-Bronnikov wormholes (l = 0) in a slowly rotating
+background expanded up to second order in rotation. We find indications that simple wormhole
+solutions such as Ellis-Bronnikov in General Relativity can be stabilized by rotation, thus favoring
+a viable traversable wormhole.
+This opens up the intriguing question whether the many other
+wormhole solutions with or without the support of exotic matter can become linearly mode stable
+when the wormhole rotates.
+Introduction.
+Wormholes have received much at-
+tention in recent years, in particular, concerning their
+possible astrophysical signatures such as gravitational
+lensing effects of wormholes [1–12], wormhole shadows
+[10, 13–17], or wormhole accretion disks and quasi-
+periodic oscillations [18–24]. In fact, wormholes may
+mimic many properties of black holes [10, 25–34].
+Therefore the study of gravitational waves emitted by
+wormholes is also of great interest.
+The observations of gravitational waves by the
+LIGO/VIRGO collaboration have provided us with in-
+valuable information about the merger of highly com-
+pact objects such as black holes and neutron stars
+(see e.g.
+[35, 36]).
+Here the ringdown phase after
+the merger events is dominated by the characteristic
+frequencies of the final objects, encoded in their quasi-
+normal modes (QNMs). Whereas the QNMs of black
+holes and neutron stars are well studied in General
+Relativity (GR) [37–39], the corresponding modes of
+wormholes have received much less attention.
+In GR classical wormholes need exotic matter in
+order to violate the energy conditions [40–50]. While
+this issue can be circumvented in several ways (for ex-
+ample, by considering wormholes in generalized grav-
+ities [50, 51] or by adding fermionic matter [52, 53]),
+another question concerns the stability of wormholes
+in GR and generalized gravities.
+For instance, the
+notorious radial instability of the static spherically
+symmetric Ellis-Bronnikov (EB) wormholes of GR has
+been known for long [32, 54–57].
+Here we present intriguing evidence that this ra-
+dial instability of EB wormholes will disappear, when
+these wormholes rotate sufficiently fast.
+However,
+unlike their static counterparts, rotating EB worm-
+∗ bahareh.azad@uni-oldenburg.de
+† jlblaz01@ucm.es
+‡ fech.scen.khoo@uni-oldenburg.de
+§ jutta.kunz@uni-oldenburg.de
+holes are known only perturbatively for slow rotation
+[58, 59] and numerically for fast rotation [60, 61]. Ad-
+ditionally, rotating EB wormholes have been inves-
+tigated in five dimensions for the case of two equal
+magnitude angular momenta, where the angular co-
+ordinates factorize [62].
+In four dimensions, a mode analysis of rotating com-
+pact objects becomes a challenging task.
+Already
+for Kerr-Newman black holes linear mode stability
+could only be provided by solving the resulting sys-
+tem of coupled partial differential equations with so-
+phisticated numerical methods [63]. Alternatively, the
+modes have recently been obtained perturbatively to
+second order in rotation, where parity mixing compli-
+cates their evaluation as opposed to first order [64].
+We here apply this recently developed second order
+formalism [64] in order to study the change of the un-
+stable radial mode of EB wormholes in the presence of
+slow rotation. We demonstrate, that the magnitude
+of the pertinent unstable eigenvalue decreases with in-
+creasing angular momentum of the wormhole, thus di-
+minishing the instability. In fact, in our perturbative
+analysis the eigenvalue vanishes at some critical value
+of the angular momentum, that slightly depends on
+the asymmetry parameter of the wormhole, and we
+argue that nonperturbatively the instability will dis-
+appear slightly earlier.
+EB Wormholes in Slow Rotation. We consider a
+phantom scalar field Φ minimally coupled to GR,
+S =
+1
+16πG
+�
+d4x√−g
+�
+R + 2∂µΦ ∂µΦ
+�
+(1)
+with field equations
+Rµν = −2∂µΦ∂νΦ ,
+∂µ∂µΦ = 0 .
+(2)
+Up to second order in rotation the background metric
+
+2
+is given by
+ds2 = −ef �
+1 + ǫ2
+r2 (h0(r) + h2(r)P2(θ))
+�
+dt2
++ e−f �
+1 + ǫ2
+r2 (b0(r) + b2(r)P2(θ))
+�
+dr2
++ e−fR2 �
+1 + ǫ2
+r2 (k0(r) + k2(r)P2(θ))
+�
+×
+�
+dθ2 + sin2 (θ) [dϕ − ǫrw(r)dt]2�
+,
+(3)
+where R2 = (r2 + r2
+0), with slow-rotation parame-
+ter ǫr ≪ 1, and the Legendre polynomial P2(θ) =
+�
+3 cos2 (θ) − 1
+�
+/2.
+The corresponding second order
+phantom scalar field has the form
+Φ = φ(r) + ǫ2
+r (φ20(r) + φ22(r)P2(θ)) .
+(4)
+In the following we choose the gauge k0(r) = 0. The
+throat of the wormhole is located at r = 0, and it has
+an area A = 4πr2
+0e
+πC
+2r0 .
+The EB wormhole background solutions are
+f(r) = C
+r0
+�
+tan−1
+� r
+r0
+�
+− π
+2
+�
+, φ(r) = Qf
+C
+, (5)
+with asymmetry parameter C = 2M0, related to
+the static background wormhole mass M0, and the
+static background phantom scalar charge Q0
+=
+�
+C2/4 + r2
+0. Here we will focus on C ≥ 0. At the
+first order in rotation, we solve to obtain
+w(r) =
+3J
+2C(C2 + r2
+0)
+�
+1 −
+�
+1 + 2C C + r
+R2
+�
+e2f
+�
+. (6)
+In the system of field equations, the differential equa-
+tions of the radial functions h0, b0, φ20 are decoupled
+from those of h2, k2, b2, φ22.
+When focusing on ra-
+dial perturbations, we only need the functions φ20, b0,
+h0. For the simpler symmetric background wormholes
+(C = 0) they are given by
+φ20(r) = 3J2
+rr3
+0
+�
+φ + π
+2 − 3r
+2r0
+−
+4
+πr0
+− 3r2 + 4r2
+0
+2R2φ
+�
+,
+b0(r) = ∆M
+r
+�
+1 − φr0
+r
+�
++ 6J2
+rr3
+0
+�r0
+r + φ
+�
+− 3J2
+r2r2
+0
+φ2 ,
+h0(r) = 3J2
+r4
+0
+�
+φ2 + π2 − 8
+2π
+φ − r2
+0
+R2
+�
+.
+(7)
+Solutions for C > 0 are more involved, and hence
+provided elsewhere [65]. The static mass M0 is cor-
+rected by ∆M = 3J2(π2 − 8)/(2πr3
+0) in the limit of
+C = 0 when r → +∞. Therefore, for C = 0, the total
+mass of the wormhole is no longer zero. In fact, it
+receives a contribution from the rotation. This echoes
+the finding in [59], that there is no massless rotating
+EB wormhole. The static scalar charge Q0 is also cor-
+rected by ∆Q = −∆M. The mass and scalar charge
+corrections for C > 0 are given elsewhere [65].
+Stability Analysis: Radial Perturbations. We
+linearly perturb the metric field g and the phantom
+field Φ up to first order in the perturbation parameter
+ǫq, in our slowly rotating (sr) background considered
+up to second order in rotation ǫ2
+r. The perturbed met-
+ric is then given by
+gµν = g(sr)
+µν
++ ǫqδhµν(t, r, θ, ϕ) ,
+(8)
+the perturbation of the metric being a combination of
+polar (P) and axial (A) perturbations: δhµν = δh(A)
+µν +
+δh(P )
+µν . The perturbed phantom field is
+Φ = Φ(sr) + ǫqδφ(P )(t, r, θ, ϕ) .
+(9)
+These perturbations can be decomposed in spherical
+harmonics Y [l, m](θ, ϕ) , with multipole numbers l and
+m [64]. When wormholes rotate the perturbations of
+even (polar) and odd (axial) parities are entangled,
+i.e. the multipole numbers l are coupled. This results
+in an infinite tower of equations carrying the index l
+that are to be summed, as is well-known for other ro-
+tating objects such as black holes and neutron stars.
+Here, we are interested in the radial-led perturbations,
+hence l = 0, m = 0 polar-led perturbations. For a
+given slowly rotating background up to second order
+in rotation, it is possible to show that these perturba-
+tions are just coupled to axial l = 1 perturbations (see
+[64]). Hence for our purposes, the metric perturbation
+can be simplified to
+δhµν = eiωt
+
+
+
+
+NY0
+0
+0
+S0Yθ
+0
+LY0
+0
+S1Yθ
+0
+0
+R2T Y0
+0
+S0Yθ S1Yθ
+0
+R2T sin2 θY0
+
+
+
+(10)
+where N, T, L, S0, S1 are radial functions,
+Y0
+=
+Y [0, 0]
+=
+1/
+√
+4π
+and Yθ
+=
+sin θ ∂θY [1, 0]
+=
+− sin2 θ
+�
+3/4π.
+The phantom perturbation on the
+other hand can be taken as
+δφ(P ) = φ1(r)Y0 .
+(11)
+With these perturbations, a consistent set of equations
+is formed, as the rotations are approximately slow and
+we work only up to second order in the rotation pa-
+rameter ǫr. More technical details of our slow-rotation
+approximation framework can be found in [64].
+Furthermore, the gauge can be fixed by taking
+S0 = φ1 = 0 to simplify the resulting system of equa-
+tions. It can be shown that the final system can be
+cast into a single second order ordinary differential
+equation for T , with algebraic relations for the rest of
+the perturbation functions. If we take T = αZ, the
+perturbation equation can be cast into a Schr¨odinger
+equation,
+∂2
+r∗Z + (ω2 − V (r))Z = 0 ,
+(12)
+
+3
+where ω is the quasi-normal mode frequency, V (r)
+is the potential, and the tortoise coordinate r∗ and
+amplitude α are given by,
+∂rr∗ = e−f �
+1 + ǫ2
+r(e−fb0 − h0)
+�
+, (13)
+∂rα
+α
+= (∂rφ)2R2 + ǫ2
+r[2e−fb0 − e−2fR4(∂rw)2/6]
+r −
+�
+R2(∂rφ)2 − r2
+0
+. (14)
+In this case, the potential is given by
+V (r) = Vstatic(r) + J2 V2(r) .
+(15)
+There is no linear term in the angular momentum J.
+Its contribution only enters at second order, and it is
+a function of the static background solutions f, φ, the
+first order function w, and the second order functions
+h0, b0, φ20. Taking C to zero gives rise to the following
+potential
+VC=0 (r) = 3r2r2
+0 + 2r4
+0
+r2R4
++
+(16)
+J2
+�
+−12φ2
+r4R2 + 12πr(R4 + r4
+0) − 6r3
+0(π2 − 8)R2
+πr3
+0r4R4
+φ +
+3r(π2 − 8)(R4 + r4
+0) + 12πr0(R2r2 + 2r4
+0)
+πr3
+0r4R4
+�
+.
+We provide the complete expression of the potential
+term for C > 0 elsewhere [65]. In Figure 1, we show
+the correction to the potential (15) of the perturbation
+system due to slow rotation, obtained with symmetric
+(C = 0) and asymmetric (C ̸= 0) background worm-
+hole configurations.
+The potential shown has been
+scaled with (r − C/2)2.
+Numerical Method. Here we are interested in un-
+stable modes, therefore we focus on perturbations for
+which the eigenvalue ω is a purely imaginary number,
+ω = iωI, such that ωI < 0. In this case, an asymptotic
+analysis of the perturbation function T (r) reveals that
+it has to decay exponentially to zero at both infinities,
+T (±∞) = 0. In practice we solve the second order
+ODE for the function T (r) together with an auxiliary
+equation E′ = 0 where E = ω2. Non-trivial regular
+solutions are obtained by fixing the amplitude of the
+perturbation function to an arbitrary real constant at
+an arbitrary point r = rc, i.e. T (rc) = constant. This
+method has been used successfully before to generate
+unstable modes of the static EB wormholes [32]. As
+an illustrative example, we show in Figure 1 a typical
+Gaussian profile of the function T (r).
+In order to generate the values of ωI, we start from
+the static limit (J = 0), fixing r0 = 1 and an arbitrary
+value of C. We reproduce the results from [32], and by
+slowly increasing the value of the angular momentum
+J, we get a relation
+ωI = ω(0)
+I
++ J2∆ω(2)
+I
+,
+(17)
+ 0
+ 5
+ 10
+ 15
+-10
+-5
+ 0
+ 5
+r0=1
+V~
+2, C=0
+V~
+2, C=0.1
+V~
+2, C=0.2
+T(r)
+r
+FIG. 1. Potential correction ˜V2 = V2(r − C/2)2 vs radial
+coordinate r for several values of C for r0 = 1, and a
+typical profile of the perturbation function T (r) (in grey).
+where ω(0)
+I
+is the unstable mode of the static EB
+wormhole, and ∆ω(2)
+I
+is the slow rotation correction
+to the mode.
+Radial Instability. We exhibit in Figure 2 the de-
+pendence of the unstable radial mode on the scaled
+and dimensionless angular momentum J/A of the EB
+wormholes, where A is the area of the throat (ob-
+tained in second order). We recall that rotating EB
+wormholes approach the extremal Kerr black hole as
+their limiting configuration for large angular momenta
+[60, 61].
+Thus the limiting case here is given by
+J/A → 1/(8π) ≈ 0.0397.
+Since fast rotating EB
+wormholes cannot exceed this limit, the slow rotation
+approximation can only make sense below this limit.
+In Figure 2 we show that, as we increase the value
+of the angular momentum, the value of |ωI| decreases
+to zero. This is because, for every value of C we have
+studied, the second order correction to the mode is
+positive, ∆ω(2)
+I
+> 0. As a result, it is always possible
+to obtain a critical value of the angular momentum,
+Jc, for which the eigenvalue reaches zero and hence
+this unstable mode disappears.
+Most importantly,
+this happens at a value of J much smaller than the
+limiting value.
+In fact, as seen in the inset of Fig-
+ure 2, the critical values of the scaled angular mo-
+mentum Jc/A decrease monotonically with C. More-
+over, for each of the values of C examined, Jc/A is
+around 50% of the limiting value, which is well within
+the regime for which the second order slow rotation
+approximation has been shown to work well (for ex-
+ample, the quadratic approximation reproduces with
+excellent precision the spectrum of QNMs of Kerr-
+Newman black holes [64]).
+However, the critical value of the angular momen-
+tum for which the wormholes become stable is pre-
+
+4
+-4
+-2
+ 0
+ 0
+ 0.02
+ 0.04
+ωI√�A
+J/A
+C=0
+C=0.2
+C=0.4
+C=0.7
+C=1.0
+ 0.015
+ 0.02
+ 0
+ 1
+Jc/A
+C
+FIG. 2. Main unstable mode ωI
+√
+A vs angular momentum
+J/A (scaled with throat area A) for several values of C.
+Inset: Critical values Jc/A vs C.
+−18
+−9
+ 0
+ 0
+ 0.006
+ 0.012
+ω2 A
+J/A
+C=0
+C=0.05
+C=0.1
+C=0.2
+FIG. 3. Main unstable mode (solid) and second unstable
+mode (dotted) vs angular momentum J/A (scaled with
+throat area A) for C = 0, 0.05, 0.1 and 0.2.
+sumably still smaller, as the following observations
+suggest. When the wormholes are set into rotation, a
+zero mode of the static wormholes changes into a sec-
+ond unstable radial mode of the slowly rotating EB
+wormholes. Following a quadratic dependence with
+a small negative coefficient ∆ω(2)
+I , this mode remains
+at first very small in magnitude.
+Subsequently, its
+magnitude increases and it crosses the corresponding
+main unstable mode not too far from its critical value
+Jc/A. We exhibit both unstable modes in Figure 3.
+Note that for the values of C displayed the crossing
+occurs around J/A ≈ 0.013, slightly above 30% of the
+limiting value.
+We do not continue both modes beyond their cross-
+ing, however, since this crossing is indicating a de-
+ficiency of the applied quadratic approximation.
+A
+non-perturbative study will allow for a far stronger
+dependence on the angular momentum than the cur-
+rent quadratic one.
+We therefore conjecture, that
+when nearing the critical value, both unstable modes
+will change much more rapidly, approach each other,
+merge and disappear. They will thus reflect the be-
+havior observed in higher dimensions in the special
+case of wormholes with equal angular momenta [62].
+Conclusions. While EB wormholes represent rather
+simple wormhole solutions they may be considered
+prototypes whose properties can teach a lot about
+wormholes in general.
+Therefore their response to
+rotation is highly remarkable. Indeed, the potential
+of rotation to stabilize previously unstable wormholes
+was suggested before (see e.g.
+[66]), but we have
+given first evidence here, that it can indeed happen
+for wormholes in four space-time dimensions, where
+rotation implies significant loss of symmetry (associ-
+ated with the technical challenge).
+As we have shown above, the notorious unstable
+mode of static EB wormholes becomes more stable,
+as slow rotation is turned on. In our second order and
+thus quadratic approximation the mode then becomes
+stable at a critical value of the angular momentum,
+which depends on the asymmetry parameter C. For
+the EB wormholes we have considered, this critical
+value is around 50% of the limiting angular momen-
+tum. Moreover, we have observed that for rotating
+wormholes a second mode appears, and similar to the
+higher dimensional case [62], it tends to merge with
+the fundamental unstable mode. Hence we have con-
+jectured the behavior of the radial instability for the
+case of a future non-perturbative study: the radial in-
+stability will disappear at even smaller values of the
+angular momentum, around 30% of the limiting value.
+Thus for sufficiently rapid rotation the wormholes will
+be stable.
+Acknowledgements.
+We gratefully acknowledge
+support by DAAD, the DFG Research Training Group
+1620 Models of Gravity, DFG project Ku612/18-1,
+FCT project PTDC/FIS-AST/3041/2020, COST Ac-
+tions CA15117 and CA16104, and MICINN project
+PID2021-125617NB-I00 “QuasiMode”.
+JLBS grate-
+fully acknowledges support from Santander-UCM
+project PR44/21-29910.
+[1] J. G. Cramer, R. L. Forward, M. S. Morris, M. Visser,
+G. Benford and G. A. Landis, Phys. Rev. D 51, 3117
+(1995)
+
+5
+[2] M. Safonova, D. F. Torres and G. E. Romero, Phys.
+Rev. D 65, 023001 (2002)
+[3] V. Perlick, Phys. Rev. D 69, 064017 (2004)
+[4] K. K. Nandi, Y. Z. Zhang and A. V. Zakharov, Phys.
+Rev. D 74, 024020 (2006)
+[5] F. Abe, Astrophys. J. 725, 787 (2010)
+[6] Y. Toki, T. Kitamura, H. Asada and F. Abe, Astro-
+phys. J. 740, 121 (2011)
+[7] K. Nakajima and H. Asada, Phys. Rev. D 85, 107501
+(2012)
+[8] N. Tsukamoto, T. Harada and K. Yajima, Phys. Rev.
+D 86, 104062 (2012)
+[9] P. K. F. Kuhfittig, Eur. Phys. J. C 74, 2818 (2014)
+[10] C. Bambi, Phys. Rev. D 87, 107501 (2013)
+[11] R. Takahashi and H. Asada, Astrophys. J. 768, L16
+(2013)
+[12] N. Tsukamoto and T. Harada, Phys. Rev. D 95,
+024030 (2017)
+[13] P. G. Nedkova, V. K. Tinchev and S. S. Yazadjiev,
+Phys. Rev. D 88, 124019 (2013)
+[14] T. Ohgami and N. Sakai, Phys. Rev. D 91, 124020
+(2015)
+[15] R. Shaikh, Phys. Rev. D 98, 024044 (2018)
+[16] G. Gyulchev, P. Nedkova, V. Tinchev and S. Yazad-
+jiev, Eur. Phys. J. C 78 , 544 (2018)
+[17] M. Guerrero, G. J. Olmo, D. Rubiera-Garcia and
+D. G´omez S´aez-Chill´on, Phys. Rev. D 105, no.8,
+084057 (2022)
+[18] T. Harko, Z. Kovacs and F. S. N. Lobo, Phys. Rev. D
+78, 084005 (2008)
+[19] T. Harko, Z. Kovacs and F. S. N. Lobo, Phys. Rev. D
+79, 064001 (2009)
+[20] C. Bambi, Phys. Rev. D 87, 084039 (2013)
+[21] M. Zhou, A. Cardenas-Avendano, C. Bambi, B. Klei-
+haus and J. Kunz, Phys. Rev. D 94, 024036 (2016)
+[22] F. Lamy, E. Gourgoulhon, T. Paumard and F. H. Vin-
+cent, Class. Quant. Grav. 35, 115009 (2018)
+[23] E. Deligianni, J. Kunz, P. Nedkova, S. Yazadjiev and
+R. Zheleva, Phys. Rev. D 104, 024048 (2021)
+[24] E. Deligianni, B. Kleihaus, J. Kunz, P. Nedkova and
+S. Yazadjiev, Phys. Rev. D 104, 064043 (2021)
+[25] T. Damour and S. N. Solodukhin, Phys. Rev. D 76,
+024016 (2007)
+[26] M. Azreg-Ainou, JCAP 1507, 037 (2015)
+[27] V. Dzhunushaliev, V. Folomeev, B. Kleihaus and
+J. Kunz, JCAP 1608, no. 08, 030 (2016)
+[28] V. Cardoso, E. Franzin and P. Pani, Phys. Rev. Lett.
+116, 171101 (2016); Erratum: [Phys. Rev. Lett. 117,
+089902 (2016)]
+[29] R. A. Konoplya and A. Zhidenko, JCAP 1612, 043
+(2016)
+[30] K. K. Nandi, R. N. Izmailov, A. A. Yanbekov and
+A. A. Shayakhmetov, Phys. Rev. D 95, 104011 (2017)
+[31] P. Bueno, P. A. Cano, F. Goelen, T. Hertog and
+B. Vercnocke, Phys. Rev. D 97, 024040 (2018)
+[32] J. L. Bl´azquez-Salcedo, X. Y. Chew and J. Kunz,
+Phys. Rev. D 98 no.4, 044035 (2018)
+[33] B. Azad, F. Loran and A. Mostafazadeh, Eur. Phys.
+J. C 80 (2020) no.12, 1097
+[34] B. Azad, J. L. Bl´azquez-Salcedo, X. Y. Chew, J. Kunz
+and D. h. Yeom, [arXiv:2212.12601 [gr-qc]]
+[35] B. P. Abbott et al. [LIGO Scientific and Virgo], Phys.
+Rev. Lett. 116, no.6, 061102 (2016)
+[36] B. P. Abbott et al. [LIGO Scientific and Virgo], Phys.
+Rev. Lett. 119 , no.16, 161101 (2017)
+[37] K. D. Kokkotas and B. G. Schmidt, Living Rev. Rel.
+2, 2 (1999)
+[38] E. Berti, V. Cardoso and A. O. Starinets, Class.
+Quant. Grav. 26, 163001 (2009)
+[39] R. A. Konoplya and A. Zhidenko, Rev. Mod. Phys.
+83, 793 (2011)
+[40] H. G. Ellis, J. Math. Phys. 14, 104 (1973)
+[41] H. G. Ellis, Gen. Rel. Grav. 10, 105 (1979)
+[42] K. A. Bronnikov, Acta Phys. Polon. B 4, 251 (1973)
+[43] M. S. Morris, K. S. Thorne and U. Yurtsever, Phys.
+Rev. Lett. 61, 1446-1449 (1988)
+[44] M. S. Morris and K. S. Thorne, Am. J. Phys. 56, 395
+(1988)
+[45] M. Visser, Woodbury, USA: AIP (1995)
+[46] D. Hochberg, A. Popov and S. V. Sushkov, Phys. Rev.
+Lett. 78, 2050 (1997)
+[47] D. Hochberg and M. Visser, Phys. Rev. Lett. 81, 746
+(1998)
+[48] M. Visser, S. Kar and N. Dadhich, Phys. Rev. Lett.
+90, 201102 (2003)
+[49] K. A. Bronnikov and J. C. Fabris, Phys. Rev. Lett.
+96, 251101 (2006)
+[50] M. Alcubierre and F. S. N. Lobo, Fundam. Theor.
+Phys. 189 (2017)
+[51] P. Kanti, B. Kleihaus and J. Kunz, Phys. Rev. Lett.
+107, 271101 (2011)
+[52] J. L. Bl´azquez-Salcedo, C. Knoll and E. Radu, Phys.
+Rev. Lett. 126, 101102 (2021)
+[53] R. A. Konoplya and A. Zhidenko, Phys. Rev. Lett.
+128, 091104 (2022)
+[54] H. a. Shinkai and S. A. Hayward, Phys. Rev. D 66,
+044005 (2002)
+[55] J. A. Gonzalez, F. S. Guzman and O. Sarbach, Class.
+Quant. Grav. 26, 015010 (2009)
+[56] J. A. Gonzalez, F. S. Guzman and O. Sarbach, Class.
+Quant. Grav. 26, 015011 (2009)
+[57] F. Cremona, F. Pirotta and L. Pizzocchero, Gen. Rel.
+Grav. 51, 19 (2019)
+[58] P. E. Kashargin and S. V. Sushkov, Grav. Cosmol.
+14, 80 (2008)
+[59] P. E. Kashargin and S. V. Sushkov, Phys. Rev. D 78,
+064071 (2008)
+[60] B. Kleihaus and J. Kunz, Phys. Rev. D 90, 121503
+(2014)
+[61] X. Y. Chew, B. Kleihaus and J. Kunz, Phys. Rev. D
+94, 104031 (2016)
+[62] V. Dzhunushaliev, V. Folomeev, B. Kleihaus, J. Kunz
+and E. Radu, Phys. Rev. D 88, 124028 (2013)
+[63] O. J. C. Dias, M. Godazgar and J. E. Santos, Phys.
+Rev. Lett. 114, 151101 (2015)
+[64] J.
+L.
+Bl´azquez-Salcedo
+and
+F.
+S.
+Khoo,
+[arXiv:2212.00054 [gr-qc]]
+[65] B. Azad, J. L. Bl´azquez-Salcedo, F. S. Khoo and
+
+6
+J. Kunz, in preparation
+[66] T. Matos and D. Nunez, Class. Quant. Grav. 23, 4485
+(2006)
+
diff --git a/qNE4T4oBgHgl3EQfvg2M/content/tmp_files/load_file.txt b/qNE4T4oBgHgl3EQfvg2M/content/tmp_files/load_file.txt
new file mode 100644
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@@ -0,0 +1,560 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf,len=559
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='05243v1  [gr-qc]  12 Jan 2023  Are slowly rotating Ellis-Bronnikov wormholes stable?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  Bahareh Azad1,∗ Jose Luis Bl´azquez-Salcedo2,† Fech Scen Khoo1,‡ and Jutta Kunz1§  1Institut f¨ur Physik, Universit¨at Oldenburg, Postfach 2503, D-26111 Oldenburg, Germany  2Departamento de F´ısica Te´orica and IPARCOS,  Facultad de Ciencias F´ısicas, Universidad Complutense de Madrid, Spain  (Dated: January 16, 2023)  We investigate the radial perturbations of Ellis-Bronnikov wormholes (l = 0) in a slowly rotating  background expanded up to second order in rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' We find indications that simple wormhole  solutions such as Ellis-Bronnikov in General Relativity can be stabilized by rotation, thus favoring  a viable traversable wormhole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  This opens up the intriguing question whether the many other  wormhole solutions with or without the support of exotic matter can become linearly mode stable  when the wormhole rotates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  Wormholes have received much at-  tention in recent years, in particular, concerning their  possible astrophysical signatures such as gravitational  lensing effects of wormholes [1–12], wormhole shadows  [10, 13–17], or wormhole accretion disks and quasi-  periodic oscillations [18–24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' In fact, wormholes may  mimic many properties of black holes [10, 25–34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  Therefore the study of gravitational waves emitted by  wormholes is also of great interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  The observations of gravitational waves by the  LIGO/VIRGO collaboration have provided us with in-  valuable information about the merger of highly com-  pact objects such as black holes and neutron stars  (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  [35, 36]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  Here the ringdown phase after  the merger events is dominated by the characteristic  frequencies of the final objects, encoded in their quasi-  normal modes (QNMs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Whereas the QNMs of black  holes and neutron stars are well studied in General  Relativity (GR) [37–39], the corresponding modes of  wormholes have received much less attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  In GR classical wormholes need exotic matter in  order to violate the energy conditions [40–50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' While  this issue can be circumvented in several ways (for ex-  ample, by considering wormholes in generalized grav-  ities [50, 51] or by adding fermionic matter [52, 53]),  another question concerns the stability of wormholes  in GR and generalized gravities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  For instance, the  notorious radial instability of the static spherically  symmetric Ellis-Bronnikov (EB) wormholes of GR has  been known for long [32, 54–57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  Here we present intriguing evidence that this ra-  dial instability of EB wormholes will disappear, when  these wormholes rotate sufficiently fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  However,  unlike their static counterparts, rotating EB worm-  ∗ bahareh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='azad@uni-oldenburg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='de  † jlblaz01@ucm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='es  ‡ fech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='scen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='khoo@uni-oldenburg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='de  § jutta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='kunz@uni-oldenburg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='de  holes are known only perturbatively for slow rotation  [58, 59] and numerically for fast rotation [60, 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Ad-  ditionally, rotating EB wormholes have been inves-  tigated in five dimensions for the case of two equal  magnitude angular momenta, where the angular co-  ordinates factorize [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  In four dimensions, a mode analysis of rotating com-  pact objects becomes a challenging task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  Already  for Kerr-Newman black holes linear mode stability  could only be provided by solving the resulting sys-  tem of coupled partial differential equations with so-  phisticated numerical methods [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Alternatively, the  modes have recently been obtained perturbatively to  second order in rotation, where parity mixing compli-  cates their evaluation as opposed to first order [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  We here apply this recently developed second order  formalism [64] in order to study the change of the un-  stable radial mode of EB wormholes in the presence of  slow rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' We demonstrate, that the magnitude  of the pertinent unstable eigenvalue decreases with in-  creasing angular momentum of the wormhole, thus di-  minishing the instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' In fact, in our perturbative  analysis the eigenvalue vanishes at some critical value  of the angular momentum, that slightly depends on  the asymmetry parameter of the wormhole, and we  argue that nonperturbatively the instability will dis-  appear slightly earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  EB Wormholes in Slow Rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' We consider a  phantom scalar field Φ minimally coupled to GR,  S =  1  16πG  �  d4x√−g  �  R + 2∂µΦ ∂µΦ  �  (1)  with field equations  Rµν = −2∂µΦ∂νΦ ,  ∂µ∂µΦ = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  (2)  Up to second order in rotation the background metric  2  is given by  ds2 = −ef �  1 + ǫ2  r2 (h0(r) + h2(r)P2(θ))  �  dt2  + e−f �  1 + ǫ2  r2 (b0(r) + b2(r)P2(θ))  �  dr2  + e−fR2 �  1 + ǫ2  r2 (k0(r) + k2(r)P2(θ))  �  ×  �  dθ2 + sin2 (θ) [dϕ − ǫrw(r)dt]2�  ,  (3)  where R2 = (r2 + r2  0), with slow-rotation parame-  ter ǫr ≪ 1, and the Legendre polynomial P2(θ) =  �  3 cos2 (θ) − 1  �  /2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  The corresponding second order  phantom scalar field has the form  Φ = φ(r) + ǫ2  r (φ20(r) + φ22(r)P2(θ)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  (4)  In the following we choose the gauge k0(r) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' The  throat of the wormhole is located at r = 0, and it has  an area A = 4πr2  0e  πC  2r0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  The EB wormhole background solutions are  f(r) = C  r0  �  tan−1  � r  r0  �  − π  2  �  , φ(r) = Qf  C  , (5)  with asymmetry parameter C = 2M0, related to  the static background wormhole mass M0, and the  static background phantom scalar charge Q0  =  �  C2/4 + r2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Here we will focus on C ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' At the  first order in rotation, we solve to obtain  w(r) =  3J  2C(C2 + r2  0)  �  1 −  �  1 + 2C C + r  R2  �  e2f  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' (6)  In the system of field equations, the differential equa-  tions of the radial functions h0, b0, φ20 are decoupled  from those of h2, k2, b2, φ22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  When focusing on ra-  dial perturbations, we only need the functions φ20, b0,  h0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' For the simpler symmetric background wormholes  (C = 0) they are given by  φ20(r) = 3J2  rr3  0  �  φ + π  2 − 3r  2r0  −  4  πr0  − 3r2 + 4r2  0  2R2φ  �  ,  b0(r) = ∆M  r  �  1 − φr0  r  �  + 6J2  rr3  0  �r0  r + φ  �  − 3J2  r2r2  0  φ2 ,  h0(r) = 3J2  r4  0  �  φ2 + π2 − 8  2π  φ − r2  0  R2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  (7)  Solutions for C > 0 are more involved, and hence  provided elsewhere [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' The static mass M0 is cor-  rected by ∆M = 3J2(π2 − 8)/(2πr3  0) in the limit of  C = 0 when r → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Therefore, for C = 0, the total  mass of the wormhole is no longer zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' In fact, it  receives a contribution from the rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' This echoes  the finding in [59], that there is no massless rotating  EB wormhole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' The static scalar charge Q0 is also cor-  rected by ∆Q = −∆M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' The mass and scalar charge  corrections for C > 0 are given elsewhere [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  Stability Analysis: Radial Perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' We  linearly perturb the metric field g and the phantom  field Φ up to first order in the perturbation parameter  ǫq, in our slowly rotating (sr) background considered  up to second order in rotation ǫ2  r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' The perturbed met-  ric is then given by  gµν = g(sr)  µν  + ǫqδhµν(t, r, θ, ϕ) ,  (8)  the perturbation of the metric being a combination of  polar (P) and axial (A) perturbations: δhµν = δh(A)  µν +  δh(P )  µν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' The perturbed phantom field is  Φ = Φ(sr) + ǫqδφ(P )(t, r, θ, ϕ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  (9)  These perturbations can be decomposed in spherical  harmonics Y [l, m](θ, ϕ) , with multipole numbers l and  m [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' When wormholes rotate the perturbations of  even (polar) and odd (axial) parities are entangled,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' the multipole numbers l are coupled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' This results  in an infinite tower of equations carrying the index l  that are to be summed, as is well-known for other ro-  tating objects such as black holes and neutron stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  Here, we are interested in the radial-led perturbations,  hence l = 0, m = 0 polar-led perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' For a  given slowly rotating background up to second order  in rotation, it is possible to show that these perturba-  tions are just coupled to axial l = 1 perturbations (see  [64]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Hence for our purposes, the metric perturbation  can be simplified to  δhµν = eiωt  \uf8eb  \uf8ec  \uf8ec  \uf8ed  NY0  0  0  S0Yθ  0  LY0  0  S1Yθ  0  0  R2T Y0  0  S0Yθ S1Yθ  0  R2T sin2 θY0  \uf8f6  \uf8f7  \uf8f7  \uf8f8(10)  where N, T, L, S0, S1 are radial functions,  Y0  =  Y [0, 0]  =  1/  √  4π  and Yθ  =  sin θ ∂θY [1, 0]  =  − sin2 θ  �  3/4π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  The phantom perturbation on the  other hand can be taken as  δφ(P ) = φ1(r)Y0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  (11)  With these perturbations, a consistent set of equations  is formed, as the rotations are approximately slow and  we work only up to second order in the rotation pa-  rameter ǫr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' More technical details of our slow-rotation  approximation framework can be found in [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  Furthermore, the gauge can be fixed by taking  S0 = φ1 = 0 to simplify the resulting system of equa-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' It can be shown that the final system can be  cast into a single second order ordinary differential  equation for T , with algebraic relations for the rest of  the perturbation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' If we take T = αZ, the  perturbation equation can be cast into a Schr¨odinger  equation,  ∂2  r∗Z + (ω2 − V (r))Z = 0 ,  (12)  3  where ω is the quasi-normal mode frequency, V (r)  is the potential, and the tortoise coordinate r∗ and  amplitude α are given by,  ∂rr∗ = e−f �  1 + ǫ2  r(e−fb0 − h0)  �  , (13)  ∂rα  α  = (∂rφ)2R2 + ǫ2  r[2e−fb0 − e−2fR4(∂rw)2/6]  r −  �  R2(∂rφ)2 − r2  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' (14)  In this case, the potential is given by  V (r) = Vstatic(r) + J2 V2(r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  (15)  There is no linear term in the angular momentum J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  Its contribution only enters at second order, and it is  a function of the static background solutions f, φ, the  first order function w, and the second order functions  h0, b0, φ20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Taking C to zero gives rise to the following  potential  VC=0 (r) = 3r2r2  0 + 2r4  0  r2R4  +  (16)  J2  �  −12φ2  r4R2 + 12πr(R4 + r4  0) − 6r3  0(π2 − 8)R2  πr3  0r4R4  φ +  3r(π2 − 8)(R4 + r4  0) + 12πr0(R2r2 + 2r4  0)  πr3  0r4R4  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  We provide the complete expression of the potential  term for C > 0 elsewhere [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' In Figure 1, we show  the correction to the potential (15) of the perturbation  system due to slow rotation, obtained with symmetric  (C = 0) and asymmetric (C ̸= 0) background worm-  hole configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  The potential shown has been  scaled with (r − C/2)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  Numerical Method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Here we are interested in un-  stable modes, therefore we focus on perturbations for  which the eigenvalue ω is a purely imaginary number,  ω = iωI, such that ωI < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' In this case, an asymptotic  analysis of the perturbation function T (r) reveals that  it has to decay exponentially to zero at both infinities,  T (±∞) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' In practice we solve the second order  ODE for the function T (r) together with an auxiliary  equation E′ = 0 where E = ω2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Non-trivial regular  solutions are obtained by fixing the amplitude of the  perturbation function to an arbitrary real constant at  an arbitrary point r = rc, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' T (rc) = constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' This  method has been used successfully before to generate  unstable modes of the static EB wormholes [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' As  an illustrative example, we show in Figure 1 a typical  Gaussian profile of the function T (r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  In order to generate the values of ωI, we start from  the static limit (J = 0), fixing r0 = 1 and an arbitrary  value of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' We reproduce the results from [32], and by  slowly increasing the value of the angular momentum  J, we get a relation  ωI = ω(0)  I  + J2∆ω(2)  I  ,  (17)  0  5  10  15  10  5  0  5  r0=1  V~  2, C=0  V~  2, C=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='1  V~  2, C=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='2  T(r)  r  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Potential correction ˜V2 = V2(r − C/2)2 vs radial  coordinate r for several values of C for r0 = 1, and a  typical profile of the perturbation function T (r) (in grey).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  where ω(0)  I  is the unstable mode of the static EB  wormhole, and ∆ω(2)  I  is the slow rotation correction  to the mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  Radial Instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' We exhibit in Figure 2 the de-  pendence of the unstable radial mode on the scaled  and dimensionless angular momentum J/A of the EB  wormholes, where A is the area of the throat (ob-  tained in second order).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' We recall that rotating EB  wormholes approach the extremal Kerr black hole as  their limiting configuration for large angular momenta  [60, 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  Thus the limiting case here is given by  J/A → 1/(8π) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='0397.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  Since fast rotating EB  wormholes cannot exceed this limit, the slow rotation  approximation can only make sense below this limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  In Figure 2 we show that, as we increase the value  of the angular momentum, the value of |ωI| decreases  to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' This is because, for every value of C we have  studied, the second order correction to the mode is  positive, ∆ω(2)  I  > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' As a result, it is always possible  to obtain a critical value of the angular momentum,  Jc, for which the eigenvalue reaches zero and hence  this unstable mode disappears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  Most importantly,  this happens at a value of J much smaller than the  limiting value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  In fact, as seen in the inset of Fig-  ure 2, the critical values of the scaled angular mo-  mentum Jc/A decrease monotonically with C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' More-  over, for each of the values of C examined, Jc/A is  around 50% of the limiting value, which is well within  the regime for which the second order slow rotation  approximation has been shown to work well (for ex-  ample, the quadratic approximation reproduces with  excellent precision the spectrum of QNMs of Kerr-  Newman black holes [64]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  However, the critical value of the angular momen-  tum for which the wormholes become stable is pre-  4  4  2  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='04  ωI√�A  J/A  C=0  C=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='2  C=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='4  C=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='7  C=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='02  0  1  Jc/A  C  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Main unstable mode ωI  √  A vs angular momentum  J/A (scaled with throat area A) for several values of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  Inset: Critical values Jc/A vs C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  −18  −9  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='006  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='012  ω2 A  J/A  C=0  C=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='05  C=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='1  C=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='2  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Main unstable mode (solid) and second unstable  mode (dotted) vs angular momentum J/A (scaled with  throat area A) for C = 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  sumably still smaller, as the following observations  suggest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' When the wormholes are set into rotation, a  zero mode of the static wormholes changes into a sec-  ond unstable radial mode of the slowly rotating EB  wormholes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Following a quadratic dependence with  a small negative coefficient ∆ω(2)  I , this mode remains  at first very small in magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  Subsequently, its  magnitude increases and it crosses the corresponding  main unstable mode not too far from its critical value  Jc/A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' We exhibit both unstable modes in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  Note that for the values of C displayed the crossing  occurs around J/A ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='013, slightly above 30% of the  limiting value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  We do not continue both modes beyond their cross-  ing, however, since this crossing is indicating a de-  ficiency of the applied quadratic approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  A  non-perturbative study will allow for a far stronger  dependence on the angular momentum than the cur-  rent quadratic one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  We therefore conjecture, that  when nearing the critical value, both unstable modes  will change much more rapidly, approach each other,  merge and disappear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' They will thus reflect the be-  havior observed in higher dimensions in the special  case of wormholes with equal angular momenta [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  Conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' While EB wormholes represent rather  simple wormhole solutions they may be considered  prototypes whose properties can teach a lot about  wormholes in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  Therefore their response to  rotation is highly remarkable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Indeed, the potential  of rotation to stabilize previously unstable wormholes  was suggested before (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  [66]), but we have  given first evidence here, that it can indeed happen  for wormholes in four space-time dimensions, where  rotation implies significant loss of symmetry (associ-  ated with the technical challenge).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  As we have shown above, the notorious unstable  mode of static EB wormholes becomes more stable,  as slow rotation is turned on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' In our second order and  thus quadratic approximation the mode then becomes  stable at a critical value of the angular momentum,  which depends on the asymmetry parameter C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' For  the EB wormholes we have considered, this critical  value is around 50% of the limiting angular momen-  tum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Moreover, we have observed that for rotating  wormholes a second mode appears, and similar to the  higher dimensional case [62], it tends to merge with  the fundamental unstable mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Hence we have con-  jectured the behavior of the radial instability for the  case of a future non-perturbative study: the radial in-  stability will disappear at even smaller values of the  angular momentum, around 30% of the limiting value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  Thus for sufficiently rapid rotation the wormholes will  be stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  We gratefully acknowledge  support by DAAD, the DFG Research Training Group  1620 Models of Gravity, DFG project Ku612/18-1,  FCT project PTDC/FIS-AST/3041/2020, COST Ac-  tions CA15117 and CA16104, and MICINN project  PID2021-125617NB-I00 “QuasiMode”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  JLBS grate-  fully acknowledges support from Santander-UCM  project PR44/21-29910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Cramer, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Forward, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Morris, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Visser,  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Benford and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Landis, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' D 51, 3117  (1995)  5  [2] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Safonova, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Torres and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Romero, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' D 65, 023001 (2002)  [3] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Perlick, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' D 69, 064017 (2004)  [4] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Nandi, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Zhang and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Zakharov, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' D 74, 024020 (2006)  [5] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Abe, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' 725, 787 (2010)  [6] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Toki, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Kitamura, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Asada and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Abe, Astro-  phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' 740, 121 (2011)  [7] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Nakajima and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Asada, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' D 85, 107501  (2012)  [8] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Tsukamoto, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Harada and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Yajima, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  D 86, 104062 (2012)  [9] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Kuhfittig, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' C 74, 2818 (2014)  [10] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Bambi, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' D 87, 107501 (2013)  [11] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Takahashi and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Asada, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' 768, L16  (2013)  [12] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Tsukamoto and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Harada, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' D 95,  024030 (2017)  [13] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Nedkova, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Tinchev and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
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+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
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+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' D 76,  024016 (2007)  [26] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Azreg-Ainou, JCAP 1507, 037 (2015)  [27] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Dzhunushaliev, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
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+page_content=' 08, 030 (2016)  [28] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
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+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  116, 171101 (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
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+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' 117,  089902 (2016)]  [29] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
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+page_content=' Zhidenko, JCAP 1612, 043  (2016)  [30] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
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+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
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+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
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+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
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+page_content=' D 95, 104011 (2017)  [31] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
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+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
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+page_content='4, 044035 (2018)  [33] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Azad, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Loran and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Mostafazadeh, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' C 80 (2020) no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='12, 1097  [34] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Azad, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Bl´azquez-Salcedo, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Chew, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Kunz  and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Yeom, [arXiv:2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='12601 [gr-qc]]  [35] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' [LIGO Scientific and Virgo], Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' 116, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='6, 061102 (2016)  [36] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' [LIGO Scientific and Virgo], Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' 119 , no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='16, 161101 (2017)  [37] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Kokkotas and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Schmidt, Living Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  2, 2 (1999)  [38] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Berti, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Cardoso and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Starinets, Class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  Quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' 26, 163001 (2009)  [39] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Konoplya and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Zhidenko, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  83, 793 (2011)  [40] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Ellis, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' 14, 104 (1973)  [41] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Ellis, Gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' 10, 105 (1979)  [42] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Bronnikov, Acta Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Polon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' B 4, 251 (1973)  [43] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Morris, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Thorne and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Yurtsever, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' 61, 1446-1449 (1988)  [44] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Morris and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Thorne, Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' 56, 395  (1988)  [45] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Visser, Woodbury, USA: AIP (1995)  [46] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Hochberg, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Popov and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Sushkov, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' 78, 2050 (1997)  [47] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Hochberg and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Visser, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' 81, 746  (1998)  [48] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Visser, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Kar and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Dadhich, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  90, 201102 (2003)  [49] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Bronnikov and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Fabris, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  96, 251101 (2006)  [50] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Alcubierre and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Lobo, Fundam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' 189 (2017)  [51] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Kanti, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Kleihaus and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Kunz, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  107, 271101 (2011)  [52] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Bl´azquez-Salcedo, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Knoll and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Radu, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' 126, 101102 (2021)  [53] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
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+page_content='  128, 091104 (2022)  [54] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
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+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
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+page_content=' D 66,  044005 (2002)  [55] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
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+page_content=' Gonzalez, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Guzman and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
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+page_content=' 26, 015010 (2009)  [56] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Gonzalez, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Guzman and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Sarbach, Class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  Quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
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+page_content=' 26, 015011 (2009)  [57] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Cremona, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Pirotta and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Pizzocchero, Gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
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+page_content=' 51, 19 (2019)  [58] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Kashargin and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
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+page_content='  14, 80 (2008)  [59] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Kashargin and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Sushkov, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' D 78,  064071 (2008)  [60] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Kleihaus and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Kunz, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' D 90, 121503  (2014)  [61] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Chew, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Kleihaus and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Kunz, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' D  94, 104031 (2016)  [62] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Dzhunushaliev, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Folomeev, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Kleihaus, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Kunz  and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Radu, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' D 88, 124028 (2013)  [63] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Dias, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Godazgar and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Santos, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' 114, 151101 (2015)  [64] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  Bl´azquez-Salcedo  and  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='  Khoo,  [arXiv:2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content='00054 [gr-qc]]  [65] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Azad, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Bl´azquez-Salcedo, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Khoo and  6  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Kunz, in preparation  [66] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Matos and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
+page_content=' Nunez, Class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE4T4oBgHgl3EQfvg2M/content/2301.05243v1.pdf'}
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+Draft version January 12, 2023
+Typeset using LATEX preprint style in AASTeX631
+Kinetic equilibrium of two-dimensional force-free current sheets
+Xin An
+,1 Anton Artemyev
+,1, 2 Vassilis Angelopoulos
+,1 Andrei Runov
+,1 and
+Sergey Kamaletdinov
+2, 3
+1Department of Earth, Planetary, and Space Sciences, University of California, Los Angeles, CA, 90095, USA
+2Space Research Institute of the Russian Academy of Sciences, Moscow, 117997, Russia
+3Faculty of Physics, National Research University Higher School of Economics, Moscow, 101000, Russia
+ABSTRACT
+Force-free current sheets are local plasma structures with field-aligned electric currents
+and approximately uniform plasma pressures. Such structures, widely found throughout
+the heliosphere, are sites for plasma instabilities and magnetic reconnection, the growth
+rate of which is controlled by the structure’s current sheet configuration. Despite the
+fact that many kinetic equilibrium models have been developed for one-dimensional
+(1D) force-free current sheets, their two-dimensional (2D) counterparts, which have a
+magnetic field component normal to the current sheets, have not received sufficient at-
+tention to date. Here, using particle-in-cell simulations, we search for such 2D force-free
+current sheets through relaxation from an initial, magnetohydrodynamic equilibrium.
+Kinetic equilibria are established toward the end of our simulations, thus demonstrating
+the existence of kinetic force-free current sheets. Although the system currents in the
+late equilibrium state remain field aligned as in the initial configuration, the velocity
+distribution functions of both ions and electrons systematically evolve from their ini-
+tial drifting Maxwellians to their final time-stationary Vlasov state. The existence of
+2D force-free current sheets at kinetic equilibrium necessitates future work in discover-
+ing additional integrals of motion of the system, constructing the kinetic distribution
+functions, and eventually investigating their stability properties.
+1. INTRODUCTION
+Current sheets are spatially localized plasma structures that play an essential role in various space
+plasma systems: solar flares (Syrovatskii 1981; Parker 1994; Fleishman & Pevtsov 2018), solar wind
+turbulence (Servidio et al. 2011; Borovsky 2010; Vasko et al. 2022), boundaries of planetary magne-
+tospheres – magnetopauses (De Keyser et al. 2005), magnetotails of planets (Jackman et al. 2014;
+Achilleos 2018; Lui 2018) and comets (Cravens & Gombosi 2004; Volwerk et al. 2018; Volwerk 2018).
+Strong currents flowing within current sheets are subject to various instabilities resulting in magnetic
+field line reconnection, which converts magnetic energy to plasma heating and particle acceleration
+(e.g., Gonzalez & Parker 2016; Birn & Priest 2007).
+Corresponding author: Xin An
+phyax@ucla.edu
+arXiv:2301.04590v1  [physics.plasm-ph]  11 Jan 2023
+
+ID2
+The properties of magnetic reconnection and the dissipation rate of magnetic energy strongly de-
+pend on the current sheet configurations. The simplest magnetic field geometry is the one-dimensional
+(1D) current sheet, which is either a tangential discontinuity separating two plasmas with different
+properties or a rotational discontinuity having a finite magnetic field component, Bn, normal to the
+current sheet. The stress balance in 1D tangential discontinuities is established by the gradients
+of plasma and magnetic field pressures [see Figure 1(a) and Allanson et al. (2015); Neukirch et al.
+(2020a,b)], whereas the stress balance in 1D rotational discontinuities requires a contribution from
+the plasma dynamic pressure in order to balance the magnetic field line tension force ∝ Bn [see
+Figure 1(b) and Hudson (1970)].
+In two-dimensional (2D) current sheets, Bn ̸= 0 naturally appears (Schindler 2006); thus, the
+stress balance requires 2D plasma pressure gradients [see Figure 1(c) and Yoon & Lui (2005)] or 2D
+dynamic pressure gradients [see Figure 1(d) and Birn (1992); Nickeler & Wiegelmann (2010); Cicogna
+& Pegoraro (2015)].
+Although there are many types of current sheet configurations, they can all be categorized by two
+plasma parameters: the Alfv´en Mach number MA (i.e., the ratio of plasma flow speed to Alfv´en speed)
+and the plasma beta β (i.e., the ratio of plasma thermal pressure to magnetic field pressure). Current
+sheet configurations from different space environments are located within different domains in this
+(β, MA) space. Tangential discontinuities (Bn = 0) do not require contributions from the plasma
+dynamic pressure. These discontinuities exist in systems with either large β (if the plasma pressure
+balances the magnetic field pressure) or small β [if the current sheet configuration is magnetically
+force free without cross-field currents, J × B = 0; see Figure 1(a)]. Such current sheets are often
+observed in the solar wind (see discussion in Neugebauer 2006; Artemyev et al. 2019b), where they
+propagate with plasma flows and have MA ≈ 0 in the current sheet reference frame (see examples
+of kinetic models of such current sheet configurations in de Keyser et al. 1996; Harrison & Neukirch
+2009; Allanson et al. 2016; Neukirch et al. 2020a). Rotational discontinuities are also observed in
+the solar wind (β ∼ 1) and are characterized by plasma flow gradients with MA ∼ 1 in the current
+sheet reference frame [de Keyser et al. (e.g., 1997); Haaland et al. (e.g., 2012); Paschmann et al. (e.g.,
+2013); Artemyev et al. (e.g., 2019b); see Figure 1(b)]. Figure 1(e) shows the parameter regime of
+such current sheets in (β, MA) space, whereas Figure 2(a) shows an example of a low-β, force-free
+current sheet in the solar wind.
+Current sheets in the shocked (magnetosheath) solar wind are characterized by lower MA [Figure
+1(e)] compared to that of the solar wind. The stress balance in these current sheets is significantly
+influenced by plasma pressure gradients, i.e., compressional current sheets with both parallel and
+transverse current density components (e.g., Chaston et al. 2020; Webster et al. 2021). Another
+(β, MA) domain, with similar MA and larger β, is found in Earth’s distant magnetotail, where strong
+plasma flows may reach MA ≥ 1 and β ∈ [10, 100] (e.g., Vasko et al. 2015; Artemyev et al. 2017).
+These current sheets are mostly balanced by plasma pressure gradients. However, in the distant
+magnetotail and for β ≲ 10 there have been previous observations of force-free current sheets with
+J × B = 0 (Xu et al. 2018). Figure 1(e) shows the parameter domain of distant magnetotail current
+sheets, whereas Figure 2(b) shows an example of an almost force-free current sheet observed in the
+distant magnetotail. Both the magnetosheath and distant magnetotail current sheet configurations
+are characterized by a small but finite Bn, and thus the magnetic field line tension force ∝ Bn may be
+balanced by weak plasma flows or by plasma anisotropy. There is a series of models on such quasi-1D
+
+3
+Figure 1. Panels (a)-(d) show magnetic field lines and the main components of stress balance for different
+current sheet configurations. The coordinate system consists of the l component along the main magnetic
+field direction, reversing sign at the current sheet neutral plane (Bl = 0), the m component along the
+main current density direction corresponding to variations of Bl, and the n (normal) component along the
+spatial gradient of Bl (i.e., 4πjm/c = ∂Bl/∂rn). Panel (e) shows the typical parameter regimes of current
+sheets observed by THEMIS and ARTEMIS (Angelopoulos 2008, 2011) in the solar wind (blue), Earth’s
+magnetosheath (shocked solar wind; black), near-Earth magnetotail (red), and lunar-distance magnetotail
+(green). The black dashed box defines the parameter regime where 2D force-free current sheets are expected.
+Our dataset includes ∼ 300 solar wind current sheets, ∼ 100 magnetosheath current sheets, ∼ 100 current
+sheets in the near-Earth magnetotail, and ∼ 500 current sheets in the lunar-distance magnetotail.
+All
+current sheets were identified from THEMIS and ARTEMIS fluxgate magnetometer measurements (Auster
+et al. 2008). The selection procedure and data processing for solar wind current sheets are described in
+Artemyev et al. (2019a): ion temperatures are calculated using the OMNI dataset (King & Papitashvili
+2005; Artemyev et al. 2018), plasma flows in the current sheet reference frame are calculated as the change
+of the most variable flow component across the current sheet (see details in Artemyev et al. 2019a). The
+same criteria and data processing methods were applied to the magnetosheath (dayside) current sheets, but
+without using the OMNI data because THEMIS accurately measures magnetosheath ion temperatures in that
+region (McFadden et al. 2008). The dataset of the near-Earth magnetotail current sheets (radial distances
+∼ 10−30 Earth radii) is described in Artemyev et al. (2016). The same criteria and data processing methods
+were applied to the lunar-distance current sheets. Details of the calculation of β and MA are described in
+Artemyev et al. (2019a, 2017).
+current sheets, with Bn ̸= 0, that describe both large-β (see Burkhart et al. 1992; Sitnov et al. 2000,
+2006; Mingalev et al. 2007; Zelenyi et al. 2011) and force-free (Artemyev 2011; Mingalev et al. 2012;
+Vasko et al. 2014) current sheets.
+The distant magnetotail (β, MA) domain smoothly extends towards the lower MA with the decrease
+of the radial distance from the Earth. Near-Earth magnetotail current sheets are characterized by
+
+B, V
+(a)
+(b
+BI-nB?
+solar wind
+ln" u~
+Bnjm
+B,j
+n
+distant
+V,Bm
+B,=0
+B;=0
+B,j
+magnetotail
+- nB?
+B
+B, V
+10°
+1
+(c)
+Bnjm
+B;=0
+n
+1
+10-1
+B
+magnetosheath
+(d)
+B, V
+Bnjm
+near-Earth
+Vn Vnyi+vi Vivi
+B,=0
+n
+magnetotail
+1
+10-2
+10-1
+100
+101
+102
+B, V
+B
+14
+Figure 2. Four examples of force-free current sheets in (a) solar wind, (b) lunar-distance magnetotail,
+(c) Martian magnetotail, and (d) Jovian magnetotail. The top panels show the magnetic field in the local
+coordinate system Sonnerup & Cahill (1968). Grey curves show the magnetic field magnitudes; B ≈ constant
+indicates the force-free current sheets. Middle panels show electron and ion β profiles. Bottom panels show
+profiles of the field-aligned currents with an indication on the dominant ion species and estimations of the
+current sheet thickness in the ion inertial length. For the solar wind and Earth’s lunar-distance magnetotail,
+we used ARTEMIS magnetic field (Auster et al. 2008) and plasma (McFadden et al. 2008) measurements
+(see details of data processing procedure in Artemyev et al. 2019a). For the Martian magnetotail, we used
+MAVEN magnetic field (Connerney et al. 2015) and plasma (McFadden et al. 2015; Halekas et al. 2015)
+measurements (see details of data processing procedure in Artemyev et al. 2017). For the Jovian magnetotail,
+we used Juno magnetic field (Connerney et al. 2017a,b) and plasma (McComas et al. 2017; Kim et al. 2020)
+measurements (see details of data processing procedure in Artemyev et al. 2023).
+slightly higher β > 100 and are quantified by the plasma pressure contribution to the stress balance
+(Runov et al. 2006; Artemyev et al. 2011; Petrukovich et al. 2015). The important difference between
+solar wind/magnetosheath and magnetotail current sheets is their magnetic field line configuration;
+solar wind current sheets may be considered as 1D discontinuities, whereas the magnetotail current
+sheets are 2D [Figures 1(a-d)]. Therefore, current sheet models with Bn ̸= 0 and low MA should
+include 2D plasma pressure gradients balancing the magnetic field line tension force (see examples
+of such models in Schindler & Birn 2002; Birn et al. 2004; Yoon & Lui 2005; Sitnov & Merkin 2016).
+In the (β, MA) space, the large-β domain can be described by kinetic models of current sheets
+with Bn = 0 (1D models) and Bn ̸= 0 (2D models with plasma pressure gradients). Conversely, the
+low-β domain with dominant field-aligned currents (force-free current sheets) can be described only
+by 1D kinetic models with Bn = 0. Large MA, low-β (1D rotational discontinuities with Bn ̸= 0)
+and low MA, low-β (2D force-free current sheet) domains have been analyzed only by fluid models
+(e.g., Cowley 1978; Hilmer & Voigt 1987; Tassi et al. 2008; Lukin et al. 2018). The class of such
+2D force-free current sheets is not limited to observations in the solar wind and distant Earth’s
+
+B
+B
+B
+Bn
+βe~βi
+ot&0t
+15d
+5d
+~7d;5
+magnetotail, but also includes current sheets observed in the cold plasma of the Martian magnetotail
+(e.g., DiBraccio et al. 2015; Artemyev et al. 2017) and in the low-density Jovian magnetotail (e.g.,
+Behannon et al. 1981; Artemyev et al. 2014). Figures 2(c,d) show examples of force-free current
+sheets observed in the Martian and Jovian magnetotails. There are not enough known integrals of
+motion to describe distribution functions of charged particles in such current sheet configurations (see
+discussion in Lukin et al. 2022). Consequently, the absence of kinetic models of force-free current
+sheets with Bn ̸= 0 significantly hinders the analysis of their stability, the process responsible for the
+magnetic reconnection onset and particle acceleration.
+In this study, we develop a 2D kinetic force-free current sheet model. In Section 2, we describe
+a magnetohydrodynamic (MHD) model of 2D force-free current sheets. In Section 3, we initialize
+self-consistent, particle-in-cell (PIC) simulations using currents and magnetic fields from the MHD
+equilibrium. We load particle distributions using drifting Maxwellians, which initially do not nec-
+essarily satisfy the time-stationary Vlasov equation. In Section 4, we obtain kinetic equilibria by
+evolving these particle distribution functions using the self-consistent PIC simulations and demon-
+strate the existence of kinetic equilibria for 2D force-free current sheets. We describe the equilibrium
+distribution functions and the difference between ion- and electron-dominated 2D current sheets. In
+Section 5, we summarize the results and discuss their applications.
+2. MHD EQUILIBRIA OF FORCE-FREE CURRENT SHEETS
+We consider the general case of 2D force-free equilibrium in which all quantities vary with coor-
+dinates x and z only (i.e., ∂/∂y = 0), and, as is customary in this definition, the pressure gradient
+force contribution to the force balance is negligible, i.e., ∇P ∼ 0. The divergence-free magnetic field
+has the form (e.g., Low & Wolfson 1988)
+B =
+�
+−∂A
+∂z , By, ∂A
+∂x
+�
+,
+(1)
+where Aey is the vector potential in the y direction, and By is the magnetic field in the y direction.
+The current density is
+J = c
+4π∇ × B = c
+4π
+�
+−∂By
+∂z , −∇2A, ∂By
+∂x
+�
+,
+(2)
+where c is the speed of light, and ∇2 = ∂2/∂x2 + ∂2/∂z2 is the Laplacian. The force-free condition,
+J × B = 0, is equivalent to J = αB, where α = α(x, z) is a scalar function. Comparing Equations
+(1) and (2), the force-free condition requires that By is a function of A only, and that By satisfies
+∇2A + By(A)dBy
+dA = 0,
+(3)
+and the coefficient α is (c/4π)dBy/dA. Once By(A) is given, we can solve Equation (3) for A(x, z)
+with appropriate boundary conditions.
+As shown in Figure 3, for balancing force-free current sheets with low thermal and dynamic pressures
+(e.g., low β/low Mach number current sheets in the solar wind, magnetosheath, and lunar-distance
+magnetotail; see Figure 1), the magnetic pressure B2
+y plays an role analogous to the thermal pres-
+sure. Motivated by the functional form of pressure p ∝ exp [2A/(λB0)] describing thermal-pressure
+balanced current sheets (Lembege & Pellat 1982), we adopt
+By(A) = B0 exp
+� A
+λB0
+�
+,
+(4)
+
+6
+for force-free current sheets, so that Equation (3) describing magnetic field lines in the x-z plane
+(i.e., contours of A) resembles that of the Lembege-Pellat current sheet (e.g., Tassi et al. 2008; Lukin
+et al. 2018):
+∇2A = −B0
+λ exp
+� 2A
+λB0
+�
+,
+(5)
+where λ is the current sheet thickness, and B0 is the magnetic field at large |z|.
+JyBz/c = ∂p
+∂x
+JyBx/c = ∂p
+∂z
+JyBz/c = JzBy/c = ∂
+∂x
+B2
+y
+8π
+JyBx/c = JxBy/c = ∂
+∂z
+B2
+y
+8π
+Near-Earth magnetotail
+Mixed states
+Solar wind; Lunar-distance magnetotail
+Figure 3. Force balance of current sheets in different space environments. At one end of the spectrum, rep-
+resenting current sheets in the near-Earth magnetotail, magnetic tension force JyBz/c and magnetic pressure
+force JyBx/c ≈ ∂(B2
+x/8π)/∂z are balanced by thermal pressure gradients ∂p/∂x and ∂p/∂z, respectively. At
+the other end of the spectrum, representing current sheets in the solar wind and lunar-distance magnetotail,
+JyBz/c and JyBx/c are balanced by shears of B2
+y, i.e., ∂(B2
+y/8π)/∂x and ∂(B2
+y/8π)/∂z, respectively. There
+may be a continuous spectrum of current sheets between these two ends (Yoon et al. 2023), in which JyBz/c
+and JyBx/c are balanced by a mixture of thermal pressure gradients and shears of B2
+y.
+Because the scale length along current sheets is much larger than that across current sheets, we
+assume A = A(εx, z) to be weakly nonuniform in the x direction (ε being a small parameter).
+Equation (5) can be approximated as
+∂2A
+∂z2 = −B0
+λ exp
+� 2A
+λB0
+�
+,
+(6)
+which is accurate up to order ε. The boundary condition is
+∂A
+∂z
+����
+z=0
+= 0,
+A
+��
+z=0 = εB0x,
+(7)
+which is equivalent to Bx(z = 0) = 0 and Bz(z = 0) = εB0. The solution of A is
+A = −λB0 ln
+�
+cosh
+�
+z
+λF(x)
+�
+· F(x)
+�
+,
+(8)
+where F(x) = exp (−εx/λ). Thus the three components of the magnetic field are
+Bx = B0 tanh
+�
+z
+λF(x)
+�
+F −1(x),
+By = B0
+�
+cosh
+�
+z
+λF(x)
+�
+· F(x)
+�−1
+,
+Bz = εB0
+�
+1 −
+z
+λF(x) tanh
+�
+z
+λF(x)
+��
+,
+(9)
+
+7
+where Bz ̸= 0 describes rotational discontinuities. The current density is
+J = αB = cB
+4πλ
+�
+cosh
+�
+z
+λF(x)
+�
+· F(x)
+�−1
+.
+(10)
+The direction of magnetic field rotates from pointing in +x at the z > 0 half-space (z/λ ≫ 1) to
+pointing in +y around the neutral (equatorial) plane (−1 ≲ z/λ ≲ 1), and further to pointing in
+−x at the z < 0 half-space (z/λ ≪ −1). The magnitude of magnetic field, (B2
+x + B2
+y + B2
+z)1/2 =
+B0[F −1(x) + O(ε)], has a weak dependence on x and is roughly a constant at a given x location.
+The magnitude of current density, (J2
+x + J2
+y + J2
+z )1/2 ∝ cosh−1[z/(λF(x))], is concentrated in a layer
+|z| < λF(x).
+3. COMPUTATIONAL SETUP
+Searching for kinetic equilibria of 2D force-free current sheets, we initialize our simulated current
+sheets from the results of an MHD equilibrium. Importantly, this may not satisfy the time-stationary
+Vlasov equation because the initial particle distributions are simply drifting Maxwellians. We sim-
+ulate their relaxation toward a certain kinetic equilibrium (if any) based on a massively parallel
+PIC code (Pritchett 2001, 2005). Our simulations have two dimensions (x, z) in configuration space
+and three dimensions (vx, vy, vz) in velocity space. The results are presented in normalized units:
+magnetic fields are normalized to B0, lengths to the ion inertial length di = c/(4πn0e2/mi)1/2,
+time to the reciprocal of the ion gyrofrequency ω−1
+ci
+= mic/(eB0), velocities to the Alfv´en ve-
+locity vA = B0/(4πn0mi)1/2, electric fields to vAB0/c, and energies to miv2
+A.
+Here n0 is the
+plasma density, mi is the ion mass, and e is the elementary charge. The computational domain is
+[−64 ≤ x/di ≤ 0] × [−8 ≤ z/di ≤ 8] with a cell length ∆x = di/32. The time step is ∆t = 0.001 ω−1
+ci .
+The ion-to-electron mass ratio is mi/me = 100. The normalized speed of light is c/vA = 20, which
+gives the ratio of electron plasma frequency to electron gyrofrequency ωpe/ωce = 2. The reference
+density n0 is represented by 1929 particles. The total number of particles is 1.6 × 109 including ions
+and electrons in each simulation.
+Figure 4 shows the initial magnetic field configuration and current density, which are determined
+by Equations (9) and (10) using the input parameters ε = 0.04 and λ = 2di. There are a few degrees
+of freedom in choosing the initial particle distribution functions: (1) While the current density is
+known a priori, the proportion of ion and electron currents is left unspecified; (2) Electron and ion
+temperatures and plasma β vary across different space environments (e.g., Figure 2); (3) Specific
+forms of the velocity distribution functions are left undetermined. To this end, we choose initial
+electron and ion distribution functions as drifting Maxwellians
+fs(v, x, z) =
+n0
+(2πTs/ms)3/2 exp
+�
+−ms (v − vds(x, z))2
+2Ts
+�
+,
+(11)
+where the subscript s = i, e represents ions and electrons, vds(x, z) is the position-dependent drift
+velocity, and Ts is the temperature. Note that Ti, Te, and n0 are constants throughout the domain so
+that the initial plasma pressure n0(Ti +Te) is a constant, as required by the force-free condition. The
+simulations allow either ions or electrons to be the main current carrier, while keeping the relative drift
+between them commensurate with the current density [Figures 4(c), 4(d), and 4(e)]. Additionally, we
+vary particle temperatures to search for kinetic equilibrium in different plasma beta regimes, where
+
+8
+the plasma beta is β = 2(Ti + Te)/miv2
+A. The detailed parameters for particle distribution functions
+in the six runs are listed in Table 1. These parameters cover the β and Ti/Te ranges of force-free
+current sheets observed in the solar wind and planetary magnetotails (see Figure 2).
+Figure 4. MHD equilibrium of a force-free current sheet. (a) Magnetic field in the x direction Bx. (b)
+Magnetic field in the y direction By. (c) Current density in the x direction Jx. (d) Current density in the y
+direction Jy. (e) Current density in the parallel direction J∥.
+In our simulations, the electromagnetic fields are advanced in time by integrating Maxwell’s equa-
+tions using a leapfrog scheme. These fields are stored on the Yee grid. The relativistic equations of
+motion for ions and electrons are integrated in time using a leapfrog scheme with a standard Boris
+push for velocity update. The conservation of charge is ensured by applying a Poisson correction to
+the electric fields (Marder 1987; Langdon 1992).
+For particles crossing the x boundaries, we take advantage of the symmetry between z > 0 and
+z < 0 [Figure 4], so that particles exiting the system at a location z with velocity (vx, vy, vz) are
+reinjected into the system at the conjugate location −z with velocity (−vx, vy, vz). This is equivalent
+to an open boundary condition for particles because the injected particle distribution matches that
+at one cell interior to the boundary at all simulation times. At the z boundaries, particles striking
+the boundary are reflected into the system with vz = −vz.
+
+(a)
+0.8
+5
+0.4
+B
+0
+0.0
+-5
+-0.8
+(q)
+0.8
+5
+0.4
+0
+0.0
+-0.8
+(c)
+0.2
+5
+(Vaou)/
+0.1
+0
+0.0
+-0.1
+-0.2
+(p)
+0.4
+5
+0.2
+0
+0.0
+-0.2
+(e)
+0.4
+5
+0.2
+0
+0.0
+7
+-0.2
+-5
+-0.4
+0
+-20
+-40
+-60
+x/di9
+Run NO.
+vdi
+vde
+βi = 2Ti/miv2
+A
+βe = 2Te/miv2
+A
+β = βi + βe
+1A
+J/(n0e)
+0
+0.05
+0.05
+0.1
+1B
+0
+−J/(n0e)
+0.05
+0.05
+0.1
+2A
+J/(n0e)
+0
+5/6
+1/6
+1
+2B
+0
+−J/(n0e)
+5/6
+1/6
+1
+3A
+J/(n0e)
+0
+25/3
+5/3
+10
+3B
+0
+−J/(n0e)
+25/3
+5/3
+10
+Table 1. Parameters for particle distribution functions in the six PIC runs. The first digit in Run NO.,
+‘1’, ‘2’, ‘3’, denotes three plasma betas, β = 0.1, 1, 10, respectively. The second digit in Run NO., ‘A’, ‘B’,
+denotes current carriers as ions and electrons, respectively. The current density J comes from Equation (10),
+and is visualized in Figures 4(c) and 4(d).
+For fields at the x boundaries, the guard values of the tangential magnetic fields are determined by
+δBn
+y,g = δBn
+y,i1,
+δBn
+z,g = δBn−1
+z,i1 (2 − ∆x/c∆t) + δBn−2
+z,i2 (∆x/c∆t − 1),
+(12)
+where the superscript indicates the time level, and the subscripts g, i1, i2 indicate the guard point,
+first interior point and second interior point, respectively. This boundary condition for δBz ensures
+that the magnetic flux can freely cross the x boundaries (Pritchett 2005). The guard values of the
+normal electric field δEx at the x boundaries are determined by δEn
+x,g = δEn
+x,i1. Similarly, at the z
+boundaries, the two components of the tangential magnetic fields in the guard point are determined
+by δBn
+x,g = δBn
+x,i1 and δBn
+y,g = δBn
+y,i1; The normal electric field δEz in the guard point are determined
+by δEn
+z,g = δEn
+z,i1. Unlike the driven simulations that adds magnetic flux to the simulation box, no
+external Ey is applied at the z boundaries.
+4. RESULTS
+We track the evolution of the initially force-free current sheets in the six runs up to t = 180 ω−1
+ci ,
+at which time the macroscopic states (e.g., electric and magnetic fields, currents, pressures) of the
+system are quasi-stationary. Below we examine if the configurations satisfy J × B = 0 at the end
+of the simulations. We further investigate how the particle velocity distributions deviate from the
+initial Maxwellians and whether or not the system reaches a kinetic equilibrium.
+4.1. Initially ion-dominated current sheets
+When ions initially carry entirely the (field-aligned) currents (Runs 1A, 2A, and 3A), electrons are
+accelerated by transient parallel electric fields to form field-aligned currents [e.g., Figure 5(b)], while
+ions are slightly decelerated by such electric fields [e.g., comparing Figures 5(a) and 4(e)]. These
+electron currents can be a significant fraction (e.g., ≳ 1/3) of ion currents in the off-equatorial region
+(1 < |z| < 5). Almost all currents remain field aligned [see J⊥ ∼ 0 in Figure 5(c)], implying that a
+force-free configuration may indeed be obtained.
+During the relaxation of ion-dominated current sheets, electrostatic fields are generated and remain
+present in the late equilibrium states. These electrostatic fields are perpendicular to the magnetic
+field: At |z| > 0, they points away from the equatorial plane [Figure 6(a)]; Near the equator, they
+points in the −x direction (i.e., tailward) [Figure 6(b)]. Here the Ex component is much weaker than
+the Ez component. The electrostatic fields arise due to the decoupling of the unmagnetized ions
+
+10
+Figure 5. Current density for the ion-dominated force-free current sheet with plasma beta β = 1 in Run
+2A. The snapshot is taken at t = 180 ω−1
+ci
+in the simulation. (a) Ion field-aligned currents. (b) Electron
+field-aligned currents. (c) Total perpendicular currents.
+and magnetized electrons (Schindler et al. 2012), which can be derived from the ordering among ion
+thermal gyroradius, current sheet thickness, and electron thermal gyroradius
+ρi : λ : ρe = di
+�
+βi
+2 : 2di : di
+�
+βe
+2
+me
+mi
+=
+�
+βi
+8 : 1 :
+�
+βe
+8
+me
+mi
+.
+(13)
+Taking Run 2A for example, we have ρi : λ : ρe = 0.32 : 1 : 0.01. As the plasma beta is lowered, the
+electrostatic field decreases – due to stronger magnetization of ions, which is shown in Figures 7 and
+8 below.
+To show the detailed force balance of these current sheet configurations, we decompose the forces
+in parallel and perpendicular directions with respect to local magnetic fields. The force balance in
+the parallel direction is rapidly established because particles move freely along field lines. The unit
+vectors in the two perpendicular directions are defined as e⊥1 = ˆz × ˆb and e⊥2 = ˆb × (ˆz × ˆb), where
+ˆb and ˆz are the unit vectors along the magnetic field and z direction, respectively. It is worth noting
+that e⊥1 is roughly along the ±y direction far above/below the equator, and along the −x direction
+at the equator, while e⊥2 is roughly along the +z direction all over the domain.
+At the equator, the electrostatic fields, E⊥1 (i.e., along the −x direction), cause a drift cE⊥1×b/|B|
+of both ions and electrons [Figures 7(b-c), 7(h-i), and 7(n-o)], which does not produce net currents
+[Figures 7(a), 7(g), and 7(m)]. In addition, the pressure gradient (∇ · P)⊥1 is about zero in the
+two cases, with β = 0.1 [Run 1A; see Figure 7(a)] and β = 1 [Run 2A; see Figure 7(g)]. Thus, we
+obtain (J × B)⊥1/c = (∇ · P)⊥1 = 0 in the e⊥1 direction. In the case with β = 10, however, the
+current sheet gets compressed and rarefied periodically in the x direction. Correspondingly, (∇·P)⊥1
+oscillates around zero and acts as a restoring force [Run 3A; see Figure 7(m)]. Such oscillations are
+
+5
+(a)
+0.2
+J ill
+0
+-5
+0.1
+5
+(q)
+J(enovA)
+1p/2
+Jell
+0
+0.0
+-5
+5
+(c)
+F-0.1
+1p/2
+J1
+0
+-0.2
+-5
+-20
+-25
+-30
+-35
+-40
+-45
+x/di11
+Figure 6. Electrostatic field for the ion-dominated force-free current sheet with plasma beta β = 1 in Run
+2A. The snapshot is taken at t = 180 ω−1
+ci
+in the simulation. (a) Electric field in the z direction Ez. (b)
+Electric field in the x direction Ex. The magnitude of Ex is much smaller than that of Ez.
+still localized around the equilibrium (J×B)⊥1/c = (∇·P)⊥1 = 0 when performing a spatial average
+over x.
+Along the e⊥2 direction (i.e., roughly the z direction), both the pressure gradient (∇ · P)⊥2 and
+Lorentz force (J×B)⊥2/c vanish [Figures 8(a), 8(g), 8(m)]. The dominant component of the electro-
+static fields, E⊥2 (along the ±z directions), leads to a drift cE⊥2×b/|B| (roughly in the +y direction)
+of both ions and electrons [Figures 8(b-c), 8(h-i), and 8(n-o)], which does not give net currents. All
+aforementioned perpendicular drift velocities are small, when compared to ion and electron parallel
+flow velocities that carry currents [e.g., Figure 5].
+4.2. Initially electron-dominated current sheets
+When electrons initially carry entirely the (field-aligned) currents (Runs 1B, 2B, and 3B), there is no
+change in the macroscopic states of the system at the end of simulations [e.g., Figure 9]: The currents
+remain field aligned, and are carried by electrons only. Compared to the ion-dominated current sheets,
+the relaxation of electron-dominated current sheets does not alter the macroscopic states: neither
+the proportion of electron and ion currents is changed, nor do the electrostatic fields develop. In
+all three cases with different plasma betas, the system remains in a force-free configuration, with no
+plasma pressure gradients [Figures 7(d-f), 7(j-l), 7(p-r), 8(d-f), 8(j-l), 8(p-r)]. Despite the system
+being identical to the initial ones from the fluid point of view, the underlying velocity distribution
+functions evolve substantially from the initial Maxwellians to reach the Vlasov equilibrium, shown
+below.
+4.3. Reaching kinetic equilibrium or not?
+Since the force-free equilibria are reestablished toward the end of simulations for both ion- and
+electron-dominated current sheets from the fluid point of view, a logical next step would be to
+further examine if such current sheets reach the kinetic equilibrium. Because all the currents are
+field aligned, we integrate the full distribution function fs(x, z, v∥, v⊥, φ, t) of species s to obtain the
+
+0.06
+(a)
+5
+0.04
+()/
+0.02
+0
+0.00
+-0.02
+-0.04
+-5
+-0.06
+(b)
+5
+0.004
+cEx/(vABo)
+0.002
+0
+0.000
+-0.002
+-0.004
+-5
+-20
+-25
+-30
+-35
+-40
+-45
+x/di12
+Figure 7. Force balance in the e⊥1 = ˆz × ˆb direction at the equator for different plasma betas. Note e⊥2 is
+along the −x direction at the equator. All the snapshots are taken at t = 180 ω−1
+ci
+in the simulations. The
+volume forces are averaged in the range −1 ≤ z/di ≤ 1. The rows from top to bottom represent Runs 1A,
+1B, 2A, 2B, 3A, and 3B, respectively. The columns from left to right represent the force balance from the
+point view of single fluid, ion fluid, and electron fluid, respectively.
+reduced distribution gs(x, v∥, t) at the equator as a function of x, v∥, and t:
+gs(x, v∥, t) =
+� di
+−di
+dz
+� ∞
+0
+dv⊥
+� 2π
+0
+dφ fs(x, z, v∥, v⊥, φ, t),
+(14)
+where v∥ and v⊥ are the parallel and perpendicular velocities, respectively, and φ is the gyrophase.
+Figure 10 shows the difference between the reduced distributions at the end of simulations and the
+initial Maxwellians, ∆gs = gs(x, v∥, t = 180 ω−1
+ci ) − gs(x, v∥, t = 0), from Run 2A as an example. The
+
+(J×B)11/c
+-(V.P)11
+ngE11
+SUM
+All
+Ions
+Electrons
+0.01
+0.00
+-0.01
+β= 0.1
+0.01
+0.00
+-0.01
+0.01
+0.00
+-0.01
+=
+0.01
+B
+0.00
+-0.01
+0.01
+0.00
+-0.01
+β=10
+0.01
+0.00
+-0.01
+-20
+-40
+-20
+-40
+-20
+-40
+x/di
+x/di
+x/di13
+Figure 8.
+Force balance in the e⊥2 = ˆb × (ˆz × ˆb) direction for different plasma betas.
+Note e⊥2 is
+approximately along the +z direction. All the snapshots are taken at t = 180 ω−1
+ci
+in the simulations. The
+volume forces are averaged in the range −16 ≤ x/di ≤ −12. The format is the same as Figure 7.
+ion phase space density is transported toward velocities both above and below the mean ion flow
+velocity [Figure 10(a)]. Consequently, the ion velocity distribution becomes wider and less peaked,
+compared to the initial Maxwellian [Figure 10(b)]. The mean ion flow velocity is slowed more on
+the earthward side than the tailward side [see the inset of Figure 10(a)]. Correspondingly, electrons
+are accelerated on average in the v∥ < 0 direction to develop an electron current to compensate the
+reduction of the ion current [see the inset of Figure 10(c)]. Such changes of mean flow velocities are
+not apparent in the phase space density plot because the thermal velocities are much larger than
+
+(J ×B) 12/c
+-(V.P) 12
+ngE 12
+SUM
+All
+Ions
+Electrons
+0.05
+0.00
+-0.05
+β= 0.1
+0.05
+0.00
+-0.05
+0.05
+0.00
+-0.05
+0.05
+0.00
+-0.05
+0.05
+0.00
+-0.05
+β= 10
+0.05
+0.00
+-0.05
+-5
+0
+5
+0
+5
+-5
+0
+5
+zldi
+zldi
+zldi14
+Figure 9. Current density for the electron-dominated force-free current sheet with plasma beta β = 1 in
+Run 2B. The snapshot is taken at t = 180 ω−1
+ci in the simulation. (a) Ion field-aligned currents. (b) Electron
+field-aligned currents. (c) Total perpendicular currents.
+the mean flow velocities. The peak of the electron phase space density is transported toward larger
+velocities in both v∥ > 0 and v∥ < 0 [Figures 10(c) and 10(d)].
+The macroscopic states of electron-dominated current sheets do not differ from their initial config-
+urations, as seen in Figure 9. The electron reduced distribution function ge(x, v∥), however, shows
+systematic deviations from the initial Maxwellians, as displayed in Figure 11. Such deviations are
+similar to those in ion-dominated current sheets. The peak of the electron velocity distributions is
+reduced and redistributed toward larger velocities of v∥ > 0 and v∥ < 0 (but without changes in mean
+electron flow velocities) [Figures 11(c) and 11(d)]. The deviation of the ion velocity distribution from
+the initial Maxwellian is small [Figure 11(b)], but is required to reach the Vlasov equilibrium.
+To quantify the convergence of the system toward kinetic equilibrium, we define the metric
+Gs(t) =
+� ∞
+−∞
+dv∥
+��
+gs(x, v∥, t) − gs(x, v∥, 0)
+�2�
+x ,
+(15)
+where ⟨·⟩x denotes the average over spatial coordinate x. This metric is chosen in such a way that
+the non-Maxwellian features of gs(x, v∥, t) are properly accounted for. During the evolution of the
+system, it is expected that Gs(t) experiences significant changes at the beginning of the simulation
+and eventually reaches a steady state, if a kinetic equilibrium is established. Figure 12 shows the
+absolute values of the time derivative |∂Gs/∂t| for both ion- and electron-dominated current sheets
+with different plasma betas. All runs end up with |∂Gs/∂t| ≲ 0.01, except for Run 1A (i.e., the
+ion-dominated current sheet with β = 0.1). For the same plasma beta, electron-dominated current
+sheets reach the kinetic equilibrium faster than ion-dominated current sheets. In the latter case,
+the system simply takes more time to redistribute the currents between unmagnetized ions and
+
+5
+(a)
+0.2
+1p/2
+Jill
+0
+-5
+0.1
+(b)
+5
+JI(enovA)
+Jell
+0
+0.0
+-5
+(c)
+--0.1
+5
+JI
+0
+F-0.2
+-5
+-20
+-25
+-30
+-35
+-40
+-45
+x/di15
+Figure 10. Phase space distributions for the ion-dominated force-free current sheet with plasma beta β = 1
+in Run 2A. (a) The difference between the reduced ion distribution and the initial Maxwellian ∆gi. The
+inset plot shows the mean ion flow velocity as a function of x at t = 0 (gray line) and t = 180 ω−1
+ci
+(black
+line). (b) A cut of the final ion distribution gi(x, v∥, t = 180 ω−1
+ci ) (black line), gi(x, v∥, t = 0) (gray line),
+and their difference ∆gi multiplied by 5, for better visibility (red line) between −16 ≤ x/di ≤ −12. (c) The
+different between the reduced electron distribution and the initial Maxwellian ∆ge. The inset plot shows
+the mean electron flow velocity as a function of x at t = 0 (gray line) and t = 180 ω−1
+ci
+(black line). (d) A
+cut of the final electron distribution ge(x, v∥, t = 180 ω−1
+ci ) (black line), ge(x, v∥, t = 0) (gray line), and their
+difference ∆ge multiplied by 5, for better visibility (red line) between −16 ≤ x/di ≤ −12.
+magnetized electrons. For both types (ion- and electron-dominated) of current sheets, those with
+higher plasma β evolve toward kinetic equilibrium faster, because the current sheets with higher
+plasma β can support larger transient electric fields to redistribute particles more rapidly in phase
+space toward the equilibrium.
+At present, we cannot simulate further in time the relaxation of the low-β, ion-dominated current
+sheet in Run 1A toward the final kinetic equilibrium, because the current sheet becomes unstable at
+a later time (most likely due to boundary conditions). However, the progression of the relaxation for
+different beta values and the behavior of Run 1A up to t · ωci ∼ 10 suggests that it also conforms to
+the explanatory model presented herein. The distribution functions at that time provide a sufficiently
+good representation of the Vlasov equilibrium to use in future modeling.
+5. CONCLUSION AND DISCUSSION
+In summary, we demonstrate that kinetic equilibria of 2D force-free current sheets exist for different
+proportions of ion and electron currents in various plasma betas. When initial currents are carried
+purely by ions, field-aligned electron currents are developed by transient parallel electric fields, while
+
+X10-1
+5.0
+1.0
+<v>
+(a)
+(b)
+gi
+1.0
+(0=1)<)
+2.5
+gi(t = 0)
+△gil(no/vA)
+0.5
+0.5
+5△gi
+0.0
+0.0
+-0.5
+0.0
+0.25
+-2.5 
+0.00
+-1.0
+-20
+-40
+-5.0
+-0.5
+-2.5 0.0
+2.5
+X10-2
+20
+0.2
+<v >
+(c)
+(d)
+ge
+2
+(0=↓)<Il )
+10
+ge(t = 0)
+Agel(no/vA)
+1
+0.1
+5△ge
+Va/ II
+0
+0
+0.0
+0.0
+-10 
+-0.1
+-2
+-20
+-40
+-20
+-0.1
+-20
+-40
+-10
+0
+10
+x/di
+VIl /VA16
+Figure 11. Phase space distributions for the electron-dominated force-free current sheet with plasma beta
+β = 1 in Run 2B. The format is the same as Figure 10.
+perpendicular electrostatic fields are generated due to unmagnetized ions and magnetized electrons.
+When initial currents are carried exclusively by electrons, the macroscopic state of the system remains
+unchanged from its initial state, described by the MHD model. In both scenarios, the electron and
+ion distribution functions at the late equilibrium states show systematic deviations from the initial
+drifting Maxwellians. These deviations occur in order to satisfy the time-stationary Vlasov equation.
+The existence of a kinetic equilibrium for 2D force-free current sheets indicates that there is at
+least one hidden symmetry in the system, implying the existence of an additional integral of motion
+(i.e., an additional invariant). Our system has five dimensions ND = 5 in the phase space (2D in
+the coordinate space and 3D in the velocity space). The two known invariants are the total energy
+H, and the y component of the canonical momentum Py = mvy + eA/c. The degrees of freedom
+of the system are Nf = ND − NI, where NI is the number of invariants. The particle phase space
+densities at equilibrium are constructed as a function of such invariants of motion. To fully describe
+such kinetic equilibrium, which we now know it does exist, we must have NI > Nf, or equivalently,
+NI > ND/2. Thus, NI is at least 3 (in our case) and there is at least one hidden symmetry. Starting
+from particle trajectory data in the equilibrium electromagnetic fields, future research on this topic
+could make use of machine learning models to help find the number of invariants or even discover
+the analytic formula of these invariants (e.g., Liu & Tegmark 2021; Liu et al. 2022).
+6. ACKNOWLEDGMENTS
+. This work was supported by NASA grants 80NSSC20K1788, 80NSSC22K0752, and NAS5-02099.
+We acknowledge high-performance computing support from Cheyenne (doi:10.5065/D6RX99HX) pro-
+
+X10-2
+5.0
+0.75
+<v=>
+(a)
+(b)
+6
+gi
+<vμ>(t =0)
+2.5
+4
+0.50
+gi(t = 0)
+2
+50△gi
+Va/ ll/1
+0.0
+0.25 
+0.01
+2
+-2.5 
+0.00
+0.00
+-0.01
+-6
+-20
+-40
+-5.0
+-0.25
+-2.5 0.0
+2.5
+X10-2
+20
+0.2
+<v>
+3
+(c)
+(d)
+ge
+(0=↓)<Il )
+2
+10
+ge(t = 0)
+(Vayou)/98
+0.1 
+1
+5△ge
+Va/ II
+0
+0
+0.00
+0.0
+-10 
+-0.25
+2
+-20
+-40
+-20
+-0.1
+-20
+-40
+-10
+0
+10
+x/di
+VI /VA17
+Figure 12. Evolution of force-free current sheets toward kinetic equilibrium with different plasma betas.
+The x-axis has two scales, a linear scale in 0 ≤ tω−1
+ci ≤ 10 and a log scale in 10 ≤ tω−1
+ci ≤ 180. The y axis
+denotes |∂G/∂t|. The rows from top to bottom represent Runs 1A, 1B, 2A, 2B, 3A, and 3B, respectively.
+The grey line |∂G/∂t| = 0.01 in each panel is drawn as a reference.
+vided by NCAR’s Computational and Information Systems Laboratory, sponsored by the National
+Science Foundation (Computational and Information Systems Laboratory 2019).
+REFERENCES
+Achilleos, N. 2018, in Electric Currents in
+Geospace and Beyond, ed. A. Keiling,
+O. Marghitu, & M. Wheatland, Vol. 235,
+127–138, doi: 10.1002/9781119324522.ch8
+Allanson, O., Neukirch, T., Troscheit, S., &
+Wilson, F. 2016, Journal of Plasma Physics, 82,
+905820306, doi: 10.1017/S0022377816000519
+Allanson, O., Neukirch, T., Wilson, F., &
+Troscheit, S. 2015, Physics of Plasmas, 22,
+102116, doi: 10.1063/1.4934611
+Angelopoulos, V. 2008, SSRv, 141, 5,
+doi: 10.1007/s11214-008-9336-1
+—. 2011, SSRv, 165, 3,
+doi: 10.1007/s11214-010-9687-2
+Artemyev, A. V. 2011, Physics of Plasmas, 18,
+022104, doi: 10.1063/1.3552141
+Artemyev, A. V., Angelopoulos, V., Halekas, J. S.,
+et al. 2017, J. Geophys. Res., 122, 5404,
+doi: 10.1002/2017JA024078
+Artemyev, A. V., Angelopoulos, V., & McTiernan,
+J. M. 2018, Journal of Geophysical Research
+(Space Physics), 123, 9955,
+doi: 10.1029/2018JA025904
+Artemyev, A. V., Angelopoulos, V., & Runov, A.
+2016, J. Geophys. Res., 121, 4017,
+doi: 10.1002/2016JA022480
+Artemyev, A. V., Angelopoulos, V., & Vasko, I. Y.
+2019a, Journal of Geophysical Research (Space
+Physics), 124, 3858, doi: 10.1029/2019JA026597
+
+Ions
+Electrons
+100
+(a)
+G
+10-1
+M
+β= 0.1
+100
+(b)
+G
+10-
+100
+(c)
+10
+B
+100
+(d)
+10-
+10-2
+100
+(e)
+G
+10-
+10
+=
+100
+(f)
+10
+10
+0
+5
+10
+100
+toci18
+Artemyev, A. V., Angelopoulos, V., Vasko, I.-Y.,
+et al. 2019b, Geophys. Res. Lett., 46,
+1185–1194, doi: 10.1029/2018GL079906
+Artemyev, A. V., Ma, Q., Ebert, R. W., Zhang,
+X. J., & Allegrini, F. 2023, Force-free current
+sheets in the Jovian magnetodisk: the key role
+of electron field-aligned anisotropy, arXiv,
+doi: 10.48550/ARXIV.2301.03731
+Artemyev, A. V., Petrukovich, A. A., Nakamura,
+R., & Zelenyi, L. M. 2011, J. Geophys. Res.,
+116, A0923, doi: 10.1029/2011JA016801
+Artemyev, A. V., Vasko, I. Y., & Kasahara, S.
+2014, Planet. Space Sci., 96, 133,
+doi: 10.1016/j.pss.2014.03.012
+Auster, H. U., Glassmeier, K. H., Magnes, W.,
+et al. 2008, SSRv, 141, 235,
+doi: 10.1007/s11214-008-9365-9
+Behannon, K. W., Burlaga, L. F., & Ness, N. F.
+1981, J. Geophys. Res., 86, 8385,
+doi: 10.1029/JA086iA10p08385
+Birn, J. 1992, J. Geophys. Res., 97, 16817,
+doi: 10.1029/92JA01527
+Birn, J., & Priest, E. R. 2007, Reconnection of
+magnetic fields : magnetohydrodynamics and
+collisionless theory and observations, ed. Birn,
+J. & Priest, E. R.
+Birn, J., Schindler, K., & Hesse, M. 2004, J.
+Geophys. Res., 109, 2217,
+doi: 10.1029/2003JA010303
+Borovsky, J. E. 2010, Physical Review Letters,
+105, 111102,
+doi: 10.1103/PhysRevLett.105.111102
+Burkhart, G. R., Drake, J. F., Dusenbery, P. B.,
+& Speiser, T. W. 1992, J. Geophys. Res., 97,
+13799, doi: 10.1029/92JA00495
+Chaston, C. C., Bonnell, J. W., Wygant, J. R.,
+Reeves, G. D., & Baker, D. N. 2020,
+Geophys. Res. Lett., 47, e86318,
+doi: 10.1029/2019GL086318
+Cicogna, G., & Pegoraro, F. 2015, Physics of
+Plasmas, 22, 022520, doi: 10.1063/1.4913580
+Computational and Information Systems
+Laboratory. 2019, Cheyenne: HPE/SGI ICE XA
+System (University Community Computing),
+Boulder, CO: National Center for Atmospheric
+Research. https://doi.org/10.5065/D6RX99HX
+Connerney, J. E. P., Espley, J., Lawton, P., et al.
+2015, SSRv, 195, 257,
+doi: 10.1007/s11214-015-0169-4
+Connerney, J. E. P., Benn, M., Bjarno, J. B.,
+et al. 2017a, SSRv, 213, 39,
+doi: 10.1007/s11214-017-0334-z
+Connerney, J. E. P., Adriani, A., Allegrini, F.,
+et al. 2017b, Science, 356, 826,
+doi: 10.1126/science.aam5928
+Cowley, S. W. H. 1978, Planetary and Space
+Science, 26, 1037,
+doi: 10.1016/0032-0633(78)90028-4
+Cravens, T. E., & Gombosi, T. I. 2004, Advances
+in Space Research, 33, 1968,
+doi: 10.1016/j.asr.2003.07.053
+de Keyser, J., Roth, M., Lemaire, J., et al. 1996,
+SoPh, 166, 415, doi: 10.1007/BF00149407
+de Keyser, J., Roth, M., Tsurutani, B. T., Ho,
+C. M., & Phillips, J. L. 1997, A&A, 321, 945
+De Keyser, J., Dunlop, M. W., Owen, C. J., et al.
+2005, SSRv, 118, 231,
+doi: 10.1007/s11214-005-3834-1
+DiBraccio, G. A., Espley, J. R., Gruesbeck, J. R.,
+et al. 2015, Geophys. Res. Lett., 42, 8828,
+doi: 10.1002/2015GL065248
+Fleishman, G. D., & Pevtsov, A. A. 2018, in
+Electric Currents in Geospace and Beyond, ed.
+A. Keiling, O. Marghitu, & M. Wheatland, Vol.
+235, 43–65, doi: 10.1002/9781119324522.ch3
+Gonzalez, W., & Parker, E. 2016, Magnetic
+Reconnection, Vol. 427,
+doi: 10.1007/978-3-319-26432-5
+Haaland, S., Sonnerup, B., & Paschmann, G.
+2012, Annales Geophysicae, 30, 867,
+doi: 10.5194/angeo-30-867-2012
+Halekas, J. S., Taylor, E. R., Dalton, G., et al.
+2015, SSRv, 195, 125,
+doi: 10.1007/s11214-013-0029-z
+Harrison, M. G., & Neukirch, T. 2009, Physical
+Review Letters, 102, 135003,
+doi: 10.1103/PhysRevLett.102.135003
+Hilmer, R. V., & Voigt, G. 1987, J. Geophys. Res.,
+92, 8660, doi: 10.1029/JA092iA08p08660
+Hudson, P. D. 1970, Planet. Space Sci., 18, 1611,
+doi: 10.1016/0032-0633(70)90036-X
+Jackman, C. M., Arridge, C. S., Andr´e, N., et al.
+2014, SSRv, 182, 85,
+doi: 10.1007/s11214-014-0060-8
+Kim, T. K., Ebert, R. W., Valek, P. W., et al.
+2020, Journal of Geophysical Research (Space
+Physics), 125, e27696,
+doi: 10.1029/2019JA027696
+
+19
+King, J. H., & Papitashvili, N. E. 2005,
+J. Geophys. Res., 110, A02104,
+doi: 10.1029/2004JA010649
+Langdon, A. B. 1992, Computer Physics
+Communications, 70, 447
+Lembege, B., & Pellat, R. 1982, Physics of Fluids,
+25, 1995, doi: 10.1063/1.863677
+Liu, Z., Madhavan, V., & Tegmark, M. 2022,
+arXiv preprint arXiv:2203.12610
+Liu, Z., & Tegmark, M. 2021, Physical Review
+Letters, 126, 180604
+Low, B., & Wolfson, R. 1988, The Astrophysical
+Journal, 324, 574
+Lui, A. T. Y. 2018, in Washington DC American
+Geophysical Union Geophysical Monograph
+Series, Vol. 235, Electric Currents in Geospace
+and Beyond, ed. A. Keiling, O. Marghitu, &
+M. Wheatland, 155–175,
+doi: 10.1002/9781119324522.ch10
+Lukin, A., Artemyev, A., Vainchtein, D., &
+Petrukovich, A. 2022, Regimes of charged
+particle dynamics in current sheets: the machine
+learning approach.
+https://arxiv.org/abs/2211.03787
+Lukin, A. S., Vasko, I., Artemyev, A., & Yushkov,
+E. 2018, Physics of Plasmas, 25, 012906,
+doi: 10.1063/1.5016178
+Marder, B. 1987, Journal of Computational
+Physics, 68, 48
+McComas, D. J., Alexander, N., Allegrini, F.,
+et al. 2017, SSRv, 213, 547,
+doi: 10.1007/s11214-013-9990-9
+McFadden, J. P., Carlson, C. W., Larson, D.,
+et al. 2008, SSRv, 141, 277,
+doi: 10.1007/s11214-008-9440-2
+McFadden, J. P., Kortmann, O., Curtis, D., et al.
+2015, SSRv, 195, 199,
+doi: 10.1007/s11214-015-0175-6
+Mingalev, O. V., Mingalev, I. V., Malova, K. V.,
+& Zelenyi, L. M. 2007, Plasma Physics Reports,
+33, 942, doi: 10.1134/S1063780X07110062
+Mingalev, O. V., Mingalev, I. V., Mel’nik, M. N.,
+et al. 2012, Plasma Physics Reports, 38, 300,
+doi: 10.1134/S1063780X12030063
+Neugebauer, M. 2006, J. Geophys. Res., 111,
+A04103, doi: 10.1029/2005JA011497
+Neukirch, T., Vasko, I. Y., Artemyev, A. V., &
+Allanson, O. 2020a, ApJ, 891, 86,
+doi: 10.3847/1538-4357/ab7234
+Neukirch, T., Wilson, F., & Allanson, O. 2020b,
+Journal of Plasma Physics, 86, 825860302,
+doi: 10.1017/S0022377820000604
+Nickeler, D. H., & Wiegelmann, T. 2010, Annales
+Geophysicae, 28, 1523,
+doi: 10.5194/angeo-28-1523-2010
+Parker, E. N. 1994, Spontaneous current sheets in
+magnetic fields: with applications to stellar
+x-rays. International Series in Astronomy and
+Astrophysics, Vol. 1. New York : Oxford
+University Press, 1994., 1
+Paschmann, G., Haaland, S., Sonnerup, B., &
+Knetter, T. 2013, Annales Geophysicae, 31, 871,
+doi: 10.5194/angeo-31-871-2013
+Petrukovich, A. A., Artemyev, A. V., Vasko, I. Y.,
+Nakamura, R., & Zelenyi, L. M. 2015, SSRv,
+188, 311, doi: 10.1007/s11214-014-0126-7
+Pritchett, P. 2005, Journal of Geophysical
+Research: Space Physics, 110
+Pritchett, P. L. 2001, J. Geophys. Res., 106,
+25961, doi: 10.1029/2001JA000016
+—. 2005, J. Geophys. Res., 110, 5209,
+doi: 10.1029/2004JA010948
+Runov, A., Sergeev, V. A., Nakamura, R., et al.
+2006, Annales Geophysicae, 24, 247
+Schindler, K. 2006, Physics of Space Plasma
+Activity, ed. Schindler, K. (Cambridge
+University Press), doi: 10.2277/0521858976
+Schindler, K., & Birn, J. 2002, J. Geophys. Res.,
+107, 1193, doi: 10.1029/2001JA000304
+Schindler, K., Birn, J., & Hesse, M. 2012, Physics
+of Plasmas, 19, 082904, doi: 10.1063/1.4747162
+Servidio, S., Dmitruk, P., Greco, A., et al. 2011,
+Nonlinear Processes in Geophysics, 18, 675,
+doi: 10.5194/npg-18-675-2011
+Sitnov, M. I., & Merkin, V. G. 2016,
+J. Geophys. Res., 121, 7664,
+doi: 10.1002/2016JA023001
+Sitnov, M. I., Swisdak, M., Guzdar, P. N., &
+Runov, A. 2006, J. Geophys. Res., 111, 8204,
+doi: 10.1029/2005JA011517
+Sitnov, M. I., Zelenyi, L. M., Malova, H. V., &
+Sharma, A. S. 2000, J. Geophys. Res., 105,
+13029, doi: 10.1029/1999JA000431
+Sonnerup, B. U. ¨O., & Cahill, Jr., L. J. 1968,
+J. Geophys. Res., 73, 1757,
+doi: 10.1029/JA073i005p01757
+Syrovatskii, S. I. 1981, Annual review of
+astronomy and astrophysics, 19, 163,
+doi: 10.1146/annurev.aa.19.090181.001115
+
+20
+Tassi, E., Pegoraro, F., & Cicogna, G. 2008,
+Physics of Plasmas, 15, 092113,
+doi: 10.1063/1.2988338
+Vasko, I. Y., Alimov, K., Phan, T., et al. 2022,
+ApJL, 926, L19, doi: 10.3847/2041-8213/ac4fc4
+Vasko, I. Y., Artemyev, A. V., Petrukovich, A. A.,
+& Malova, H. V. 2014, Annales Geophysicae,
+32, 1349, doi: 10.5194/angeo-32-1349-2014
+Vasko, I. Y., Petrukovich, A. A., Artemyev, A. V.,
+Nakamura, R., & Zelenyi, L. M. 2015,
+J. Geophys. Res., 120, 8663,
+doi: 10.1002/2015JA021633
+Volwerk, M. 2018, in Electric Currents in
+Geospace and Beyond, ed. A. Keiling,
+O. Marghitu, & M. Wheatland, Vol. 235,
+513–533, doi: 10.1002/9781119324522.ch30
+Volwerk, M., Goetz, C., Richter, I., et al. 2018,
+A&A, 614, A10,
+doi: 10.1051/0004-6361/201732198
+Webster, L., Vainchtein, D., & Artemyev, A. 2021,
+SoPh, 296, 87, doi: 10.1007/s11207-021-01824-2
+Xu, S., Runov, A., Artemyev, A., Angelopoulos,
+V., & Lu, Q. 2018, Geophys. Res. Lett., 45,
+4610, doi: 10.1029/2018GL077902
+Yoon, P. H., & Lui, A. T. Y. 2005, J. Geophys.
+Res., 110, 1202, doi: 10.1029/2003JA010308
+Yoon, Y. D., Wendel, D. E., & Yun, G. S. 2023,
+Nature Communications, 14, 139,
+doi: 10.1038/s41467-023-35821-9
+Zelenyi, L. M., Malova, H. V., Artemyev, A. V.,
+Popov, V. Y., & Petrukovich, A. A. 2011,
+Plasma Physics Reports, 37, 118,
+doi: 10.1134/S1063780X1102005X
+
diff --git a/s9E3T4oBgHgl3EQfjgr4/content/tmp_files/load_file.txt b/s9E3T4oBgHgl3EQfjgr4/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf,len=1227
+page_content='Draft version January 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2023  Typeset using LATEX preprint style in AASTeX631  Kinetic equilibrium of two-dimensional force-free current sheets  Xin An  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1 Anton Artemyev  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2 Vassilis Angelopoulos  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1 Andrei Runov  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1 and  Sergey Kamaletdinov  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 3  1Department of Earth,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Planetary,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' and Space Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' University of California,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Los Angeles,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' CA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 90095,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' USA  2Space Research Institute of the Russian Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Moscow,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 117997,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Russia  3Faculty of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' National Research University Higher School of Economics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Moscow,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 101000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Russia  ABSTRACT  Force-free current sheets are local plasma structures with field-aligned electric currents  and approximately uniform plasma pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Such structures, widely found throughout  the heliosphere, are sites for plasma instabilities and magnetic reconnection, the growth  rate of which is controlled by the structure’s current sheet configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Despite the  fact that many kinetic equilibrium models have been developed for one-dimensional  (1D) force-free current sheets, their two-dimensional (2D) counterparts, which have a  magnetic field component normal to the current sheets, have not received sufficient at-  tention to date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Here, using particle-in-cell simulations, we search for such 2D force-free  current sheets through relaxation from an initial, magnetohydrodynamic equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  Kinetic equilibria are established toward the end of our simulations, thus demonstrating  the existence of kinetic force-free current sheets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Although the system currents in the  late equilibrium state remain field aligned as in the initial configuration, the velocity  distribution functions of both ions and electrons systematically evolve from their ini-  tial drifting Maxwellians to their final time-stationary Vlasov state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' The existence of  2D force-free current sheets at kinetic equilibrium necessitates future work in discover-  ing additional integrals of motion of the system, constructing the kinetic distribution  functions, and eventually investigating their stability properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' INTRODUCTION  Current sheets are spatially localized plasma structures that play an essential role in various space  plasma systems: solar flares (Syrovatskii 1981;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Parker 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Fleishman & Pevtsov 2018), solar wind  turbulence (Servidio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Borovsky 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Vasko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2022), boundaries of planetary magne-  tospheres – magnetopauses (De Keyser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2005), magnetotails of planets (Jackman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  Achilleos 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Lui 2018) and comets (Cravens & Gombosi 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Volwerk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Volwerk 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  Strong currents flowing within current sheets are subject to various instabilities resulting in magnetic  field line reconnection, which converts magnetic energy to plasma heating and particle acceleration  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Gonzalez & Parker 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Birn & Priest 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  Corresponding author: Xin An  phyax@ucla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='edu  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='04590v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='plasm-ph]  11 Jan 2023  ID2  The properties of magnetic reconnection and the dissipation rate of magnetic energy strongly de-  pend on the current sheet configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' The simplest magnetic field geometry is the one-dimensional  (1D) current sheet, which is either a tangential discontinuity separating two plasmas with different  properties or a rotational discontinuity having a finite magnetic field component, Bn, normal to the  current sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' The stress balance in 1D tangential discontinuities is established by the gradients  of plasma and magnetic field pressures [see Figure 1(a) and Allanson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Neukirch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  (2020a,b)], whereas the stress balance in 1D rotational discontinuities requires a contribution from  the plasma dynamic pressure in order to balance the magnetic field line tension force ∝ Bn [see  Figure 1(b) and Hudson (1970)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  In two-dimensional (2D) current sheets, Bn ̸= 0 naturally appears (Schindler 2006);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' thus, the  stress balance requires 2D plasma pressure gradients [see Figure 1(c) and Yoon & Lui (2005)] or 2D  dynamic pressure gradients [see Figure 1(d) and Birn (1992);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Nickeler & Wiegelmann (2010);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Cicogna  & Pegoraro (2015)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  Although there are many types of current sheet configurations, they can all be categorized by two  plasma parameters: the Alfv´en Mach number MA (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', the ratio of plasma flow speed to Alfv´en speed)  and the plasma beta β (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', the ratio of plasma thermal pressure to magnetic field pressure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Current  sheet configurations from different space environments are located within different domains in this  (β, MA) space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Tangential discontinuities (Bn = 0) do not require contributions from the plasma  dynamic pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' These discontinuities exist in systems with either large β (if the plasma pressure  balances the magnetic field pressure) or small β [if the current sheet configuration is magnetically  force free without cross-field currents, J × B = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' see Figure 1(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Such current sheets are often  observed in the solar wind (see discussion in Neugebauer 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Artemyev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2019b), where they  propagate with plasma flows and have MA ≈ 0 in the current sheet reference frame (see examples  of kinetic models of such current sheet configurations in de Keyser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Harrison & Neukirch  2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Allanson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Neukirch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2020a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Rotational discontinuities are also observed in  the solar wind (β ∼ 1) and are characterized by plasma flow gradients with MA ∼ 1 in the current  sheet reference frame [de Keyser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', 1997);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Haaland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', 2012);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Paschmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=',  2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Artemyev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', 2019b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' see Figure 1(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Figure 1(e) shows the parameter regime of  such current sheets in (β, MA) space, whereas Figure 2(a) shows an example of a low-β, force-free  current sheet in the solar wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  Current sheets in the shocked (magnetosheath) solar wind are characterized by lower MA [Figure  1(e)] compared to that of the solar wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' The stress balance in these current sheets is significantly  influenced by plasma pressure gradients, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', compressional current sheets with both parallel and  transverse current density components (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Chaston et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Webster et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Another  (β, MA) domain, with similar MA and larger β, is found in Earth’s distant magnetotail, where strong  plasma flows may reach MA ≥ 1 and β ∈ [10, 100] (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Vasko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Artemyev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  These current sheets are mostly balanced by plasma pressure gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' However, in the distant  magnetotail and for β ≲ 10 there have been previous observations of force-free current sheets with  J × B = 0 (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Figure 1(e) shows the parameter domain of distant magnetotail current  sheets, whereas Figure 2(b) shows an example of an almost force-free current sheet observed in the  distant magnetotail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Both the magnetosheath and distant magnetotail current sheet configurations  are characterized by a small but finite Bn, and thus the magnetic field line tension force ∝ Bn may be  balanced by weak plasma flows or by plasma anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' There is a series of models on such quasi-1D  3  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Panels (a)-(d) show magnetic field lines and the main components of stress balance for different  current sheet configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' The coordinate system consists of the l component along the main magnetic  field direction, reversing sign at the current sheet neutral plane (Bl = 0), the m component along the  main current density direction corresponding to variations of Bl, and the n (normal) component along the  spatial gradient of Bl (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', 4πjm/c = ∂Bl/∂rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Panel (e) shows the typical parameter regimes of current  sheets observed by THEMIS and ARTEMIS (Angelopoulos 2008, 2011) in the solar wind (blue), Earth’s  magnetosheath (shocked solar wind;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' black), near-Earth magnetotail (red), and lunar-distance magnetotail  (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' The black dashed box defines the parameter regime where 2D force-free current sheets are expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  Our dataset includes ∼ 300 solar wind current sheets, ∼ 100 magnetosheath current sheets, ∼ 100 current  sheets in the near-Earth magnetotail, and ∼ 500 current sheets in the lunar-distance magnetotail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  All  current sheets were identified from THEMIS and ARTEMIS fluxgate magnetometer measurements (Auster  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' The selection procedure and data processing for solar wind current sheets are described in  Artemyev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' (2019a): ion temperatures are calculated using the OMNI dataset (King & Papitashvili  2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Artemyev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2018), plasma flows in the current sheet reference frame are calculated as the change  of the most variable flow component across the current sheet (see details in Artemyev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2019a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' The  same criteria and data processing methods were applied to the magnetosheath (dayside) current sheets, but  without using the OMNI data because THEMIS accurately measures magnetosheath ion temperatures in that  region (McFadden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' The dataset of the near-Earth magnetotail current sheets (radial distances  ∼ 10−30 Earth radii) is described in Artemyev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' The same criteria and data processing methods  were applied to the lunar-distance current sheets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Details of the calculation of β and MA are described in  Artemyev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' (2019a, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  current sheets, with Bn ̸= 0, that describe both large-β (see Burkhart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Sitnov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2000,  2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Mingalev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Zelenyi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2011) and force-free (Artemyev 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Mingalev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  Vasko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2014) current sheets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  The distant magnetotail (β, MA) domain smoothly extends towards the lower MA with the decrease  of the radial distance from the Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Near-Earth magnetotail current sheets are characterized by  B, V  (a)  (b  BI-nB?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  solar wind  ln" u~  Bnjm  B,j  n  distant  V,Bm  B,=0  B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='=0  B,j  magnetotail  nB?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  B  B, V  10°  1  (c)  Bnjm  B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='=0  n  1  10-1  B  magnetosheath  (d)  B, V  Bnjm  near-Earth  Vn Vnyi+vi Vivi  B,=0  n  magnetotail  1  10-2  10-1  100  101  102  B, V  B  14  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Four examples of force-free current sheets in (a) solar wind, (b) lunar-distance magnetotail,  (c) Martian magnetotail, and (d) Jovian magnetotail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' The top panels show the magnetic field in the local  coordinate system Sonnerup & Cahill (1968).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Grey curves show the magnetic field magnitudes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' B ≈ constant  indicates the force-free current sheets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Middle panels show electron and ion β profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Bottom panels show  profiles of the field-aligned currents with an indication on the dominant ion species and estimations of the  current sheet thickness in the ion inertial length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' For the solar wind and Earth’s lunar-distance magnetotail,  we used ARTEMIS magnetic field (Auster et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2008) and plasma (McFadden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2008) measurements  (see details of data processing procedure in Artemyev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2019a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' For the Martian magnetotail, we used  MAVEN magnetic field (Connerney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2015) and plasma (McFadden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Halekas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2015)  measurements (see details of data processing procedure in Artemyev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' For the Jovian magnetotail,  we used Juno magnetic field (Connerney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2017a,b) and plasma (McComas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2020)  measurements (see details of data processing procedure in Artemyev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  slightly higher β > 100 and are quantified by the plasma pressure contribution to the stress balance  (Runov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Artemyev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Petrukovich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' The important difference between  solar wind/magnetosheath and magnetotail current sheets is their magnetic field line configuration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  solar wind current sheets may be considered as 1D discontinuities, whereas the magnetotail current  sheets are 2D [Figures 1(a-d)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Therefore, current sheet models with Bn ̸= 0 and low MA should  include 2D plasma pressure gradients balancing the magnetic field line tension force (see examples  of such models in Schindler & Birn 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Birn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Yoon & Lui 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Sitnov & Merkin 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  In the (β, MA) space, the large-β domain can be described by kinetic models of current sheets  with Bn = 0 (1D models) and Bn ̸= 0 (2D models with plasma pressure gradients).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Conversely, the  low-β domain with dominant field-aligned currents (force-free current sheets) can be described only  by 1D kinetic models with Bn = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Large MA, low-β (1D rotational discontinuities with Bn ̸= 0)  and low MA, low-β (2D force-free current sheet) domains have been analyzed only by fluid models  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Cowley 1978;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Hilmer & Voigt 1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Tassi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Lukin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' The class of such  2D force-free current sheets is not limited to observations in the solar wind and distant Earth’s  B  B  B  Bn  βe~βi  ot&0t  15d  5d  ~7d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='5  magnetotail, but also includes current sheets observed in the cold plasma of the Martian magnetotail  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', DiBraccio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Artemyev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2017) and in the low-density Jovian magnetotail (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=',  Behannon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 1981;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Artemyev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Figures 2(c,d) show examples of force-free current  sheets observed in the Martian and Jovian magnetotails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' There are not enough known integrals of  motion to describe distribution functions of charged particles in such current sheet configurations (see  discussion in Lukin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Consequently, the absence of kinetic models of force-free current  sheets with Bn ̸= 0 significantly hinders the analysis of their stability, the process responsible for the  magnetic reconnection onset and particle acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  In this study, we develop a 2D kinetic force-free current sheet model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' In Section 2, we describe  a magnetohydrodynamic (MHD) model of 2D force-free current sheets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' In Section 3, we initialize  self-consistent, particle-in-cell (PIC) simulations using currents and magnetic fields from the MHD  equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' We load particle distributions using drifting Maxwellians, which initially do not nec-  essarily satisfy the time-stationary Vlasov equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' In Section 4, we obtain kinetic equilibria by  evolving these particle distribution functions using the self-consistent PIC simulations and demon-  strate the existence of kinetic equilibria for 2D force-free current sheets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' We describe the equilibrium  distribution functions and the difference between ion- and electron-dominated 2D current sheets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' In  Section 5, we summarize the results and discuss their applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' MHD EQUILIBRIA OF FORCE-FREE CURRENT SHEETS  We consider the general case of 2D force-free equilibrium in which all quantities vary with coor-  dinates x and z only (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', ∂/∂y = 0), and, as is customary in this definition, the pressure gradient  force contribution to the force balance is negligible, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', ∇P ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' The divergence-free magnetic field  has the form (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Low & Wolfson 1988)  B =  �  −∂A  ∂z , By, ∂A  ∂x  �  ,  (1)  where Aey is the vector potential in the y direction, and By is the magnetic field in the y direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  The current density is  J = c  4π∇ × B = c  4π  �  −∂By  ∂z , −∇2A, ∂By  ∂x  �  ,  (2)  where c is the speed of light, and ∇2 = ∂2/∂x2 + ∂2/∂z2 is the Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' The force-free condition,  J × B = 0, is equivalent to J = αB, where α = α(x, z) is a scalar function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Comparing Equations  (1) and (2), the force-free condition requires that By is a function of A only, and that By satisfies  ∇2A + By(A)dBy  dA = 0,  (3)  and the coefficient α is (c/4π)dBy/dA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Once By(A) is given, we can solve Equation (3) for A(x, z)  with appropriate boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  As shown in Figure 3, for balancing force-free current sheets with low thermal and dynamic pressures  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', low β/low Mach number current sheets in the solar wind, magnetosheath, and lunar-distance  magnetotail;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' see Figure 1), the magnetic pressure B2  y plays an role analogous to the thermal pres-  sure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Motivated by the functional form of pressure p ∝ exp [2A/(λB0)] describing thermal-pressure  balanced current sheets (Lembege & Pellat 1982), we adopt  By(A) = B0 exp  � A  λB0  �  ,  (4)  6  for force-free current sheets, so that Equation (3) describing magnetic field lines in the x-z plane  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', contours of A) resembles that of the Lembege-Pellat current sheet (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Tassi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Lukin  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2018):  ∇2A = −B0  λ exp  � 2A  λB0  �  ,  (5)  where λ is the current sheet thickness, and B0 is the magnetic field at large |z|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  JyBz/c = ∂p  ∂x  JyBx/c = ∂p  ∂z  JyBz/c = JzBy/c = ∂  ∂x  B2  y  8π  JyBx/c = JxBy/c = ∂  ∂z  B2  y  8π  Near-Earth magnetotail  Mixed states  Solar wind;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Lunar-distance magnetotail  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Force balance of current sheets in different space environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' At one end of the spectrum, rep-  resenting current sheets in the near-Earth magnetotail, magnetic tension force JyBz/c and magnetic pressure  force JyBx/c ≈ ∂(B2  x/8π)/∂z are balanced by thermal pressure gradients ∂p/∂x and ∂p/∂z, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' At  the other end of the spectrum, representing current sheets in the solar wind and lunar-distance magnetotail,  JyBz/c and JyBx/c are balanced by shears of B2  y, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', ∂(B2  y/8π)/∂x and ∂(B2  y/8π)/∂z, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' There  may be a continuous spectrum of current sheets between these two ends (Yoon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2023), in which JyBz/c  and JyBx/c are balanced by a mixture of thermal pressure gradients and shears of B2  y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  Because the scale length along current sheets is much larger than that across current sheets, we  assume A = A(εx, z) to be weakly nonuniform in the x direction (ε being a small parameter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  Equation (5) can be approximated as  ∂2A  ∂z2 = −B0  λ exp  � 2A  λB0  �  ,  (6)  which is accurate up to order ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' The boundary condition is  ∂A  ∂z  ����  z=0  = 0,  A  ��  z=0 = εB0x,  (7)  which is equivalent to Bx(z = 0) = 0 and Bz(z = 0) = εB0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' The solution of A is  A = −λB0 ln  �  cosh  �  z  λF(x)  �  F(x)  �  ,  (8)  where F(x) = exp (−εx/λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Thus the three components of the magnetic field are  Bx = B0 tanh  �  z  λF(x)  �  F −1(x),  By = B0  �  cosh  �  z  λF(x)  �  F(x)  �−1  ,  Bz = εB0  �  1 −  z  λF(x) tanh  �  z  λF(x)  ��  ,  (9)  7  where Bz ̸= 0 describes rotational discontinuities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' The current density is  J = αB = cB  4πλ  �  cosh  �  z  λF(x)  �  F(x)  �−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  (10)  The direction of magnetic field rotates from pointing in +x at the z > 0 half-space (z/λ ≫ 1) to  pointing in +y around the neutral (equatorial) plane (−1 ≲ z/λ ≲ 1), and further to pointing in  −x at the z < 0 half-space (z/λ ≪ −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' The magnitude of magnetic field, (B2  x + B2  y + B2  z)1/2 =  B0[F −1(x) + O(ε)], has a weak dependence on x and is roughly a constant at a given x location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  The magnitude of current density, (J2  x + J2  y + J2  z )1/2 ∝ cosh−1[z/(λF(x))], is concentrated in a layer  |z| < λF(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' COMPUTATIONAL SETUP  Searching for kinetic equilibria of 2D force-free current sheets, we initialize our simulated current  sheets from the results of an MHD equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Importantly, this may not satisfy the time-stationary  Vlasov equation because the initial particle distributions are simply drifting Maxwellians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' We sim-  ulate their relaxation toward a certain kinetic equilibrium (if any) based on a massively parallel  PIC code (Pritchett 2001, 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Our simulations have two dimensions (x, z) in configuration space  and three dimensions (vx, vy, vz) in velocity space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' The results are presented in normalized units:  magnetic fields are normalized to B0, lengths to the ion inertial length di = c/(4πn0e2/mi)1/2,  time to the reciprocal of the ion gyrofrequency ω−1  ci  = mic/(eB0), velocities to the Alfv´en ve-  locity vA = B0/(4πn0mi)1/2, electric fields to vAB0/c, and energies to miv2  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  Here n0 is the  plasma density, mi is the ion mass, and e is the elementary charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' The computational domain is  [−64 ≤ x/di ≤ 0] × [−8 ≤ z/di ≤ 8] with a cell length ∆x = di/32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' The time step is ∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='001 ω−1  ci .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  The ion-to-electron mass ratio is mi/me = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' The normalized speed of light is c/vA = 20, which  gives the ratio of electron plasma frequency to electron gyrofrequency ωpe/ωce = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' The reference  density n0 is represented by 1929 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' The total number of particles is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='6 × 109 including ions  and electrons in each simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  Figure 4 shows the initial magnetic field configuration and current density, which are determined  by Equations (9) and (10) using the input parameters ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='04 and λ = 2di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' There are a few degrees  of freedom in choosing the initial particle distribution functions: (1) While the current density is  known a priori, the proportion of ion and electron currents is left unspecified;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' (2) Electron and ion  temperatures and plasma β vary across different space environments (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Figure 2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' (3) Specific  forms of the velocity distribution functions are left undetermined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' To this end, we choose initial  electron and ion distribution functions as drifting Maxwellians  fs(v, x, z) =  n0  (2πTs/ms)3/2 exp  �  −ms (v − vds(x, z))2  2Ts  �  ,  (11)  where the subscript s = i, e represents ions and electrons, vds(x, z) is the position-dependent drift  velocity, and Ts is the temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Note that Ti, Te, and n0 are constants throughout the domain so  that the initial plasma pressure n0(Ti +Te) is a constant, as required by the force-free condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' The  simulations allow either ions or electrons to be the main current carrier, while keeping the relative drift  between them commensurate with the current density [Figures 4(c), 4(d), and 4(e)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Additionally, we  vary particle temperatures to search for kinetic equilibrium in different plasma beta regimes, where  8  the plasma beta is β = 2(Ti + Te)/miv2  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' The detailed parameters for particle distribution functions  in the six runs are listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' These parameters cover the β and Ti/Te ranges of force-free  current sheets observed in the solar wind and planetary magnetotails (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' MHD equilibrium of a force-free current sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' (a) Magnetic field in the x direction Bx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' (b)  Magnetic field in the y direction By.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' (c) Current density in the x direction Jx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' (d) Current density in the y  direction Jy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' (e) Current density in the parallel direction J∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  In our simulations, the electromagnetic fields are advanced in time by integrating Maxwell’s equa-  tions using a leapfrog scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' These fields are stored on the Yee grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' The relativistic equations of  motion for ions and electrons are integrated in time using a leapfrog scheme with a standard Boris  push for velocity update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' The conservation of charge is ensured by applying a Poisson correction to  the electric fields (Marder 1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Langdon 1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  For particles crossing the x boundaries, we take advantage of the symmetry between z > 0 and  z < 0 [Figure 4], so that particles exiting the system at a location z with velocity (vx, vy, vz) are  reinjected into the system at the conjugate location −z with velocity (−vx, vy, vz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' This is equivalent  to an open boundary condition for particles because the injected particle distribution matches that  at one cell interior to the boundary at all simulation times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' At the z boundaries, particles striking  the boundary are reflected into the system with vz = −vz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  (a)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='8  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='4  B  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='0  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='8  (q)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='8  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='4  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='8  (c)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='2  5  (Vaou)/  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='2  (p)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='4  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='2  (e)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='4  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='0  7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='2  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='4  0  20  40  60  x/di9  Run NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  vdi  vde  βi = 2Ti/miv2  A  βe = 2Te/miv2  A  β = βi + βe  1A  J/(n0e)  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1  1B  0  −J/(n0e)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1  2A  J/(n0e)  0  5/6  1/6  1  2B  0  −J/(n0e)  5/6  1/6  1  3A  J/(n0e)  0  25/3  5/3  10  3B  0  −J/(n0e)  25/3  5/3  10  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Parameters for particle distribution functions in the six PIC runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' The first digit in Run NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=',  ‘1’, ‘2’, ‘3’, denotes three plasma betas, β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1, 1, 10, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' The second digit in Run NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', ‘A’, ‘B’,  denotes current carriers as ions and electrons, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' The current density J comes from Equation (10),  and is visualized in Figures 4(c) and 4(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  For fields at the x boundaries, the guard values of the tangential magnetic fields are determined by  δBn  y,g = δBn  y,i1,  δBn  z,g = δBn−1  z,i1 (2 − ∆x/c∆t) + δBn−2  z,i2 (∆x/c∆t − 1),  (12)  where the superscript indicates the time level, and the subscripts g, i1, i2 indicate the guard point,  first interior point and second interior point, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' This boundary condition for δBz ensures  that the magnetic flux can freely cross the x boundaries (Pritchett 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' The guard values of the  normal electric field δEx at the x boundaries are determined by δEn  x,g = δEn  x,i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Similarly, at the z  boundaries, the two components of the tangential magnetic fields in the guard point are determined  by δBn  x,g = δBn  x,i1 and δBn  y,g = δBn  y,i1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' The normal electric field δEz in the guard point are determined  by δEn  z,g = δEn  z,i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Unlike the driven simulations that adds magnetic flux to the simulation box, no  external Ey is applied at the z boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' RESULTS  We track the evolution of the initially force-free current sheets in the six runs up to t = 180 ω−1  ci ,  at which time the macroscopic states (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', electric and magnetic fields, currents, pressures) of the  system are quasi-stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Below we examine if the configurations satisfy J × B = 0 at the end  of the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' We further investigate how the particle velocity distributions deviate from the  initial Maxwellians and whether or not the system reaches a kinetic equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Initially ion-dominated current sheets  When ions initially carry entirely the (field-aligned) currents (Runs 1A, 2A, and 3A), electrons are  accelerated by transient parallel electric fields to form field-aligned currents [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Figure 5(b)], while  ions are slightly decelerated by such electric fields [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', comparing Figures 5(a) and 4(e)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' These  electron currents can be a significant fraction (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', ≳ 1/3) of ion currents in the off-equatorial region  (1 < |z| < 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Almost all currents remain field aligned [see J⊥ ∼ 0 in Figure 5(c)], implying that a  force-free configuration may indeed be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  During the relaxation of ion-dominated current sheets, electrostatic fields are generated and remain  present in the late equilibrium states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' These electrostatic fields are perpendicular to the magnetic  field: At |z| > 0, they points away from the equatorial plane [Figure 6(a)];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Near the equator, they  points in the −x direction (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', tailward) [Figure 6(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Here the Ex component is much weaker than  the Ez component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' The electrostatic fields arise due to the decoupling of the unmagnetized ions  10  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Current density for the ion-dominated force-free current sheet with plasma beta β = 1 in Run  2A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' The snapshot is taken at t = 180 ω−1  ci  in the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' (a) Ion field-aligned currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' (b) Electron  field-aligned currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' (c) Total perpendicular currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  and magnetized electrons (Schindler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2012), which can be derived from the ordering among ion  thermal gyroradius, current sheet thickness, and electron thermal gyroradius  ρi : λ : ρe = di  �  βi  2 : 2di : di  �  βe  2  me  mi  =  �  βi  8 : 1 :  �  βe  8  me  mi  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  (13)  Taking Run 2A for example, we have ρi : λ : ρe = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='32 : 1 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' As the plasma beta is lowered, the  electrostatic field decreases – due to stronger magnetization of ions, which is shown in Figures 7 and  8 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  To show the detailed force balance of these current sheet configurations, we decompose the forces  in parallel and perpendicular directions with respect to local magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' The force balance in  the parallel direction is rapidly established because particles move freely along field lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' The unit  vectors in the two perpendicular directions are defined as e⊥1 = ˆz × ˆb and e⊥2 = ˆb × (ˆz × ˆb), where  ˆb and ˆz are the unit vectors along the magnetic field and z direction, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' It is worth noting  that e⊥1 is roughly along the ±y direction far above/below the equator, and along the −x direction  at the equator, while e⊥2 is roughly along the +z direction all over the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  At the equator, the electrostatic fields, E⊥1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', along the −x direction), cause a drift cE⊥1×b/|B|  of both ions and electrons [Figures 7(b-c), 7(h-i), and 7(n-o)], which does not produce net currents  [Figures 7(a), 7(g), and 7(m)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' In addition, the pressure gradient (∇ · P)⊥1 is about zero in the  two cases, with β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1 [Run 1A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' see Figure 7(a)] and β = 1 [Run 2A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' see Figure 7(g)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Thus, we  obtain (J × B)⊥1/c = (∇ · P)⊥1 = 0 in the e⊥1 direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' In the case with β = 10, however, the  current sheet gets compressed and rarefied periodically in the x direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Correspondingly, (∇·P)⊥1  oscillates around zero and acts as a restoring force [Run 3A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' see Figure 7(m)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Such oscillations are  5  (a)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='2  J ill  0  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1  5  (q)  J(enovA)  1p/2  Jell  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='0  5  5  (c)  F-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1  1p/2  J1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='2  5  20  25  30  35  40  45  x/di11  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Electrostatic field for the ion-dominated force-free current sheet with plasma beta β = 1 in Run  2A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' The snapshot is taken at t = 180 ω−1  ci  in the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' (a) Electric field in the z direction Ez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' (b)  Electric field in the x direction Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' The magnitude of Ex is much smaller than that of Ez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  still localized around the equilibrium (J×B)⊥1/c = (∇·P)⊥1 = 0 when performing a spatial average  over x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  Along the e⊥2 direction (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', roughly the z direction), both the pressure gradient (∇ · P)⊥2 and  Lorentz force (J×B)⊥2/c vanish [Figures 8(a), 8(g), 8(m)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' The dominant component of the electro-  static fields, E⊥2 (along the ±z directions), leads to a drift cE⊥2×b/|B| (roughly in the +y direction)  of both ions and electrons [Figures 8(b-c), 8(h-i), and 8(n-o)], which does not give net currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' All  aforementioned perpendicular drift velocities are small, when compared to ion and electron parallel  flow velocities that carry currents [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Figure 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Initially electron-dominated current sheets  When electrons initially carry entirely the (field-aligned) currents (Runs 1B, 2B, and 3B), there is no  change in the macroscopic states of the system at the end of simulations [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Figure 9]: The currents  remain field aligned, and are carried by electrons only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Compared to the ion-dominated current sheets,  the relaxation of electron-dominated current sheets does not alter the macroscopic states: neither  the proportion of electron and ion currents is changed, nor do the electrostatic fields develop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' In  all three cases with different plasma betas, the system remains in a force-free configuration, with no  plasma pressure gradients [Figures 7(d-f), 7(j-l), 7(p-r), 8(d-f), 8(j-l), 8(p-r)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Despite the system  being identical to the initial ones from the fluid point of view, the underlying velocity distribution  functions evolve substantially from the initial Maxwellians to reach the Vlasov equilibrium, shown  below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Reaching kinetic equilibrium or not?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  Since the force-free equilibria are reestablished toward the end of simulations for both ion- and  electron-dominated current sheets from the fluid point of view, a logical next step would be to  further examine if such current sheets reach the kinetic equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Because all the currents are  field aligned, we integrate the full distribution function fs(x, z, v∥, v⊥, φ, t) of species s to obtain the  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='06  (a)  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='04  ()/  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='02  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='04  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='06  (b)  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='004  cEx/(vABo)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='002  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='004  5  20  25  30  35  40  45  x/di12  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Force balance in the e⊥1 = ˆz × ˆb direction at the equator for different plasma betas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Note e⊥2 is  along the −x direction at the equator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' All the snapshots are taken at t = 180 ω−1  ci  in the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' The  volume forces are averaged in the range −1 ≤ z/di ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' The rows from top to bottom represent Runs 1A,  1B, 2A, 2B, 3A, and 3B, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' The columns from left to right represent the force balance from the  point view of single fluid, ion fluid, and electron fluid, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  reduced distribution gs(x, v∥, t) at the equator as a function of x, v∥, and t:  gs(x, v∥, t) =  � di  −di  dz  � ∞  0  dv⊥  � 2π  0  dφ fs(x, z, v∥, v⊥, φ, t),  (14)  where v∥ and v⊥ are the parallel and perpendicular velocities, respectively, and φ is the gyrophase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  Figure 10 shows the difference between the reduced distributions at the end of simulations and the  initial Maxwellians, ∆gs = gs(x, v∥, t = 180 ω−1  ci ) − gs(x, v∥, t = 0), from Run 2A as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' The  (J×B)11/c  (V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='P)11  ngE11  SUM  All  Ions  Electrons  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='01  β= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='01  =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='01  B  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='01  β=10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='01  20  40  20  40  20  40  x/di  x/di  x/di13  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  Force balance in the e⊥2 = ˆb × (ˆz × ˆb) direction for different plasma betas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  Note e⊥2 is  approximately along the +z direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' All the snapshots are taken at t = 180 ω−1  ci  in the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' The  volume forces are averaged in the range −16 ≤ x/di ≤ −12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' The format is the same as Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  ion phase space density is transported toward velocities both above and below the mean ion flow  velocity [Figure 10(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Consequently, the ion velocity distribution becomes wider and less peaked,  compared to the initial Maxwellian [Figure 10(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' The mean ion flow velocity is slowed more on  the earthward side than the tailward side [see the inset of Figure 10(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Correspondingly, electrons  are accelerated on average in the v∥ < 0 direction to develop an electron current to compensate the  reduction of the ion current [see the inset of Figure 10(c)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Such changes of mean flow velocities are  not apparent in the phase space density plot because the thermal velocities are much larger than  (J ×B) 12/c  (V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='P) 12  ngE 12  SUM  All  Ions  Electrons  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='05  β= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='05  β= 10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='05  5  0  5  0  5  5  0  5  zldi  zldi  zldi14  Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Current density for the electron-dominated force-free current sheet with plasma beta β = 1 in  Run 2B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' The snapshot is taken at t = 180 ω−1  ci in the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' (a) Ion field-aligned currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' (b) Electron  field-aligned currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' (c) Total perpendicular currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  the mean flow velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' The peak of the electron phase space density is transported toward larger  velocities in both v∥ > 0 and v∥ < 0 [Figures 10(c) and 10(d)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  The macroscopic states of electron-dominated current sheets do not differ from their initial config-  urations, as seen in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' The electron reduced distribution function ge(x, v∥), however, shows  systematic deviations from the initial Maxwellians, as displayed in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Such deviations are  similar to those in ion-dominated current sheets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' The peak of the electron velocity distributions is  reduced and redistributed toward larger velocities of v∥ > 0 and v∥ < 0 (but without changes in mean  electron flow velocities) [Figures 11(c) and 11(d)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' The deviation of the ion velocity distribution from  the initial Maxwellian is small [Figure 11(b)], but is required to reach the Vlasov equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  To quantify the convergence of the system toward kinetic equilibrium, we define the metric  Gs(t) =  � ∞  −∞  dv∥  ��  gs(x, v∥, t) − gs(x, v∥, 0)  �2�  x ,  (15)  where ⟨·⟩x denotes the average over spatial coordinate x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' This metric is chosen in such a way that  the non-Maxwellian features of gs(x, v∥, t) are properly accounted for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' During the evolution of the  system, it is expected that Gs(t) experiences significant changes at the beginning of the simulation  and eventually reaches a steady state, if a kinetic equilibrium is established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Figure 12 shows the  absolute values of the time derivative |∂Gs/∂t| for both ion- and electron-dominated current sheets  with different plasma betas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' All runs end up with |∂Gs/∂t| ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='01, except for Run 1A (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', the  ion-dominated current sheet with β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' For the same plasma beta, electron-dominated current  sheets reach the kinetic equilibrium faster than ion-dominated current sheets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' In the latter case,  the system simply takes more time to redistribute the currents between unmagnetized ions and  5  (a)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='2  1p/2  Jill  0  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1  (b)  5  JI(enovA)  Jell  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='0  5  (c)  --0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1  5  JI  0  F-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='2  5  20  25  30  35  40  45  x/di15  Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Phase space distributions for the ion-dominated force-free current sheet with plasma beta β = 1  in Run 2A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' (a) The difference between the reduced ion distribution and the initial Maxwellian ∆gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' The  inset plot shows the mean ion flow velocity as a function of x at t = 0 (gray line) and t = 180 ω−1  ci  (black  line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' (b) A cut of the final ion distribution gi(x, v∥, t = 180 ω−1  ci ) (black line), gi(x, v∥, t = 0) (gray line),  and their difference ∆gi multiplied by 5, for better visibility (red line) between −16 ≤ x/di ≤ −12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' (c) The  different between the reduced electron distribution and the initial Maxwellian ∆ge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' The inset plot shows  the mean electron flow velocity as a function of x at t = 0 (gray line) and t = 180 ω−1  ci  (black line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' (d) A  cut of the final electron distribution ge(x, v∥, t = 180 ω−1  ci ) (black line), ge(x, v∥, t = 0) (gray line), and their  difference ∆ge multiplied by 5, for better visibility (red line) between −16 ≤ x/di ≤ −12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  magnetized electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' For both types (ion- and electron-dominated) of current sheets, those with  higher plasma β evolve toward kinetic equilibrium faster, because the current sheets with higher  plasma β can support larger transient electric fields to redistribute particles more rapidly in phase  space toward the equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  At present, we cannot simulate further in time the relaxation of the low-β, ion-dominated current  sheet in Run 1A toward the final kinetic equilibrium, because the current sheet becomes unstable at  a later time (most likely due to boundary conditions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' However, the progression of the relaxation for  different beta values and the behavior of Run 1A up to t · ωci ∼ 10 suggests that it also conforms to  the explanatory model presented herein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' The distribution functions at that time provide a sufficiently  good representation of the Vlasov equilibrium to use in future modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' CONCLUSION AND DISCUSSION  In summary, we demonstrate that kinetic equilibria of 2D force-free current sheets exist for different  proportions of ion and electron currents in various plasma betas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' When initial currents are carried  purely by ions, field-aligned electron currents are developed by transient parallel electric fields, while  X10-1  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='0  <v>  (a)  (b)  gi  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='0  (0=1)<)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='5  gi(t = 0)  △gil(no/vA)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='5  5△gi  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='25  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='0  20  40  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='5  X10-2  20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='2  <v >  (c)  (d)  ge  2  (0=↓)<Il )  10  ge(t = 0)  Agel(no/vA)  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1  5△ge  Va/ II  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
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+page_content='1  20  40  10  0  10  x/di  VIl /VA16  Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Phase space distributions for the electron-dominated force-free current sheet with plasma beta  β = 1 in Run 2B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' The format is the same as Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  perpendicular electrostatic fields are generated due to unmagnetized ions and magnetized electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  When initial currents are carried exclusively by electrons, the macroscopic state of the system remains  unchanged from its initial state, described by the MHD model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' In both scenarios, the electron and  ion distribution functions at the late equilibrium states show systematic deviations from the initial  drifting Maxwellians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' These deviations occur in order to satisfy the time-stationary Vlasov equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  The existence of a kinetic equilibrium for 2D force-free current sheets indicates that there is at  least one hidden symmetry in the system, implying the existence of an additional integral of motion  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', an additional invariant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Our system has five dimensions ND = 5 in the phase space (2D in  the coordinate space and 3D in the velocity space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' The two known invariants are the total energy  H, and the y component of the canonical momentum Py = mvy + eA/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' The degrees of freedom  of the system are Nf = ND − NI, where NI is the number of invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' The particle phase space  densities at equilibrium are constructed as a function of such invariants of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' To fully describe  such kinetic equilibrium, which we now know it does exist, we must have NI > Nf, or equivalently,  NI > ND/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Thus, NI is at least 3 (in our case) and there is at least one hidden symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Starting  from particle trajectory data in the equilibrium electromagnetic fields, future research on this topic  could make use of machine learning models to help find the number of invariants or even discover  the analytic formula of these invariants (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Liu & Tegmark 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' ACKNOWLEDGMENTS  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' This work was supported by NASA grants 80NSSC20K1788, 80NSSC22K0752, and NAS5-02099.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  We acknowledge high-performance computing support from Cheyenne (doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='5065/D6RX99HX) pro-  X10-2  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
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+page_content='1  20  40  10  0  10  x/di  VI /VA17  Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Evolution of force-free current sheets toward kinetic equilibrium with different plasma betas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  The x-axis has two scales, a linear scale in 0 ≤ tω−1  ci ≤ 10 and a log scale in 10 ≤ tω−1  ci ≤ 180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' The y axis  denotes |∂G/∂t|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' The rows from top to bottom represent Runs 1A, 1B, 2A, 2B, 3A, and 3B, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  The grey line |∂G/∂t| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='01 in each panel is drawn as a reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  vided by NCAR’s Computational and Information Systems Laboratory, sponsored by the National  Science Foundation (Computational and Information Systems Laboratory 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  REFERENCES  Achilleos, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2018, in Electric Currents in  Geospace and Beyond, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Keiling,  O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Marghitu, & M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Wheatland, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 235,  127–138, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1002/9781119324522.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='ch8  Allanson, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Neukirch, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Troscheit, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', &  Wilson, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2016, Journal of Plasma Physics, 82,  905820306, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1017/S0022377816000519  Allanson, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Neukirch, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Wilson, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', &  Troscheit, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2015, Physics of Plasmas, 22,  102116, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1063/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='4934611  Angelopoulos, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2008, SSRv, 141, 5,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1007/s11214-008-9336-1  —.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2011, SSRv, 165, 3,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1007/s11214-010-9687-2  Artemyev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2011, Physics of Plasmas, 18,  022104, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1063/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='3552141  Artemyev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Angelopoulos, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Halekas, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=',  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2017, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', 122, 5404,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1002/2017JA024078  Artemyev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Angelopoulos, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', & McTiernan,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2018, Journal of Geophysical Research  (Space Physics), 123, 9955,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1029/2018JA025904  Artemyev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Angelopoulos, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', & Runov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  2016, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
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+page_content=', 121, 4017,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1002/2016JA022480  Artemyev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Angelopoulos, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', & Vasko, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  2019a, Journal of Geophysical Research (Space  Physics), 124, 3858, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1029/2019JA026597  Ions  Electrons  100  (a)  G  10-1  M  β= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1  100  (b)  G  10-  100  (c)  10  B  100  (d)  10-  10-2  100  (e)  G  10-  10  =  100  (f)  10  10  0  5  10  100  toci18  Artemyev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Angelopoulos, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Vasko, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=',  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2019b, Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', 46,  1185–1194, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1029/2018GL079906  Artemyev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Ma, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Ebert, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Zhang,  X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', & Allegrini, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2023, Force-free current  sheets in the Jovian magnetodisk: the key role  of electron field-aligned anisotropy, arXiv,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='48550/ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='03731  Artemyev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Petrukovich, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Nakamura,  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', & Zelenyi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2011, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=',  116, A0923, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1029/2011JA016801  Artemyev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Vasko, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', & Kasahara, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  2014, Planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Space Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', 96, 133,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='pss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='012  Auster, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Glassmeier, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Magnes, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=',  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2008, SSRv, 141, 235,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1007/s11214-008-9365-9  Behannon, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Burlaga, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', & Ness, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  1981, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', 86, 8385,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1029/JA086iA10p08385  Birn, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 1992, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', 97, 16817,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1029/92JA01527  Birn, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', & Priest, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2007, Reconnection of  magnetic fields : magnetohydrodynamics and  collisionless theory and observations, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Birn,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' & Priest, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  Birn, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Schindler, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', & Hesse, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2004, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', 109, 2217,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1029/2003JA010303  Borovsky, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2010, Physical Review Letters,  105, 111102,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='111102  Burkhart, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Drake, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Dusenbery, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=',  & Speiser, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 1992, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', 97,  13799, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1029/92JA00495  Chaston, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Bonnell, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Wygant, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=',  Reeves, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', & Baker, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2020,  Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', 47, e86318,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1029/2019GL086318  Cicogna, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', & Pegoraro, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2015, Physics of  Plasmas, 22, 022520, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1063/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='4913580  Computational and Information Systems  Laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2019, Cheyenne: HPE/SGI ICE XA  System (University Community Computing),  Boulder, CO: National Center for Atmospheric  Research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='5065/D6RX99HX  Connerney, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Espley, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Lawton, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  2015, SSRv, 195, 257,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1007/s11214-015-0169-4  Connerney, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Benn, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Bjarno, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=',  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2017a, SSRv, 213, 39,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1007/s11214-017-0334-z  Connerney, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Adriani, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Allegrini, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=',  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2017b, Science, 356, 826,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1126/science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='aam5928  Cowley, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 1978, Planetary and Space  Science, 26, 1037,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1016/0032-0633(78)90028-4  Cravens, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', & Gombosi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2004, Advances  in Space Research, 33, 1968,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='asr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='053  de Keyser, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Roth, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Lemaire, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 1996,  SoPh, 166, 415, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1007/BF00149407  de Keyser, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Roth, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Tsurutani, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Ho,  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', & Phillips, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 1997, A&A, 321, 945  De Keyser, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Dunlop, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Owen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  2005, SSRv, 118, 231,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1007/s11214-005-3834-1  DiBraccio, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Espley, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Gruesbeck, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=',  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2015, Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', 42, 8828,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1002/2015GL065248  Fleishman, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', & Pevtsov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2018, in  Electric Currents in Geospace and Beyond, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Keiling, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Marghitu, & M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Wheatland, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  235, 43–65, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1002/9781119324522.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='ch3  Gonzalez, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', & Parker, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2016, Magnetic  Reconnection, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 427,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1007/978-3-319-26432-5  Haaland, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Sonnerup, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', & Paschmann, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  2012, Annales Geophysicae, 30, 867,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='5194/angeo-30-867-2012  Halekas, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Taylor, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Dalton, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  2015, SSRv, 195, 125,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1007/s11214-013-0029-z  Harrison, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', & Neukirch, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2009, Physical  Review Letters, 102, 135003,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='135003  Hilmer, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', & Voigt, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 1987, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=',  92, 8660, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1029/JA092iA08p08660  Hudson, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 1970, Planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Space Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', 18, 1611,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1016/0032-0633(70)90036-X  Jackman, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Arridge, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Andr´e, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  2014, SSRv, 182, 85,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1007/s11214-014-0060-8  Kim, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Ebert, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Valek, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  2020, Journal of Geophysical Research (Space  Physics), 125, e27696,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1029/2019JA027696  19  King, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', & Papitashvili, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2005,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', 110, A02104,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1029/2004JA010649  Langdon, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 1992, Computer Physics  Communications, 70, 447  Lembege, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', & Pellat, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 1982, Physics of Fluids,  25, 1995, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1063/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='863677  Liu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Madhavan, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', & Tegmark, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2022,  arXiv preprint arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='12610  Liu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', & Tegmark, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2021, Physical Review  Letters, 126, 180604  Low, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', & Wolfson, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 1988, The Astrophysical  Journal, 324, 574  Lui, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2018, in Washington DC American  Geophysical Union Geophysical Monograph  Series, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 235, Electric Currents in Geospace  and Beyond, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Keiling, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Marghitu, &  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Wheatland, 155–175,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1002/9781119324522.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='ch10  Lukin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Artemyev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Vainchtein, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', &  Petrukovich, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2022, Regimes of charged  particle dynamics in current sheets: the machine  learning approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='org/abs/2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='03787  Lukin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Vasko, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Artemyev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', & Yushkov,  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2018, Physics of Plasmas, 25, 012906,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1063/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='5016178  Marder, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 1987, Journal of Computational  Physics, 68, 48  McComas, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Alexander, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Allegrini, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=',  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2017, SSRv, 213, 547,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1007/s11214-013-9990-9  McFadden, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Carlson, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Larson, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=',  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2008, SSRv, 141, 277,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1007/s11214-008-9440-2  McFadden, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Kortmann, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Curtis, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  2015, SSRv, 195, 199,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1007/s11214-015-0175-6  Mingalev, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Mingalev, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Malova, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=',  & Zelenyi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2007, Plasma Physics Reports,  33, 942, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1134/S1063780X07110062  Mingalev, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Mingalev, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Mel’nik, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=',  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2012, Plasma Physics Reports, 38, 300,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1134/S1063780X12030063  Neugebauer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2006, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', 111,  A04103, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1029/2005JA011497  Neukirch, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Vasko, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Artemyev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', &  Allanson, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2020a, ApJ, 891, 86,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='3847/1538-4357/ab7234  Neukirch, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Wilson, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', & Allanson, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2020b,  Journal of Plasma Physics, 86, 825860302,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1017/S0022377820000604  Nickeler, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', & Wiegelmann, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2010, Annales  Geophysicae, 28, 1523,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='5194/angeo-28-1523-2010  Parker, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 1994, Spontaneous current sheets in  magnetic fields: with applications to stellar  x-rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' International Series in Astronomy and  Astrophysics, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' New York : Oxford  University Press, 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', 1  Paschmann, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Haaland, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Sonnerup, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', &  Knetter, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2013, Annales Geophysicae, 31, 871,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='5194/angeo-31-871-2013  Petrukovich, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Artemyev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Vasko, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=',  Nakamura, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', & Zelenyi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2015, SSRv,  188, 311, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1007/s11214-014-0126-7  Pritchett, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2005, Journal of Geophysical  Research: Space Physics, 110  Pritchett, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2001, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', 106,  25961, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1029/2001JA000016  —.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2005, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', 110, 5209,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1029/2004JA010948  Runov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Sergeev, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Nakamura, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  2006, Annales Geophysicae, 24, 247  Schindler, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2006, Physics of Space Plasma  Activity, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Schindler, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' (Cambridge  University Press), doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='2277/0521858976  Schindler, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', & Birn, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2002, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=',  107, 1193, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1029/2001JA000304  Schindler, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Birn, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', & Hesse, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2012, Physics  of Plasmas, 19, 082904, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1063/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='4747162  Servidio, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Dmitruk, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Greco, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2011,  Nonlinear Processes in Geophysics, 18, 675,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='5194/npg-18-675-2011  Sitnov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', & Merkin, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2016,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', 121, 7664,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1002/2016JA023001  Sitnov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Swisdak, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Guzdar, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', &  Runov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2006, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', 111, 8204,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1029/2005JA011517  Sitnov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Zelenyi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Malova, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', &  Sharma, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2000, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', 105,  13029, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1029/1999JA000431  Sonnerup, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' ¨O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', & Cahill, Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 1968,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', 73, 1757,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1029/JA073i005p01757  Syrovatskii, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 1981, Annual review of  astronomy and astrophysics, 19, 163,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1146/annurev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='aa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='090181.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='001115  20  Tassi, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Pegoraro, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', & Cicogna, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2008,  Physics of Plasmas, 15, 092113,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1063/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='2988338  Vasko, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Alimov, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Phan, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2022,  ApJL, 926, L19, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='3847/2041-8213/ac4fc4  Vasko, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Artemyev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Petrukovich, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=',  & Malova, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2014, Annales Geophysicae,  32, 1349, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='5194/angeo-32-1349-2014  Vasko, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Petrukovich, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Artemyev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=',  Nakamura, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', & Zelenyi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2015,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', 120, 8663,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1002/2015JA021633  Volwerk, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2018, in Electric Currents in  Geospace and Beyond, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Keiling,  O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Marghitu, & M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Wheatland, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 235,  513–533, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1002/9781119324522.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='ch30  Volwerk, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Goetz, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Richter, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2018,  A&A, 614, A10,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1051/0004-6361/201732198  Webster, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Vainchtein, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', & Artemyev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2021,  SoPh, 296, 87, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1007/s11207-021-01824-2  Xu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Runov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Artemyev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Angelopoulos,  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', & Lu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2018, Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', 45,  4610, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1029/2018GL077902  Yoon, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', & Lui, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2005, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='  Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', 110, 1202, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1029/2003JA010308  Yoon, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Wendel, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', & Yun, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2023,  Nature Communications, 14, 139,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1038/s41467-023-35821-9  Zelenyi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Malova, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', Artemyev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=',  Popov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=', & Petrukovich, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content=' 2011,  Plasma Physics Reports, 37, 118,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
+page_content='1134/S1063780X1102005X' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQfjgr4/content/2301.04590v1.pdf'}
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+RESEARCH
+Open Access
+A metagenomics roadmap to the
+uncultured genome diversity in hypersaline
+soda lake sediments
+Charlotte D. Vavourakis1
+, Adrian-Stefan Andrei2†, Maliheh Mehrshad2†, Rohit Ghai2, Dimitry Y. Sorokin3,4
+and Gerard Muyzer1*
+Abstract
+Background: Hypersaline soda lakes are characterized by extreme high soluble carbonate alkalinity. Despite the
+high pH and salt content, highly diverse microbial communities are known to be present in soda lake brines but
+the microbiome of soda lake sediments received much less attention of microbiologists. Here, we performed metagenomic
+sequencing on soda lake sediments to give the first extensive overview of the taxonomic diversity found in these complex,
+extreme environments and to gain novel physiological insights into the most abundant, uncultured prokaryote lineages.
+Results: We sequenced five metagenomes obtained from four surface sediments of Siberian soda lakes with a pH 10 and
+a salt content between 70 and 400 g L−1. The recovered 16S rRNA gene sequences were mostly from Bacteria, even in
+the salt-saturated lakes. Most OTUs were assigned to uncultured families. We reconstructed 871 metagenome-assembled
+genomes (MAGs) spanning more than 45 phyla and discovered the first extremophilic members of the Candidate Phyla
+Radiation (CPR). Five new species of CPR were among the most dominant community members. Novel dominant
+lineages were found within previously well-characterized functional groups involved in carbon, sulfur, and nitrogen
+cycling. Moreover, key enzymes of the Wood-Ljungdahl pathway were encoded within at least four bacterial phyla
+never previously associated with this ancient anaerobic pathway for carbon fixation and dissimilation, including the
+Actinobacteria.
+Conclusions: Our first sequencing effort of hypersaline soda lake sediment metagenomes led to two important
+advances. First, we showed the existence and obtained the first genomes of haloalkaliphilic members of the CPR
+and several hundred other novel prokaryote lineages. The soda lake CPR is a functionally diverse group, but the most
+abundant organisms in this study are likely fermenters with a possible role in primary carbon degradation. Second,
+we found evidence for the presence of the Wood-Ljungdahl pathway in many more taxonomic groups than those
+encompassing known homo-acetogens, sulfate-reducers, and methanogens. Since only few environmental metagenomics
+studies have targeted sediment microbial communities and never to this extent, we expect that our findings are relevant
+not only for the understanding of haloalkaline environments but can also be used to set targets for future studies on marine
+and freshwater sediments.
+Keywords: Soda lake sediments, Metagenomics, Haloalkaliphilic extremophiles, Candidate Phyla Radiation, Wood-Ljungdahl
+pathway
+* Correspondence: G.Muijzer@uva.nl
+†Adrian-Stefan Andrei and Maliheh Mehrshad contributed equally to this
+work.
+1Microbial Systems Ecology, Department of Freshwater and Marine Ecology,
+Institute for Biodiversity and Ecosystem Dynamics, Faculty of Science,
+University of Amsterdam, Postbus 94248, 1090, GE, Amsterdam, the
+Netherlands
+Full list of author information is available at the end of the article
+© The Author(s). 2018 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0
+International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and
+reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to
+the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver
+(http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
+Vavourakis et al. Microbiome  (2018) 6:168 
+https://doi.org/10.1186/s40168-018-0548-7
+
+MicrobiomeBackground
+Soda lakes are evaporative, athallasic salt lakes with low cal-
+cium and magnesium concentrations and a high-alkaline
+pH up to 11 buffered by dissolved (bi-) carbonate ions [1].
+They are constrained to arid regions across the globe,
+mainly the tropical East African Rift Valley [2], the Libyan
+Desert [3], the deserts in California and Nevada [4], and the
+dry steppe belt of Central Asia that spans to southern Si-
+beria, north-eastern Mongolia, and Inner Mongolia in
+China [1]. On top of the extreme salinity and alkaline pH,
+the Eurasian soda lakes experience extreme seasonal
+temperature differences, causing highly unstable water re-
+gimes and fluctuating salinities [5]. Yet, soda lakes harbor
+diverse communities of haloalkaliphilic microbes, mostly
+prokaryotes that are well adapted to survive and grow in
+these extreme environments and consist of similar func-
+tional groups in soda lakes around the world [1, 2, 6]. The
+relative abundance of different groups is typically governed
+by the salinity of the brine [1, 7, 8], and microbial-mediated
+nutrient
+cycles
+become
+partially
+hampered
+only
+at
+salt-saturating conditions [1].
+So far, all characterized prokaryotic lineages cultured
+from soda lakes comprise over 70 different species within
+more than 30 genera [1, 6, 9, 10]. From these, only a lim-
+ited number of genomes have been sequenced today,
+mostly from chemolithoautotrophic sulfur-oxidizing bac-
+teria belonging to the genus Thioalkalivibrio (class Gam-
+maproteobacteria) [1, 11, 12]. It is well established that
+metagenomics enables the recovery of genomes and the
+identification of novel genetic diversity where culturing ef-
+forts fail [13, 14]. In recent years, next-generation sequen-
+cing has recovered a massive number of genomes from
+previously unknown groups of prokaryotes [15, 16],
+including a strikingly large and diverse group called
+“Candidate Phyla Radiation” (CPR), only distantly related
+to other cultured bacterial lineages [17]. Previously, we
+conducted a metagenomics study on soda lakes and re-
+constructed novel genomes from uncultured Bacteroidetes
+and “Candidatus Nanohaloarchaeaota” living in hypersa-
+line Siberian soda brines [7]. Here, we turned our atten-
+tion to the far more complex prokaryotic communities
+living in the sediments of the hypersaline soda lakes from
+the same region. We give a broad overview of all the
+taxonomic groups sequenced and focus on the metabolic
+diversity found in the reconstructed genomes of the most
+abundant, uncultured organisms.
+Results
+Overall prokaryote community structure
+The salinities from the studied soda lakes ranged from
+moderately hypersaline (between 70 and 110 g L−1) to
+salt-saturated (400 g L−1 salt). The soluble carbonate al-
+kalinity was in the molar range, and the pH in all lakes
+was around ten (see Additional file 1: Table S1). To give
+an overview of the overall prokaryotic community com-
+position in each of the samples, we looked at the taxo-
+nomic classification of 16S rRNA genes recovered both
+by amplicon sequencing and direct metagenomics se-
+quencing (Fig. 1, see also Additional file 2: Figure S1;
+Additional file 3). The prokaryotic communities of all
+five sediment samples were highly diverse and consisted
+mostly of uncultured taxonomic groups. Bacteria were
+more abundant than Archaea, regardless of the salinity
+of the overlaying brine [7] (Fig. 1). Euryarchaeota were
+the second and third largest group in the sediments of
+the two salt-saturated lakes comprising ~ 10 and ~ 20%
+of the 16S rRNA genes in the metagenomes. Most
+Euryarchaeota-related OTUs detected by amplicon se-
+quencing belonged either to the uncultured Thermoplas-
+mata group KTK 4A (SILVA classification) or the genera
+Halohasta and Halorubrum (class Halobacteria). In ac-
+cordance with cultivation-dependent studies [6], most
+OTUs assigned to methanogens were from the class
+Methanomicrobia,
+especially
+the
+lithotrophic
+genus
+Methanocalculus (up to ~ 3%) and the methylotrophic
+genus Methanosalsum (Additional file 3).
+The varying ratio of the three dominant bacterial groups,
+Firmicutes, Bacteroidetes (including the newly proposed
+phyla
+Rhodothermaeota
+and
+Balneolaeota
+[18]),
+and
+Gammaproteobacteria, showed no clear trend in relation to
+the salinity in the lakes, but when Firmicutes were domin-
+ant, Bacteroidetes were less abundant and vice versa. Most
+Firmicutes belonged to the order Clostridales. Uncultured
+members from the family Syntrophomonadaceae had a
+relative abundance of more than 5% in all five metagen-
+omes and comprised in two lakes even ~ 11–20% of the
+recovered amplicon sequences. The second most abundant
+Firmicutes order was Halanaerobiales, particularly the
+genus Halanaerobium (family Halanaerobiaceae) and un-
+cultured members of the Halobacteroidaceae. The majority
+of Bacteroidetes-related OTUs could not be assigned down
+to the genus level. The uncultured ML635J-40 aquatic
+group (order Bacteroidales) comprised at least 5% of all five
+prokaryotic communities. This group has been previously
+found to be abundant in Mono Lake [4] (a soda lake) and
+in an anoxic bioreactor degrading cyanobacterial biomass
+under haloalkaline conditions [19]. Two other highly abun-
+dant (up to ~ 8%) uncultured groups from the class Balneo-
+lia (proposed new phylum Balneolaeota [18]) were also
+detected in other soda lakes before [3, 4]. Within the Gam-
+maproteobacteria, the genus Thioalkalivibrio was abundant
+(~ 3% of the total community), but also uncultured
+members of HOC36 were prevailing at moderate salinities.
+Members of the Deltaproteobacteria, Alphaproteobacteria,
+and Chloroflexi comprised up to ~ 10% of the detected 16S
+rRNA gene in some of the metagenomes. The GIF9 family
+of the class Dehalococcoidia was among the top three most
+abundant OTUs in two lakes. The extremely salt-tolerant
+Vavourakis et al. Microbiome  (2018) 6:168 
+Page 2 of 18
+
+and alkaliphilic genera Desulfonatronobacter (order Desulfo-
+bacterales) and Desulfonatronospira (order Desulfovibrio-
+nales)
+were
+the
+dominant
+Deltaproteobacteria.
+Highly
+abundant OTUs, within the Actinobacteria belonged to the
+class Nitriliruptoria and within the Alphaproteobacteria to
+the family Rhodobacteraceae and the genus Roseibaca. The
+important nitrifying genus Nitrobacter (Alphaproteobacteria)
+was present in only one of the lakes with moderate salinity
+(Additional file 3).
+Some bacterial top-level taxa appeared less dominant
+(< 5%) from the 16S rRNA genes recovered from the
+metagenomes but were represented mainly by a single
+highly abundant OTU in the amplicon sequences, in-
+cluding the haloalkaliphilic genus Truepera within the
+phylum Deinococcus-Thermus, the genus Spirochaeata
+within the phylum Spirochaetes, the family BSN166
+within the phylum Ignavibacteriae, the BD2–11 terres-
+trial group within the Gemmatimonadetes, and the
+WCHB1–41
+order
+within
+the
+Verrucomicrobia.
+All
+OTUs
+within
+the
+Thermotogae
+and
+Lentisphaerae
+belonged to uncultured genera from the family Kosmoto-
+gaceae and Oligosphaeraceae, respectively. All Tenericu-
+tes-related OTUs belonged to the class Mollicutes, and
+especially the order NB1-n was dominant. In contrast,
+the phylum Planctomycetes was relatively diverse, with
+at least 11 different genus-level OTUs spread over four
+class-level groups.
+High-throughput genome recovery
+We obtained 717 medium-quality (≥ 50% complete,
+< 10% contamination) and 154 near-complete (≥ 90%
+complete, < 5% contamination) metagenome-assembled
+genomes (MAGs) across three major prokaryote groups:
+Archaea, Bacteria, and CPR (see Additional file 4 and
+Additional file 2: Figure S2). Figures 2 and 3 show the
+top-level phylogeny of all MAGs based on 16 ribosomal
+proteins. The reference database used contains a repre-
+sentative for each major prokaryote lineage [17]. We
+a
+b
+Fig. 1 Abundant prokaryotic groups in five hypersaline soda lake sediments. a Relative abundance of the top-level taxa (those with ≥ 1% abundance
+in at least one dataset) based on 16S rRNA reads in unassembled metagenomic datasets. b Relative abundance of the 16S rRNA OTUs (those with sum
+of abundance in all datasets ≥ 3%) recovered by amplicon sequencing assigned where possible down to the genus-level. Three of the assessed soda
+lakes have a moderate salinity (70–110 g L−1), two are salt-saturated (400 g L− 1)
+Vavourakis et al. Microbiome  (2018) 6:168 
+Page 3 of 18
+
+colored the different phyla from which we obtained a
+MAG
+in
+alternate
+blue
+and
+orange
+colors,
+and
+highlighted the MAGs obtained here in a darker shade.
+Many MAGs belonged to uncultured groups commonly
+detected in soda lake 16S rRNA gene surveys, over 100
+MAGs still belonged to candidate prokaryote phyla and
+divisions that to our knowledge were never detected be-
+fore in soda lakes, including CPR. Although only few
+MAGs had near-complete 16S rRNA genes, in most
+cases we were able to link available taxonomic gene an-
+notations and ribosomal protein phylogeny to the SILVA
+taxonomy of the OTUs assigned to the amplicon se-
+quences, while cross-checking the abundance profiles of
+both MAGs (Additional file 5) and OTUs.
+The soda lake CPR recovered from the metagenomes was
+restricted to a few distinct phyla within the Parcubacteria
+group, mostly affiliating with “Candidatus Nealsonbacteria”
+and “Ca. Zambryskibacteria” [15] (Fig. 2). The first group of
+MAGs encompassed four different branches in our riboso-
+mal protein tree, suggesting a high-phylogenetic diversity,
+with 33 putative new species sampled here (ANI and con-
+DNA matrices given in Additional file 6). The “Ca. Zambrys-
+kibacteria-”related MAGs consisted of at least five new
+species. Few MAGs were recovered from CPR groups also
+detected by amplicon sequencing (see Additional file 2:
+Figure S1), namely the “Ca. Dojkabacteria” (former WS6),
+“Ca. Saccharibacteria” (former TM7), CPR2, and “Ca.
+Katanobacteria” (former WWE3).
+Fig. 2 Maximum-likelihood phylogeny of the CPR and archaeal MAGs based on 16 ribosomal proteins. The archaeal tree is unrooted. The CPR tree is rooted
+to the Wirthbacteria. Alternate orange and blue colors show phyla/classes from which we obtained MAGs (labeled as “Phyla present”). Reconstructed MAGs of
+this study are highlighted by darker shades (labeled as “MAG present”). Phyla/classes for which there was no representative in the reconstructed MAGs of this
+study are shown as gray cartoons (labeled as “Phyla not present”), and the numerical labels are represented at the bottom. Colored circles at the nodes show
+confidence percentage of the bootstraps analysis (100×)
+Vavourakis et al. Microbiome  (2018) 6:168 
+Page 4 of 18
+
+Most archaeal MAGs belonged to the phylum Euryarch-
+aeota and the abundant classes Halobacteria, Methanomi-
+crobia, and Thermoplasmata (related to OTU KTK 4A)
+within. In addition, three Thermoplasmata-related MAGs
+that encoded for the key enzyme for methanogenesis
+(methyl-coenzyme M reductase, mcr) affiliated with refer-
+ence genomes from Methanomassilicoccales, the seventh
+order of methanogens have been recovered [20, 21].
+Another MCR-encoding MAG was closely related to the
+latest
+discovered
+group
+of
+poly-extremophilic,
+methyl-reducing methanogens from hypersaline lakes
+from the class Methanonatronarchaeia [9] (related to
+OTU ST-12K10A). We recovered also one MAG from the
+class Methanobacteria and a high-quality MAG from the
+WCHA1–57
+group
+(“Candidatus
+Methanofastidiosa”
+[22]) in the candidate division WSA2 (Arc I). Several
+MAGs were recovered from the DPANN archaeal
+groups “Ca. Diapherotrites,” “Ca. Aenigmarchaeota,”
+(see Additional file 2: Figure S3) and “Ca. Woesearch-
+aeota” (former Deep Sea Hydrothermal Vent Group 6,
+DHVEG-6). Although we did not reconstruct any
+reasonable-sized MAGs from the TACK superphylum,
+we found several 16S rRNA genes on the assembled
+contigs that affiliated to the Thaumarchaeota (see
+Additional file 1: Table S2).
+Nearly every known bacterial phylum had an extremo-
+philic lineage sampled from our hypersaline soda lake
+sediments (Fig. 3). In most cases, the soda lake lineages
+clearly formed separate branches appearing as sister
+groups to known reference lineages. The highest genome
+recovery was from the same top-level taxonomic groups
+that were also abundant in our 16S rRNA gene analysis.
+From the Verrucomicrobia, most MAGs belonged to the
+order WCHB1-41 (16S rRNA gene identity 92–100%).
+However, in our ribosomal protein tree, they branched
+within the phylum Lentisphaerae. Sixteen Tenericutes
+MAGs from at least 12 different species (Additional file 6)
+were closely related to the NB1-n group of Mollicutes.
+Based on the recovered genome size and encoded meta-
+bolic potential, these organisms are free-living anaerobic
+fermenters of simple sugars, similar to what has recently
+been
+proposed
+for
+“Candidatus
+Izimaplasma”
+[23].
+Fig. 3 Maximum-likelihood phylogeny of the bacterial MAGs (CPR excluded) based on 16 ribosomal proteins. Alternate orange and blue colors show phyla/
+classes from which we obtained MAGs (labeled as “Phyla present”). Reconstructed MAGs of this study are highlighted by darker shades (labeled as “MAG
+present”). Phyla/classes for which there was no representative in the reconstructed MAGs of this study are shown as gray cartoons (labeled as “Phyla not
+present”), and the numerical labels are represented at the bottom. Colored circles at the nodes show confidence percentage of the bootstraps analysis (100×)
+Vavourakis et al. Microbiome  (2018) 6:168 
+Page 5 of 18
+
+Several MAGs belonged to the candidate phyla “Ca.
+Omnitrophica,” “Ca. Atribacteria,” and “Ca. Acetother-
+mia” (former OP1), which were moderately abundant
+also in some sediment (see Additional file 2: Figure S1).
+For the latter phylum, we suspect that four MAGs were
+more closely related to ca. div. WS1 and “Ca. Lindow-
+bacteria” for which only few reference genomes are
+currently available in NCBI (see Additional file 2:
+Figure S4). Due to a high-sequencing coverage, we also
+managed to reconstruct several MAGs from rare Bacteria
+(< 100 amplicon sequences detected, see Additional file 2:
+Figure S1), including the phyla “Ca. Hydrogenedentes,”
+“Ca. Cloacimonetes,” ca. div. BRC1, Elusimicrobia, Caldi-
+serica, and “Ca. Latescibacteria.” The MAGs from the
+latter phylum were more closely related to the recently
+proposed phylum “Ca. Handelsmanbacteria” [15]. Two
+additional MAGs with 16S rRNA gene fragments with
+94–95% identity to the class MD2898-B26 (Nitrospinae)
+were more likely members of ca. div. KSB3 (proposed
+“Ca. Moduliflexus” [24], see Additional file 2: Figure S5).
+Draft genomes of haloalkaliphilic CPR
+Strikingly, members of the CPR related to “Ca. Nealson-
+bacteria” and “Ca. Vogelbacteria” were among the top
+5% of abundant organisms in the surface sediments of
+the soda lakes, especially those with moderate salinity
+(Fig. 4). Like most members of the CPR, the MAGs of
+the four most abundant “Ca. Nealsonbacteria” seem to
+be anaerobic fermenters [25]. They lacked a complete
+TCA cycle and most complexes from the oxidative elec-
+tron transfer chain, except for the subunit F of a
+NADH-quinone oxidoreductase (complex I, nuoF, nuoG,
+nuoA) and coxB genes (complex II). All CPR MAGs had
+a near-complete glycolysis pathway (Embden-Meyerhof-
+Parnas) encoded, but pentose phosphate pathways were
+severely truncated. The commonly encoded F- and
+V-type ATPase can establish a membrane potential for
+symporter-antiporters by utilizing the ATP formed by
+substrate-level phosphorylation during fermentation. All
+CPR have V-type ATPases that can translocate Na+ in
+addition to H+ (see Additional file 2: Figure S6), while
+only two members of the “Ca. Falkowbacteria” had puta-
+tive Na+-coupled F-type ATPases (see Additional file 2:
+Figure S7). The coupling of ATP hydrolysis to sodium
+translocation is advantageous to maintain pH homeosta-
+sis in alkaline environments. Interestingly, with only two
+exceptions [26, 27], all CPR genomes recovered from
+other environments with neutral pH were reported to
+encode only F-type ATPases [28–32]. One low-abundant
+MAG affiliated to “Ca. Peregrinibacteria” contained both
+the
+large
+subunit
+of
+a
+RuBisCO
+(type
+II/III,
+see
+Additional file 2: Figure S8) and a putative phosphoribu-
+lokinase (PRK, K00855) encoded in the same contig.
+This is remarkable because PRK homologs were not
+previously identified among CPR, and RuBisCo form II/
+III was inferred to function in a nucleoside salvage path-
+way [33]. One “Ca. Saccharibacteria” MAG encoded for
+a putative channelrhodopsin (see Additional file 2:
+Figure S9). This is the first rhodopsin found among the
+CPR and suggests that this enigmatic group of organ-
+isms may have acquired evolutionary adaptations to a
+life in sunlit surface environments.
+A previous study showed that most CPR has coccoid
+cell morphotypes with a monoderm cell envelope resem-
+bling those from Gram-positives and Archaea but with a
+distinct S-layer [34]. Thick peptidoglycans coated with
+acidic surface polymers such as teichoic acids help pro-
+tect the cells of Gram-positives against reactive hydroxyl
+ions in highly alkaline environments [35] (Fig. 5a). All
+soda lake CPR had indeed the capability for peptidogly-
+can biosynthesis, but we found proteins typical for
+Gram-negatives for the biosynthesis of lipopolysaccha-
+rides (see Additional file 1: Table S3), homologous to the
+inner membrane proteins of type II secretion systems
+and
+to
+several
+proteins
+associated
+to
+the
+outer
+membrane and peptidoglycan, including OmpA.
+It remains to be determined whether the soda lake
+CPR also lacks an outer membrane and perhaps anchor
+lipopolysaccharides, S-layer proteins, and lipoproteins to
+the inner cell membrane or peptidoglycan. We also
+found gene encoding cardiolipin and squalene synthases.
+Increased levels of cardiolipin and the presence of squa-
+lene make the cytoplasmic membrane less leaky for
+protons [36]. In addition, cation/proton exchangers are
+known to play a crucial role for pH homeostasis in alka-
+liphilic prokaryotes as they help acidify the cytoplasm
+during the extrusion of cations [35]. Putative Na+/H+
+exchangers of the Nha-type and multi-subunit Mnh-type
+were found only within a few soda lake CPR. Secondary
+active transport of K+ might be mediated in most soda
+lake CPR by KefB (COG0475)/kch Kef-type, glutathione-
+dependent K+ transport systems, with or without H+
+antiport (67,68).
+Various soda lake CPR had an acidic proteome, with
+pI curves resembling those found in extremely halophilic
+Bacteria. Intracellular proteins enriched in acidic amino
+acids might be an adaptation to a “salt-in” strategy, i.e.,
+maintaining high intracellular potassium (K+) concentra-
+tions to keep osmotic balance [7, 37] (Fig. 5b; see
+Additional file 2: Figure S10). Such a strategy is energet-
+ically favorable over de novo synthesis or import of
+osmolytes such as ectoine and glycine betaine. We did
+not find genes for the synthesis of organic osmolytes and
+homologs of ABC-type transporters for primary active
+uptake of proline/glycine betaine which were encoded
+only in one MAG (Fig. 5a). For the “Ca. Nealsonbac-
+teria” and “Ca. Vogelbacteria,” the salt-in strategy might
+be a unique feature for the soda lake species explaining
+Vavourakis et al. Microbiome  (2018) 6:168 
+Page 6 of 18
+
+their high abundance in the hypersaline soda lake sedi-
+ments, as we did not found an acidic proteome pre-
+dicted from genomes obtained from other non-saline
+environments (See Additional file 2: Figure S11). The
+uptake of K+ ions remains enigmatic for most soda lake
+CPR. Low-affinity Trk-type K+ uptake transporters (gen-
+erally with symport of H+) (67,68) were encoded only by
+a limited number of MAGs. We found three MAGs
+Fig. 4 Relative abundance and metabolic potential of the dominant species. Abundance values, expressed as reads per kilobase of MAG per gigabase
+of mapped reads (RPKG), were averaged for the top ten abundant MAGs from each dataset that were (likely) the same species (Additional file 5,
+Additional file 6). Population genomes were ranked by their “salinity preference scores”: those recruiting relatively more from moderate salinity datasets
+(cold colors) are drawn to the top, from high salinity datasets (warm colors) to the bottom. The metabolic potential derived from functional marker
+genes (Additional file 7) is depicted by the colored symbols. CBB = Calvin-Benson-Bassham cycle, DNRA = dissimilatory nitrite reduction to ammonia,
+fix. = fixation, red. = reduction, ox. = oxidation, dis. = disproportionation
+Vavourakis et al. Microbiome  (2018) 6:168 
+Page 7 of 18
+
+a
+b
+Fig. 5 (See legend on next page.)
+Vavourakis et al. Microbiome  (2018) 6:168 
+Page 8 of 18
+
+encoding for Kdp-type sensor kinases (kdpD) but no
+corresponding genes for the response regulator (kdpE)
+or for Kdp-ATPases that function as the inducible, high-
+affinity K+ transporters in other Bacteria (67,68). Finally,
+mechanosensitive ion channels (mscS, mscL) and ABC-
+type multidrug transport systems (AcrAB, ccmA, EmrA,
+MdlB, NorM) and sodium efflux permeases (NatB) were
+encoded in almost every MAG. The first might rapidly
+restore the turgor pressure under fluctuating salinity
+conditions by releasing cytoplasmic ions [38].
+Novel abundant groups involved in sulfur, nitrogen, and
+carbon cycles
+A new species of Thioalkalivibrio (family Ectothiorhodospir-
+aceae) was by far the most abundant in the sediments of
+the two salt-saturated lakes (Fig. 4). In the sediment of
+Bitter-1, also a purple sulfur bacterium from the same fam-
+ily was highly abundant. It was closely related to Halorho-
+dospira, a genus also frequently cultured from hypersaline
+soda lakes [1]. None of the abundant Ectothiorhodospira-
+ceae spp. had already a species-representative genome
+sequenced (Additional file 6). The potential of the Thioalk-
+alivibrio spp. for chemolithotrophic sulfur oxidation was
+evident (Additional file 7; see Additional file 8: Information
+S1). Interestingly, the encoded nitrogen metabolisms were
+quite versatile. While Thioalkalivibrio sp. 1 had the poten-
+tial for nitrate reduction to nitrite, Thioalkalivibrio sp. 2
+might perform dissimilatory nitrite reduction to ammonia
+(DNRA; see Additional file 2: Figure S12).
+Two
+deltaproteobacterial
+lineages
+of
+dissimilatory
+sulfate-reducing bacteria (SRB) were highly abundant in
+the soda lake sediment of Bitter-1. One MAG from the
+family Desulfobacteraceae (order Desulfobacterales) is
+the first genome from the genus Desulfonatronobacter. It
+encodes the genes for complete sulfate reduction to sul-
+fide using various electron donors, as well as for the
+complete oxidation of volatile fatty acids and alcohols, a
+unique
+feature
+for
+the
+genus
+Desulfonatronobacter
+among haloalkaliphilic SRB [10] (see Additional file 8:
+Information S2). Fumarate and nitrite (DNRA, NrfAH)
+could be used as alternative electron acceptors. The sec-
+ond dominant lineage was a new species from the genus
+Desulfonatronospira (family Desulfohalobiaceae, order
+Desulfovibrionales). Like other members of this genus, it
+had the potential to reduce or disproportionate partially
+reduced sulfur compounds. In addition, it could also use
+nitrite as an alternative electron acceptor (NrfAH) [6].
+A novel lineage of gammaproteobacterial SOB was
+highly abundant in the sediments of the moderately hy-
+persaline Cock Soda Lake. It appeared as a sister group of
+the family Xanthomonadaceae in the ribosomal protein
+tree. This heterotrophic organism could conserve energy
+through aerobic respiration. It might detoxify sulfide by
+oxidizing it to elemental sulfur (sqr) with subsequent re-
+duction or disproportionation of the polysulfides (psrA)
+chemically formed from the sulfur. It also encoded the po-
+tential for DNRA (nrfA and napC). Genes likely involved
+in sulfide detoxification (sqr and psrA) were found also in
+several other abundant MAGs of heterotrophs, including
+one new abundant species from the family of Nitrilirup-
+toraceae (class Nitriliruptoria, phylum Actinobacteria).
+We found a wide variety of carbohydrate-active enzymes
+in these MAGs, such as cellulases (GH1 family) in
+addition to genes for glycolysis and TCA cycle and a
+chlorophyll/bacteriochlorophyll a/b synthase (bchG gene).
+The latter was also found in other Actinobacteria from the
+genus Rubrobacter [39]. No evidence was found for
+nitrile-degrading potential.
+A second novel, uncultured lineage of Gammaproteo-
+bacteria that was highly abundant at moderate salinities
+branched in our ribosomal protein tree as a sister group
+to the family Halothiobacillaceae. The MAGs encoded
+for a versatile metabolism typical for purple non-sulfur
+bacteria. The MAGs contained puf genes, bch genes,
+genes for carotenoid biosynthesis (not shown), and a
+Calvin cycle for photoautotrophic growth. Alternatively,
+energy may be conserved through aerobic respiration,
+while acetate and proprionate could be taken up via an
+acetate permease (actP) and further used for acetyl-CoA
+biosynthesis and carbon assimilation. Since the sqr gene
+was present, but no dsr or sox genes, the organism
+might oxidize sulfide only to elemental sulfur. One bin
+contained also nifDKH genes suggesting putative diazo-
+trophy, as well as a coenzyme F420 hydrogenase suggest-
+ing photoproduction of hydrogen [40].
+The abundant Euryarchaeota organism showed a clear
+preference for higher salinities. We obtained one highly
+abundant MAG from the class Thermoplasmata that
+encoded a full-length 16S rRNA gene only distantly re-
+lated (91,2% identity, e value 0) to that of a member of
+the KTK 4A group found in a hypersaline endoevaporitic
+microbial mat [8]. The abundant soda lake organism is
+likely a new genus and species. All KTK 4A-related
+MAGs found here encoded for similar heterotrophic,
+fermentative
+metabolisms,
+with
+the
+potential
+for
+(See figure on previous page.)
+Fig. 5 Potential mechanisms for regulating the intracellular pH and cytoplasmic ion content in different CPR phyla. a Membrane transporters,
+channels, and lipids. Peptidoglycan is depicted in gray and S-layer proteins in cyan. b Predicted isoelectric points (bin width 0.2) for the coding
+sequences of MAGs. A representative proteome is depicted for each phylum for which several members had a pronounced acidic peak (see also
+Additional file 2: Figure S11)
+Vavourakis et al. Microbiome  (2018) 6:168 
+Page 9 of 18
+
+anaerobic formate and CO oxidation. The KTK 4A
+might be also primary degraders since they encoded pu-
+tative cellulases (CAZY-families GH1, GH5) and chiti-
+nases (GH18). Interestingly, half of the MAGs encoded a
+putative
+chlorophyll/bacteriochlorophyll
+a/b
+synthase
+(bchG), which is highly unusual for Archaea. Although
+little can be inferred from the presence of only one
+marker gene, a functional bchG was previously also
+found in Crenarchaeota [41]. The remaining two highly
+abundant Euryarchaeota-related MAGs belonged to a
+new species of Halorubrum (Additional file 6).
+Key genes of the Wood-Ljungdahl pathway found in
+novel phylogenetic groups
+More than 50 MAGs were related to the family Syntro-
+phomonadaceae (class Clostridia, phylum Firmicutes)
+based on ribosomal protein phylogeny. All 16S rRNA
+gene sequences found in the MAGS had 86–95% iden-
+tity to sequences obtained from uncultured organisms
+related to the genus Dethiobacter. While an isolated
+strain of Dethiobacter alkaliphilus is a facultative auto-
+troph
+that
+respires
+thiosulfate,
+elemental
+sulfur
+or
+polysulfides with hydrogen as an electron donor [42] or
+disproportionates
+sulfur
+[43],
+other
+haloalkaliphilic
+members
+of
+the
+Syntrophomonadaceae
+are
+reverse
+acetogens, oxidizing acetate in syntrophy with a hydro-
+genotrophic partner [44]. Two populations (different
+species, Additional file 6) were especially abundant in
+Cock Soda Lake (Fig. 4). They encoded for a full
+CODH/ACS complex, the key enzyme for the reductive
+acetyl-CoA or Wood-Ljungdahl pathway (WL) and a
+complete
+Eastern
+branch
+for
+CO2
+conversion
+to
+5-methyl-tetrahydrofolate (Additional file 9) [45, 46].
+Acetogens use the WL to reduce CO2 to acetyl-CoA,
+which can be fixed into the cell or used to conserve en-
+ergy via acetogenesis. Syntrophic acetate oxidizers, some
+sulfate reducing bacteria and aceticlastic methanogens
+run the WL in reverse. Syntrophomonadaceae sp. 2
+encoded for a putative thiosulfate/polysulfide reductase
+as well as a phosphotransacetylase (pta) and an acetate
+kinase (ack) for the ATP-dependent conversion of acet-
+ate to acetyl-CoA. Although alternative pathways for the
+latter interconversion can exist, this second species has
+the complete potential for (reversed) acetogenesis.
+Highly remarkable was the presence of a bacterial-type
+CODH/ACS
+complex
+and
+a
+near-complete
+eastern
+branch of the WL in a highly abundant species in Cock
+Soda Lake from the family Coriobacteriaceae (phylum
+Actinobacteria). This prompted us to scan all 871 MAGs
+for the presence of acsB encoding for the beta-subunit
+of the oxido-reductase module of CODH/ACS. We con-
+firmed an encoded
+(near)-complete
+WL in several
+additional organisms belonging to phylogenetic groups
+not
+previously
+associated
+with
+this
+pathway
+[46]
+(Additional file 9). We removed the Coriobacteriaceae
+acsB genes from the final dataset to construct a phylo-
+genetic tree since they were < 500 aa (Fig. 6) but found
+seven MAGs from the OPB41 class within the Actino-
+bacteria (16S rRNA gene fragment identity 94–96%).
+The eastern branch of WL can function independently
+in folate-dependent C1 metabolism [45], but the pres-
+ence of the Western-branch in a phylum that comprises
+mostly aerobic isolates is very surprising. The WL in
+combination with the potential for acetate to acetyl-CoA
+interconversion (pta/ack) and a glycolysis pathway were
+also present in the soda lake MAGs from the phyla “Ca.
+Handelsmanbacteria,” “Ca. Atribacteria” (latter branched
+within the “Ca. Acetothermia”), and the class LD1-PA32
+(Chlamydiae), suggesting all these uncultured organisms
+might be heterotrophic acetogens. However, it should be
+noted that a PFOR typically connecting glycolysis to the
+WL was only encoded in the LD1-PA32 MAGs. More-
+over, from the genetic make-up alone, it cannot be
+excluded that acetate is activated, and the WL run in
+reverse for syntrophic acetate oxidation. Finally, the
+novel acsB genes from soda lake Halanaerobiaceae,
+Natranaerobiaceae, and Halobacteroidaceae (Firmicutes)
+and from Brocadiaceae and Planctomycetaceae (Plancto-
+mycetes) disrupt the previously proposed dichotomy
+between Terrabacteria and Gracilicutes bacterial groups
+unifying 16S rRNA and acsB gene phylogenies [46] and
+suggest a far more complex evolutionary history of the
+WL pathway than previously anticipated.
+Discussion
+Extensive
+classical
+microbiology
+efforts
+have
+been
+already undertaken to explore the unique extremophilic
+microbial communities inhabiting soda lakes. These un-
+covered the presence of most of the functional groups
+participating in carbon, nitrogen, sulfur, and minor
+element cycling at haloalkaline conditions. The main re-
+sults of this work are summarized in several recent re-
+views [1, 6, 47, 48]. Since most microbes, including
+those living in soda lakes, still evade all cultivation ef-
+forts, a very effective way to discover new microbes and
+assess their physiology based on their genetic repertoire
+is either through single cell genomics or by directly se-
+quenced environmental DNA. This exploratory metage-
+nomics
+study
+performed
+on
+soda
+lake
+sediments
+effectively overcame the existing cultivation bottleneck.
+First, we expanded the known diversity of CPR consider-
+ably with the first genomes of poly-extremophiles sam-
+pled from soda lake sediments. Although the presence of
+16S rRNA genes from CPR in marine sediments and hy-
+persaline microbial mats was previously shown [34],
+until now, CPR MAGs were mainly obtained from deep,
+subsurface environments [15, 26, 29, 32, 49–52], and hu-
+man microbiota [30]. Despite being highly abundant
+Vavourakis et al. Microbiome  (2018) 6:168 
+Page 10 of 18
+
+100 %
+90-100 %
+70-90 %
+50-70 %
+some MAGs
+all MAGs
+Bootstraps
+Genes present
+Glycolysis (EMP)
+PFOR
+WL-Eastern branch
+H4MPT
+TH4
+WL-Western branch
+CODH/ACS
+Acetogenesis/ 
+acetate activation 
+(pta/ack)
+0.4
+PVC group (Chlamydiae LD1-PA32)
+Syntrophorhabdus aromaticivorans
+PVC group bacterium CSSed11_184
+Aerophobetes bacterium SCGC_AAA255-F10
+Ca. Acetothermia
+Ca. Handelsmanbacteria
+Planctomycetaceae
+Anaerolineae
+Firmicutes
+Brocadiaceae
+Planctomycetes
+Methanomassiliicoccales
+Halobacteroidaceae
+Natranaerobiaceae
+Methanomicrobiales
+Desulfonatronospira
+Firmicutes
+Dehalococcoidia
+Armatimonadetes bacterium CSP1-3
+Deltaproteobacteria
+Thermodesulfobacteria
+Desulfobulbaceae
+Halanaerobiaceae
+Nitrospirae
+Actinobacteria (OPB41)
+Fig. 6 Maximum likelihood phylogeny of the bacterial-type acetyl-coA synthases (acsB) found in the MAGs. Only sequences ≥ 500 aa
+were included. Lineages for which we discovered the Wood-Ljungdahl (WL) in this study are highlighted in orange, and the presence
+of genes in the respective MAGs related to WL, glycolysis, pyruvate, and acetate conversion is indicated by the colored symbols (see
+also Additional file 9: Dataset S6). Additional lineages found in this study are marked in blue. The three was rooted according to [46].
+Circles at the nodes show confidence percentage of the bootstraps analysis (100×). EMP = Embden-Meyerhof-Parnas, PFOR = pyruvate:ferredoxin
+oxidoreductase complex, pta = phosphotransacetylase gene, ack = acetate kinase gene, H4MPT = tetrahydromethanopterin-linked pathway, TH4 =
+tetrahydrofolate pathway, CODH/ACS = carbon monoxide dehydrogenase/acetyl-CoA synthase. PVC group bacterium CSSed11_184 is likely a member
+of the WCHB1-41 class within the Verrucomicrobia
+Vavourakis et al. Microbiome  (2018) 6:168 
+Page 11 of 18
+
+here, CPR went unnoticed in previous amplicon sequen-
+cing studies. This might be due to the fact that many
+CPR representatives have random inserts of various
+length in their 16S rRNA genes or due to primer mis-
+matches [29, 34]. This illustrates also that direct metage-
+nomics should not only be preferred over amplicon
+sequencing to infer functional potential, but the former
+is far more effective for the discovery of novel organ-
+isms. Second, we obtained many more genomes from
+“traditional” bacterial phyla such as the Planctomycetes
+and Chloroflexi, as well as candidate phyla, for which no
+soda lake isolates, hence no genomes were previously
+obtained. Third, even within the sulfur cycle, the most
+active and frequently studied element cycle in soda lakes
+[1], we found considerable metabolic novelty. Finally, we
+found the Wood-Ljungdahl pathway in several novel
+phyla, not closely related to any known acetogens,
+methanogens, or sulfate-reducing bacteria [46]. The lat-
+ter shows that our sequencing recovery effort has also
+significantly contributed to the discovery of metabolic
+novelty within various prokaryote phylogenetic groups.
+Salinity is often considered to be the major factor
+shaping prokaryote community composition in diverse
+habitats [53, 54]. Extreme halophilic Euryarchaeota
+seem to be always the dominant group in salt-saturated
+hypersaline brines, both those with neutral or alkaline
+pH [1, 7, 37]. Here, we found that although these
+haloarchaea are still relatively more abundant in the sed-
+iments exposed to brines with salt-saturating conditions,
+the clear majority of microbes in all investigated hyper-
+saline soda lake sediments are Bacteria. It could be
+hypothesized that the sediment is a hide-out for the
+extreme alkalinity and salinity governing the water
+column, and that sediment stratification, especially in
+the anoxic part, offers plenty of opportunities for niche
+diversification. On the other hand, it should no longer
+be a surprise that soda lakes are such productive and
+biodiverse
+systems.
+First,
+it
+has
+been
+previously
+elaborated that soda lake organisms are exposed to
+approximately half the osmotic pressure in sodium
+carbonate-dominated
+brines
+compared
+to
+sodium
+chloride-dominated brines with the same Na+ molarity
+[47]. Second, nitrogen limitation in the community can
+be overcome when many members contribute to the
+fixation of atmospheric N2, and various forms of organic
+nitrogen are efficiently recycled. The soda lakes exam-
+ined in this study were also eutrophic, and sulfur com-
+pounds were abundant. Sulfide is also far less toxic at
+high pH as it mostly occurs in the form of bisulfide
+(HS−). Besides the evident high metabolic and taxo-
+nomic diversity of dissimilatory sulfur-cycling bacteria, a
+diverse heterotrophic community can be sustained com-
+prising both generalist and very specialized carbon de-
+graders. Less eutrophic soda lakes might not suffer from
+carbon
+limitation
+either,
+due
+to
+a
+presence
+of
+high-bicarbonate concentrations. These effectively elim-
+inate the inorganic carbon limitation for primary pro-
+ducers who are highly active in soda lakes, especially
+Cyanobacteria [55, 56]. Third, light that penetrates the
+surface of the sediment seems to stimulate oxygenic and
+anoxygenic phototrophic growth. Moreover, various het-
+erotrophs, such as the rhodopsin-containing haloarchaea
+and Bacteroidetes, have the option to tap into this un-
+limited energy source for example to help sustain the
+costly maintenance of osmotic balance. Unexpectedly,
+we even found the first rhodopsin encoded by a member
+of the CPR. Fourth, tight syntrophic relations, as pro-
+posed for CPR members and Syntrophomonadaceae
+spp., might be the solution to successful growth in an
+energetically challenging environment.
+Since our metagenomes are snapshots in time and space,
+the failure to reconstruct specific MAGs gives no conclu-
+sive evidence for the absence of certain microbial-mediated
+element transformation in hypersaline soda lake sediments.
+Additionally, technical limitations of the assembly and bin-
+ning of highly micro-diverse genome populations might
+hamper genome recovery [57]. More importantly, the
+abundance of a specific microbe is not necessarily corre-
+lated to the importance of its performance in an ecosystem.
+Many metabolic capacities are redundant, and often key
+transformations are reserved for a few rare organisms that
+might proliferate for a short time-span when specific condi-
+tions allow for it. For example, although no MAGs were re-
+covered from chemolithoautotrophic nitrifiers [58], we did
+detect a Nitrobacter-related OTU by amplicon sequencing
+and assembled 16S rRNA genes from Thaumarchaeota,
+suggesting bacterial and archaeal nitrifiers are present in
+the surface sediments of soda lakes at very low abundance.
+Finally, the method of DNA isolation might impact the
+community composition apparent in the final metagenome
+sequenced. Environmental samples contain complex mix-
+tures of different organisms, and it is impossible to find a
+protocol where the DNA from every single organism is ex-
+tracted as efficiently without compromising the final quality
+of the extracted DNA. However, since we find all the im-
+portant taxonomic and functional groups known from pre-
+vious cultivation-dependent studies back in either our
+amplicon sequencing datasets or our directly sequenced
+metagenomes, we are confident that the community com-
+position and the MAGs presented here are representative
+for the microbiomes of the soda lake sediments in the
+Kulunda Steppe.
+Conclusion
+Years of intensive microbiological research on soda lakes
+seem to have paid off, since many of the described gen-
+era we could detect here have a cultured representative
+from soda lakes. However, as many of the abundant
+Vavourakis et al. Microbiome  (2018) 6:168 
+Page 12 of 18
+
+lineages and groups found in soda lake sediments are
+still uncultured, metagenomics proved to be a helpful
+tool to gain primary insights in the potential physiology
+and ecology of these poly-extremophilic prokaryotes.
+We reconstructed the first genomes for many of such
+organisms and proposed new functional roles for the
+most abundant ones. Future studies should provide
+more in depth analyses of these genomes, especially
+from the less abundant organisms that might perform
+key ecological processes, such as methanogens and nitri-
+fiers. In addition, they should focus on gaining physio-
+logical culture-based evidence or proof for in situ
+activity for the abundant organisms described here. The
+key metabolic insights provided by this metagenomics
+study can lead to the design of new cultivation strategies.
+In general, sediment communities are far more complex
+than those found in the corresponding water column
+[53, 59] and are therefore often considered too complex
+for efficient metagenomic analysis. Many of the novel
+lineages found here may therefore have related neutro-
+philic lineages in marine and freshwater sediments that
+await discovery. We demonstrate here that, by providing
+sufficient sequencing depth, the “state of the art metage-
+nomics toolbox” can effectively be used on sediments as
+well.
+Methods
+Site description and sample collection
+The top 10 cm sediments from four hypersaline, eutrophic
+soda lakes located in the Kulunda Steppe (south-western
+Siberia, Altai, Russia) were sampled in July of 2010 and
+2011. General features and exact location of the sampled
+soda lakes are summarized in Additional file 1: Table S1; a
+map of the area was published previously [5]. Cock Soda
+Lake (a stand-alone lake, sampled both in 2010 and 2011)
+and Tanatar-3 (Tanatar system) were moderately hypersa-
+line (~ 100 g L−1) with sandy sediment, while Tanatar-1
+and Bitter-1 (Bitter system) were salt-saturated (400 g L−1)
+with sulfide-rich sapropel sediments underlined by rock
+trona deposits [7, 60]. Especially, Bitter-1 harbors a very
+active microbial community, probably due to its high-
+organic and -mineral content. Surface sediments were col-
+lected by a plastic corer into sterile glass containers and
+transported to the laboratory in a cooler.
+DNA isolation, 16S rRNA amplicon, and metagenomic
+sequencing
+The colloidal fraction of each sediment sample (~ 10%
+of 50 g) was separated from the course sandy fraction by
+several short (30–60 s) low-speed (1–2,000 rpm in
+50 mL Falcon tubes) centrifugation steps and washed
+with 1–2 M NaCl solution. The pelleted colloidal sedi-
+ment fraction was first subjected to 3 cycles of freezing
+in liquid nitrogen/thawing, then re-suspended in 0.1 M
+Tris (pH 8)/10 mM EDTA, and then subjected to harsh
+bead beating treatment. Next, the samples were incu-
+bated with lysozyme (15 mg/mL) for 2 h at 37 °C
+followed by a SDS (10% w/v) and proteinase K (10 μg/
+mL) treatment for 30 min. at 45 °C. High molecular
+weight DNA was isolated using phenol/chloroform ex-
+traction, quality-checked, and sequenced as described
+previously [7]. Direct high-throughput sequencing of the
+DNA was performed on an Illumina HiSeq 2000 plat-
+form to generate 150 b paired-end reads. Amplification
+of the V4-V6 region of prokaryote 16S rRNA genes
+using barcoded 926F-1392R primers, amplicon purifica-
+tion, quantification, and Roche (454)-sequencing was
+performed together in a batch with brine samples from
+the same sampling campaigns. Barcodes and adapter se-
+quences were removed from de-multiplexed amplicon
+sequence reads and analyzed with the automated NGS
+analysis pipeline of the SILVA rRNA gene database pro-
+ject [61] (SILVAngs 1.3, database release version 128)
+using default parameters. The OTUs (97% identity)
+assigned down to the genus level were only considered
+when they had a relative abundance ≥ 0.1% in at least
+one of the five datasets.
+Processing metagenomics reads, assembly, binning, and
+post-binning
+Metagenomic raw reads were quality trimmed using
+Sickle [62] (version 1.33), and only reads ≥ 21 b were
+retained. The prokaryotic community structure at taxo-
+nomic top levels was extrapolated from ten million ran-
+domly sampled singletons from each dataset. Candidate
+16S rRNA fragments > 90 b were identified [63] and
+compared against the SILVA SSU database 128 (blastn,
+min. length 90, min. identity 80%, e value 1e-5). To ver-
+ify that the microbial community composition was in-
+deed
+mostly
+prokaryotic,
+we
+did
+a
+more
+general
+screening of the metagenomics reads that identified also
+candidate 18S rRNA fragments > 90 b (see Additional
+file 1: Tables S4-S5). The complete trimmed read sets
+were assembled into contigs ≥ 1 kb with MEGAHIT [64]
+(v1.0.3–6-gc3983f9) using paired-end mode, k min = 21,
+k max = 131, k step = 10. Genes were predicted using
+Prodigal [65] (v.2.6.2) and RNAs with rna_hmm3 [66]
+and tRNAscan-SE [67]. Assembled 16S rRNA sequences
+were compared to a manually curated version from the
+SILVA SSU database (e value ≥ 1e-5). Predicted protein
+sequences
+were
+annotated
+against
+KEGG
+with
+GhostKOALA (genus_prokaryotes + family_eukaryotes
++ viruses) [68]. Marker genes for central metabolic
+pathways and key environmental element transforma-
+tions were identified based on K number assignments
+[15, 69–71].
+Contigs ≥ 2.5 kb were binned with METABAT [72]
+(superspecific mode) based on differential coverage
+Vavourakis et al. Microbiome  (2018) 6:168 
+Page 13 of 18
+
+values obtained by mapping all five trimmed readsets to
+all five contig sets with Bowtie2 [73]. The bins were sub-
+jected to post-binning (an overview of the workflow is
+given in Additional file 2: Figure S13). Bins were
+assessed with lineage-specific single copy genes using
+CheckM [74] and further processed with the metage-
+nomics workflow in Anvi’o [75] (v2.3.2). Since Candidate
+Phyla Radiant (CPR) is not included in the CheckM ref-
+erence trees and are likely to have low-genome com-
+pleteness, we used an existing training file of 797 CPR
+genomes to identify putative CPR bins [76]. Bins with
+CheckM-completeness ≥ 50% (884 out of 1778) and an
+additional four CPR bins were further processed. Coding
+sequences
+were
+annotated
+for
+taxonomy
+against
+NCBI-nr (July, 2017) with USEARCH [77] (5.2.32) to
+verify that most hits in each bin were to prokaryotic ref-
+erences. Phage or viral contigs were manually removed.
+Genome
+contamination (redundancy)
+was estimated
+based on marker sets of universal single copy genes
+identified for Bacteria [30] and Archaea [78] as imple-
+mented in Anvi’o. Genome coverage was obtained by
+mapping trimmed reads with BBMap [79] v36.x (kfilter
+31, subfilter 15, maxindel 80). Bins with ≥ 5% redun-
+dancy were further refined with Anvi’o using circle phy-
+lograms
+(guide
+trees
+tnf-cov:
+euclidian
+ward)
+and
+scanned again for CPR. Post-binning resulted in a total
+of 2499 metagenome-assembled genomes (MAGs), of
+which 871 were either medium-quality genome drafts
+(CheckM estimated completeness ≥ 50% and contamin-
+ation ≤ 10% [80], Additional file 4) or lower quality draft
+genomes from CPR.
+Phylogeny of the MAGs was assessed based on 16
+single-copy ribosomal proteins and representative refer-
+ence genomes of major prokaryote lineages across the
+tree of life [17]. Individual ribosomal proteins in our
+MAGs were identified by K number assignments. Only
+ribosomal proteins ≥ 80 aa were considered. Initial
+maximum-likelihood (ML) trees were constructed to de-
+termine which organisms belonged to the Archaea, Bac-
+teria, or CPR with FastTree 2 [81] (WAG + CAT). Final
+separate trees for the three distant evolutionary groups
+were constructed in the same manner. Each ribosomal
+protein set was aligned separately with MAFFT [82]
+(v7.055b, − auto) and concatenated only if a MAG
+encoded at least 8 out of 16 proteins. For all trees, a
+100× posterior bootstraps
+analysis
+was
+performed.
+Phylogenetic trees were visualized together with gen-
+ome statistics and abundance information using iTOL
+[83]. We cross-checked the taxonomic assignments
+based on the phylogeny of the ribosomal protein cas-
+sette
+with
+the
+top
+hit
+contig annotations
+against
+NCBI-nr and with the reference lineage obtained with
+CheckM. Lastly, we manually corrected the MAGs for
+misplaced 16S rRNA genes. The final trees presented
+in the manuscript were redrawn using FigTree v1.4.3
+[84].
+Detailed genome analyses
+CPR
+MAGs
+were
+re-annotated
+more
+thoroughly:
+genes were predicted with Prokka [85], and functional
+predictions were performed by running InterProScan
+5 locally on the supplied COG, CDD, TIGRFAMs,
+HAMAP, Pfam, and SMART databases [86]. BLAST
+Koala was used for KEGG pathway predictions [68].
+To find putative carbohydrate-active enzymes in all
+final MAGs, we used the web-resource dbCAN [87]
+to annotate all predicted proteins ≥ 80 aa against
+CAZy [88].
+To identify the top ten abundant MAGs from each re-
+spective dataset, ten million randomly sampled single-
+tons were mapped onto each MAG with a cut-off of 95%
+identity in minimum of 50 bases. Coverage values were
+additionally normalized for genome size and expressed
+as reads per kilobase of sequence per gigabase of
+mapped reads (RPKG) [89]. A positive score (from 871
+to 1) was assigned to each MAG according to the rank-
+ing of the summed RPKG of MAGs in the high-salinity
+datasets (B1Sed10 and T1Sed) and a negative score ac-
+cording to the ranking of the summed RPKGs in the
+moderate salinity datasets (CSSed10, CSSed11, T3Se
+d10). Both scores were summed to get a “salinity prefer-
+ence score” with MAGs recruiting preferably from high
+salinity datasets on the positive end, moderate salinity
+datasets in the negative end, and those without prefer-
+ence in the middle.
+We determined species delineation for the most
+abundant MAGs and their closest reference genomes
+(NCBI-nr) by Average Nucleotide Identity (ANI) and
+conserved DNA-matrices, as follows [90]: ANI ≥ 95%,
+conDNA ≥ 69% = same species, ANI ≥ 95%, condDNA
+< 69% = might be same species, ANI < 95%, condDNA
+< 69% = different species. Single gene trees based on
+maximum
+likelihood
+were
+constructed
+with
+un-
+trimmed alignments (MAFFT, L-INS-i model) and
+FastTree 2 (WAG + CAT, increased accuracy, -spr4
+-mlacc 2 -slownni) using 100× bootstraps. References
+were pulled from eggNOG (v4.5.1) [91] and supple-
+mented
+with
+sequences
+from
+NCBI-nr
+or
+refined
+according to [7, 33, 46, 92–94]. The curated MAGs
+were
+scanned
+for
+the
+presence
+of
+rhodopsin
+sequences with the hmmsearch software [95] and a
+profile
+hidden
+Markov
+model
+(HMM)
+of
+the
+bacteriorhodopsin-like protein family (Pfam accession
+number
+PF01036).
+The
+identified
+sequences
+with
+significant similarity were aligned together with a
+curated database composed of a collection of type-1
+rhodopsins, using MAFFT (L-INS-i accuracy model)
+[82]. This protein alignment was further utilized to
+Vavourakis et al. Microbiome  (2018) 6:168 
+Page 14 of 18
+
+construct a maximum likelihood tree with 100× boot-
+strap with FastTree 2 [81]. All other genes were
+identified using the KEGG annotation.
+Additional files
+Additional file 1: Table S1. General features of the four sampled soda
+lakes at time of sampling. Table S2. SILVA classification of the 16S rRNA
+gene sequences found in all ≥1 kb contigs of five soda sediment
+metagenomic datasets. Table S3. Enzymes involved in lipopolysaccharide
+biosynthesis found among different members of the CPR. Table S4.
+Sub-kingdom classification of candidate SSU rRNA gene fragments
+found in subsamples of 10 million random forward reads from the
+five soda sediment metagenomes. Table S5. Top-level taxonomic
+classification of the 18S rRNA gene fragments found in subsamples
+of 10 million random forward reads from the five soda sediment
+metagenomes. Table S6. Description of the metagenomic datasets,
+NCBI Sequence Read Archive (SRA) accession numbers and general
+statistics of the assembled contigs. (PDF 740 kb)
+Additional file 2: Figure S1. Taxonomic fingerprints determined by 16S
+rRNA gene amplicon sequencing. Figure S2. Genome statistics of the
+871 MAGs. Figure S3. Phylogeny of MAGs belonging to “Candidatus
+Aenigmarchaeota” and “Ca. Nanohaloarchaeota”. Figure S4. Phylogeny of
+MAGs related to “Candidatus Acetothermia”, candidate division WS1 and
+“Candidatus Lindowbacteria”. Figure S5. Phylogeny of MAGs related to
+candidate division KSB3 and “Candidatus Schekmanbacteria”. Figure S6.
+Multiple sequence alignment of the V-type ATPase subunits K. Figure S7.
+Multiple sequence alignment of the F-type ATPase subunits c. Figure S8.
+Maximum likelihood tree of the large subunits of RuBisCo and RubisCo-
+like proteins. Figure S9. Maximum likelihood tree of the putative
+rhodopsins. Figure S10. Predicted isoelectric points (pI) profiles of all
+MAGs from CPR members. Figure S11. Predicted isoelectric points
+profiles for members of the “Ca. Nealsonbacteria” and “Ca. Vogelbacteria”.
+Figure S12. Multiple sequence alignment of the dissimilatory
+cytochrome c nitrite reductases (nrfA/TvNiR, K03385). Figure S13.
+Overview of the post-binning workflow used for genome recovery.
+(PDF 6548 kb)
+Additional file 3: Dataset S1. Relative abundance of the OTUs assigned
+to the genus-level within the Archaea, Bacteria and organelles from
+Eukaryota detected by 16S rRNA gene amplicon sequencing. The OTUs
+with less than 0.1% abundance accross all five datasets are not shown.
+The names of highly abundant genera (≥1% in at least one of the data-
+sets) are shown in bold. (XLSX 24 kb)
+Additional file 4: Dataset S2. Organism names, statistics and general
+description incl. Completeness and contamination estimates, phylogeny
+and DDBJ/EMBL/Genbank accession numbers of the metagenome
+assembled genomes (MAGs) described in this paper. All submitted
+versions described in this paper are version XXXX01000000. Size =
+recovered genome size, Completeness (Compl1), contamination (Cont),
+strain heterogenity (Str het) and Taxon CheckM were inferred from
+lineage-specific marker sets and a reference tree build with CheckM [74].
+Additional completeness (compl2) and redundancy (red) estimates were
+inferred based on the presence of universal single copy genes for Bacteria
+and Archaea [75]. Decision and confidence intervals from the Candidate
+Phyla Radiation (CPR) scan [75] are given, as well as the taxonomy of the
+besthit in SILVA when 16S rRNA genes were present. Phylum/class 16
+ribosomal proteins is the taxonomy derived from our ribosomal protein
+trees (see main text: Figs. 2 and 3). OTU gives the inferred link of a
+population genome with our 16S rRNA gene amplicon dataset
+(Additional file 3). (XLSX 253 kb)
+Additional file 5: Dataset S3. Estimated abundance and derived
+salinity preference from each MAG in each metagenomic dataset
+expressed as Reads per Kilobase of MAG per Gigabase of mapped reads
+(RPKG) and “salinity preference score” (see Methods section), basis for
+Fig. 4. (XLSX 143 kb)
+Additional file 6: Dataset S4. Average Nucleotide Identity (ANI) and
+conserved DNA (condna) matrices to determine species delineation
+between the most abundant MAGs shown in Fig. 4, closely related
+(less abundant) MAGs and NCBI reference genomes. Decision matrix
+shows: 1 = same species, − 1 = might be same species, 0 = different
+species (see Methods section). (XLSX 1161 kb)
+Additional file 7: Dataset S5. Sheet 1 Presence and absence of marker
+genes and putative carbohydrate-active enzymes in the MAGs to infer putative
+roles in C, N and S element cycles based on K-number assignments and CAZy
+annotations. Sheet 2 Summary basis for Fig. 4. (XLSX 41 kb)
+Additional file 8: Information S1. More detailed description of the
+main metabolisms encoded by Thioalkalivibrio-related MAGs.
+Information S2 More detailed description of the main metabolisms
+encoded by Deltaproteobacterial-related MAGs. (PDF 219 kb)
+Additional file 9: Dataset 6. Sheet 1 shows the MAGs positive for the
+marker gene acsB (K14138) encoding an acetyl-CoA synthase (ACS). The
+basis for Fig. 6, namely presence and absence of key genes involved in
+the Wood-Ljungdahly pathway, acetogenesis, methanogenesis, glycolysis
+and pyruvate to CO2 conversion is shown for each MAG. Sheet 2 shows
+the MAGs positive for the marker gene cdhC (K00193) encoding for the
+beta subunit of an acetyl-CoA decarboxylase synthase complex. While
+acsB and cdhC correspond roughly to the Bacterial-type and Archaeal-
+type (methanogens) enzymes with the same function, we found few
+discrepancies between marker gene and genome phylogeny within the
+Methanomassiliicoccaceae and Chloroflexi. (XLSX 52 kb)
+Acknowledgments
+We thank Dr. Nikolai Chernych for his technical assistance during the
+isolation and purification of metagenomics DNA. We also thank the
+Department of Energy Joint Genome Institute for sequencing the
+metagenomes.
+Funding
+CDV and GM were supported by the ERC Advanced Grant PARASOL (no. 322551).
+A-SA and RG were supported by the research grant 17-04828S from the Grant
+Agency of the Czech Republic. MM was supported by the Czech Academy of
+Sciences (Postdoc program PPPLZ application number L200961651). DYS was
+supported by the SIAM/Gravitation Program (Dutch Ministry of Education and
+Science, grant 24002002) and by the Russian Science Foundation (grant 16–14-
+00121). Sequencing was performed by the U.S. Department of Energy Joint
+Genome Institute, a DOE Office of Science User Facility, as part of the Community
+Sequencing Program (contract no. DE-AC02- 05CH11231).
+Availability of data and materials
+The raw sequence reads of the five metagenomes have been deposited to
+the NCBI Sequence Read Archive (see Additional file 1: Table S6 for accession
+numbers and read and contig statistics). The final 871 MAGs described in this
+paper have been deposited as Whole Genome Shotgun projects at DDBJ/
+EMBL/GenBank, and accession numbers are listed in Additional file 4
+(BioProject ID PRJNA434545). All versions described in this paper are version
+XXXX01000000. The cleaned and dereplicated amplicon sequence datasets
+are available in FigShare (https://figshare.com/s/7684627445e3621aba24).
+Maximum likelihood trees based on the concatenated alignment of 16
+ribosomal proteins, basis for Figs. 2 and 3, in newick format (.tre file) and
+complementary datasets (used to plot completeness, contamination,
+genome recovery size, G + C mol% and RPKG in iTOL), as well as K number
+assignments for the predicted proteins of all MAGs (KEGG-orthologues,
+Ghost Koala) and the fully annotated CPR MAGs supporting the conclusions
+of this article are also available in FigShare (https://figshare.com/s/
+7684627445e3621aba24).
+Authors’ contributions
+GM and DYS initiated this study and were responsible for the fieldwork,
+sample preparation, and sequencing effort. CDV conceptualized the research
+goals under supervision of DYS and GM, and performed the bioinformatics
+analysis under close guidance of A-SA and RG. CDV is the primary author of
+this manuscript. MM, RG, and CDV prepared the main figures. All authors
+read and approved the final manuscript.
+Ethics approval and consent to participate
+Not applicable.
+Vavourakis et al. Microbiome  (2018) 6:168 
+Page 15 of 18
+
+Consent for publication
+Not applicable.
+Competing interests
+The authors declare that they have no competing interests.
+Publisher’s Note
+Springer Nature remains neutral with regard to jurisdictional claims in
+published maps and institutional affiliations.
+Author details
+1Microbial Systems Ecology, Department of Freshwater and Marine Ecology,
+Institute for Biodiversity and Ecosystem Dynamics, Faculty of Science,
+University of Amsterdam, Postbus 94248, 1090, GE, Amsterdam, the
+Netherlands. 2Department of Aquatic Microbial Ecology, Institute of
+Hydrobiology, Biology Centre CAS, Na Sadkach 7, 370 05 Ceske Budejovice,
+Czech Republic. 3Winogradsky Institute of Microbiology, Research Centre of
+Biotechnology, Russian Academy of Sciences, 60 let Oktyabrya pr-t, 7, bld. 2,
+Moscow, Russian Federation117312. 4Environmental Biotechnology,
+Department of Biotechnology, Delft University of Technology, Van der
+Maasweg 9, 2629, HZ, Delft, the Netherlands.
+Received: 23 June 2018 Accepted: 3 September 2018
+References
+1.
+Sorokin DY, Berben T, Melton ED, Overmars L, Vavourakis CD, Muyzer G.
+Microbial diversity and biogeochemical cycling in soda lakes. Extremophiles.
+2014;18:791–809.
+2.
+Oduor SO, Kotut K. Soda lakes of the East African Rift System: the past, the
+present and the future. In: Schagerl M, editor. Soda lakes of East Africa.
+Berlin: Springer; 2016. p. 365–74.
+3.
+Mesbah NM, Abou-El-Ela SH, Wiegel J. Novel and unexpected prokaryotic
+diversity in water and sediments of the alkaline, hypersaline lakes of the
+Wadi An Natrun, Egypt. Microb Ecol. 2007;54:598–617.
+4.
+Humayoun SB, Bano N, James T, Hollibaugh JT. Depth distribution of
+microbial diversity in Mono Lake, a meromictic soda lake in California. Appl
+Environ Microbiol. 2003;69:1030–42.
+5.
+Foti MJ, Sorokin DY, Zacharova EE, Pimenov NV, Kuenen JG, Muyzer G.
+Bacterial diversity and activity along a salinity gradient in soda lakes of the
+Kulunda Steppe (Altai, Russia). Extremophiles. 2008;12:133–45.
+6.
+Sorokin DY. Anaerobic haloalkaliphiles. eLS. 2017; https://doi.org/10.1002/
+9780470015902.a0027654.
+7.
+Vavourakis CD, Ghai R, Rodriguez-Valera F, Sorokin DY, Tringe SG,
+Hugenholtz P, et al. Metagenomic insights into the uncultured diversity and
+physiology of microbes in four hypersaline soda lake brines. Front Microbiol.
+2016;7:211.
+8.
+Sørensen KB, Canfield DE, Oren A. Salinity responses of benthic microbial
+communities in a solar saltern (Eilat, Israel). Appl Environ Microbiol. 2004;70:
+1608–16.
+9.
+Sorokin DY, Makarova KS, Abbas B, Ferrer M, Golyshin PN, Galinski EA, et al.
+Discovery of extremely halophilic, methyl-reducing euryarchaea provides
+insights into the evolutionary origin of methanogenesis. Nat Microbiol.
+2017;2:17081.
+10.
+Sorokin DY, Chernyh NA, Poroshina MN. Desulfonatronobacter acetoxydans
+sp. nov.,: a first acetate-oxidizing, extremely salt-tolerant alkaliphilic SRB from
+a hypersaline soda lake. Extremophiles. 2015;19:899–907.
+11.
+Ahn A-C, Meier-Kolthoff JP, Overmars L, Richter M, Woyke T, Sorokin DY,
+et al. Genomic diversity within the haloalkaliphilic genus Thioalkalivibrio.
+PLoS One. 2017;12:e0173517.
+12.
+Sorokin DY, Kuenen JG. Haloalkaliphilic sulfur-oxidizing bacteria in soda
+lakes. FEMS Microbiol Rev. 2005;29:685–702.
+13.
+Albertsen M, Hugenholtz P, Skarshewski A, Nielsen KL, Tyson GW, Nielsen
+PH. Genome sequences of rare, uncultured bacteria obtained by differential
+coverage binning of multiple metagenomes. Nat Biotechnol. 2013;31:533–8.
+14.
+Qin J, Li R, Raes J, Arumugam M, Burgdorf KS, Manichanh C, et al. A human
+gut microbial gene catalogue established by metagenomic sequencing.
+Nature. 2010;464:59–65.
+15.
+Anantharaman K, Brown CT, Hug LA, Sharon I, Castelle CJ, Probst AJ, et al.
+Thousands of microbial genomes shed light on interconnected
+biogeochemical processes in an aquifer system. Nat Commun. 2016;7:13219.
+16.
+Parks DH, Rinke C, Chuvochina M, Chaumeil P-A, Woodcroft BJ, Evans PN,
+et al. Recovery of nearly 8,000 metagenome-assembled genomes
+substantially expands the tree of life. Nat Microbiol. 2017;2:1533–42.
+17.
+Hug LA, Baker BJ, Anantharaman K, Brown CT, Probst AJ, Castelle CJ, et al. A
+new view of the tree of life. Nat Microbiol. 2016;1:16048.
+18.
+Hahnke RL, Meier-Kolthoff JP, García-López M, Mukherjee S, Huntemann M,
+Ivanova NN, et al. Genome-based taxonomic classification of Bacteroidetes.
+Front Microbiol. 2016;7:2003.
+19.
+Nolla-Ardevol V, Strous M, Tegetmeyer HE. Anaerobic digestion of the
+microalga Spirulina at extreme alkaline conditions: biogas production,
+metagenome and metatranscriptome. Front Microbiol. 2015;6:597.
+20.
+Borrel G, Parisot N, Harris HM, Peyretaillade E, Gaci N, Tottey W, et al.
+Comparative genomics highlights the unique biology of
+Methanomassiliicoccales, a Thermoplasmatales-related seventh order
+of methanogenic archaea that encodes pyrrolysine. BMC Genomics.
+2014;15:679.
+21.
+Sorokin DY, Abbas B, Geleijnse M, Pimenov NV, Sukhacheva MV, van
+Loosdrecht MCM. Methanogenesis at extremely haloalkaline conditions in the
+soda lakes of Kulunda Steppe (Altai, Russia). FEMS Microbiol Ecol. 2015;91:4.
+22.
+Nobu MK, Narihiro T, Kuroda K, Mei R, Liu WT. Chasing the elusive
+Euryarchaeota class WSA2: genomes reveal a uniquely fastidious methyl-
+reducing methanogen. ISME J. 2016;10:2478–87.
+23.
+Skennerton CT, Haroon MF, Briegel A, Shi J, Jensen GJ, Tyson GW, et al.
+Phylogenomic analysis of Candidatus “Izimaplasma” species: free-living
+representatives from a Tenericutes clade found in methane seeps. ISME J.
+2016;10:2679–92.
+24.
+Sekiguchi Y, Ohashi A, Parks DH, Yamauchi T, Tyson GW, Hugenholtz P. First
+genomic insights into members of a candidate bacterial phylum
+responsible for wastewater bulking. PeerJ. 2015;3:e740.
+25.
+Wrighton KC, Thomas BC, Sharon I, Miller CS, Castelle CJ, VerBerkmoes NC,
+et al. Fermentation, hydrogen, and sulfur metabolism in multiple
+uncultivated bacterial phyla. Science. 2012;337:1661–5.
+26.
+León-Zayas R, Peoples L, Biddle JF, Podell S, Novotny M, Cameron J, et al.
+The metabolic potential of the single cell genomes obtained from the
+Challenger Deep, Mariana Trench within the candidate superphylum
+Parcubacteria (OD1). Environ Microbiol. 2017;19:2769–84.
+27.
+Castelle CJ, Brown CT, Thomas BC, Williams KH, Banfield JF. Unusual
+respiratory capacity and nitrogen metabolism in a Parcubacterium (OD1) of
+the Candidate Phyla Radiation. Sci Rep. 2017;7:40101.
+28.
+Anantharaman K, Brown CT, Burstein D, Castelle CJ, Probst AJ, Thomas BC,
+et al. Analysis of five complete genome sequences for members of the class
+Peribacteria in the recently recognized Peregrinibacteria bacterial phylum.
+PeerJ. 2016;4:e1607.
+29.
+Brown CT, Hug LA, Thomas BC, Sharon I, Castelle CJ, Singh A, et al. Unusual
+biology across a group comprising more than 15% of domain Bacteria.
+Nature. 2015;523:208–11.
+30.
+Campbell JH, O ‘donoghue P, Campbell AG, Schwientek P, Sczyrba A,
+Woyke T, et al. UGA is an additional glycine codon in uncultured SR1
+bacteria from the human microbiota. 2013; doi:https://doi.org/10.1073/pnas.
+1303090110.
+31.
+Hanke A, Hamann E, Sharma R, Geelhoed JS, Hargesheimer T, Kraft B, et al.
+Recoding of the stop codon UGA to glycine by a BD1-5/SN-2 bacterium
+and niche partitioning between Alpha- and Gammaproteobacteria in a tidal
+sediment microbial community naturally selected in a laboratory chemostat.
+Front Microbiol. 2014;5:231.
+32.
+Kantor RS, Wrighton KC, Handley KM, Sharon I, Hug LA, Castelle CJ, et al.
+Small genomes and sparse metabolisms of sediment-associated bacteria
+from four candidate phyla. MBio. 2013;4:1–11.
+33.
+Wrighton KC, Castelle CJ, Varaljay VA, Satagopan S, Brown CT, Wilkins MJ,
+et al. RubisCO of a nucleoside pathway known from Archaea is found in
+diverse uncultivated phyla in bacteria. ISME J. 2016;10:2702–14.
+34.
+Luef B, Frischkorn KR, Wrighton KC, Holman HYN, Birarda G, Thomas BC,
+et al. Diverse uncultivated ultra-small bacterial cells in groundwater. Nat
+Commun. 2015;6:1–8.
+35.
+Krulwich TA, Sachs G, Padan E. Molecular aspects of bacterial pH sensing
+and homeostasis. Nat Rev Microbiol. 2011;9:330–43.
+36.
+Hauß T, Dante S, Dencher NA, Haines TH. Squalane is in the midplane of
+the lipid bilayer: implications for its function as a proton permeability
+barrier. Biochim Biophys Acta Bioenerg. 2002;1556:149–54.
+37.
+Oren A. Life at high salt concentrations, intracellular KCl concentrations, and
+acidic proteomes. Front Microbiol. 2013;4:315.
+Vavourakis et al. Microbiome  (2018) 6:168 
+Page 16 of 18
+
+Published online: 19 September 201838.
+Levina N. Protection of Escherichia coli cells against extreme turgor by
+activation of MscS and MscL mechanosensitive channels: identification of
+genes required for MscS activity. EMBO J. 1999;18:1730–7.
+39.
+Gupta RS, Khadka B. Evidence for the presence of key chlorophyll-
+biosynthesis-related proteins in the genus Rubrobacter (phylum
+Actinobacteria) and its implications for the evolution and origin of
+photosynthesis. Photosynth Res. 2016;127:201–18.
+40.
+Basak N, Das D. The prospect of purple non-sulfur (PNS) photosynthetic
+bacteria for hydrogen production:the present state of the art. World J
+Microbiol Biotechnol. 2007;23:31–42.
+41.
+Meng J, Wang F, Wang F, Zheng Y, Peng X, Zhou H, et al. An uncultivated
+crenarchaeota contains functional bacteriochlorophyll a synthase. ISME J.
+2009;3:106–16.
+42.
+Sorokin DY, Tourova TP, Mußmann M, Muyzer G. Dethiobacter alkaliphilus
+gen. nov. sp. nov., and Desulfurivibrio alkaliphilus gen. nov. sp. nov.: two novel
+representatives of reductive sulfur cycle from soda lakes. Extremophiles.
+2008;12:431–9.
+43.
+Poser A, Lohmayer R, Vogt C. Extremophiles KK-, 2013 U. Disproportionation
+of elemental sulfur by haloalkaliphilic bacteria from soda lakes.
+Extremophiles. 2013;17:1003–12.
+44.
+Sorokin DY, Abbas B, Tourova TP, Bumazhkin BK, Kolganova TV, Muyzer G.
+Sulfate-dependent acetate oxidation under extremely natron-alkaline
+conditions by syntrophic associations from hypersaline soda lakes.
+Microbiology. 2014;160(Pt_4):723–32.
+45.
+Ragsdale SW. Enzymology of the Wood-Ljungdahl pathway of acetogenesis.
+Ann N Y Acad Sci. 2008;1125:129–36.
+46.
+Adam PS, Borrel G, Gribaldo S. Evolutionary history of carbon monoxide
+dehydrogenase/acetyl-CoA synthase, one of the oldest enzymatic
+complexes. Proc Natl Acad Sci. 2018;115:E1166–73.
+47.
+Sorokin DY, Banciu HL, Muyzer G. Functional microbiology of soda lakes.
+Curr Opin Microbiol. 2015;25:88–96.
+48.
+Grant WD, Jones BE. Bacteria, Archaea and viruses of soda lakes. In: Schagerl
+M, editor. Soda lakes of East Africa. Berlin: Springer; 2016. p. 97–147.
+49.
+Bruno A, Sandionigi A, Rizzi E, Bernasconi M, Vicario S, Galimberti A, et al.
+Exploring the under-investigated “microbial dark matter” of drinking water
+treatment plants. Sci Rep. 2017;7:1–7.
+50.
+Danczak RE, Johnston MD, Kenah C, Slattery M, Wrighton KC, Wilkins MJ.
+Members of the Candidate Phyla Radiation are functionally differentiated by
+carbon- and nitrogen-cycling capabilities. Microbiome. 2017;5:112.
+51.
+Hu P, Tom L, Singh A, Thomas BC, Baker BJ, Piceno YM, et al. Genome-
+resolved metagenomic analysis reveals roles for candidate phyla and other
+microbial community members in biogeochemical transformations in oil
+reservoirs. MBio. 2016;7:e01669–15.
+52.
+Probst AJ, Castelle CJ, Singh A, Brown CT, Anantharaman K, Sharon I, et al.
+Genomic resolution of a cold subsurface aquifer community provides
+metabolic insights for novel microbes adapted to high CO2 concentrations.
+Environ Microbiol. 2017;19:459–74.
+53.
+Lozupone CA, Knight R. Global patterns in bacterial diversity. Proc Natl Acad
+Sci. 2007;104:11436–40.
+54.
+Thompson LR, Sanders JG, McDonald D, Amir A, Ladau J, Locey KJ, et al. A
+communal catalogue reveals Earth’s multiscale microbial diversity. Nature.
+2017;551:457.
+55.
+Samylina OS, Sapozhnikov FV, Gainanova OY, Ryabova AV, Nikitin MA,
+Sorokin DY. Algo-bacterial communities of the Kulunda steppe (Altai region,
+Russia) Soda Lakes. Microbiology. 2014;83:849–60.
+56.
+Krienitz L, Schagerl M. Tiny and tough: microphytes of east African soda lakes. In:
+Schagerl M, editor. Soda lakes of East Africa. Berlin: Springer; 2016. p. 149–77.
+57.
+Nelson WC, Maezato Y, Wu Y-W, Romine MF, Lindemann SR. Identification
+and resolution of microdiversity through metagenomic sequencing of
+parallel consortia. Appl Environ Microbiol. 2015;82:255–67.
+58.
+Hansel C. Small but mighty: how minor components drive major
+biogeochemical cycles. Environ Microbiol Rep. 2017;9:8–10.
+59.
+Zinger L, Amaral-Zettler LA, Fuhrman JA, Horner-Devine MC, Huse SM,
+Welch DBM, et al. Global patterns of bacterial beta-diversity in seafloor and
+seawater ecosystems. PLoS One. 2011;6:e24570.
+60.
+Isachenko BL. Chloride sulfate and soda lakes of Kulunda steppe and its
+biogenic processes. In: Selected works, vol. 2. Leningrad: Academy of
+Sciences USSR; 1951. p. 143–62.
+61.
+Quast C, Pruesse E, Yilmaz P, Gerken J, Schweer T, Yarza P, et al. The SILVA
+ribosomal RNA gene database project: improved data processing and web-
+based tools. Nucleic Acids Res. 2012;41:D590–6.
+62.
+Joshi NA, Fass JN. Sickle: a sliding-window, adaptive, quality-based trimming
+tool for FastQ files (Version 1.33). 2011.
+63.
+Ghai R, Pašić L, Fernández AB, Martin-Cuadrado A-B, Mizuno CM, McMahon
+KD, et al. New abundant microbial groups in aquatic hypersaline
+environments. Sci Rep. 2011;1:135.
+64.
+Li D, Liu CM, Luo R, Sadakane K, Lam TW. MEGAHIT: An ultra-fast single-
+node solution for large and complex metagenomics assembly via succinct
+de Bruijn graph. Bioinformatics. 2015;31:1674–6.
+65.
+Hyatt D, Chen G-L, LoCascio PF, Land ML, Larimer FW, Hauser LJ. Prodigal:
+prokaryotic gene recognition and translation initiation site identification.
+BMC Bioinformatics. 2010;11:119.
+66.
+Huang Y, Li W, Finn PW, Perkins DL. Ribosomal RNA identification in
+metagenomic and metatranscriptomic datasets. In: De Bruijn FJ,
+editor. Handbook of Molecular Microbial Ecology I. Hoboken: Wiley;
+2011. p. 387–91.
+67.
+Lowe TM, Eddy SR. tRNAscan-SE: a program for improved detection of
+transfer RNA genes in genomic sequence. Nucleic Acids Res. 1997;25:
+955–64.
+68.
+Kanehisa M, Sato Y, Morishima K. BlastKOALA and GhostKOALA: KEGG tools
+for functional characterization of genome and metagenome sequences. J
+Mol Biol. 2016;428:726–31.
+69.
+Lauro FM, Demaere MZ, Yau S, Brown MV, Ng C, Wilkins D, et al. An
+integrative study of a meromictic lake ecosystem in Antarctica. ISME J. 2010;
+5:879–95.
+70.
+Hernsdorf AW, Amano Y, Miyakawa K, Ise K, Suzuki Y, Anantharaman K, et al.
+Potential for microbial H2 and metal transformations associated with novel
+bacteria and archaea in deep terrestrial subsurface sediments. Nat Publ Gr.
+2017;11:1915–29.
+71.
+Llorens-Marès T, Yooseph S, Goll J, Hoffman J, Vila-Costa M, Borrego CM,
+et al. Connecting biodiversity and potential functional role in modern
+euxinic environments by microbial metagenomics. ISME J. 2015;9:1648–61.
+72.
+Kang DD, Froula J, Egan R, Wang Z. MetaBAT, an efficient tool for accurately
+reconstructing single genomes from complex microbial communities. PeerJ.
+2015;3:e1165.
+73.
+Langmead B, Salzberg SL. Fast gapped-read alignment with bowtie 2. Nat
+Methods. 2012;9:357–9.
+74.
+Parks DH, Imelfort M, Skennerton CT, Hugenholtz P, Tyson GW. CheckM:
+assessing the quality of microbial genomes recovered from isolates, single
+cells, and metagenomes. Genome Res. 2015;25:1043–55.
+75.
+Eren AM, Esen ÖC, Quince C, Vineis JH, Morrison HG, Sogin ML, et al. Anvi’o:
+an advanced analysis and visualization platform for ‘omics data. PeerJ. 2015;
+3:e1319.
+76.
+Eren AM, Delmot TO. Predicting CPR genomes in metagenomic bins. http://
+merenlab.org/2016/04/17/predicting-CPR-Genomes/.
+77.
+Edgar RC. Search and clustering orders of magnitude faster than BLAST.
+Bioinformatics. 2010;26:2460–1.
+78.
+Rinke C, Schwientek P, Sczyrba A, Ivanova NN, Anderson IJ, Cheng JF, et al.
+Insights into the phylogeny and coding potential of microbial dark matter.
+Nature. 2013;499:431–7.
+79.
+Bushnell B. BBMap short read aligner. 2016.
+80.
+Bowers RM, Kyrpides NC, Stepanauskas R, Harmon-Smith M, Doud D, Reddy
+TBK, et al. Minimum information about a single amplified genome (MISAG)
+and a metagenome-assembled genome (MIMAG) of bacteria and archaea.
+Nat Biotechnol. 2017;35:725–31.
+81.
+Price MN, Dehal PS, Arkin AP. FastTree 2--approximately maximum-
+likelihood trees for large alignments. PLoS One. 2010;5:e9490.
+82.
+Katoh K, Standley DM. MAFFT multiple sequence alignment software
+version 7: improvements in performance and usability. Mol Biol Evol. 2013;
+30:772–80.
+83.
+Letunic I, Bork P. Interactive tree of life (iTOL) v3: an online tool for the
+display and annotation of phylogenetic and other trees. Nucleic Acids Res.
+2016;44:W242–5.
+84.
+FigTree. http://tree.bio.ed.ac.uk/software/figtree/.
+85.
+Seemann T. Prokka: rapid prokaryotic genome annotation. Bioinformatics.
+2014;30:2068–9.
+86.
+Jones P, Binns D, Chang H-Y, Fraser M, Li W, McAnulla C, et al. InterProScan
+5: genome-scale protein function classification. Bioinformatics. 2014;30:
+1236–40.
+87.
+Yin Y, Mao X, Yang J, Chen X, Mao F, Xu Y. dbCAN: a web resource for
+automated carbohydrate-active enzyme annotation. Nucleic Acids Res. 2012;
+40:W445–51.
+Vavourakis et al. Microbiome  (2018) 6:168 
+Page 17 of 18
+
+88.
+Cantarel BL, Coutinho PM, Rancurel C, Bernard T, Lombard V, Henrissat B.
+The Carbohydrate-Active EnZymes database (CAZy): an expert resource for
+glycogenomics. Nucleic Acids Res. 2009;37:D233–8.
+89.
+Nayfach S, Pollard KS. Average genome size estimation improves
+comparative metagenomics and sheds light on the functional ecology of
+the human microbiome. Genome Biol. 2015;16:1–18.
+90.
+Goris J, Konstantinidis KT, Klappenbach JA, Coenye T, Vandamme P, Tiedje
+JM. DNA-DNA hybridization values and their relationship to whole-genome
+sequence similarities. Int J Syst Evol Microbiol. 2007;57:81–91.
+91.
+Huerta-Cepas J, Szklarczyk D, Forslund K, Cook H, Heller D, Walter MC, et al.
+eggNOG 4.5: a hierarchical orthology framework with improved functional
+annotations for eukaryotic, prokaryotic and viral sequences. Nucleic Acids
+Res. 2016;44:D286–93.
+92.
+Tikhonova TV, Slutsky A, Antipov AN, Boyko KM, Polyakov KM, Sorokin DY,
+et al. Molecular and catalytic properties of a novel cytochrome c nitrite
+reductase from nitrate-reducing haloalkaliphilic sulfur-oxidizing bacterium
+Thioalkalivibrio nitratireducens. Biochim Biophys Acta - Proteins Proteomics.
+2006;1764:715–23.
+93.
+Tikhonova T, Tikhonov A, Trofimov A, Polyakov K, Boyko K, Cherkashin E,
+et al. Comparative structural and functional analysis of two octaheme nitrite
+reductases from closely related Thioalkalivibrio species. FEBS J. 2012;279:
+4052–61.
+94.
+Tabita FR, Hanson TE, Li H, Satagopan S, Singh J, Chan S. Function, structure,
+and evolution of the RuBisCO-like proteins and their RuBisCO homologs.
+Microbiol Mol Biol Rev. 2007;71:576–99.
+95.
+Eddy SR. Accelerated profile HMM searches. PLoS Comput Biol. 2011;7:
+e1002195.
+Vavourakis et al. Microbiome  (2018) 6:168 
+Page 18 of 18
+
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+page_content='RESEARCH Open Access A metagenomics roadmap to the uncultured genome diversity in hypersaline soda lake sediments Charlotte D.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Vavourakis1 , Adrian-Stefan Andrei2†, Maliheh Mehrshad2†, Rohit Ghai2, Dimitry Y.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Sorokin3,4 and Gerard Muyzer1* Abstract Background: Hypersaline soda lakes are characterized by extreme high soluble carbonate alkalinity.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Despite the high pH and salt content, highly diverse microbial communities are known to be present in soda lake brines but the microbiome of soda lake sediments received much less attention of microbiologists.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Here, we performed metagenomic sequencing on soda lake sediments to give the first extensive overview of the taxonomic diversity found in these complex, extreme environments and to gain novel physiological insights into the most abundant, uncultured prokaryote lineages.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Results: We sequenced five metagenomes obtained from four surface sediments of Siberian soda lakes with a pH 10 and a salt content between 70 and 400 g L−1.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' The recovered 16S rRNA gene sequences were mostly from Bacteria, even in the salt-saturated lakes.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Most OTUs were assigned to uncultured families.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' We reconstructed 871 metagenome-assembled genomes (MAGs) spanning more than 45 phyla and discovered the first extremophilic members of the Candidate Phyla Radiation (CPR).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Five new species of CPR were among the most dominant community members.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='  Novel dominant lineages were found within previously well-characterized functional groups involved in carbon, sulfur, and nitrogen cycling.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Moreover, key enzymes of the Wood-Ljungdahl pathway were encoded within at least four bacterial phyla never previously associated with this ancient anaerobic pathway for carbon fixation and dissimilation, including the Actinobacteria.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Conclusions: Our first sequencing effort of hypersaline soda lake sediment metagenomes led to two important advances.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' First, we showed the existence and obtained the first genomes of haloalkaliphilic members of the CPR and several hundred other novel prokaryote lineages.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' The soda lake CPR is a functionally diverse group, but the most abundant organisms in this study are likely fermenters with a possible role in primary carbon degradation.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Second, we found evidence for the presence of the Wood-Ljungdahl pathway in many more taxonomic groups than those encompassing known homo-acetogens, sulfate-reducers, and methanogens.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Since only few environmental metagenomics studies have targeted sediment microbial communities and never to this extent, we expect that our findings are relevant not only for the understanding of haloalkaline environments but can also be used to set targets for future studies on marine and freshwater sediments.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='  Keywords: Soda lake sediments, Metagenomics, Haloalkaliphilic extremophiles, Candidate Phyla Radiation, Wood-Ljungdahl pathway * Correspondence: G.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='Muijzer@uva.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='nl †Adrian-Stefan Andrei and Maliheh Mehrshad contributed equally to this work.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 1Microbial Systems Ecology, Department of Freshwater and Marine Ecology, Institute for Biodiversity and Ecosystem Dynamics, Faculty of Science, University of Amsterdam, Postbus 94248, 1090, GE, Amsterdam, the Netherlands Full list of author information is available at the end of the article © The Author(s).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2018 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
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+page_content='org/publicdomain/zero/1.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='0/) applies to the data made available in this article, unless otherwise stated.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Vavourakis et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Microbiome  (2018) 6:168 https://doi.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='org/10.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='1186/s40168-018-0548-7  MicrobiomeBackground Soda lakes are evaporative, athallasic salt lakes with low cal- cium and magnesium concentrations and a high-alkaline pH up to 11 buffered by dissolved (bi-) carbonate ions [1].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' They are constrained to arid regions across the globe, mainly the tropical East African Rift Valley [2], the Libyan Desert [3], the deserts in California and Nevada [4], and the dry steppe belt of Central Asia that spans to southern Si- beria, north-eastern Mongolia, and Inner Mongolia in China [1].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' On top of the extreme salinity and alkaline pH, the Eurasian soda lakes experience extreme seasonal temperature differences, causing highly unstable water re- gimes and fluctuating salinities [5].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Yet, soda lakes harbor diverse communities of haloalkaliphilic microbes, mostly prokaryotes that are well adapted to survive and grow in these extreme environments and consist of similar func- tional groups in soda lakes around the world [1, 2, 6].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' The relative abundance of different groups is typically governed by the salinity of the brine [1, 7, 8], and microbial-mediated nutrient cycles become partially hampered only at salt-saturating conditions [1].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' So far, all characterized prokaryotic lineages cultured from soda lakes comprise over 70 different species within more than 30 genera [1, 6, 9, 10].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='  From these, only a lim- ited number of genomes have been sequenced today, mostly from chemolithoautotrophic sulfur-oxidizing bac- teria belonging to the genus Thioalkalivibrio (class Gam- maproteobacteria) [1, 11, 12].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' It is well established that metagenomics enables the recovery of genomes and the identification of novel genetic diversity where culturing ef- forts fail [13, 14].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' In recent years, next-generation sequen- cing has recovered a massive number of genomes from previously unknown groups of prokaryotes [15, 16], including a strikingly large and diverse group called “Candidate Phyla Radiation” (CPR), only distantly related to other cultured bacterial lineages [17].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Previously, we conducted a metagenomics study on soda lakes and re- constructed novel genomes from uncultured Bacteroidetes and “Candidatus Nanohaloarchaeaota” living in hypersa- line Siberian soda brines [7].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Here, we turned our atten- tion to the far more complex prokaryotic communities living in the sediments of the hypersaline soda lakes from the same region.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' We give a broad overview of all the taxonomic groups sequenced and focus on the metabolic diversity found in the reconstructed genomes of the most abundant, uncultured organisms.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Results Overall prokaryote community structure The salinities from the studied soda lakes ranged from moderately hypersaline (between 70 and 110 g L−1) to salt-saturated (400 g L−1 salt).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='  The soluble carbonate al- kalinity was in the molar range, and the pH in all lakes was around ten (see Additional file 1: Table S1).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' To give an overview of the overall prokaryotic community com- position in each of the samples, we looked at the taxo- nomic classification of 16S rRNA genes recovered both by amplicon sequencing and direct metagenomics se- quencing (Fig.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 1, see also Additional file 2: Figure S1;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Additional file 3).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' The prokaryotic communities of all five sediment samples were highly diverse and consisted mostly of uncultured taxonomic groups.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Bacteria were more abundant than Archaea, regardless of the salinity of the overlaying brine [7] (Fig.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 1).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Euryarchaeota were the second and third largest group in the sediments of the two salt-saturated lakes comprising ~ 10 and ~ 20% of the 16S rRNA genes in the metagenomes.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Most Euryarchaeota-related OTUs detected by amplicon se- quencing belonged either to the uncultured Thermoplas- mata group KTK 4A (SILVA classification) or the genera Halohasta and Halorubrum (class Halobacteria).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' In ac- cordance with cultivation-dependent studies [6], most OTUs assigned to methanogens were from the class Methanomicrobia, especially the lithotrophic genus Methanocalculus (up to ~ 3%) and the methylotrophic genus Methanosalsum (Additional file 3).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='  The varying ratio of the three dominant bacterial groups, Firmicutes, Bacteroidetes (including the newly proposed phyla Rhodothermaeota and Balneolaeota [18]), and Gammaproteobacteria, showed no clear trend in relation to the salinity in the lakes, but when Firmicutes were domin- ant, Bacteroidetes were less abundant and vice versa.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Most Firmicutes belonged to the order Clostridales.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Uncultured members from the family Syntrophomonadaceae had a relative abundance of more than 5% in all five metagen- omes and comprised in two lakes even ~ 11–20% of the recovered amplicon sequences.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' The second most abundant Firmicutes order was Halanaerobiales, particularly the genus Halanaerobium (family Halanaerobiaceae) and un- cultured members of the Halobacteroidaceae.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' The majority of Bacteroidetes-related OTUs could not be assigned down to the genus level.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' The uncultured ML635J-40 aquatic group (order Bacteroidales) comprised at least 5% of all five prokaryotic communities.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' This group has been previously found to be abundant in Mono Lake [4] (a soda lake) and in an anoxic bioreactor degrading cyanobacterial biomass under haloalkaline conditions [19].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Two other highly abun- dant (up to ~ 8%) uncultured groups from the class Balneo- lia (proposed new phylum Balneolaeota [18]) were also detected in other soda lakes before [3, 4].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='  Within the Gam- maproteobacteria, the genus Thioalkalivibrio was abundant (~ 3% of the total community), but also uncultured members of HOC36 were prevailing at moderate salinities.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Members of the Deltaproteobacteria, Alphaproteobacteria, and Chloroflexi comprised up to ~ 10% of the detected 16S rRNA gene in some of the metagenomes.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' The GIF9 family of the class Dehalococcoidia was among the top three most abundant OTUs in two lakes.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' The extremely salt-tolerant Vavourakis et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Microbiome  (2018) 6:168 Page 2 of 18  and alkaliphilic genera Desulfonatronobacter (order Desulfo- bacterales) and Desulfonatronospira (order Desulfovibrio- nales) were the dominant Deltaproteobacteria.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Highly abundant OTUs, within the Actinobacteria belonged to the class Nitriliruptoria and within the Alphaproteobacteria to the family Rhodobacteraceae and the genus Roseibaca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' The important nitrifying genus Nitrobacter (Alphaproteobacteria) was present in only one of the lakes with moderate salinity (Additional file 3).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Some bacterial top-level taxa appeared less dominant (< 5%) from the 16S rRNA genes recovered from the metagenomes but were represented mainly by a single highly abundant OTU in the amplicon sequences, in- cluding the haloalkaliphilic genus Truepera within the phylum Deinococcus-Thermus, the genus Spirochaeata within the phylum Spirochaetes, the family BSN166 within the phylum Ignavibacteriae, the BD2–11 terres- trial group within the Gemmatimonadetes, and the WCHB1–41 order within the Verrucomicrobia.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' All OTUs within the Thermotogae and Lentisphaerae belonged to uncultured genera from the family Kosmoto- gaceae and Oligosphaeraceae, respectively.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' All Tenericu- tes-related OTUs belonged to the class Mollicutes, and especially the order NB1-n was dominant.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' In contrast, the phylum Planctomycetes was relatively diverse, with at least 11 different genus-level OTUs spread over four class-level groups.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='  High-throughput genome recovery We obtained 717 medium-quality (≥ 50% complete, < 10% contamination) and 154 near-complete (≥ 90% complete, < 5% contamination) metagenome-assembled genomes (MAGs) across three major prokaryote groups: Archaea, Bacteria, and CPR (see Additional file 4 and Additional file 2: Figure S2).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Figures 2 and 3 show the top-level phylogeny of all MAGs based on 16 ribosomal proteins.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' The reference database used contains a repre- sentative for each major prokaryote lineage [17].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' We a b Fig.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 1 Abundant prokaryotic groups in five hypersaline soda lake sediments.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' a Relative abundance of the top-level taxa (those with ≥ 1% abundance in at least one dataset) based on 16S rRNA reads in unassembled metagenomic datasets.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' b Relative abundance of the 16S rRNA OTUs (those with sum of abundance in all datasets ≥ 3%) recovered by amplicon sequencing assigned where possible down to the genus-level.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Three of the assessed soda lakes have a moderate salinity (70–110 g L−1), two are salt-saturated (400 g L− 1) Vavourakis et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Microbiome  (2018) 6:168 Page 3 of 18  colored the different phyla from which we obtained a MAG in alternate blue and orange colors, and highlighted the MAGs obtained here in a darker shade.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Many MAGs belonged to uncultured groups commonly detected in soda lake 16S rRNA gene surveys, over 100 MAGs still belonged to candidate prokaryote phyla and divisions that to our knowledge were never detected be- fore in soda lakes, including CPR.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Although only few MAGs had near-complete 16S rRNA genes, in most cases we were able to link available taxonomic gene an- notations and ribosomal protein phylogeny to the SILVA taxonomy of the OTUs assigned to the amplicon se- quences, while cross-checking the abundance profiles of both MAGs (Additional file 5) and OTUs.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' The soda lake CPR recovered from the metagenomes was restricted to a few distinct phyla within the Parcubacteria group, mostly affiliating with “Candidatus Nealsonbacteria” and “Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Zambryskibacteria” [15] (Fig.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' The first group of MAGs encompassed four different branches in our riboso- mal protein tree, suggesting a high-phylogenetic diversity, with 33 putative new species sampled here (ANI and con- DNA matrices given in Additional file 6).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' The “Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Zambrys- kibacteria-”related MAGs consisted of at least five new species.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Few MAGs were recovered from CPR groups also detected by amplicon sequencing (see Additional file 2: Figure S1), namely the “Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Dojkabacteria” (former WS6), “Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Saccharibacteria” (former TM7), CPR2, and “Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Katanobacteria” (former WWE3).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Fig.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='  2 Maximum-likelihood phylogeny of the CPR and archaeal MAGs based on 16 ribosomal proteins.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' The archaeal tree is unrooted.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' The CPR tree is rooted to the Wirthbacteria.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Alternate orange and blue colors show phyla/classes from which we obtained MAGs (labeled as “Phyla present”).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Reconstructed MAGs of this study are highlighted by darker shades (labeled as “MAG present”).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Phyla/classes for which there was no representative in the reconstructed MAGs of this study are shown as gray cartoons (labeled as “Phyla not present”), and the numerical labels are represented at the bottom.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Colored circles at the nodes show confidence percentage of the bootstraps analysis (100×) Vavourakis et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Microbiome  (2018) 6:168 Page 4 of 18  Most archaeal MAGs belonged to the phylum Euryarch- aeota and the abundant classes Halobacteria, Methanomi- crobia, and Thermoplasmata (related to OTU KTK 4A) within.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' In addition, three Thermoplasmata-related MAGs that encoded for the key enzyme for methanogenesis (methyl-coenzyme M reductase, mcr) affiliated with refer- ence genomes from Methanomassilicoccales, the seventh order of methanogens have been recovered [20, 21].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Another MCR-encoding MAG was closely related to the latest discovered group of poly-extremophilic, methyl-reducing methanogens from hypersaline lakes from the class Methanonatronarchaeia [9] (related to OTU ST-12K10A).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' We recovered also one MAG from the class Methanobacteria and a high-quality MAG from the WCHA1–57 group (“Candidatus Methanofastidiosa” [22]) in the candidate division WSA2 (Arc I).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Several MAGs were recovered from the DPANN archaeal groups “Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Diapherotrites,” “Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Aenigmarchaeota,” (see Additional file 2: Figure S3) and “Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Woesearch- aeota” (former Deep Sea Hydrothermal Vent Group 6, DHVEG-6).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Although we did not reconstruct any reasonable-sized MAGs from the TACK superphylum, we found several 16S rRNA genes on the assembled contigs that affiliated to the Thaumarchaeota (see Additional file 1: Table S2).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Nearly every known bacterial phylum had an extremo- philic lineage sampled from our hypersaline soda lake sediments (Fig.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 3).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='  In most cases, the soda lake lineages clearly formed separate branches appearing as sister groups to known reference lineages.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' The highest genome recovery was from the same top-level taxonomic groups that were also abundant in our 16S rRNA gene analysis.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' From the Verrucomicrobia, most MAGs belonged to the order WCHB1-41 (16S rRNA gene identity 92–100%).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' However, in our ribosomal protein tree, they branched within the phylum Lentisphaerae.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Sixteen Tenericutes MAGs from at least 12 different species (Additional file 6) were closely related to the NB1-n group of Mollicutes.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Based on the recovered genome size and encoded meta- bolic potential, these organisms are free-living anaerobic fermenters of simple sugars, similar to what has recently been proposed for “Candidatus Izimaplasma” [23].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Fig.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 3 Maximum-likelihood phylogeny of the bacterial MAGs (CPR excluded) based on 16 ribosomal proteins.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Alternate orange and blue colors show phyla/ classes from which we obtained MAGs (labeled as “Phyla present”).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Reconstructed MAGs of this study are highlighted by darker shades (labeled as “MAG present”).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Phyla/classes for which there was no representative in the reconstructed MAGs of this study are shown as gray cartoons (labeled as “Phyla not present”), and the numerical labels are represented at the bottom.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Colored circles at the nodes show confidence percentage of the bootstraps analysis (100×) Vavourakis et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Microbiome  (2018) 6:168 Page 5 of 18  Several MAGs belonged to the candidate phyla “Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Omnitrophica,” “Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Atribacteria,” and “Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Acetother- mia” (former OP1), which were moderately abundant also in some sediment (see Additional file 2: Figure S1).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' For the latter phylum, we suspect that four MAGs were more closely related to ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' div.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' WS1 and “Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Lindow- bacteria” for which only few reference genomes are currently available in NCBI (see Additional file 2: Figure S4).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Due to a high-sequencing coverage, we also managed to reconstruct several MAGs from rare Bacteria (< 100 amplicon sequences detected, see Additional file 2: Figure S1), including the phyla “Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Hydrogenedentes,” “Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Cloacimonetes,” ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' div.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' BRC1, Elusimicrobia, Caldi- serica, and “Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Latescibacteria.” The MAGs from the latter phylum were more closely related to the recently proposed phylum “Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Handelsmanbacteria” [15].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Two additional MAGs with 16S rRNA gene fragments with 94–95% identity to the class MD2898-B26 (Nitrospinae) were more likely members of ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' div.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' KSB3 (proposed “Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Moduliflexus” [24], see Additional file 2: Figure S5).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Draft genomes of haloalkaliphilic CPR Strikingly, members of the CPR related to “Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Nealson- bacteria” and “Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Vogelbacteria” were among the top 5% of abundant organisms in the surface sediments of the soda lakes, especially those with moderate salinity (Fig.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 4).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Like most members of the CPR, the MAGs of the four most abundant “Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Nealsonbacteria” seem to be anaerobic fermenters [25].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='  They lacked a complete TCA cycle and most complexes from the oxidative elec- tron transfer chain, except for the subunit F of a NADH-quinone oxidoreductase (complex I, nuoF, nuoG, nuoA) and coxB genes (complex II).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' All CPR MAGs had a near-complete glycolysis pathway (Embden-Meyerhof- Parnas) encoded, but pentose phosphate pathways were severely truncated.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' The commonly encoded F- and V-type ATPase can establish a membrane potential for symporter-antiporters by utilizing the ATP formed by substrate-level phosphorylation during fermentation.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' All CPR have V-type ATPases that can translocate Na+ in addition to H+ (see Additional file 2: Figure S6), while only two members of the “Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Falkowbacteria” had puta- tive Na+-coupled F-type ATPases (see Additional file 2: Figure S7).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' The coupling of ATP hydrolysis to sodium translocation is advantageous to maintain pH homeosta- sis in alkaline environments.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Interestingly, with only two exceptions [26, 27], all CPR genomes recovered from other environments with neutral pH were reported to encode only F-type ATPases [28–32].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' One low-abundant MAG affiliated to “Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Peregrinibacteria” contained both the large subunit of a RuBisCO (type II/III, see Additional file 2: Figure S8) and a putative phosphoribu- lokinase (PRK, K00855) encoded in the same contig.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' This is remarkable because PRK homologs were not previously identified among CPR, and RuBisCo form II/ III was inferred to function in a nucleoside salvage path- way [33].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' One “Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='  Saccharibacteria” MAG encoded for a putative channelrhodopsin (see Additional file 2: Figure S9).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' This is the first rhodopsin found among the CPR and suggests that this enigmatic group of organ- isms may have acquired evolutionary adaptations to a life in sunlit surface environments.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' A previous study showed that most CPR has coccoid cell morphotypes with a monoderm cell envelope resem- bling those from Gram-positives and Archaea but with a distinct S-layer [34].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Thick peptidoglycans coated with acidic surface polymers such as teichoic acids help pro- tect the cells of Gram-positives against reactive hydroxyl ions in highly alkaline environments [35] (Fig.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 5a).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' All soda lake CPR had indeed the capability for peptidogly- can biosynthesis, but we found proteins typical for Gram-negatives for the biosynthesis of lipopolysaccha- rides (see Additional file 1: Table S3), homologous to the inner membrane proteins of type II secretion systems and to several proteins associated to the outer membrane and peptidoglycan, including OmpA.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' It remains to be determined whether the soda lake CPR also lacks an outer membrane and perhaps anchor lipopolysaccharides, S-layer proteins, and lipoproteins to the inner cell membrane or peptidoglycan.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' We also found gene encoding cardiolipin and squalene synthases.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Increased levels of cardiolipin and the presence of squa- lene make the cytoplasmic membrane less leaky for protons [36].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='  In addition, cation/proton exchangers are known to play a crucial role for pH homeostasis in alka- liphilic prokaryotes as they help acidify the cytoplasm during the extrusion of cations [35].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Putative Na+/H+ exchangers of the Nha-type and multi-subunit Mnh-type were found only within a few soda lake CPR.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Secondary active transport of K+ might be mediated in most soda lake CPR by KefB (COG0475)/kch Kef-type, glutathione- dependent K+ transport systems, with or without H+ antiport (67,68).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Various soda lake CPR had an acidic proteome, with pI curves resembling those found in extremely halophilic Bacteria.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Intracellular proteins enriched in acidic amino acids might be an adaptation to a “salt-in” strategy, i.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='e.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=', maintaining high intracellular potassium (K+) concentra- tions to keep osmotic balance [7, 37] (Fig.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 5b;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' see Additional file 2: Figure S10).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Such a strategy is energet- ically favorable over de novo synthesis or import of osmolytes such as ectoine and glycine betaine.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' We did not find genes for the synthesis of organic osmolytes and homologs of ABC-type transporters for primary active uptake of proline/glycine betaine which were encoded only in one MAG (Fig.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 5a).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' For the “Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Nealsonbac- teria” and “Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Vogelbacteria,” the salt-in strategy might be a unique feature for the soda lake species explaining Vavourakis et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Microbiome  (2018) 6:168 Page 6 of 18  their high abundance in the hypersaline soda lake sedi- ments, as we did not found an acidic proteome pre- dicted from genomes obtained from other non-saline environments (See Additional file 2: Figure S11).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' The uptake of K+ ions remains enigmatic for most soda lake CPR.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Low-affinity Trk-type K+ uptake transporters (gen- erally with symport of H+) (67,68) were encoded only by a limited number of MAGs.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' We found three MAGs Fig.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 4 Relative abundance and metabolic potential of the dominant species.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Abundance values, expressed as reads per kilobase of MAG per gigabase of mapped reads (RPKG), were averaged for the top ten abundant MAGs from each dataset that were (likely) the same species (Additional file 5, Additional file 6).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Population genomes were ranked by their “salinity preference scores”: those recruiting relatively more from moderate salinity datasets (cold colors) are drawn to the top, from high salinity datasets (warm colors) to the bottom.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' The metabolic potential derived from functional marker genes (Additional file 7) is depicted by the colored symbols.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' CBB = Calvin-Benson-Bassham cycle, DNRA = dissimilatory nitrite reduction to ammonia, fix.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' = fixation, red.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' = reduction, ox.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' = oxidation, dis.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' = disproportionation Vavourakis et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Microbiome  (2018) 6:168 Page 7 of 18  a b Fig.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 5 (See legend on next page.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=') Vavourakis et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Microbiome  (2018) 6:168 Page 8 of 18  encoding for Kdp-type sensor kinases (kdpD) but no corresponding genes for the response regulator (kdpE) or for Kdp-ATPases that function as the inducible, high- affinity K+ transporters in other Bacteria (67,68).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Finally, mechanosensitive ion channels (mscS, mscL) and ABC- type multidrug transport systems (AcrAB, ccmA, EmrA, MdlB, NorM) and sodium efflux permeases (NatB) were encoded in almost every MAG.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' The first might rapidly restore the turgor pressure under fluctuating salinity conditions by releasing cytoplasmic ions [38].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Novel abundant groups involved in sulfur, nitrogen, and carbon cycles A new species of Thioalkalivibrio (family Ectothiorhodospir- aceae) was by far the most abundant in the sediments of the two salt-saturated lakes (Fig.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 4).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' In the sediment of Bitter-1, also a purple sulfur bacterium from the same fam- ily was highly abundant.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' It was closely related to Halorho- dospira, a genus also frequently cultured from hypersaline soda lakes [1].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' None of the abundant Ectothiorhodospira- ceae spp.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' had already a species-representative genome sequenced (Additional file 6).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' The potential of the Thioalk- alivibrio spp.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' for chemolithotrophic sulfur oxidation was evident (Additional file 7;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' see Additional file 8: Information S1).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Interestingly, the encoded nitrogen metabolisms were quite versatile.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' While Thioalkalivibrio sp.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 1 had the poten- tial for nitrate reduction to nitrite, Thioalkalivibrio sp.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='  2 might perform dissimilatory nitrite reduction to ammonia (DNRA;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' see Additional file 2: Figure S12).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Two deltaproteobacterial lineages of dissimilatory sulfate-reducing bacteria (SRB) were highly abundant in the soda lake sediment of Bitter-1.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' One MAG from the family Desulfobacteraceae (order Desulfobacterales) is the first genome from the genus Desulfonatronobacter.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' It encodes the genes for complete sulfate reduction to sul- fide using various electron donors, as well as for the complete oxidation of volatile fatty acids and alcohols, a unique feature for the genus Desulfonatronobacter among haloalkaliphilic SRB [10] (see Additional file 8: Information S2).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Fumarate and nitrite (DNRA, NrfAH) could be used as alternative electron acceptors.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' The sec- ond dominant lineage was a new species from the genus Desulfonatronospira (family Desulfohalobiaceae, order Desulfovibrionales).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Like other members of this genus, it had the potential to reduce or disproportionate partially reduced sulfur compounds.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' In addition, it could also use nitrite as an alternative electron acceptor (NrfAH) [6].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' A novel lineage of gammaproteobacterial SOB was highly abundant in the sediments of the moderately hy- persaline Cock Soda Lake.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' It appeared as a sister group of the family Xanthomonadaceae in the ribosomal protein tree.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' This heterotrophic organism could conserve energy through aerobic respiration.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='  It might detoxify sulfide by oxidizing it to elemental sulfur (sqr) with subsequent re- duction or disproportionation of the polysulfides (psrA) chemically formed from the sulfur.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' It also encoded the po- tential for DNRA (nrfA and napC).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Genes likely involved in sulfide detoxification (sqr and psrA) were found also in several other abundant MAGs of heterotrophs, including one new abundant species from the family of Nitrilirup- toraceae (class Nitriliruptoria, phylum Actinobacteria).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' We found a wide variety of carbohydrate-active enzymes in these MAGs, such as cellulases (GH1 family) in addition to genes for glycolysis and TCA cycle and a chlorophyll/bacteriochlorophyll a/b synthase (bchG gene).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' The latter was also found in other Actinobacteria from the genus Rubrobacter [39].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' No evidence was found for nitrile-degrading potential.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' A second novel, uncultured lineage of Gammaproteo- bacteria that was highly abundant at moderate salinities branched in our ribosomal protein tree as a sister group to the family Halothiobacillaceae.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' The MAGs encoded for a versatile metabolism typical for purple non-sulfur bacteria.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' The MAGs contained puf genes, bch genes, genes for carotenoid biosynthesis (not shown), and a Calvin cycle for photoautotrophic growth.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Alternatively, energy may be conserved through aerobic respiration, while acetate and proprionate could be taken up via an acetate permease (actP) and further used for acetyl-CoA biosynthesis and carbon assimilation.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='  Since the sqr gene was present, but no dsr or sox genes, the organism might oxidize sulfide only to elemental sulfur.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' One bin contained also nifDKH genes suggesting putative diazo- trophy, as well as a coenzyme F420 hydrogenase suggest- ing photoproduction of hydrogen [40].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' The abundant Euryarchaeota organism showed a clear preference for higher salinities.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' We obtained one highly abundant MAG from the class Thermoplasmata that encoded a full-length 16S rRNA gene only distantly re- lated (91,2% identity, e value 0) to that of a member of the KTK 4A group found in a hypersaline endoevaporitic microbial mat [8].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' The abundant soda lake organism is likely a new genus and species.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' All KTK 4A-related MAGs found here encoded for similar heterotrophic, fermentative metabolisms, with the potential for (See figure on previous page.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=') Fig.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 5 Potential mechanisms for regulating the intracellular pH and cytoplasmic ion content in different CPR phyla.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' a Membrane transporters, channels, and lipids.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Peptidoglycan is depicted in gray and S-layer proteins in cyan.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' b Predicted isoelectric points (bin width 0.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='2) for the coding sequences of MAGs.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' A representative proteome is depicted for each phylum for which several members had a pronounced acidic peak (see also Additional file 2: Figure S11) Vavourakis et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Microbiome  (2018) 6:168 Page 9 of 18  anaerobic formate and CO oxidation.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' The KTK 4A might be also primary degraders since they encoded pu- tative cellulases (CAZY-families GH1, GH5) and chiti- nases (GH18).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Interestingly, half of the MAGs encoded a putative chlorophyll/bacteriochlorophyll a/b synthase (bchG), which is highly unusual for Archaea.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Although little can be inferred from the presence of only one marker gene, a functional bchG was previously also found in Crenarchaeota [41].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' The remaining two highly abundant Euryarchaeota-related MAGs belonged to a new species of Halorubrum (Additional file 6).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Key genes of the Wood-Ljungdahl pathway found in novel phylogenetic groups More than 50 MAGs were related to the family Syntro- phomonadaceae (class Clostridia, phylum Firmicutes) based on ribosomal protein phylogeny.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' All 16S rRNA gene sequences found in the MAGS had 86–95% iden- tity to sequences obtained from uncultured organisms related to the genus Dethiobacter.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' While an isolated strain of Dethiobacter alkaliphilus is a facultative auto- troph that respires thiosulfate, elemental sulfur or polysulfides with hydrogen as an electron donor [42] or disproportionates sulfur [43], other haloalkaliphilic members of the Syntrophomonadaceae are reverse acetogens, oxidizing acetate in syntrophy with a hydro- genotrophic partner [44].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Two populations (different species, Additional file 6) were especially abundant in Cock Soda Lake (Fig.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 4).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='  They encoded for a full CODH/ACS complex, the key enzyme for the reductive acetyl-CoA or Wood-Ljungdahl pathway (WL) and a complete Eastern branch for CO2 conversion to 5-methyl-tetrahydrofolate (Additional file 9) [45, 46].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Acetogens use the WL to reduce CO2 to acetyl-CoA, which can be fixed into the cell or used to conserve en- ergy via acetogenesis.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Syntrophic acetate oxidizers, some sulfate reducing bacteria and aceticlastic methanogens run the WL in reverse.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Syntrophomonadaceae sp.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2 encoded for a putative thiosulfate/polysulfide reductase as well as a phosphotransacetylase (pta) and an acetate kinase (ack) for the ATP-dependent conversion of acet- ate to acetyl-CoA.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Although alternative pathways for the latter interconversion can exist, this second species has the complete potential for (reversed) acetogenesis.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Highly remarkable was the presence of a bacterial-type CODH/ACS complex and a near-complete eastern branch of the WL in a highly abundant species in Cock Soda Lake from the family Coriobacteriaceae (phylum Actinobacteria).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' This prompted us to scan all 871 MAGs for the presence of acsB encoding for the beta-subunit of the oxido-reductase module of CODH/ACS.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' We con- firmed an encoded (near)-complete WL in several additional organisms belonging to phylogenetic groups not previously associated with this pathway [46] (Additional file 9).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' We removed the Coriobacteriaceae acsB genes from the final dataset to construct a phylo- genetic tree since they were < 500 aa (Fig.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='  6) but found seven MAGs from the OPB41 class within the Actino- bacteria (16S rRNA gene fragment identity 94–96%).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' The eastern branch of WL can function independently in folate-dependent C1 metabolism [45], but the pres- ence of the Western-branch in a phylum that comprises mostly aerobic isolates is very surprising.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' The WL in combination with the potential for acetate to acetyl-CoA interconversion (pta/ack) and a glycolysis pathway were also present in the soda lake MAGs from the phyla “Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Handelsmanbacteria,” “Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Atribacteria” (latter branched within the “Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Acetothermia”), and the class LD1-PA32 (Chlamydiae), suggesting all these uncultured organisms might be heterotrophic acetogens.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' However, it should be noted that a PFOR typically connecting glycolysis to the WL was only encoded in the LD1-PA32 MAGs.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' More- over, from the genetic make-up alone, it cannot be excluded that acetate is activated, and the WL run in reverse for syntrophic acetate oxidation.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Finally, the novel acsB genes from soda lake Halanaerobiaceae, Natranaerobiaceae, and Halobacteroidaceae (Firmicutes) and from Brocadiaceae and Planctomycetaceae (Plancto- mycetes) disrupt the previously proposed dichotomy between Terrabacteria and Gracilicutes bacterial groups unifying 16S rRNA and acsB gene phylogenies [46] and suggest a far more complex evolutionary history of the WL pathway than previously anticipated.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='  Discussion Extensive classical microbiology efforts have been already undertaken to explore the unique extremophilic microbial communities inhabiting soda lakes.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' These un- covered the presence of most of the functional groups participating in carbon, nitrogen, sulfur, and minor element cycling at haloalkaline conditions.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' The main re- sults of this work are summarized in several recent re- views [1, 6, 47, 48].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Since most microbes, including those living in soda lakes, still evade all cultivation ef- forts, a very effective way to discover new microbes and assess their physiology based on their genetic repertoire is either through single cell genomics or by directly se- quenced environmental DNA.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' This exploratory metage- nomics study performed on soda lake sediments effectively overcame the existing cultivation bottleneck.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' First, we expanded the known diversity of CPR consider- ably with the first genomes of poly-extremophiles sam- pled from soda lake sediments.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Although the presence of 16S rRNA genes from CPR in marine sediments and hy- persaline microbial mats was previously shown [34], until now, CPR MAGs were mainly obtained from deep, subsurface environments [15, 26, 29, 32, 49–52], and hu- man microbiota [30].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Despite being highly abundant Vavourakis et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Microbiome  (2018) 6:168 Page 10 of 18  100 % 90-100 % 70-90 % 50-70 % some MAGs all MAGs Bootstraps Genes present Glycolysis (EMP) PFOR WL-Eastern branch H4MPT TH4 WL-Western branch CODH/ACS Acetogenesis/ acetate activation (pta/ack) 0.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='4 PVC group (Chlamydiae LD1-PA32) Syntrophorhabdus aromaticivorans PVC group bacterium CSSed11_184 Aerophobetes bacterium SCGC_AAA255-F10 Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Acetothermia Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Handelsmanbacteria Planctomycetaceae Anaerolineae Firmicutes Brocadiaceae Planctomycetes Methanomassiliicoccales Halobacteroidaceae Natranaerobiaceae Methanomicrobiales Desulfonatronospira Firmicutes Dehalococcoidia Armatimonadetes bacterium CSP1-3 Deltaproteobacteria Thermodesulfobacteria Desulfobulbaceae Halanaerobiaceae Nitrospirae Actinobacteria (OPB41) Fig.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 6 Maximum likelihood phylogeny of the bacterial-type acetyl-coA synthases (acsB) found in the MAGs.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Only sequences ≥ 500 aa were included.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Lineages for which we discovered the Wood-Ljungdahl (WL) in this study are highlighted in orange, and the presence of genes in the respective MAGs related to WL, glycolysis, pyruvate, and acetate conversion is indicated by the colored symbols (see also Additional file 9: Dataset S6).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Additional lineages found in this study are marked in blue.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' The three was rooted according to [46].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Circles at the nodes show confidence percentage of the bootstraps analysis (100×).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='  EMP = Embden-Meyerhof-Parnas, PFOR = pyruvate:ferredoxin oxidoreductase complex, pta = phosphotransacetylase gene, ack = acetate kinase gene, H4MPT = tetrahydromethanopterin-linked pathway, TH4 = tetrahydrofolate pathway, CODH/ACS = carbon monoxide dehydrogenase/acetyl-CoA synthase.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' PVC group bacterium CSSed11_184 is likely a member of the WCHB1-41 class within the Verrucomicrobia Vavourakis et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Microbiome  (2018) 6:168 Page 11 of 18  here, CPR went unnoticed in previous amplicon sequen- cing studies.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' This might be due to the fact that many CPR representatives have random inserts of various length in their 16S rRNA genes or due to primer mis- matches [29, 34].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' This illustrates also that direct metage- nomics should not only be preferred over amplicon sequencing to infer functional potential, but the former is far more effective for the discovery of novel organ- isms.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Second, we obtained many more genomes from “traditional” bacterial phyla such as the Planctomycetes and Chloroflexi, as well as candidate phyla, for which no soda lake isolates, hence no genomes were previously obtained.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Third, even within the sulfur cycle, the most active and frequently studied element cycle in soda lakes [1], we found considerable metabolic novelty.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Finally, we found the Wood-Ljungdahl pathway in several novel phyla, not closely related to any known acetogens, methanogens, or sulfate-reducing bacteria [46].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' The lat- ter shows that our sequencing recovery effort has also significantly contributed to the discovery of metabolic novelty within various prokaryote phylogenetic groups.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Salinity is often considered to be the major factor shaping prokaryote community composition in diverse habitats [53, 54].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Extreme halophilic Euryarchaeota seem to be always the dominant group in salt-saturated hypersaline brines, both those with neutral or alkaline pH [1, 7, 37].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='  Here, we found that although these haloarchaea are still relatively more abundant in the sed- iments exposed to brines with salt-saturating conditions, the clear majority of microbes in all investigated hyper- saline soda lake sediments are Bacteria.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' It could be hypothesized that the sediment is a hide-out for the extreme alkalinity and salinity governing the water column, and that sediment stratification, especially in the anoxic part, offers plenty of opportunities for niche diversification.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' On the other hand, it should no longer be a surprise that soda lakes are such productive and biodiverse systems.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' First, it has been previously elaborated that soda lake organisms are exposed to approximately half the osmotic pressure in sodium carbonate-dominated brines compared to sodium chloride-dominated brines with the same Na+ molarity [47].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Second, nitrogen limitation in the community can be overcome when many members contribute to the fixation of atmospheric N2, and various forms of organic nitrogen are efficiently recycled.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' The soda lakes exam- ined in this study were also eutrophic, and sulfur com- pounds were abundant.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Sulfide is also far less toxic at high pH as it mostly occurs in the form of bisulfide (HS−).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Besides the evident high metabolic and taxo- nomic diversity of dissimilatory sulfur-cycling bacteria, a diverse heterotrophic community can be sustained com- prising both generalist and very specialized carbon de- graders.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='  Less eutrophic soda lakes might not suffer from carbon limitation either, due to a presence of high-bicarbonate concentrations.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' These effectively elim- inate the inorganic carbon limitation for primary pro- ducers who are highly active in soda lakes, especially Cyanobacteria [55, 56].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Third, light that penetrates the surface of the sediment seems to stimulate oxygenic and anoxygenic phototrophic growth.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Moreover, various het- erotrophs, such as the rhodopsin-containing haloarchaea and Bacteroidetes, have the option to tap into this un- limited energy source for example to help sustain the costly maintenance of osmotic balance.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Unexpectedly, we even found the first rhodopsin encoded by a member of the CPR.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Fourth, tight syntrophic relations, as pro- posed for CPR members and Syntrophomonadaceae spp.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=', might be the solution to successful growth in an energetically challenging environment.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Since our metagenomes are snapshots in time and space, the failure to reconstruct specific MAGs gives no conclu- sive evidence for the absence of certain microbial-mediated element transformation in hypersaline soda lake sediments.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Additionally, technical limitations of the assembly and bin- ning of highly micro-diverse genome populations might hamper genome recovery [57].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' More importantly, the abundance of a specific microbe is not necessarily corre- lated to the importance of its performance in an ecosystem.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='  Many metabolic capacities are redundant, and often key transformations are reserved for a few rare organisms that might proliferate for a short time-span when specific condi- tions allow for it.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' For example, although no MAGs were re- covered from chemolithoautotrophic nitrifiers [58], we did detect a Nitrobacter-related OTU by amplicon sequencing and assembled 16S rRNA genes from Thaumarchaeota, suggesting bacterial and archaeal nitrifiers are present in the surface sediments of soda lakes at very low abundance.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Finally, the method of DNA isolation might impact the community composition apparent in the final metagenome sequenced.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Environmental samples contain complex mix- tures of different organisms, and it is impossible to find a protocol where the DNA from every single organism is ex- tracted as efficiently without compromising the final quality of the extracted DNA.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' However, since we find all the im- portant taxonomic and functional groups known from pre- vious cultivation-dependent studies back in either our amplicon sequencing datasets or our directly sequenced metagenomes, we are confident that the community com- position and the MAGs presented here are representative for the microbiomes of the soda lake sediments in the Kulunda Steppe.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Conclusion Years of intensive microbiological research on soda lakes seem to have paid off, since many of the described gen- era we could detect here have a cultured representative from soda lakes.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='  However, as many of the abundant Vavourakis et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Microbiome  (2018) 6:168 Page 12 of 18  lineages and groups found in soda lake sediments are still uncultured, metagenomics proved to be a helpful tool to gain primary insights in the potential physiology and ecology of these poly-extremophilic prokaryotes.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' We reconstructed the first genomes for many of such organisms and proposed new functional roles for the most abundant ones.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Future studies should provide more in depth analyses of these genomes, especially from the less abundant organisms that might perform key ecological processes, such as methanogens and nitri- fiers.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' In addition, they should focus on gaining physio- logical culture-based evidence or proof for in situ activity for the abundant organisms described here.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' The key metabolic insights provided by this metagenomics study can lead to the design of new cultivation strategies.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' In general, sediment communities are far more complex than those found in the corresponding water column [53, 59] and are therefore often considered too complex for efficient metagenomic analysis.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Many of the novel lineages found here may therefore have related neutro- philic lineages in marine and freshwater sediments that await discovery.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' We demonstrate here that, by providing sufficient sequencing depth, the “state of the art metage- nomics toolbox” can effectively be used on sediments as well.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='  Methods Site description and sample collection The top 10 cm sediments from four hypersaline, eutrophic soda lakes located in the Kulunda Steppe (south-western Siberia, Altai, Russia) were sampled in July of 2010 and 2011.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' General features and exact location of the sampled soda lakes are summarized in Additional file 1: Table S1;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' a map of the area was published previously [5].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Cock Soda Lake (a stand-alone lake, sampled both in 2010 and 2011) and Tanatar-3 (Tanatar system) were moderately hypersa- line (~ 100 g L−1) with sandy sediment, while Tanatar-1 and Bitter-1 (Bitter system) were salt-saturated (400 g L−1) with sulfide-rich sapropel sediments underlined by rock trona deposits [7, 60].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Especially, Bitter-1 harbors a very active microbial community, probably due to its high- organic and -mineral content.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Surface sediments were col- lected by a plastic corer into sterile glass containers and transported to the laboratory in a cooler.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' DNA isolation, 16S rRNA amplicon, and metagenomic sequencing The colloidal fraction of each sediment sample (~ 10% of 50 g) was separated from the course sandy fraction by several short (30–60 s) low-speed (1–2,000 rpm in 50 mL Falcon tubes) centrifugation steps and washed with 1–2 M NaCl solution.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' The pelleted colloidal sedi- ment fraction was first subjected to 3 cycles of freezing in liquid nitrogen/thawing, then re-suspended in 0.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='1 M Tris (pH 8)/10 mM EDTA, and then subjected to harsh bead beating treatment.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='  Next, the samples were incu- bated with lysozyme (15 mg/mL) for 2 h at 37 °C followed by a SDS (10% w/v) and proteinase K (10 μg/ mL) treatment for 30 min.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' at 45 °C.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' High molecular weight DNA was isolated using phenol/chloroform ex- traction, quality-checked, and sequenced as described previously [7].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Direct high-throughput sequencing of the DNA was performed on an Illumina HiSeq 2000 plat- form to generate 150 b paired-end reads.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Amplification of the V4-V6 region of prokaryote 16S rRNA genes using barcoded 926F-1392R primers, amplicon purifica- tion, quantification, and Roche (454)-sequencing was performed together in a batch with brine samples from the same sampling campaigns.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Barcodes and adapter se- quences were removed from de-multiplexed amplicon sequence reads and analyzed with the automated NGS analysis pipeline of the SILVA rRNA gene database pro- ject [61] (SILVAngs 1.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='3, database release version 128) using default parameters.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' The OTUs (97% identity) assigned down to the genus level were only considered when they had a relative abundance ≥ 0.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='1% in at least one of the five datasets.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Processing metagenomics reads, assembly, binning, and post-binning Metagenomic raw reads were quality trimmed using Sickle [62] (version 1.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='33), and only reads ≥ 21 b were retained.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' The prokaryotic community structure at taxo- nomic top levels was extrapolated from ten million ran- domly sampled singletons from each dataset.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='  Candidate 16S rRNA fragments > 90 b were identified [63] and compared against the SILVA SSU database 128 (blastn, min.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' length 90, min.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' identity 80%, e value 1e-5).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' To ver- ify that the microbial community composition was in- deed mostly prokaryotic, we did a more general screening of the metagenomics reads that identified also candidate 18S rRNA fragments > 90 b (see Additional file 1: Tables S4-S5).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' The complete trimmed read sets were assembled into contigs ≥ 1 kb with MEGAHIT [64] (v1.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='0.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='3–6-gc3983f9) using paired-end mode, k min = 21, k max = 131, k step = 10.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Genes were predicted using Prodigal [65] (v.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='2.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='6.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='2) and RNAs with rna_hmm3 [66] and tRNAscan-SE [67].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Assembled 16S rRNA sequences were compared to a manually curated version from the SILVA SSU database (e value ≥ 1e-5).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Predicted protein sequences were annotated against KEGG with GhostKOALA (genus_prokaryotes + family_eukaryotes + viruses) [68].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Marker genes for central metabolic pathways and key environmental element transforma- tions were identified based on K number assignments [15, 69–71].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Contigs ≥ 2.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='5 kb were binned with METABAT [72] (superspecific mode) based on differential coverage Vavourakis et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Microbiome  (2018) 6:168 Page 13 of 18  values obtained by mapping all five trimmed readsets to all five contig sets with Bowtie2 [73].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' The bins were sub- jected to post-binning (an overview of the workflow is given in Additional file 2: Figure S13).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Bins were assessed with lineage-specific single copy genes using CheckM [74] and further processed with the metage- nomics workflow in Anvi’o [75] (v2.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='3.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='2).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Since Candidate Phyla Radiant (CPR) is not included in the CheckM ref- erence trees and are likely to have low-genome com- pleteness, we used an existing training file of 797 CPR genomes to identify putative CPR bins [76].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Bins with CheckM-completeness ≥ 50% (884 out of 1778) and an additional four CPR bins were further processed.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Coding sequences were annotated for taxonomy against NCBI-nr (July, 2017) with USEARCH [77] (5.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='2.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='32) to verify that most hits in each bin were to prokaryotic ref- erences.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Phage or viral contigs were manually removed.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Genome contamination (redundancy) was estimated based on marker sets of universal single copy genes identified for Bacteria [30] and Archaea [78] as imple- mented in Anvi’o.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Genome coverage was obtained by mapping trimmed reads with BBMap [79] v36.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='x (kfilter 31, subfilter 15, maxindel 80).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Bins with ≥ 5% redun- dancy were further refined with Anvi’o using circle phy- lograms (guide trees tnf-cov: euclidian ward) and scanned again for CPR.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='  Post-binning resulted in a total of 2499 metagenome-assembled genomes (MAGs), of which 871 were either medium-quality genome drafts (CheckM estimated completeness ≥ 50% and contamin- ation ≤ 10% [80], Additional file 4) or lower quality draft genomes from CPR.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Phylogeny of the MAGs was assessed based on 16 single-copy ribosomal proteins and representative refer- ence genomes of major prokaryote lineages across the tree of life [17].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Individual ribosomal proteins in our MAGs were identified by K number assignments.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Only ribosomal proteins ≥ 80 aa were considered.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Initial maximum-likelihood (ML) trees were constructed to de- termine which organisms belonged to the Archaea, Bac- teria, or CPR with FastTree 2 [81] (WAG + CAT).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Final separate trees for the three distant evolutionary groups were constructed in the same manner.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Each ribosomal protein set was aligned separately with MAFFT [82] (v7.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='055b, − auto) and concatenated only if a MAG encoded at least 8 out of 16 proteins.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' For all trees, a 100× posterior bootstraps analysis was performed.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Phylogenetic trees were visualized together with gen- ome statistics and abundance information using iTOL [83].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' We cross-checked the taxonomic assignments based on the phylogeny of the ribosomal protein cas- sette with the top hit contig annotations against NCBI-nr and with the reference lineage obtained with CheckM.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Lastly, we manually corrected the MAGs for misplaced 16S rRNA genes.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='  The final trees presented in the manuscript were redrawn using FigTree v1.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='4.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='3 [84].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Detailed genome analyses CPR MAGs were re-annotated more thoroughly: genes were predicted with Prokka [85], and functional predictions were performed by running InterProScan 5 locally on the supplied COG, CDD, TIGRFAMs, HAMAP, Pfam, and SMART databases [86].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' BLAST Koala was used for KEGG pathway predictions [68].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' To find putative carbohydrate-active enzymes in all final MAGs, we used the web-resource dbCAN [87] to annotate all predicted proteins ≥ 80 aa against CAZy [88].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' To identify the top ten abundant MAGs from each re- spective dataset, ten million randomly sampled single- tons were mapped onto each MAG with a cut-off of 95% identity in minimum of 50 bases.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Coverage values were additionally normalized for genome size and expressed as reads per kilobase of sequence per gigabase of mapped reads (RPKG) [89].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' A positive score (from 871 to 1) was assigned to each MAG according to the rank- ing of the summed RPKG of MAGs in the high-salinity datasets (B1Sed10 and T1Sed) and a negative score ac- cording to the ranking of the summed RPKGs in the moderate salinity datasets (CSSed10, CSSed11, T3Se d10).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Both scores were summed to get a “salinity prefer- ence score” with MAGs recruiting preferably from high salinity datasets on the positive end, moderate salinity datasets in the negative end, and those without prefer- ence in the middle.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='  We determined species delineation for the most abundant MAGs and their closest reference genomes (NCBI-nr) by Average Nucleotide Identity (ANI) and conserved DNA-matrices, as follows [90]: ANI ≥ 95%, conDNA ≥ 69% = same species, ANI ≥ 95%, condDNA < 69% = might be same species, ANI < 95%, condDNA < 69% = different species.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Single gene trees based on maximum likelihood were constructed with un- trimmed alignments (MAFFT, L-INS-i model) and FastTree 2 (WAG + CAT, increased accuracy, -spr4 -mlacc 2 -slownni) using 100× bootstraps.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' References were pulled from eggNOG (v4.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='5.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='1) [91] and supple- mented with sequences from NCBI-nr or refined according to [7, 33, 46, 92–94].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' The curated MAGs were scanned for the presence of rhodopsin sequences with the hmmsearch software [95] and a profile hidden Markov model (HMM) of the bacteriorhodopsin-like protein family (Pfam accession number PF01036).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' The identified sequences with significant similarity were aligned together with a curated database composed of a collection of type-1 rhodopsins, using MAFFT (L-INS-i accuracy model) [82].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' This protein alignment was further utilized to Vavourakis et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Microbiome  (2018) 6:168 Page 14 of 18  construct a maximum likelihood tree with 100× boot- strap with FastTree 2 [81].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' All other genes were identified using the KEGG annotation.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Additional files Additional file 1: Table S1.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' General features of the four sampled soda lakes at time of sampling.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Table S2.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' SILVA classification of the 16S rRNA gene sequences found in all ≥1 kb contigs of five soda sediment metagenomic datasets.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Table S3.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Enzymes involved in lipopolysaccharide biosynthesis found among different members of the CPR.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Table S4.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Sub-kingdom classification of candidate SSU rRNA gene fragments found in subsamples of 10 million random forward reads from the five soda sediment metagenomes.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Table S5.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Top-level taxonomic classification of the 18S rRNA gene fragments found in subsamples of 10 million random forward reads from the five soda sediment metagenomes.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Table S6.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Description of the metagenomic datasets, NCBI Sequence Read Archive (SRA) accession numbers and general statistics of the assembled contigs.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' (PDF 740 kb) Additional file 2: Figure S1.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Taxonomic fingerprints determined by 16S rRNA gene amplicon sequencing.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Figure S2.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Genome statistics of the 871 MAGs.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Figure S3.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Phylogeny of MAGs belonging to “Candidatus Aenigmarchaeota” and “Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Nanohaloarchaeota”.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Figure S4.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Phylogeny of MAGs related to “Candidatus Acetothermia”, candidate division WS1 and “Candidatus Lindowbacteria”.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Figure S5.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Phylogeny of MAGs related to candidate division KSB3 and “Candidatus Schekmanbacteria”.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Figure S6.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='  Multiple sequence alignment of the V-type ATPase subunits K.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Figure S7.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Multiple sequence alignment of the F-type ATPase subunits c.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Figure S8.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Maximum likelihood tree of the large subunits of RuBisCo and RubisCo- like proteins.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Figure S9.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Maximum likelihood tree of the putative rhodopsins.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Figure S10.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Predicted isoelectric points (pI) profiles of all MAGs from CPR members.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Figure S11.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Predicted isoelectric points profiles for members of the “Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Nealsonbacteria” and “Ca.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Vogelbacteria”.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Figure S12.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Multiple sequence alignment of the dissimilatory cytochrome c nitrite reductases (nrfA/TvNiR, K03385).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Figure S13.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Overview of the post-binning workflow used for genome recovery.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' (PDF 6548 kb) Additional file 3: Dataset S1.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Relative abundance of the OTUs assigned to the genus-level within the Archaea, Bacteria and organelles from Eukaryota detected by 16S rRNA gene amplicon sequencing.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' The OTUs with less than 0.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='1% abundance accross all five datasets are not shown.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' The names of highly abundant genera (≥1% in at least one of the data- sets) are shown in bold.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' (XLSX 24 kb) Additional file 4: Dataset S2.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Organism names, statistics and general description incl.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Completeness and contamination estimates, phylogeny and DDBJ/EMBL/Genbank accession numbers of the metagenome assembled genomes (MAGs) described in this paper.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' All submitted versions described in this paper are version XXXX01000000.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='  Size = recovered genome size, Completeness (Compl1), contamination (Cont), strain heterogenity (Str het) and Taxon CheckM were inferred from lineage-specific marker sets and a reference tree build with CheckM [74].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Additional completeness (compl2) and redundancy (red) estimates were inferred based on the presence of universal single copy genes for Bacteria and Archaea [75].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Decision and confidence intervals from the Candidate Phyla Radiation (CPR) scan [75] are given, as well as the taxonomy of the besthit in SILVA when 16S rRNA genes were present.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Phylum/class 16 ribosomal proteins is the taxonomy derived from our ribosomal protein trees (see main text: Figs.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2 and 3).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' OTU gives the inferred link of a population genome with our 16S rRNA gene amplicon dataset (Additional file 3).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' (XLSX 253 kb) Additional file 5: Dataset S3.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Estimated abundance and derived salinity preference from each MAG in each metagenomic dataset expressed as Reads per Kilobase of MAG per Gigabase of mapped reads (RPKG) and “salinity preference score” (see Methods section), basis for Fig.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 4.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' (XLSX 143 kb) Additional file 6: Dataset S4.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Average Nucleotide Identity (ANI) and conserved DNA (condna) matrices to determine species delineation between the most abundant MAGs shown in Fig.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 4, closely related (less abundant) MAGs and NCBI reference genomes.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Decision matrix shows: 1 = same species, − 1 = might be same species, 0 = different species (see Methods section).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' (XLSX 1161 kb) Additional file 7: Dataset S5.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='  Sheet 1 Presence and absence of marker genes and putative carbohydrate-active enzymes in the MAGs to infer putative roles in C, N and S element cycles based on K-number assignments and CAZy annotations.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Sheet 2 Summary basis for Fig.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 4.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' (XLSX 41 kb) Additional file 8: Information S1.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' More detailed description of the main metabolisms encoded by Thioalkalivibrio-related MAGs.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Information S2 More detailed description of the main metabolisms encoded by Deltaproteobacterial-related MAGs.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' (PDF 219 kb) Additional file 9: Dataset 6.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Sheet 1 shows the MAGs positive for the marker gene acsB (K14138) encoding an acetyl-CoA synthase (ACS).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' The basis for Fig.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 6, namely presence and absence of key genes involved in the Wood-Ljungdahly pathway, acetogenesis, methanogenesis, glycolysis and pyruvate to CO2 conversion is shown for each MAG.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Sheet 2 shows the MAGs positive for the marker gene cdhC (K00193) encoding for the beta subunit of an acetyl-CoA decarboxylase synthase complex.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' While acsB and cdhC correspond roughly to the Bacterial-type and Archaeal- type (methanogens) enzymes with the same function, we found few discrepancies between marker gene and genome phylogeny within the Methanomassiliicoccaceae and Chloroflexi.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' (XLSX 52 kb) Acknowledgments We thank Dr.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Nikolai Chernych for his technical assistance during the isolation and purification of metagenomics DNA.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' We also thank the Department of Energy Joint Genome Institute for sequencing the metagenomes.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='  Funding CDV and GM were supported by the ERC Advanced Grant PARASOL (no.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 322551).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' A-SA and RG were supported by the research grant 17-04828S from the Grant Agency of the Czech Republic.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' MM was supported by the Czech Academy of Sciences (Postdoc program PPPLZ application number L200961651).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' DYS was supported by the SIAM/Gravitation Program (Dutch Ministry of Education and Science, grant 24002002) and by the Russian Science Foundation (grant 16–14- 00121).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Sequencing was performed by the U.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='S.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Department of Energy Joint Genome Institute, a DOE Office of Science User Facility, as part of the Community Sequencing Program (contract no.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' DE-AC02- 05CH11231).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Availability of data and materials The raw sequence reads of the five metagenomes have been deposited to the NCBI Sequence Read Archive (see Additional file 1: Table S6 for accession numbers and read and contig statistics).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' The final 871 MAGs described in this paper have been deposited as Whole Genome Shotgun projects at DDBJ/ EMBL/GenBank, and accession numbers are listed in Additional file 4 (BioProject ID PRJNA434545).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' All versions described in this paper are version XXXX01000000.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' The cleaned and dereplicated amplicon sequence datasets are available in FigShare (https://figshare.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='com/s/7684627445e3621aba24).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Maximum likelihood trees based on the concatenated alignment of 16 ribosomal proteins, basis for Figs.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='  2 and 3, in newick format (.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='tre file) and complementary datasets (used to plot completeness, contamination, genome recovery size, G + C mol% and RPKG in iTOL), as well as K number assignments for the predicted proteins of all MAGs (KEGG-orthologues, Ghost Koala) and the fully annotated CPR MAGs supporting the conclusions of this article are also available in FigShare (https://figshare.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='com/s/ 7684627445e3621aba24).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Authors’ contributions GM and DYS initiated this study and were responsible for the fieldwork, sample preparation, and sequencing effort.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' CDV conceptualized the research goals under supervision of DYS and GM, and performed the bioinformatics analysis under close guidance of A-SA and RG.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' CDV is the primary author of this manuscript.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' MM, RG, and CDV prepared the main figures.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' All authors read and approved the final manuscript.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Ethics approval and consent to participate Not applicable.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Vavourakis et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Microbiome  (2018) 6:168 Page 15 of 18  Consent for publication Not applicable.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Competing interests The authors declare that they have no competing interests.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Author details 1Microbial Systems Ecology, Department of Freshwater and Marine Ecology, Institute for Biodiversity and Ecosystem Dynamics, Faculty of Science, University of Amsterdam, Postbus 94248, 1090, GE, Amsterdam, the Netherlands.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2Department of Aquatic Microbial Ecology, Institute of Hydrobiology, Biology Centre CAS, Na Sadkach 7, 370 05 Ceske Budejovice, Czech Republic.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 3Winogradsky Institute of Microbiology, Research Centre of Biotechnology, Russian Academy of Sciences, 60 let Oktyabrya pr-t, 7, bld.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2, Moscow, Russian Federation117312.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 4Environmental Biotechnology, Department of Biotechnology, Delft University of Technology, Van der Maasweg 9, 2629, HZ, Delft, the Netherlands.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Received: 23 June 2018 Accepted: 3 September 2018 References 1.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Sorokin DY, Berben T, Melton ED, Overmars L, Vavourakis CD, Muyzer G.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Microbial diversity and biogeochemical cycling in soda lakes.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Extremophiles.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2014;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='18:791–809.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Oduor SO, Kotut K.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Soda lakes of the East African Rift System: the past, the present and the future.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' In: Schagerl M, editor.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Soda lakes of East Africa.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Berlin: Springer;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2016.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' p.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 365–74.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 3.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Mesbah NM, Abou-El-Ela SH, Wiegel J.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='  Novel and unexpected prokaryotic diversity in water and sediments of the alkaline, hypersaline lakes of the Wadi An Natrun, Egypt.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Microb Ecol.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2007;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='54:598–617.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 4.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Humayoun SB, Bano N, James T, Hollibaugh JT.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Depth distribution of microbial diversity in Mono Lake, a meromictic soda lake in California.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Appl Environ Microbiol.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2003;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='69:1030–42.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 5.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Foti MJ, Sorokin DY, Zacharova EE, Pimenov NV, Kuenen JG, Muyzer G.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Bacterial diversity and activity along a salinity gradient in soda lakes of the Kulunda Steppe (Altai, Russia).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Extremophiles.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2008;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='12:133–45.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 6.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Sorokin DY.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Anaerobic haloalkaliphiles.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' eLS.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2017;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' https://doi.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='org/10.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='1002/ 9780470015902.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='a0027654.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 7.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Vavourakis CD, Ghai R, Rodriguez-Valera F, Sorokin DY, Tringe SG, Hugenholtz P, et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Metagenomic insights into the uncultured diversity and physiology of microbes in four hypersaline soda lake brines.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Front Microbiol.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2016;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='7:211.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 8.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Sørensen KB, Canfield DE, Oren A.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Salinity responses of benthic microbial communities in a solar saltern (Eilat, Israel).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Appl Environ Microbiol.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2004;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='70: 1608–16.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 9.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Sorokin DY, Makarova KS, Abbas B, Ferrer M, Golyshin PN, Galinski EA, et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Discovery of extremely halophilic, methyl-reducing euryarchaea provides insights into the evolutionary origin of methanogenesis.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Nat Microbiol.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2017;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='2:17081.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 10.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Sorokin DY, Chernyh NA, Poroshina MN.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Desulfonatronobacter acetoxydans sp.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' nov.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=',: a first acetate-oxidizing, extremely salt-tolerant alkaliphilic SRB from a hypersaline soda lake.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Extremophiles.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='  2015;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='19:899–907.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 11.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Ahn A-C, Meier-Kolthoff JP, Overmars L, Richter M, Woyke T, Sorokin DY, et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Genomic diversity within the haloalkaliphilic genus Thioalkalivibrio.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' PLoS One.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2017;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='12:e0173517.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 12.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Sorokin DY, Kuenen JG.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Haloalkaliphilic sulfur-oxidizing bacteria in soda lakes.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' FEMS Microbiol Rev.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2005;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='29:685–702.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 13.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Albertsen M, Hugenholtz P, Skarshewski A, Nielsen KL, Tyson GW, Nielsen PH.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Genome sequences of rare, uncultured bacteria obtained by differential coverage binning of multiple metagenomes.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Nat Biotechnol.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2013;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='31:533–8.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 14.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Qin J, Li R, Raes J, Arumugam M, Burgdorf KS, Manichanh C, et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' A human gut microbial gene catalogue established by metagenomic sequencing.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Nature.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2010;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='464:59–65.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 15.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Anantharaman K, Brown CT, Hug LA, Sharon I, Castelle CJ, Probst AJ, et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Thousands of microbial genomes shed light on interconnected biogeochemical processes in an aquifer system.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Nat Commun.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2016;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='7:13219.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 16.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Parks DH, Rinke C, Chuvochina M, Chaumeil P-A, Woodcroft BJ, Evans PN, et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Recovery of nearly 8,000 metagenome-assembled genomes substantially expands the tree of life.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Nat Microbiol.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2017;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='2:1533–42.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 17.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Hug LA, Baker BJ, Anantharaman K, Brown CT, Probst AJ, Castelle CJ, et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' A new view of the tree of life.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Nat Microbiol.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2016;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='1:16048.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 18.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Hahnke RL, Meier-Kolthoff JP, García-López M, Mukherjee S, Huntemann M, Ivanova NN, et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Genome-based taxonomic classification of Bacteroidetes.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Front Microbiol.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2016;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='7:2003.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 19.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Nolla-Ardevol V, Strous M, Tegetmeyer HE.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='  Anaerobic digestion of the microalga Spirulina at extreme alkaline conditions: biogas production, metagenome and metatranscriptome.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Front Microbiol.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2015;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='6:597.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 20.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Borrel G, Parisot N, Harris HM, Peyretaillade E, Gaci N, Tottey W, et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Comparative genomics highlights the unique biology of Methanomassiliicoccales, a Thermoplasmatales-related seventh order of methanogenic archaea that encodes pyrrolysine.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' BMC Genomics.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2014;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='15:679.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 21.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Sorokin DY, Abbas B, Geleijnse M, Pimenov NV, Sukhacheva MV, van Loosdrecht MCM.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Methanogenesis at extremely haloalkaline conditions in the soda lakes of Kulunda Steppe (Altai, Russia).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' FEMS Microbiol Ecol.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2015;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='91:4.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 22.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Nobu MK, Narihiro T, Kuroda K, Mei R, Liu WT.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Chasing the elusive Euryarchaeota class WSA2: genomes reveal a uniquely fastidious methyl- reducing methanogen.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' ISME J.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2016;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='10:2478–87.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 23.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Skennerton CT, Haroon MF, Briegel A, Shi J, Jensen GJ, Tyson GW, et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Phylogenomic analysis of Candidatus “Izimaplasma” species: free-living representatives from a Tenericutes clade found in methane seeps.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' ISME J.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2016;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='10:2679–92.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 24.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Sekiguchi Y, Ohashi A, Parks DH, Yamauchi T, Tyson GW, Hugenholtz P.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' First genomic insights into members of a candidate bacterial phylum responsible for wastewater bulking.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' PeerJ.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2015;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='3:e740.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 25.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Wrighton KC, Thomas BC, Sharon I, Miller CS, Castelle CJ, VerBerkmoes NC, et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Fermentation, hydrogen, and sulfur metabolism in multiple uncultivated bacterial phyla.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Science.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2012;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='337:1661–5.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 26.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='  León-Zayas R, Peoples L, Biddle JF, Podell S, Novotny M, Cameron J, et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' The metabolic potential of the single cell genomes obtained from the Challenger Deep, Mariana Trench within the candidate superphylum Parcubacteria (OD1).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Environ Microbiol.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2017;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='19:2769–84.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 27.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Castelle CJ, Brown CT, Thomas BC, Williams KH, Banfield JF.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Unusual respiratory capacity and nitrogen metabolism in a Parcubacterium (OD1) of the Candidate Phyla Radiation.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Sci Rep.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2017;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='7:40101.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 28.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Anantharaman K, Brown CT, Burstein D, Castelle CJ, Probst AJ, Thomas BC, et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Analysis of five complete genome sequences for members of the class Peribacteria in the recently recognized Peregrinibacteria bacterial phylum.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' PeerJ.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2016;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='4:e1607.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 29.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Brown CT, Hug LA, Thomas BC, Sharon I, Castelle CJ, Singh A, et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Unusual biology across a group comprising more than 15% of domain Bacteria.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Nature.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2015;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='523:208–11.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 30.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Campbell JH, O ‘donoghue P, Campbell AG, Schwientek P, Sczyrba A, Woyke T, et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' UGA is an additional glycine codon in uncultured SR1 bacteria from the human microbiota.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2013;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' doi:https://doi.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='org/10.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='1073/pnas.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 1303090110.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 31.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Hanke A, Hamann E, Sharma R, Geelhoed JS, Hargesheimer T, Kraft B, et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Recoding of the stop codon UGA to glycine by a BD1-5/SN-2 bacterium and niche partitioning between Alpha- and Gammaproteobacteria in a tidal sediment microbial community naturally selected in a laboratory chemostat.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Front Microbiol.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2014;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='5:231.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 32.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='  Kantor RS, Wrighton KC, Handley KM, Sharon I, Hug LA, Castelle CJ, et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Small genomes and sparse metabolisms of sediment-associated bacteria from four candidate phyla.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' MBio.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2013;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='4:1–11.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 33.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Wrighton KC, Castelle CJ, Varaljay VA, Satagopan S, Brown CT, Wilkins MJ, et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' RubisCO of a nucleoside pathway known from Archaea is found in diverse uncultivated phyla in bacteria.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' ISME J.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2016;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='10:2702–14.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 34.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Luef B, Frischkorn KR, Wrighton KC, Holman HYN, Birarda G, Thomas BC, et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Diverse uncultivated ultra-small bacterial cells in groundwater.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Nat Commun.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2015;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='6:1–8.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 35.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Krulwich TA, Sachs G, Padan E.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Molecular aspects of bacterial pH sensing and homeostasis.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Nat Rev Microbiol.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2011;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='9:330–43.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 36.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Hauß T, Dante S, Dencher NA, Haines TH.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Squalane is in the midplane of the lipid bilayer: implications for its function as a proton permeability barrier.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Biochim Biophys Acta Bioenerg.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2002;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='1556:149–54.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 37.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Oren A.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Life at high salt concentrations, intracellular KCl concentrations, and acidic proteomes.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Front Microbiol.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2013;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='4:315.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Vavourakis et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Microbiome  (2018) 6:168 Page 16 of 18  Published online: 19 September 201838.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Levina N.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Protection of Escherichia coli cells against extreme turgor by activation of MscS and MscL mechanosensitive channels: identification of genes required for MscS activity.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' EMBO J.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 1999;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='18:1730–7.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 39.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Gupta RS, Khadka B.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Evidence for the presence of key chlorophyll- biosynthesis-related proteins in the genus Rubrobacter (phylum Actinobacteria) and its implications for the evolution and origin of photosynthesis.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Photosynth Res.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2016;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='127:201–18.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 40.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Basak N, Das D.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' The prospect of purple non-sulfur (PNS) photosynthetic bacteria for hydrogen production:the present state of the art.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' World J Microbiol Biotechnol.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2007;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='23:31–42.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 41.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Meng J, Wang F, Wang F, Zheng Y, Peng X, Zhou H, et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' An uncultivated crenarchaeota contains functional bacteriochlorophyll a synthase.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' ISME J.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2009;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='3:106–16.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 42.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Sorokin DY, Tourova TP, Mußmann M, Muyzer G.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Dethiobacter alkaliphilus gen.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' nov.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' sp.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' nov.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=', and Desulfurivibrio alkaliphilus gen.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' nov.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' sp.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' nov.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=': two novel representatives of reductive sulfur cycle from soda lakes.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Extremophiles.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2008;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='12:431–9.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 43.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Poser A, Lohmayer R, Vogt C.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Extremophiles KK-, 2013 U.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Disproportionation of elemental sulfur by haloalkaliphilic bacteria from soda lakes.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Extremophiles.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2013;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='17:1003–12.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 44.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Sorokin DY, Abbas B, Tourova TP, Bumazhkin BK, Kolganova TV, Muyzer G.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Sulfate-dependent acetate oxidation under extremely natron-alkaline conditions by syntrophic associations from hypersaline soda lakes.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Microbiology.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='  2014;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='160(Pt_4):723–32.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 45.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Ragsdale SW.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Enzymology of the Wood-Ljungdahl pathway of acetogenesis.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Ann N Y Acad Sci.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2008;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='1125:129–36.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 46.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Adam PS, Borrel G, Gribaldo S.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Evolutionary history of carbon monoxide dehydrogenase/acetyl-CoA synthase, one of the oldest enzymatic complexes.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Proc Natl Acad Sci.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2018;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='115:E1166–73.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 47.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Sorokin DY, Banciu HL, Muyzer G.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Functional microbiology of soda lakes.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Curr Opin Microbiol.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2015;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='25:88–96.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 48.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Grant WD, Jones BE.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Bacteria, Archaea and viruses of soda lakes.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' In: Schagerl M, editor.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Soda lakes of East Africa.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Berlin: Springer;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2016.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' p.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 97–147.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 49.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Bruno A, Sandionigi A, Rizzi E, Bernasconi M, Vicario S, Galimberti A, et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Exploring the under-investigated “microbial dark matter” of drinking water treatment plants.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Sci Rep.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2017;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='7:1–7.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 50.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Danczak RE, Johnston MD, Kenah C, Slattery M, Wrighton KC, Wilkins MJ.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Members of the Candidate Phyla Radiation are functionally differentiated by carbon- and nitrogen-cycling capabilities.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Microbiome.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2017;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='5:112.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 51.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Hu P, Tom L, Singh A, Thomas BC, Baker BJ, Piceno YM, et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Genome- resolved metagenomic analysis reveals roles for candidate phyla and other microbial community members in biogeochemical transformations in oil reservoirs.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' MBio.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2016;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='7:e01669–15.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 52.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Probst AJ, Castelle CJ, Singh A, Brown CT, Anantharaman K, Sharon I, et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Genomic resolution of a cold subsurface aquifer community provides metabolic insights for novel microbes adapted to high CO2 concentrations.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Environ Microbiol.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='  2017;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='19:459–74.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 53.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Lozupone CA, Knight R.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Global patterns in bacterial diversity.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Proc Natl Acad Sci.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2007;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='104:11436–40.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 54.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Thompson LR, Sanders JG, McDonald D, Amir A, Ladau J, Locey KJ, et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' A communal catalogue reveals Earth’s multiscale microbial diversity.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Nature.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2017;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='551:457.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 55.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Samylina OS, Sapozhnikov FV, Gainanova OY, Ryabova AV, Nikitin MA, Sorokin DY.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Algo-bacterial communities of the Kulunda steppe (Altai region, Russia) Soda Lakes.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Microbiology.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2014;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='83:849–60.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 56.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Krienitz L, Schagerl M.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Tiny and tough: microphytes of east African soda lakes.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' In: Schagerl M, editor.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Soda lakes of East Africa.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Berlin: Springer;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2016.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' p.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 149–77.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 57.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Nelson WC, Maezato Y, Wu Y-W, Romine MF, Lindemann SR.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Identification and resolution of microdiversity through metagenomic sequencing of parallel consortia.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Appl Environ Microbiol.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2015;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='82:255–67.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 58.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Hansel C.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Small but mighty: how minor components drive major biogeochemical cycles.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Environ Microbiol Rep.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2017;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='9:8–10.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 59.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Zinger L, Amaral-Zettler LA, Fuhrman JA, Horner-Devine MC, Huse SM, Welch DBM, et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Global patterns of bacterial beta-diversity in seafloor and seawater ecosystems.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' PLoS One.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2011;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='6:e24570.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 60.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Isachenko BL.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Chloride sulfate and soda lakes of Kulunda steppe and its biogenic processes.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' In: Selected works, vol.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Leningrad: Academy of Sciences USSR;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 1951.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' p.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 143–62.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 61.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Quast C, Pruesse E, Yilmaz P, Gerken J, Schweer T, Yarza P, et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='  The SILVA ribosomal RNA gene database project: improved data processing and web- based tools.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Nucleic Acids Res.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2012;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='41:D590–6.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 62.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Joshi NA, Fass JN.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Sickle: a sliding-window, adaptive, quality-based trimming tool for FastQ files (Version 1.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='33).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2011.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 63.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Ghai R, Pašić L, Fernández AB, Martin-Cuadrado A-B, Mizuno CM, McMahon KD, et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' New abundant microbial groups in aquatic hypersaline environments.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Sci Rep.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2011;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='1:135.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 64.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Li D, Liu CM, Luo R, Sadakane K, Lam TW.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' MEGAHIT: An ultra-fast single- node solution for large and complex metagenomics assembly via succinct de Bruijn graph.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Bioinformatics.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2015;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='31:1674–6.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 65.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Hyatt D, Chen G-L, LoCascio PF, Land ML, Larimer FW, Hauser LJ.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Prodigal: prokaryotic gene recognition and translation initiation site identification.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' BMC Bioinformatics.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2010;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='11:119.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 66.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Huang Y, Li W, Finn PW, Perkins DL.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Ribosomal RNA identification in metagenomic and metatranscriptomic datasets.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' In: De Bruijn FJ, editor.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Handbook of Molecular Microbial Ecology I.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Hoboken: Wiley;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2011.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' p.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 387–91.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 67.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Lowe TM, Eddy SR.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' tRNAscan-SE: a program for improved detection of transfer RNA genes in genomic sequence.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Nucleic Acids Res.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 1997;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='25: 955–64.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 68.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Kanehisa M, Sato Y, Morishima K.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' BlastKOALA and GhostKOALA: KEGG tools for functional characterization of genome and metagenome sequences.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' J Mol Biol.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2016;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='428:726–31.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 69.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Lauro FM, Demaere MZ, Yau S, Brown MV, Ng C, Wilkins D, et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' An integrative study of a meromictic lake ecosystem in Antarctica.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' ISME J.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='  2010;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 5:879–95.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 70.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Hernsdorf AW, Amano Y, Miyakawa K, Ise K, Suzuki Y, Anantharaman K, et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Potential for microbial H2 and metal transformations associated with novel bacteria and archaea in deep terrestrial subsurface sediments.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Nat Publ Gr.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2017;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='11:1915–29.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 71.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Llorens-Marès T, Yooseph S, Goll J, Hoffman J, Vila-Costa M, Borrego CM, et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Connecting biodiversity and potential functional role in modern euxinic environments by microbial metagenomics.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' ISME J.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2015;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='9:1648–61.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 72.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Kang DD, Froula J, Egan R, Wang Z.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' MetaBAT, an efficient tool for accurately reconstructing single genomes from complex microbial communities.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' PeerJ.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2015;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='3:e1165.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 73.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Langmead B, Salzberg SL.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Fast gapped-read alignment with bowtie 2.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Nat Methods.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2012;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='9:357–9.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 74.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Parks DH, Imelfort M, Skennerton CT, Hugenholtz P, Tyson GW.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' CheckM: assessing the quality of microbial genomes recovered from isolates, single cells, and metagenomes.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Genome Res.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2015;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='25:1043–55.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 75.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Eren AM, Esen ÖC, Quince C, Vineis JH, Morrison HG, Sogin ML, et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Anvi’o: an advanced analysis and visualization platform for ‘omics data.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' PeerJ.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2015;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 3:e1319.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 76.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Eren AM, Delmot TO.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Predicting CPR genomes in metagenomic bins.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' http:// merenlab.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='org/2016/04/17/predicting-CPR-Genomes/.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 77.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Edgar RC.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Search and clustering orders of magnitude faster than BLAST.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Bioinformatics.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2010;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='26:2460–1.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 78.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Rinke C, Schwientek P, Sczyrba A, Ivanova NN, Anderson IJ, Cheng JF, et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='  Insights into the phylogeny and coding potential of microbial dark matter.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Nature.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2013;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='499:431–7.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 79.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Bushnell B.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' BBMap short read aligner.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2016.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 80.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Bowers RM, Kyrpides NC, Stepanauskas R, Harmon-Smith M, Doud D, Reddy TBK, et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Minimum information about a single amplified genome (MISAG) and a metagenome-assembled genome (MIMAG) of bacteria and archaea.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Nat Biotechnol.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2017;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='35:725–31.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 81.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Price MN, Dehal PS, Arkin AP.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' FastTree 2--approximately maximum- likelihood trees for large alignments.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' PLoS One.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2010;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='5:e9490.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 82.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Katoh K, Standley DM.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' MAFFT multiple sequence alignment software version 7: improvements in performance and usability.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Mol Biol Evol.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2013;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 30:772–80.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 83.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Letunic I, Bork P.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Interactive tree of life (iTOL) v3: an online tool for the display and annotation of phylogenetic and other trees.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Nucleic Acids Res.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2016;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='44:W242–5.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 84.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' FigTree.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' http://tree.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='bio.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='ed.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='ac.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='uk/software/figtree/.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 85.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Seemann T.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Prokka: rapid prokaryotic genome annotation.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Bioinformatics.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2014;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='30:2068–9.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 86.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Jones P, Binns D, Chang H-Y, Fraser M, Li W, McAnulla C, et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' InterProScan 5: genome-scale protein function classification.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Bioinformatics.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2014;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='30: 1236–40.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 87.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Yin Y, Mao X, Yang J, Chen X, Mao F, Xu Y.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' dbCAN: a web resource for automated carbohydrate-active enzyme annotation.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Nucleic Acids Res.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2012;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 40:W445–51.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Vavourakis et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Microbiome  (2018) 6:168 Page 17 of 18  88.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Cantarel BL, Coutinho PM, Rancurel C, Bernard T, Lombard V, Henrissat B.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' The Carbohydrate-Active EnZymes database (CAZy): an expert resource for glycogenomics.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Nucleic Acids Res.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2009;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='37:D233–8.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 89.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Nayfach S, Pollard KS.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Average genome size estimation improves comparative metagenomics and sheds light on the functional ecology of the human microbiome.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Genome Biol.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2015;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='16:1–18.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 90.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Goris J, Konstantinidis KT, Klappenbach JA, Coenye T, Vandamme P, Tiedje JM.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' DNA-DNA hybridization values and their relationship to whole-genome sequence similarities.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Int J Syst Evol Microbiol.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2007;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='57:81–91.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 91.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Huerta-Cepas J, Szklarczyk D, Forslund K, Cook H, Heller D, Walter MC, et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' eggNOG 4.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='5: a hierarchical orthology framework with improved functional annotations for eukaryotic, prokaryotic and viral sequences.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Nucleic Acids Res.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2016;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='44:D286–93.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 92.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Tikhonova TV, Slutsky A, Antipov AN, Boyko KM, Polyakov KM, Sorokin DY, et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Molecular and catalytic properties of a novel cytochrome c nitrite reductase from nitrate-reducing haloalkaliphilic sulfur-oxidizing bacterium Thioalkalivibrio nitratireducens.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Biochim Biophys Acta - Proteins Proteomics.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2006;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='1764:715–23.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 93.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Tikhonova T, Tikhonov A, Trofimov A, Polyakov K, Boyko K, Cherkashin E, et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Comparative structural and functional analysis of two octaheme nitrite reductases from closely related Thioalkalivibrio species.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' FEBS J.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2012;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='279: 4052–61.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 94.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='  Tabita FR, Hanson TE, Li H, Satagopan S, Singh J, Chan S.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Function, structure, and evolution of the RuBisCO-like proteins and their RuBisCO homologs.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Microbiol Mol Biol Rev.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2007;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='71:576–99.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 95.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Eddy SR.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Accelerated profile HMM searches.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' PLoS Comput Biol.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' 2011;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content='7: e1002195.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Vavourakis et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
+page_content=' Microbiome  (2018) 6:168 Page 18 of 18' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\test_0\\content\\98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A%20metagenomics%20roadmap%20to%20the%20uncultured%20genome%20diversity%20in%20hypersaline%20soda%20lake%20sediments.pdf'}
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+Efficient Enumeration of the Optimal Solutions to the Correlation
+Clustering problem
+Nejat Arınık, Rosa Figueiredo & Vincent Labatut
+January 16, 2023
+Abstract
+According to the structural balance theory, a signed graph is considered structurally balanced when
+it can be partitioned into a number of modules such that positive and negative edges are respectively
+located inside and between the modules. In practice, real-world networks are rarely structurally balanced,
+though. In this case, one may want to measure the magnitude of their imbalance, and to identify the set
+of edges causing this imbalance. The correlation clustering (CC) problem precisely consists in looking
+for the signed graph partition having the least imbalance. Recently, it has been shown that the space of
+the optimal solutions of the CC problem can be constituted of numerous and diverse optimal solutions.
+Yet, this space is difficult to explore, as the CC problem is NP-hard, and exact approaches do not scale
+well even when looking for a single optimal solution. To alleviate this issue, in this work we propose
+an efficient enumeration method allowing to retrieve the complete space of optimal solutions of the
+CC problem. It combines an exhaustive enumeration strategy with neighborhoods of varying sizes, to
+achieve computational effectiveness. Results obtained for middle-sized networks confirm the usefulness
+of our method.
+Keywords: Correlation Clustering, Enumeration Strategy, Local Search, Optimal Solution Space,
+Signed Graph, Structural Balance.
+Cite as: N. Arınık, R. Figueiredo & V. Labatut. Efficient Enumeration of the Optimal Solutions to
+the Correlation Clustering problem. Journal of Global Optimization, 2023 (forthcoming)
+1
+Introduction
+Many real-world social systems involve antagonistic relations, a feature that can be directly modeled
+through signed networks, which contain both positive and negative edges. Alternatively, in some cases,
+signed edges can be extracted from originally unsigned relations [29]. The exact semantic of these edges
+depends on the considered system: friendship/foe [35] and trust/distrust [38] in social networks, inhi-
+bition/activation in biological networks [16, 45], agreement/disagreement in political vote networks [5,
+7, 19] and so on.
+Determining the structural balance of a signed network is a key aspect in the investigation of the
+structure of polarized relations. According to the structural balance theory, a signed graph is considered
+to be balanced if it can be partitioned into two [12] or more [17] modules (i.e. clusters), such that all
+positive edges are located inside these modules, whereas negative ones lie between them.
+However, it is very rare for a real-world network to be perfectly balanced, in which case one wants
+to assess the magnitude and the causes of the imbalance.
+For a given partition, this imbalance is
+traditionally measured by counting the number of frustrated edges [12, 46], i.e. positive edges located
+in-between modules and negative ones located inside them. Computing the graph imbalance amounts
+to identifying the partition corresponding to the lowest imbalance measure over the space of all possible
+partitions. The optimal solution found exhibits the source of the imbalance, i.e., the frustrated edges.
+This minimization problem is known as the correlation clustering (CC) problem, proven to be NP-
+hard [8].
+When solving an instance of the CC problem, the standard approach in the literature is to find a
+single solution and focus the rest of the analysis on it, as if it were the only optimal solution [23, 44].
+Yet, as shown empirically, it is possible that several, and even many, alternative optimal solutions exist
+for the considered instance [6, 10, 11]. All these alternate solutions are equally relevant in terms of the
+objective function of the CC problem. Yet, they can be very different, in terms of how they partition
+the graph. One can then wonder how many solutions shape the space of optimal solutions, and if many,
+1
+arXiv:2301.05479v1  [math.OC]  13 Jan 2023
+
+Arınık et al. – Efficient Enumeration of the Optimal Solutions to the Correlation Clustering problem
+how different/diverse they are. From an application point of view, these are important questions to
+answer in order to identify how many of these solutions are more suitable to the situation at hand.
+In order to answer these questions, we recently conducted an empirical study in our previous work [6].
+Therein, we first enumerated the optimal solutions of a collection of instances of the CC problem through
+Integer Linear Programming (ILP) modeling [18]. We applied the best state-of-the-art optimal solution
+enumeration method, as proposed by Danna et al. [15] and incorporated in the commercial solver
+CPLEX [30].
+Then, we studied the obtained spaces of optimal solutions through complex network
+analysis. In that work, we found out that slightly imbalanced networks tend to have fewer and structurally
+more similar optimal solutions forming a single solution class, whereas a higher imbalance leads to many
+diverse solutions possibly forming multiple classes of solutions. Consequently, we concluded that it is
+of great importance that the decision-maker generates the full set, or as many optimal solutions as
+possible, for a given instance.
+In such an analysis, guaranteeing the completeness of the space of optimal solutions requires em-
+ploying exact approaches. However, it is well known that, due to their NP-hard nature, generic exact
+approaches able to solve most clustering problems (including the CC problem) are very costly in terms
+of execution time and do not scale well, even when looking for a single optimal solution [27]. In that
+respect, the enumeration method by Danna et al. is the best state-of-the-art enumeration method able
+to handle the CC problem so far. However, as it is a generic method designed to handle any combinato-
+rial problem formulated in ILP, it cannot take full advantage of the problem knowledge. Consequently,
+the maximum number of vertices (i.e. graph order) that we could handle with their method in our
+previous work was only 36. Moreover, the execution time of the enumeration process for such small
+graphs was very long (typically more than one day). This highlights the need for an efficient optimal
+space enumeration method for the CC problem.
+In this work, we aim to obtain this efficiency by putting the problem knowledge into the enumeration
+process. Based on the high similarity observed empirically between optimal solutions belonging to the
+same class of solutions, it integrates into an exact approach a local search mechanism, which relies on
+the use of neighborhoods of different sizes. In order to accelerate the proposed enumeration method,
+we develop pruning strategies based on optimal conditions satisfied by partial solutions.
+Our main
+contribution is the combination of local search and pruning strategies, which gives rise to an efficient
+enumeration method. We present extensive computational experiments established for the proposed
+method, relying on sparse and dense signed networks.
+The rest of the article is organized as follows. In Section 2, we review the literature related to
+the exact enumeration of all optimal solutions for the CC problem. In Section 3, we give the formal
+definition of the CC problem. Next, we introduce our enumeration method in Section 4, and detail
+our complete neighborhood search and pruning strategies in Section 5 and 6, respectively. We put
+the proposed method into practice on a selection of partially and fully connected signed networks in
+Section 7 and discuss our results in Section 8. Finally, we review our main findings in Section 9, and
+identify some perspectives for our work.
+2
+Related Work
+There are several methods proposed in the literature to solve the CC problem exactly [18, 23, 32],
+however very few were designed to enumerate the whole space of its optimal solutions. In fact, very
+few methods were proposed to enumerate all the optimal solutions of any combinatorial problem (e.g.,
+[4] for maximal covering problem). In this section, we review the existing optimal solution enumeration
+methods for the CC problem and for related problems.
+The literature provides two main approaches to enumerate all optimal solutions of a combinatorial
+problem: Branch-and-Bound and Fixed-Parameter Enumeration. In the case of the CC problem, most
+works focus on the former.
+Branch-and-Bound Approach
+The enumeration of all optimal solutions of the CC problem (like
+for any combinatorial problem) through a branch-and-bound method can be performed through two
+algorithmic methods: Integer Linear Programming (ILP) vs. ad hoc branch-and-bound programming.
+Their main difference is that the former can tackle any problem translated in the language of mathe-
+matical programming through any industrial optimization solver, whereas in the latter the construction
+of the branch-and-bound tree relies on problem-specific idiosyncrasies.
+Based on how the branch-and-bound tree is used, the ILP approach can be implemented in two
+different ways. The first one relies on a simple sequential process: a slightly modified version of the
+original problem is solved as many times as there are solutions in the optimal solution space (like in [4]
+for the maximal covering problem). At each iteration, already-found optimal solutions are sequentially
+2 / 30
+
+Arınık et al. – Efficient Enumeration of the Optimal Solutions to the Correlation Clustering problem
+added as constraints into the mathematical model, in order to exclude them during future iterations.
+However, the drawback of this approach is that a branch-and-bound tree needs to be built from scratch
+for each new optimal solution found. The second type of ILP implementation is called OneTree, and
+was proposed by Danna et al. [15]. Instead of the previous simple sequential approach, it relies on a
+more efficient two-step method that allows reducing the computational effort. As its name suggests, it
+constructs a single branch-and-bound tree. It first builds and explores the search tree in order to find
+efficiently the first optimal solution, and then enumerate all the alternative optimal solutions based on
+the same tree. This method is available in the industrial optimization solver CPLEX [30], and we explain
+in Section 4.3 how we use it for the CC problem. In our previous work [6], we used this approach for
+generating the optimal solution space of complete random graphs up to 36 vertices.
+An ad hoc B&B programming method constructs the branch-and-bound tree differently from what
+we described before. For a given clustering problem, these methods systematically construct partial
+solutions by assigning the vertices to one of the existing modules. This produces a search tree, whose
+branches correspond to assignments of vertices to modules, and whose nodes correspond to partial
+assignments.
+Similar to Danna et al. [15], Brusco and Steinly [10] propose a two-step method for
+generating all optimal CC solutions. They first identify the optimal objective function value by finding
+an optimal solution, then use the optimal value as input when building another branch-and-bound tree
+from scratch. As shown by Figueiredo & Moura [23], this method does not deal well with finding a
+single optimal solution for graphs with more than 20 vertices.
+Fixed-Parameter Enumeration Approach
+This is also an ad hoc method which requires to
+transform a part of the considered problem into a new input parameter.
+This in turn guarantees
+to bound the overall running time as a function of an input parameter.
+Assuming that this input
+parameter is small, the parameterized enumeration is efficient in practice. Damaschke [14] proposes
+an FPT (Fixed-Parameter Tractable) algorithm to enumerate all optimal solutions in a given graph for
+the Cluster Editing problem, which is equivalent to the CC problem when the input graph is complete
+and unweighted. Concretely, this FPT algorithm makes the number of frustrated edges in structural
+balance parameterized to solve the problem. However, as shown in our previous work [6], a drawback
+of this approach is that the amount of imbalance, the input parameter, can be very large. Furthermore,
+Damaschke does not provide any computational results in his theoretical work.
+The computational results presented in the works mentioned in this section identify the ILP branch-
+and-bound as the more efficient method for exactly solving the CC problem. This is explained by a
+tightened initial linear relaxation of the ILP formulation and the quality of the cuts used. In this work,
+we apply an ILP branch-and-bound method for the complete enumeration of the optimal CC solutions,
+and propose a compromise strategy between the sequential approach and the OneTree method from
+Danna et al. [15].
+3
+Correlation Clustering Problem
+Let us now introduce the notations necessary for defining the correlation clustering problem. Let G =
+(V, E) be an undirected graph, where V and E are the sets of vertices and edges, respectively. We note
+n = |V | and m = |E| the numbers of vertices (i.e. graph order) and edges, respectively. We assume
+that the graph contains no loops, i.e. edges connecting a vertex to itself. Moreover, G[S] denotes the
+subgraph induced by vertex subset S ⊂ V .
+Consider a function s : E → {+, −} that assigns a sign to each edge in E. An undirected graph G
+together with a function s is called a signed graph, denoted by G = (V, E, s). An edge (u, v) is called
+negative if s(u, v) = − and positive if s(u, v) = +. We note E− and E+ the sets of negative and
+positive edges in the signed graph, respectively. Let also define the positive graph G+ of a given signed
+graph G as the subgraph (V, E+, {+}) (i.e. same vertices, but only the positive edges). The entries
+auv of the signed adjacency matrix A associated with a signed graph G are defined in Equation 1.
+auv =
+�
+�
+�
+�
+�
+1,
+if (u, v) ∈ E+,
+−1,
+if (u, v) ∈ E−,
+0,
+otherwise.
+(1)
+Let P = {M1, ..., Mℓ} (1 ≤ ℓ ≤ n) be an ℓ-partition of V , i.e. a division of V into ℓ non-overlapping
+and non-empty subsets Mi (1 ≤ i ≤ ℓ) called modules. The partition P is called a solution of the CC
+problem for the given graph G. Given a solution P, an edge (u, v) is called internal if it is located inside a
+module, i.e., u and v belong to the same module. Likewise, an edge (u, v) is called external if it is located
+3 / 30
+
+Arınık et al. – Efficient Enumeration of the Optimal Solutions to the Correlation Clustering problem
+between any two modules, i.e., u and v belong to two different modules. Given σ ∈ {+, −}, we define the
+set of positive/negative (depending on σ) edges connecting two modules Mi, Mj ∈ P as Eσ(Mi, Mj) =
+{(u, v) | (u, v) ∈ Eσ, u ∈ Mi and v ∈ Mj}. We also define E(Mi, Mj) = E−(Mi, Mj) ∪ E+(Mi, Mj)
+as the set of all edges connecting these modules. Similarly, we note Ωσ(Mi, Mj) =
+�
+(u,v)∈Eσ(Mi,Mj)
+auv
+the signed number of positive/negative edges connecting these modules: note that this value can be
+negative if σ = −. We also define Ω(Mi, Mj) = Ω−(Mi, Mj) + Ω+(Mi, Mj) as the signed sum of the
+edges connecting the same modules. It can also be negative, if there are more negative than positive
+edges between Mi and Mj.
+The Imbalance I(P) of a partition P is defined as the total number of frustrated edges, i.e. the
+numbers of positive edges located between modules and of negative edges located inside them:
+I(P) =
+�
+1≤i<j≤ℓ
+Ω+(Mi, Mj) −
+�
+1≤i≤ℓ
+Ω−(Mi, Mi).
+(2)
+The objective of the CC problem is to minimize the number of frustrated edges. This problem can
+be formally described as follows.
+Problem 1 (CC problem). Given a signed graph G = (V, E, s), the correlation clustering problem
+consists in finding a partition P of V such that the imbalance I(P) is minimized.
+To the best of our knowledge, this NP-hard minimization problem appears under this name for the
+first time in Bansal’s paper [8], although it is addressed before in the literature (e.g. [20]).
+4
+Enumeration of Optimal CC Solutions
+Any method that enumerates optimal solutions needs to find an initial optimal solution by optimally
+solving the problem at hand. This is why we first describe an efficient method for finding an optimal
+solution for the CC problem (Section 4.1), before presenting the methods for identifying an alternative
+optimal solution (Section 4.2) and enumerating all optimal solutions (Section 4.3).
+4.1
+Finding an Initial Optimal Solution
+As largely illustrated in the literature (e.g. [18], [23]), the CC problem can be modeled by means of an
+ILP proposed for the Uncapacitated Graph Clustering problem [39]. Note that the model can handle
+weighted graphs, but we use it on unweighted ones in the context of this article.
+For each pair of vertices u, v ∈ V : u < v, a binary variable is first defined to describe the relative
+module membership of pairs of vertices, i.e.,
+xuv =
+�
+1,
+if u and v are in a same module,
+0,
+otherwise.
+(3)
+Then, the ILP formulation of the CC problem for unweighted signed graphs is written as follows:
+Min
+�
+u,v∈V :uv∈E−
+xuv +
+�
+u,v∈V :uv∈E+
+(1 − xuv)
+(4)
+s.t.
+xuv + xvr − xur ≤ 1,
+∀u < v < r ∈ V
+(5)
+xuv − xvr + xur ≤ 1,
+∀u < v < r ∈ V
+(6)
+− xuv + xvr + xur ≤ 1,
+∀u < v < r ∈ V
+(7)
+xuv ∈ {0, 1},
+∀u, v ∈ V.
+(8)
+The objective function in (4) looks for a feasible solution minimizing the imbalance defined by
+Equation (2). A feasible solution, i.e., decision variables corresponding to a partition, must satisfy the
+triangle inequalities (5), (6) and (7). Finally, (8) describes the domain constraints for the variables in
+the formulation.
+This ILP model is sufficient to find an optimal solution through an optimization solver, but it
+can be very time-consuming. One way to deal with this issue is to strengthen it through a cutting
+plane approach [40]. We use the 2-partition and the 2-chorded cycle valid inequalities as described by
+Grötschel and Wakabayashi in [28] (see Appendix A for more details). In the cutting plane approach,
+we add these two tight valid inequalities (only) during the root relaxation phase as described by Ales et
+al. [2], before proceeding to the construction of the branch-and-bound search tree. As reported in [2]
+4 / 30
+
+Arınık et al. – Efficient Enumeration of the Optimal Solutions to the Correlation Clustering problem
+for a similar clustering problem, we also verified that adopting this cut strategy improves the overall
+processing time. There are other inequalities in the literature that can further tighten the LP relaxation
+for the CC problem. These are the 2-chorded even wheel [28] inequality and those defined in [42].
+Nevertheless, in practice, together the 2-partition and the 2-chorded cycle inequalities describe a tight
+linear relaxation [28].
+Throughout this work, we denote ILP(G) as the formulation defined by Equations (4)–(8) and
+B&C(ILP(G)) as the branch-and-cut procedure described above. Also, we denote ILP+(G) as the
+strengthened ILP obtained after executing B&C(ILP(G)), i.e., the formulation obtained by adding to
+ILP(G) all the cuts generated by B&C(ILP(G)). Finally, let B&B(ILP+(G)) be the branch-and-
+bound procedure based on ILP+(G). In the following, for the sake of simplicity, the term solution
+refers to a graph partition, as well as a feasible solution to the formulation ILP+(G).
+4.2
+Finding an Alternative Optimal Solution
+An enumeration method needs to constantly find an optimal solution that would be different from those
+identified before, and stored in a set S. For this purpose, we need to define an extended model of
+ILP+(G), that we denote ILP+jump(G, S).
+Consider a partition P. Let Xp ∈ {0, 1}n×n be the representative matrix of P defined as: xp
+ij = 1
+if i and j belong to the same module in P; and xp
+ij = 0 otherwise. ILP+jump(G, S) includes the set
+of constraints defined in Equation (9) on top of ILP+(G):
+�
+u,v∈V :u<v
+|xp
+uv − xuv| > 0, ∀P ∈ S.
+(9)
+We refer the reader to [24] for the implementation details of these constraints.
+We see that this extended formulation allows us to find an alternative optimal solution other than
+the ones in S. Furthermore, given an optimal solution P ∈ S, to ensure that this alternative solution
+has the same objective value as I(P), we add the objective function in (4) as a constraint, as shown in
+Equation (10).
+�
+u,v∈V :uv∈E−
+xuv +
+�
+u,v∈V :uv∈E+
+(1 − xuv) ≤ I(P).
+(10)
+Equation (9) along with Equation (10) ensures that each feasible solution to ILP+jump(S) is
+an optimal solution to the original problem. Finally, we denote the corresponding branch-and-bound
+procedure based on formulation ILP+jump(G, S) by B&B(ILP+jump(G, S)).
+4.3
+Enumerating All Optimal Solutions
+As we have mentioned before, the CC problem can have multiple optimal solutions. In this section, we
+are interested in methods enumerating all optimal solutions of the CC problem for a given signed graph
+G. In the following, we first present OneTreeCC [15], which is the best general method to enumerate
+solutions available in the literature. It is incorporated in CPLEX [30], and we apply it to the formulation
+ILP+(G). Then, we introduce our ad hoc method EnumCC, in the aim of obtaining a faster method
+able to handle relatively larger graphs.
+We start with OneTreeCC(G), which takes as input a signed graph G. The sketch of this two-step
+method is shown in Algorithm 1. In the first step (line 2), it finds an initial optimal solution based on
+ILP+(G). We note TB&B the branch-and-bound search tree constructed during this step. The search
+tree as well as the nodes considered at this first step are stored for further examination during the second
+step. Moreover, Danna et al. completely turn off dual tightening in the branch-and-bound procedure
+in order to guarantee exhaustive enumeration. According to the authors, this fact can have a negative
+impact on performance, in terms of computational time. In the second step (line 3), OneTreeCC(G)
+enumerates the set S of all alternative optimal solutions by expanding the search tree TB&B until
+obtaining the complete set of all optimal solutions.
+To compete with OneTreeCC(G), we propose a new method for the CC problem in order to com-
+pletely enumerate the optimal partitions of a given signed graph. We call it EnumCC(G, rmax), and
+it takes as input a signed graph G and a maximum distance parameter rmax. As shown in Algorithm 2,
+it can be viewed as an improved version of the sequential approach proposed by Arthur et al. [4] for
+the Maximal Covering problem (described in Section 2). In the first step (line 2), EnumCC(G, rmax)
+obtains an initial optimal solution thanks to B&B(ILP+(G)).
+In the second step, instead of di-
+rectly "jumping" onto undiscovered optimal solutions one by one through ILP+jump(G, S) (like in
+the sequential approach), we slightly change this process. Our method first discovers the recurrent
+5 / 30
+
+Arınık et al. – Efficient Enumeration of the Optimal Solutions to the Correlation Clustering problem
+Algorithm 1: OneTreeCC(G)
+Result: The set of all optimal solutions S
+1 S = ∅
+/* 1st step
+*/
+2 Solve B&B(ILP+(G)) to obtain an optimal solution P, and let TB&B be the constructed
+branch-and-bound search tree
+/* 2nd step
+*/
+3 Expand fathomed nodes in TB&B until enumerating the set S of all alternative optimal solutions.
+neighborhood RN≤rmax(P) of the current optimal solution P (line 4), with the hope of discovering new
+optimal solutions. The recurrent neighborhood RN≤rmax(P) of an optimal solution P, is the set of
+optimal solutions reached directly or indirectly from P, depending on the maximum distance parameter
+rmax. The complete description of the Recurrent Neighborhood Search (RNS) is given in Section 5.
+Whether a new solution is found or not through RNS(G, P, rmax), the method then moves on to the
+next step, consisting in jumping onto a new solution P (line 6). These RNS and jumping phases are
+repeated, until all optimal solutions are discovered.
+Clearly, EnumCC(G, rmax) can only improve the sequential approach as used in [4], provided
+that RNS discovers at least one new solution and that its execution time is shorter than that of
+B&B(ILP+jump(G, S)). Furthermore, the choice of rmax is sensitive: it does not contribute much
+to the method when it is small, whereas the method becomes slower than the sequential approach when
+rmax is large.
+Algorithm 2: EnumCC(G, rmax)
+Result: The set of all optimal solutions S
+1 S = ∅
+/* 1st step
+*/
+2 Solve B&B(ILP+(G)) to obtain an optimal solution P
+/* 2nd step
+*/
+3 while P ̸= null do
+/* step 2.1
+*/
+4
+Apply RNS(G, P, rmax) to generate the recurrent neighborhood RN≤rmax(P) of P
+5
+S = S � RN≤rmax(P)
+/* step 2.2
+*/
+6
+Jump onto an undiscovered optimal solution P, with P /∈ S, through
+B&B(ILP+jump(G, S)) // if not possible, then P = null
+7 end
+The jumping phase in step 2.2 of EnumCC(G, rmax) can be time-consuming for two reasons. First,
+even finding an alternative solution can be costly. Second, the number of jumps, denoted by the notation
+njump(.) (here, njump(EnumCC(G, rmax))), can be very large, despite the use of the neighborhood
+RN≤rmax(P). Nonetheless, we expect to save time and compensate the time spent in the jumping
+phase thanks to RN≤rmax(P).
+In the rest of this work, we leave out the common parameter G from the notations of the men-
+tioned methods and ILP models, for the sake of convenience: OneTreeCC(), EnumCC(rmax) and
+ILP+jump(S).
+In the next section, we detail the recurrent neighborhood search RNS used in Algorithm 2.
+5
+Recurrent Neighborhood Search (RNS)
+Recurrent Neighborhood Search (RNS) is the common and undoubtedly the most important part of
+our method EnumCC. As mentioned in the previous section, if we want this method to compete with
+OneTreeCC(G), RNS needs to be efficient and fast.
+Before defining the neighborhood of a solution, i.e. of a partition, we need to select a distance
+function between partitions. In the following, we first present the edit distance (Section 5.1), then
+6 / 30
+
+Arınık et al. – Efficient Enumeration of the Optimal Solutions to the Correlation Clustering problem
+introduce the algorithmic details of the Complete Neighborhood Search (CoNS) used in our Recurrent
+Neighborhood Search (Section 5.2). Finally, we explain the main procedure for Recurrent Neighborhood
+Search (Section 5.3).
+Let us first introduce some definitions used in the rest of the article. A partition P with ℓ modules
+can be associated with one or more membership vectors. Such a vector of size n, denoted by π, defines
+a function over V → {1, 2, ..., ℓ}. For a given vertex u, π(u) denotes the label of the module of P
+that contains u, while Mπ(u) denotes the module itself. Finally, given a partition P, we define an
+r-neighborhood structure, denoted by Nr(P), as the family of partitions obtained by moving r vertices
+into different modules of P. We also define Nr(π) as the family of corresponding membership vectors
+obtained by moving r vertices.
+5.1
+Edit Distance
+In Natural Language Processing, and more generally in Computer Science, the edit distance [31] is
+defined as the minimum number of edit operations required to transform one string into another (or
+more precisely, their total cost). In the context of graph partitioning, with fixed vertices and edges,
+edit operations are used as neighborhood search operators in local search heuristics developed for graph
+problems [1, 9, 37]. Such an operation consists in moving one or more vertices (called moving vertices)
+from their current modules (called source modules) to other ones (called target modules). From the
+perspective of the membership vectors corresponding to the concerned partitions, this transforms a
+source vector into a target one.
+The number of moving vertices constitutes the cost of the edit
+operation. In this work, we are interested in edit operations whose cost is minimal.
+Definition 2 (Min-edit operation). Consider an edit operation transforming a source membership vector
+into a target membership one.
+If the set of moving vertices is the minimum set required for this
+transformation, then it is a min-edit operation. Otherwise, we call it non-min-edit operation.
+Notice that the cost of an edit operation is not always minimal. This is because two membership
+vectors, i.e. the module labels of two partitions, can be very different, but essentially suggest very similar
+module assignments for the vertices. The distinction between min-edit and non-min-edit operations is
+illustrated in Figure 1.
+Importantly, the edit distance between two membership vectors is the cost of the min-edit operation
+allowing to turn one vector into the other. The calculation of such distance between two membership
+vectors can be done in polynomial time by solving an assignment problem (see Appendix B for more
+details).
+Let us now introduce our notations related to the edit distance.
+Let πs and πt represent the
+source and target membership vectors for an edit operation, respectively. They are associated with the
+source and target partitions P s = {M s
+1, M s
+2, ..., M s
+ℓs} and P t = {M t
+1, M t
+2, ..., M t
+ℓt}, containing ℓs and
+ℓt modules, respectively. Since the edit distance is symmetric, without loss of generality, let ℓs ≤ ℓt.
+Moreover, we note πs → πt an edit operation applied onto P s to obtain P t, whose set of moving vertices
+is defined as
+s#»t
+V
+= {u | πs(u) ̸= πt(u)}. This means that the remaining vertices, called non-moving
+vertices, do not change modules, hence their module labels are the same in πs and πt. The cost of this
+edit operation is simply the number of vertices whose module has changed. We denote this cost with
+cost(πs → πt) and compute it by taking the cardinality of
+s#»t
+V . In the rest of the text, we use r for
+short whenever we are interested only in the value of cost(πs → πt). Also, a r-edit operation (resp.
+min-r-edit operation) is any edit operation (resp. min-edit operation) whose cost equals r. Consider
+an edit operation πs → πt whose moving vertices are in set
+s#»t
+V . The distinct labels of the source and
+target modules of a subset V ′ ⊂ V of vertices are denoted, respectively, by πs(V ′) = �
+u∈V ′ πs(u) and
+πt(V ′) = �
+u∈V ′ πt(u). Finally, we sometimes need to refer to several modules in an edit operation
+πs → πt. For this reason, we also define M s
+L = �
+l∈L M s
+l for any L ⊆ {1, .., ℓs} in the source partition
+P s and M t
+L = �
+l∈L M t
+l for any L ⊆ {1, .., ℓt} in the target partition P t.
+To illustrate the aforementioned concepts, let us consider the example of Figure 1. As shown in
+Figure 1a, πs = (1, 1, 2, 2, 2), where M s
+1 = {v1, v2} and M s
+2 = {v3, v4, v5}. Let us consider an edit
+operation πs → πt with the moving vertices
+s#»t
+V
+= {v2, v4} (Figure 1b). The edit operation consists
+in moving v2 into M s
+2 and v4 into M s
+1, which produces πt. Hence, we have M t
+1 =
+�
+M s
+1 \ {v2}
+�
+∪ {v4}
+and M t
+2 =
+�
+M s
+2 \ {v4}
+�
+∪ {v2}.
+The labels of the source and target distinct modules of
+s#»t
+V
+are
+πs(
+s#»t
+V ) = {1, 2} and πt(
+s#»t
+V ) = {1, 2}, respectively.
+Also, we have M s
+πs(s #»t
+V ) = M s
+1 ∪ M s
+2 and
+M t
+πt(s #»t
+V ) = M t
+1 ∪ M t
+2 for this example. The sets M s
+πs(s #»t
+V ) and M t
+πt(s #»t
+V ) could be different, if some of
+7 / 30
+
+Arınık et al. – Efficient Enumeration of the Optimal Solutions to the Correlation Clustering problem
+the moving vertices move into a module M s
+i other than M s
+πs(s #»t
+V ).
+M s
+1
+M s
+2
+v1
+v2
+v3
+v4
+v5
+(a) Source member-
+ship
+vector
+πs
+=
+(1, 1, 2, 2, 2),
+where
+M s
+1 = {v1, v2} and
+M s
+2 = {v3, v4, v5}.
+M t
+1
+M t
+2
+v1
+v4
+v3
+v2
+v5
+(b) Target membership vector πt =
+(1, 2, 2, 1, 2), where M t
+1 = {v1, v4}
+and M t
+2 = {v2, v3, v5}. The mov-
+ing vertices are shown with a purple
+border. The operation πs → πt is
+min-2-edit, where
+s#»t
+V
+= {v2, v4}.
+M t′
+1 = M t
+2
+M t′
+2 = M t
+1
+v2
+v3
+v5
+v4
+v1
+(c) Target membership vector πt′
+=
+(2, 1, 1, 2, 1), where M t′
+1
+= {v2, v3, v5}
+and M t′
+2 = {v1, v4}. The moving vertices
+are shown with a purple border.
+Since
+M t′
+1 = M t
+2 and M t′
+2 = M t
+1, πs → πt′ is
+not a min-3-edit operation, where
+s#»t′
+V
+=
+{v1, v3, v5}.
+Figure 1: Illustrative example for two edit operations applied to the same source membership vector πs,
+whose associated partition is shown in Fig. 1a. The first operation,
+s#»t
+V , is shown in Fig. 1b, and the second,
+s#»t′
+V , in Fig. 1.c. The underlying partition in P t and P t′ is the same, but the corresponding membership
+vectors πt and πt′ are different. Consequently,
+s#»t
+V
+is a min-edit operation, whereas
+s#»t′
+V
+is not.
+5.2
+Complete Neighborhood Search (CoNS)
+Our neighborhood search method CoNS(G, πs, r) takes as input a signed graph G, a membership
+vector πs associated with an optimal partition P s of ℓ modules and an edit distance r. It returns the set
+Nr(πs) of membership vectors associated with all optimal partitions, obtained by applying min-r-edit
+operations to a given membership vector πs. To ensure the completeness of set Nr(πs), CoNS(G, πs, r)
+analyses all combinations of moving vertices and their target modules. We use two pruning strategies in
+order to eliminate some combinations that do not lead to an optimal partition when an edit operation is
+applied. Both of them are described in this section, whereas the properties upon which they are based
+are detailed in Section 6. In simple terms, they are based on the relations between a set of moving
+vertices and their target modules. In the following, we present CoNS(G, πs, r) as a pipeline procedure
+consisting of four parts, as shown in Algorithm 3.
+In the first part, CoNS(G, πs, r) starts by selecting a candidate set of r moving vertices
+s#»t
+V
+among all vertices in V (line 2). The set of all possible r moving vertices is obtained by generating all
+combinations
+�V
+r
+�
+. We know that probably not every
+s#»t
+V
+leads to a min-r-edit operation and an optimal
+partition P t. To detect such cases, we apply the first pruning procedure (line 3), called internalPruning
+and depicted in Algorithm 4. This procedure checks, without the knowledge of target modules, if
+s#»t
+V
+satisfies some connectivity conditions related to the atomicity property (defined in Section 6.2) and the
+MVMO property up to three moving vertices (defined in Section 6.3). As we will see in Section 6,
+whenever one of these two conditions is not satisfied, the candidate set
+s#»t
+V
+can be pruned.
+At this point, the set of moving vertices
+s#»t
+V
+is fixed. Next, the algorithm defines step by step the
+target membership vector πt by setting the target module of each vertex in
+s#»t
+V . We handle the process
+of generating the target membership vector πt in three steps, which constitutes the second, third and
+fourth parts of Algorithm 3, in order to take advantage of our pruning strategy. The pruning strategy
+used in these parts is slightly different from that of the first part: it is based on external connections
+from
+s#»t
+V
+to
+s#»t
+V . The concerned procedure is called externalPruning and depicted in Algorithm 5. It
+verifies, based on external relations among
+s#»t
+V , if
+s#»t
+V
+satisfies the min-edit (Section 6.1), atomicity
+8 / 30
+
+Arınık et al. – Efficient Enumeration of the Optimal Solutions to the Correlation Clustering problem
+Algorithm 3: CoNS(G, πs, r).
+Input
+: signed graph G(V, E, s), source membership vector πs with ℓ module labels, edit distance r
+Output: Nr(πs)
+1 T = {1, 2, .., ℓ} // module labels in πs
+/* 1st part
+*/
+2 for each
+s#»t
+V
+∈ �V
+r
+�
+do
+3
+if internalPruning(G, πs,
+s#»t
+V ) then
+4
+go to line 2
+5
+end
+/* 2nd part
+*/
+6
+T0 = π(
+s#»t
+V ) �
+{Unknown} // initial target module labels
+7
+Trem = T \ T0 // remaining module labels
+8
+for each T ′
+0 ∈ P(�T0
+r
+�
+), s.t. πs(u) /∈ T ′
+0, ∀u ∈
+s#»t
+V
+do // P(.):
+permutations
+9
+Let πt be the partition after assigning T ′
+0 to
+s#»t
+V
+10
+if externalPruning(G, πs, πt,
+s#»t
+V ) then
+11
+go to line 8
+12
+end
+/* 3rd part
+*/
+13
+Let B be the the number of Unknown labels in πt
+14
+if B ̸= 0 then
+15
+Let
+s#»t
+V
+rem be the moving vertices, whose πt(
+s#»t
+V
+rem) = Unknown
+16
+for b ∈ {1, .., B} do
+17
+Let I be the set of all vectors of size B with elements belonging to {1, .., b}, s.t. each element for
+each vector appears once
+18
+for each I ∈ I do
+19
+Update πt with the assignment of I to πt(
+s#»t
+V
+rem)
+20
+if externalPruning(G, πs, πt,
+s#»t
+V ) then
+21
+go to line 18
+22
+end
+/* 4th part
+*/
+23
+T +
+rem = Trem ∪ {ℓ + 1, .., ℓ + b} // expand for empty modules
+24
+for each T ′
+r ∈ P(�T +
+r
+b
+�
+) do
+25
+Replace the assignment of I to
+s#»t
+V
+rem by Trem in πt
+26
+if I(πs) = I(πt) and ¬externalPruning(G, πs, πt,
+s#»t
+V ) then
+27
+Nr(πs) = Nr(πs) �
+πt
+28
+end
+29
+end
+30
+end
+31
+end
+32
+else
+33
+if I(πs) = I(πt) then
+34
+Nr(πs) = Nr(πs) �
+πt
+35
+end
+36
+end
+37
+end
+38 end
+9 / 30
+
+Arınık et al. – Efficient Enumeration of the Optimal Solutions to the Correlation Clustering problem
+Algorithm 4: internalPruning(G, πs,
+s#»t
+V )
+Input
+: signed graph G(V, E, s), source membership vector πs, moving vertices
+s#»t
+V
+Output: Boolean variable
+/* 1st part
+*/
+1 if ¬intAtomic(G, πs,
+s#»t
+V ) then // Properties 6 and 7a
+2
+return true
+3 end
+/* 2nd part
+*/
+4 if 2 ≤ |
+s#»t
+V | ≤ 3 and ¬intMVMO(G, πs,
+s#»t
+V ) then // Lemma 11.1 and Lemma 12
+5
+return true;
+6 end
+7 return false;
+(Section 6.2) and MVMO (Section 6.3) properties. As discussed in Section 6, whenever one of these
+three conditions is not satisfied, the possibility of target modules for candidate set
+s#»t
+V
+is pruned. These
+pruning strategies can be applied even when some target modules are not already decided. Therefore,
+they are invoked, whenever the target modules of
+s#»t
+V
+are updated (lines 10, 20 and 26).
+In the second part, a moving vertex is allowed to move into a module, whose label is in T0. The set
+T0, defined at line 6, contains the labels of all unique source modules of
+s#»t
+V , as well as a label Unknown.
+We use this label Unknown to indicate that the target module of a vertex is not already assigned. We
+also define Trem as the remaining modules. Notice that, from this point, a target membership vector πt
+can contain multiple vertices with target modules labeled as Unknown. In the third part, this Unknown
+label allows us to handle the other modules in Trem as target possibilities. At line 8, we consider all
+possible permutations in P(
+�T0
+r
+�
+) such that, for each vertex in
+s#»t
+V , the target and source modules are
+different.
+At this point, the externalPruning procedure can be applied for vertices whose target modules are
+already assigned. If the pruning conditions are not satisfied, the algorithm proceeds with the third part.
+Line 14 checks if no target module in πt is unknown (B = 0). If so, then a new optimal partition
+has been found, and the algorithm goes on with the final test (line 32). However, if it is not the case,
+the algorithm enters the fourth part: all the combinations in Trem (more precisely, T +
+r ) are considered
+as a possible assignment of unknown target modules. One can expect this task to be computationally
+expensive. Thus, instead of directly determining the target modules of each vertex in
+s#»t
+V
+rem, we first
+generate all the coupling scenarios of moving vertices going into the same Unknown target module.
+To handle this, let b be the number of distinct Unknown target modules (line 16), for each value
+in the range of {1, .., B}. We build all possible coupling scenarios I for the value b (line 17). We
+couple a pair of remaining moving vertices, if they move into the same Unknown target module. For
+instance, {Unknown1, Unknown1, Unknown2} indicates that the last moving vertex moves into a
+different Unknown target module than the first two. Once the target membership vector πt is enriched
+with a coupling scenario I, the external pruning is repeated in line 20. If the pruning conditions are
+not satisfied, the algorithm finally determines the target modules of
+s#»t
+V
+rem, by respecting the actual
+coupling scenario I (line 25). In the end, we add the current πs → πt into Nr(πs), if the optimality
+condition is met (line 26).
+Without the problem-specific pruning strategies (internal and the external verification of properties
+from Section 6.3), Algorithm 3 amounts to a brute force method. Although CoNS(G, πs, r) can be
+very costly (even for small values of r), the properties described in Section 6 allow us to develop a
+method whose cost is relatively lower than the brute force method. Indeed, given a membership vector
+of size n with ℓ module labels, the number of non-repetitive permutations of all the combinations of
+r moving vertices is nr × r! and the number of non-repetitive permutations of all the combinations
+of target modules is ℓr × r!. In the end, the brute force method is Θ(nr × r! × ℓr × r!), where we
+assume that ℓ > r. Computational results in Section 8.1 show that the pruning strategies reduce the
+computational cost of this procedure.
+10 / 30
+
+Arınık et al. – Efficient Enumeration of the Optimal Solutions to the Correlation Clustering problem
+Algorithm 5: externalPruning(G, πs, πt,
+s#»t
+V )
+Input
+: signed graph G(V, E, s), source membership vector πs, target membership vector πt,
+moving vertices
+s#»t
+V
+Output: Boolean variable
+/* 1st part
+*/
+1 if ¬minEdit(πs, πt,
+s#»t
+V ) or ¬extAtomic(G, πs, πt,
+s#»t
+V ) then // Properties 3, 5, 7b and 8
+2
+return true
+3 end
+/* 2nd part
+*/
+4 if 2 ≤ |
+s#»t
+V | and ¬extMVMO(G, πs, πt,
+s#»t
+V ) then // Lemmas 11.2, 12 and 13
+5
+return true
+6 end
+7 return false;
+...
+...
+...
+...
+P0
+P1
+1-edit
+kmax-edit
+2-edit
+1-edit
+2-edit
+P6
+P7
+P8 P9
+P10
+P11
+P12
+P2 P3
+P4
+P5
+kmax-edit
+(a) RNS Tree.
+P1
+P2
+P3
+P4
+V
+1
+2
+={v1,v2}
+V
+3
+4
+={v1,v2}
+V
+2
+4
+={v3}
+V
+1
+3
+={v3}
+(b) Example of duplication in RNS.
+Both P 2 and P 3 generate P 4.
+Figure 2: Illustrations for the Recurrent Neighborhood Search (RNS). In both figures, circles represent
+membership vectors associated with optimal partitions.
+5.3
+Main Procedure
+The main procedure for Recurrent Neighborhood Search (RNS) consists in applying CoNS from Sec-
+tion 5.2 in a recursive manner, in order to build a search tree called the RNS tree. This recursive
+procedure is depicted in Figure 2a. The root node of RNS tree corresponds to the initial partition P.
+All other nodes of the RNS tree constitute the set of optimal solutions reached directly or indirectly
+from P. A solution associated with a node at some level h of the RNS tree is obtained after h − 1
+applications of the CoNS procedure. The height of the RNS tree is unknown but necessarily finite, as
+we will see next.
+Method RNS(G, P, rmax), which takes as input a signed graph G, an optimal partition P and a
+maximum edit distance rmax, is detailed in Algorithm 6. It starts from the initial partition P (line 1),
+and enumerates a set of its neighbor optimal partitions up to edit distance rmax (line 4), denoted
+by RN≤rmax(P). Each partition P t associated with πt ∈ Nr(πs) is considered as a potential initial
+partition for seeking new neighbor partitions (line 8). Despite the pruning techniques which avoid some
+repetitions, it is possible that P t has been already discovered and belongs to RN≤rmax(P) (see such an
+example in Figure 2b). Line 9 checks this case, and discards P t, if required. This process is repeated in
+a recursive manner, until there is no new partition obtained.
+In the rest of this work, for the sake of convenience, we leave out the common parameter G and P s
+(or P) when referring to the mentioned methods: CoNS(r) and RNS(rmax).
+6
+Pruning Strategies
+In this section, we present the pruning strategies incorporated in the method CoNS(G, πs, r), described
+in Section 5.
+But before this, we need to introduce some additional definitions.
+Consider an edit
+operation πs → πt with moving vertices in set
+s#»t
+V , where membership vector πs (resp. πt) is associated
+11 / 30
+
+Arınık et al. – Efficient Enumeration of the Optimal Solutions to the Correlation Clustering problem
+Algorithm 6: RNS(G, P, rmax)
+Input
+: signed graph G(V, E, s), partition P, maximum edit distance rmax
+Output: RN≤rmax(P)
+1 U = {P} // a set of unprocessed partitions
+2 RN≤rmax(P) = ∅ // a set of discovered partitions
+3 while U ̸= ∅ do
+4
+for each r ∈ {1, .., rmax} do
+5
+Let P s be an element of U, where πs is an associated membership vector of P s
+6
+RN≤rmax(P) = RN≤rmax(P) � P s
+7
+Nr(πs) = CoNS(G, πs, r) // Complete Neighborhood Search
+8
+for πt ∈ Nr(πs) do // P t is the associated partition of πt
+/* memory-based duplication removal
+*/
+9
+if P t /∈ RN≤rmax(P) then
+10
+U = U � P t
+11
+end
+12
+end
+13
+end
+14 end
+with source partition P s (resp.
+target partition P t).
+Let us define #»
+V s
+πs(u) =
+s#»t
+V
+∩ M s
+πs(u) (resp.
+#»
+V t
+πs(u) =
+s#»t
+V
+∩ M t
+πs(u)) as the set of moving vertices being in the source module of vertex u in
+P s (resp.
+in P t).
+Also, let #»
+V s
+πt(u) =
+s#»t
+V
+∩ M s
+πt(u) (resp.
+#»
+V t
+πt(u) =
+s#»t
+V
+∩ M t
+πt(u)) be the set
+of moving vertices being in the target module of u in P s (resp. in P t). Finally, we define
+s#»t
+V u =
+�#»
+V s
+πs(u) ∪ #»
+V t
+πs(u) ∪ #»
+V t
+πt(u) ∪ #»
+V s
+πt(u)
+�
+\ {u} as the set of moving vertices connected to vertex u in
+�G[
+s#»t
+V ]. To illustrate the aforementioned notations related to
+s#»t
+V , we consider again the same example
+from Figure 1. For v1 ∈
+s#»t
+V , we have πs(v1) = 1 and πt(v1) = 3. Then, #»
+V s
+πs(v1) = {v1, v2} is the
+set of vertices in M s
+1 and #»
+V t
+πt(v1) = {v1} since v1 is the only moving vertex belonging to M t
+3. Also,
+#»
+V s
+πt(v1) = #»
+V s
+{3} = ∅ since there is no moving vertex in M s
+3, and #»
+V t
+πs(v1) = {v3} since v3 ∈ M t
+1. Finally,
+we get
+s#»t
+V v1 = {v2, v3}.
+6.1
+Non-Minimum Edit Operation Pruning
+When generating the set Nr(πs) in RNS, only considering min-edit operations is sufficient. Next, we
+define a property allowing to detect, in some cases, that an edit operation is not minimal. The idea
+behind it is to define two sufficient conditions on the set of moving vertices implying that there exist
+an operation of smaller cost leading to the same partition. In the following, let πs → πt be an edit
+operation with moving vertices in
+s#»t
+V .
+Property 3 (Non-min-edit operation). Let # »
+Va ⊆
+s#»t
+V
+be a subset of
+s#»t
+V . Assume that the vertices in
+# »
+Va constitute the majority part of module M s
+a (i.e. |# »
+Va| > |M s
+a \ # »
+Va|) and that they are in the same
+target module M t
+b in P t (i.e. |πt(# »
+Va)| = 1). Moreover, suppose there exists a subset of moving vertices
+#»
+Vb ⊆
+s#»t
+V , such that ∅ ⊆ #»
+Vb ⊆ M s
+b . Let us define V a (resp. V b) as M s
+a \
+s#»t
+V
+(resp. M s
+b \
+s#»t
+V ) in P s.
+Each condition in the following is a sufficient condition for πs → πt to be a non-min-edit operation:
+(a) If #»
+Vb does not move into M s
+a and |# »
+Va| > |V a| + |V b|, then πs → πt is not a min-edit operation.
+(b) If #»
+Vb moves into M s
+a and |# »
+Va| + |#»
+Vb| > |V a| + |V b|, then πs → πt is not a min-edit operation.
+Proof. Let us first handle the condition (3a), which is illustrated in Figure 3. When this condition
+is satisfied, keeping # »
+Va in M s
+a and rather moving V a (resp. V b) into M s
+b (resp. M s
+a) results in an
+r′-edit operation with r′ < r. In the case where condition (3b) is satisfied, keeping #»
+Vb in M s
+b while
+moving V a and V b results in an r′-edit operation with r′ < r. Let us take the following example to
+illustrate Property 3.b. Two modules exchange all of their vertices with each other, such that # »
+Va = M s
+a,
+12 / 30
+
+Arınık et al. – Efficient Enumeration of the Optimal Solutions to the Correlation Clustering problem
+M s
+a
+M s
+b
+# »
+Va
+V a
+#»
+Vb
+V b
+(a) Source partition P s.
+M t
+a
+M t
+b
+# »
+Va
+V a
+V b
+(b) Target partition P t.
+Figure 3: Illustrative example for Property 3a, where vertices in # »
+Va (resp.
+#»
+Vb) move from their source
+module Ms
+a (resp.
+Ms
+b ) (Fig. 3a) into target module Ms
+b (resp.
+Ms
+c , which is purposely not shown)
+(Fig. 3b). V a and V b represent the non-moving vertices.
+M1
+M2
+M3
+M4
+v1
+v2
+v3
+v4
+v1
+v2
+v3
+v4
+(a) An illustration of an r-edit operation πs → πt with
+s#»t
+V
+= {v1, v2, v3, v4}. The mem-
+bership vector πs associated with partition P s is defined by πs = (1, 1, 3, 4), whereas
+πt = (2, 2, 2, 3).
+Positive (resp.
+negative) edges between moving vertices are drawn in
+green (resp. red).
+v1
+v2
+v3
+v4
+(b) The corresponding interaction graph �G[
+s#»t
+V ] of the r-
+edit operation illustrated in Fig. 4a
+Figure 4: Illustrative example regarding the definition of an interaction subgraph, here �G[
+s#»t
+V ].
+V a = ∅, #»
+Vb = M s
+b and V b = ∅. In the end, we obtain the same two modules. Since the inequality
+|# »
+Va| + |#»
+Vb| > |V a| + |V b| holds in this particular scenario, this is not a min-edit operation.
+6.2
+Decomposable Edit Operation
+The next set of properties is related to the decomposability of an r-edit operation πs → πt. We assume
+that I(P s) = I(P t), which is in line with our objective of enumerating all optimal solutions of a given
+graph G.
+Definition 4 (Decomposable edit operation). We consider an r-edit operation πs → πt as decomposable
+if there is an intermediate membership vector πt′, associated with a partition P t′, between πs and πt
+such that:
+1. I(P s) = I(P t) = I(P t′);
+2.
+s#»t′
+V
+⊂
+s#»t
+V ; and
+3. πt′(u) = πt(u) for each u ∈
+s#»t′
+V .
+Property 5 (Atomic edit operation). We consider that an r-edit operation πs → πt is atomic if the
+condition I(P s) < I(P t′) is satisfied for any r′-edit operation P s → P t′ with r′ < r satisfying (2) and
+(3) in Definition 4.
+Proof. We assume that there is an r′-edit operation such that I(P s) = I(P t′). Then, we can construct
+an r′′-edit operation πt′ → πt satisfying (1) in Definition 4 and πt′(u) = πt(u) for each u ∈
+s#»t
+V
+\
+s#»t′
+V .
+Atomicity requires three types of connectivity conditions. We state the first one in the following
+property.
+13 / 30
+
+Arınık et al. – Efficient Enumeration of the Optimal Solutions to the Correlation Clustering problem
+Property 6 (Edge connectivity). In an atomic edit operation πs → πt, G[
+s#»t
+V ] is a single connected
+component.
+Proof. Let
+s#»t′
+V
+and
+t′ #»t
+V
+be any two proper subsets of
+s#»t
+V , such that
+s#»t′
+V ∩
+t′ #»t
+V
+= ∅ and E(
+s#»t′
+V ,
+t′ #»t
+V ) =
+∅. Then, we can construct with
+s#»t′
+V
+an r′-edit operation satisfying Definition 4.
+Nevertheless, G[
+s#»t
+V ] being connected is not a sufficient condition. The next property states that
+the signs of edges between moving vertices play a key role with respect to atomicity. In the following,
+we assume that G[
+s#»t
+V ] is connected.
+Property 7 (Spurious edge connectivity). Each condition in the following is a sufficient condition for
+πs → πt to be a decomposable edit operation:
+(a) Let S ⊆
+s#»t
+V
+be a subset such that |πs(S)| = 1 and E(S, #»
+V s
+πs(S)) ̸= ∅. Let G′ = (
+s#»t
+V , E′, w),
+where E′ = E \ E(S, #»
+V s
+πs(S)) and w(e) = 1 for each e ∈ E′. If Ω(S, #»
+V s
+πs(S)) = 0 and G′ is not
+connected, then πs → πt is decomposable.
+(b) Let S ⊆
+s#»t
+V
+be a subset such that |πt(S)| = 1 and E(S, #»
+V t
+πt(S)) ̸= ∅. Let G′ = (
+s#»t
+V , E′, w),
+where E′ = E \ E(S, #»
+V t
+πt(S)) and w(e) = 1 for each e ∈ E′. If Ω(S, #»
+V t
+πt(S)) = 0 and G′ is not
+connected, then πs → πt is decomposable.
+Proof. Straightforwardly, we can imagine S as a contracted vertex v in both cases, such that the
+vertices in S and the edges between them are replaced by a single vertex v. Since Ω(S, #»
+V s
+πs(S)) = 0
+(Ω(S, #»
+V t
+πt(S)) = 0), the edges between v and #»
+V s
+πs(v) (resp. #»
+V t
+πt(v)) do not play a role (i.e., as if they
+did not exist). Therefore, the proof is the same as for Property 6, with subsets S and
+s#»t
+V
+\ S.
+The last connectivity condition allows to verify if a moving vertex depends on the simultaneous
+movement of a subset of other vertices in
+s#»t
+V . It can be checked through the concept of interaction
+subgraph �G[
+s#»t
+V ].
+We define the interaction subgraph defined for the edit operation πs → πt as
+�G[
+s#»t
+V ] = (
+s#»t
+V , �E), where �E = {(u, v) | u, v ∈
+s#»t
+V and (u, v) ∈ E
+and (πs(u) = πs(v) ∨ πt(u) =
+πs(v) ∨ πs(u) = πt(v) ∨ πt(u) = πt(v))}. Its extraction from its original graph G is illustrated in
+Figure 4.
+Property 8 (Interaction connectivity). Consider an atomic edit operation πs → πt. The interaction
+subgraph �G[
+s#»t
+V ] is a single connected component.
+Proof. Assume that �G[
+s#»t
+V ] is not connected (e.g.
+Figure 4).
+This implies that there is a subset
+s#»t′
+V
+⊆
+s#»t
+V , such that �E(
+s#»t′
+V ,
+s#»t
+V
+\
+s#»t′
+V ) = ∅. Let πs → πt′ be an edit operation with moving vertex
+set
+s#»t′
+V , which satisfies (2)-(3) in Definition 4. When moving a vertex u ∈
+s#»t′
+V
+from M s
+πs(u) to M s
+πt′(u)
+the contribution of vertex u to the imbalance is
+Ω({u}, #»
+V s
+πs(u)) − Ω({u}, #»
+V s
+πt′(u)) + Ω({u}, #»
+V t′
+πt′(u)) − Ω({u}, #»
+V t′
+πs(u))
+2
+.
+(11)
+From the definition of �E and from �E(
+s#»t′
+V ,
+s#»t
+V \
+s#»t′
+V ) = ∅, there is no vertex v ∈
+s#»t
+V \
+s#»t′
+V
+which
+contributes to the contribution of vertex u, as shown in Equation (11). Therefore, Equation (1) in
+Definition 4 is also satisfied for πs → πt′.
+For the sake of simplicity, in Algorithm 4, we define the function intAtomic(G, πs,
+s#»t
+V ) which is
+used to indicate whether πs → πt satisfies Properties 6 and 7a. Notice that we do so when the target
+modules of
+s#»t
+V
+are not known. Likewise, in Algorithm 5, we define the function extAtomic(G, πs, πt,
+s#»t
+V ), which indicates whether πs → πt satisfies Properties 3, 5, 7b and 8. Notice that we do so when
+the target modules of
+s#»t
+V
+are partially or completely known, i.e., in lines 10, 20 and 26 of Algorithm 3.
+6.3
+Multiple Vertex Moves between Optima (MVMO) Property
+In this section, we introduce our last family of properties, which rely on the assumption that both
+partitions associated to an edit operation are optimal.
+14 / 30
+
+Arınık et al. – Efficient Enumeration of the Optimal Solutions to the Correlation Clustering problem
+Property 9 (MVMO on weighted signed networks). Consider an atomic edit operation πs → πt with
+r > 1, where P s and P t are optimal. The following property holds for each u ∈
+s#»t
+V :
+γleft
+u
+> γright
+u
+,
+(12)
+where
+γleft
+u
+= Ω({u}, #»
+V s
+πs(u)) − Ω({u}, #»
+V s
+πt(u)),
+(13)
+γright
+u
+= −Ω({u}, #»
+V t
+πt(u)) + Ω({u}, #»
+V t
+πs(u)).
+(14)
+(15)
+Proof. Let πs → πt be an atomic min-r-edit operation applied on Ps to obtain Pt. First, we recall a
+simple condition satisfied by any optimal partition for the CC problem [13] when moving a single vertex,
+i.e. when r = 1. Given an optimal partition Ps, the placement of any vertex u in its module M s
+πs(u)
+is also optimal with respect to any other modules. This means that any vertex u maximizes (resp.
+minimizes) its number of positive (resp. negative) edges in module M s
+πs(u) of Ps, so that moving u to
+any other module does not improve the objective function value. We can write this condition as
+Ω({u}, M s
+πs(u)) ≥ Ω({u}, M s
+πt(u)).
+(16)
+Now, when we move more than one vertex in πs → πt, i.e. if r > 1, this condition becomes
+Ω({u}, M s
+πs(u)) > Ω({u}, M s
+πt(u)),
+(17)
+for each u ∈
+s#»t
+V , due to the atomicity assumption. Next, in order to take into account the existence of
+other moving vertices in the source and target modules of vertex u, Equation (17) can be rewritten as
+γleft
+u
+>
+s#»t
+∆u, where
+s#»t
+∆u = Ω({u}, M s
+πt(u) \ #»
+V s
+πt(u)) − Ω({u}, M s
+πs(u) \ #»
+V s
+πs(u)). Similarly, considering
+the atomic min-r-edit operation P t → P s, we have also −
+t#»s
+∆u > γright
+u
+, where
+t#»s
+∆u = Ω({u}, M t
+πs(u)\
+#»
+V t
+πs(u)) − Ω({u}, M t
+πt(u) \ #»
+V t
+πt(u)). Since both
+s#»t
+∆u and
+t#»s
+∆u are defined on relations of vertex u with
+non-moving vertices in P s and P t, we have
+s#»t
+∆u = −
+t#»s
+∆u. Finally, by rewriting the previous equations
+we obtain γleft
+u
+>
+s#»t
+∆u = −
+t#»s
+∆u > γright
+u
+.
+Corollary 10 (MVMO on unweighted signed networks). The inequality γleft
+u
+− γright
+u
+≥ 2 holds for
+each vertex u, on top of Property 9.
+Proof. The proof is straightforward from the proof of Property 9.
+Figure 5 can be used to illustrate Property 9 and Corollary 10. For instance, for vertex v1, the
+equation in Property 9 becomes
+a12 − a13 > a15 − a12 − a14.
+(18)
+Similarly, for vertex v3, the equation in Property 9 becomes
+−a34 − a35 > a13 + a23 + a34.
+(19)
+6.4
+Tractable Cases of the MVMO Property
+Notice that the MVMO property requires πs and πt to be known, which makes this property not very
+useful for computational purposes (in Algorithm 3). In this section, we analyze the MVMO property
+when only partial information of πt is known (lines 10 and 20 in Algorithm 3).
+We start by analysing some special relational patterns for 2-edit and 3-edit operations. Lemma 11
+focuses on 2-edit operations. Recall that �E = {(u, v) ∈ E | u, v ∈
+s#»t
+V
+∧ (πs(u) = πs(v) ∨ πt(u) =
+πs(v) ∨ πs(u) = πt(v) ∨ πt(u) = πt(v))}.
+Lemma 11 (MVMO 2-edit). Consider an atomic 2-edit operation πs → πt, where
+s#»t
+V
+= {u, v}. Since
+it is an atomic edit operation, we have (u, v) ∈ �E. Then
+1. If (πs(u) = πs(v)) ∨ (πt(u) = πt(v)), then auv > 0.
+15 / 30
+
+Arınık et al. – Efficient Enumeration of the Optimal Solutions to the Correlation Clustering problem
+M1
+M2
+M3
+v1
+v2
+v3
+v4
+v5
+v1
+v2
+v4
+v3
+v5
+a12
+a15
+a34
+Figure 5: Illustrative example, where five vertices v1, v2, v3, v4 and v5 move from their source modules
+to their target ones. Dashed circles indicate the moving vertices, when they are in their target modules
+after the edit operation. Note that we do not show non-moving vertices for the sake of simplicity. Positive
+(resp. negative) edges between moving vertices are drawn in green (resp. red).
+a)
+d)
+e)
+b)
+c)
+u
+v
+u
+v
+u
+v
+u
+u
+v
+v
+Figure 6: All 2-edit operation scenarios, where (u, v) ∈ �E. Note that only scenarios (6a) and (6b) are
+atomic 2-edit operations, with auv > 0 and auv < 0, respectively.
+2. If (πs(u) = πt(v)) ∨ (πt(u) = πs(v)), then auv < 0.
+Proof. From Property 6, we have (u, v) ∈ E. In this proof, we use an exhausting strategy: moving
+vertices u and v, with (u, v) ∈ �E, can be in one of the five scenarios presented in Figure 6. Since
+Corollary 10 holds for each vertex u ∈
+s#»t
+V , only scenarios (6a) and (6b) are atomic. Therefore, auv > 0
+for scenario (a) whereas auv < 0 for scenario (b):
+a) We have (γleft
+u
+= auv) > (γright
+u
+= −auv) and (γleft
+v
+= auv) > (γright
+v
+= −auv). We see that
+auv must be positive. Since Corollary 10 is satisfied with auv > 0, it is atomic.
+b) We have (γleft
+u
+= −auv) > (γright
+u
+= auv) and (γleft
+v
+= −auv) > (γright
+v
+= auv). We see that
+auv must be negative. Since Corollary 10 is satisfied with auv < 0, it is atomic.
+c) We have (γleft
+u
+= 0) > (γright
+u
+= −auv) and (γleft
+v
+= 0) > (γright
+v
+= −auv). We see that auv
+must be positive and this satisfies Property 9. However, Corollary 10 is not satisfied, which makes
+this edit operation decomposable.
+d) We have (γleft
+u
+= auv) > (γright
+u
+= 0) and (γleft
+v
+= auv) > (γright
+v
+= 0). We see that auv must
+be positive and this satisfies Property 9. However, Corollary 10 is not satisfied, which makes this
+edit operation decomposable.
+e) We have (γleft
+u
+= −auv) > (γright
+u
+= 0) and (γleft
+v
+= 0) > (γright
+v
+= −auv). We see that auv
+must be negative and this satisfies Property 9. However, Corollary 10 is not satisfied, which makes
+this edit operation decomposable.
+Next, we focus on 3-edit operations. We verify some cases in which Lemma 11 is also valid.
+Lemma 12 (MVMO 3-edit). Consider an atomic r-edit operation πs → πt with r = 3, where
+s#»t
+V
+=
+{u, v, z}. Since it is an atomic edit operation, we have (u, v), (u, z), (v, z) ∈ �E. Lemma 11 holds true
+for each pair (u, v) of vertices, with two exceptions:
+16 / 30
+
+Arınık et al. – Efficient Enumeration of the Optimal Solutions to the Correlation Clustering problem
+b)
+c)
+a)
+d)
+e)
+u
+u
+u
+v
+v
+v
+z
+z
+z
+u
+v
+z
+u
+v
+z
+Figure 7: Five of the possible 3-edit operation scenarios.
+The full list can be found in the Appendix
+(Figure 12). In this figure, all edit operations are atomic.
+1. all vertices in
+s#»t
+V
+are in the same source module and are moved into the same target module
+(Figure 7a),
+2. only two of
+s#»t
+V
+are in the same source module and are moved into the source module of the third
+moving vertex. Reciprocally, the third moving vertex moves into the source module of the others’
+source module (Figure 7b).
+Proof. Without these two exceptions, G[
+s#»t
+V ] forms a triangle. Nevertheless, in these two exceptions,
+G[
+s#»t
+V ] can form a path, and, as will see next, this path ensures that moving vertices from the same
+source module are positively connected. Similar to Lemma 11, the proof is based on all possible 17
+scenarios of three moving vertices with (u, v), (u, z), (v, z) ∈ �E. Some of those scenarios are shown in
+Figure 7, and we provide the full list in the Appendix (Figure 12). The proof is straightforward when
+one adapts Corollary 10 to those scenarios. We verify below only those in Figure 7. The full list of
+verifications is found in Appendix C.1.
+a) We have (γleft
+u
+= auv + auz) > (γright
+u
+= −auv − auz), (γleft
+v
+= auv + avz) > (γright
+v
+=
+−auv − avz) and (γleft
+z
+= auz + avz) > (γright
+z
+= −auz − avz). We see that auv, auz and avz
+cannot be negative. Since Corollary 10 is satisfied with a positive path formed in G[
+s#»t
+V ] for each
+vertex u ∈
+s#»t
+V , it is atomic.
+b) We have (γleft
+u
+= auv − auz) > (γright
+u
+= −auv + auz), (γleft
+v
+= auv − avz) > (γright
+v
+=
+−auv + avz) and (γleft
+z
+= −auz − avz) > (γright
+z
+= auz + avz). We see that auv (resp. auz
+and avz) cannot be negative (resp. positive). Since Corollary 10 is satisfied with a path formed
+in G[
+s#»t
+V ], it is atomic.
+c) We have (γleft
+u
+= 0) > (γright
+u
+= −auv − auz), (γleft
+v
+= 0) > (γright
+v
+= −auv − avz) and
+(γleft
+z
+= 0) > (γright
+z
+= −auz − avz). We see that auv, auz and avz must be positive. Since
+Corollary 10 is satisfied with a positive triangle formed in G[
+s#»t
+V ], it is atomic.
+d) We have (γleft
+u
+= auv) > (γright
+u
+= −auv − auz), (γleft
+v
+= auv) > (γright
+v
+= −auv − avz) and
+(γleft
+z
+= 0) > (γright
+z
+= −auz − avz). We see that auv, auz and avz must be positive. Since
+Corollary 10 is satisfied with a positive triangle formed in G[
+s#»t
+V ], it is atomic.
+e) We have (γleft
+u
+= auv −auz) > (γright
+u
+= 0), (γleft
+v
+= auv) > (γright
+v
+= −avz) and (γleft
+z
+= 0) >
+(γright
+z
+= auz − avz). We see that auv and avz (resp. auz) must be positive (resp. negative).
+Since Corollary 10 is satisfied with a triangle formed in G[
+s#»t
+V ], it is atomic.
+Unlike Lemma 12, we cannot completely generalize Lemma 11 for more than three moving vertices.
+Nevertheless, there are some circumstances, where Lemma 11 is still valid for a subset of
+s#»t
+V , and this
+is formalized in Lemma 13.
+Lemma 13 (MVMO r-edit). Consider an atomic r-edit operation πs → πt with r ≥ 4 and a vertex
+u ∈
+s#»t
+V . If 2 ≤ |u ∪
+s#»t
+V u| ≤ 3, Lemma 11 holds true for each pair (u, v) with v ∈
+s#»t
+V u.
+Proof. The condition of 2 ≤ |u∪
+s#»t
+V u| ≤ 3 ensures that subset u∪
+s#»t
+V u represents one of the scenarios in
+2- or 3-edit operation. We note that two exceptional scenarios mentioned in Lemma 12 are not of interest
+here, because they necessarily satisfy the definition of atomic edit operation, hence |u ∪
+s#»t
+V u| > 3.
+For the sake of simplicity, in Algorithm 4, we define the function intMVMO(G, πs,
+s#»t
+V ) which
+indicates whether πs → πt satisfies Lemmas 11.1 and 12, when the target modules of
+s#»t
+V
+are not
+known. Likewise, in Algorithm 5 we define the function extMVMO(G, πs, πt,
+s#»t
+V ), which indicates
+17 / 30
+
+Arınık et al. – Efficient Enumeration of the Optimal Solutions to the Correlation Clustering problem
+whether πs → πt satisfies Lemmas 11.2, 12 and 13, when the target modules of
+s#»t
+V
+are partially or
+completely known (10, 20 and 26 of Algorithm 3).
+7
+Dataset
+This section is dedicated to the description of the dataset used in our experiments. As mentioned in the
+introduction, in this article we focus on unweighted graphs as we have done in [6]. Application-wise,
+unweighted signed networks fit certain modeling situations and methodological choices. Indeed, in some
+works, binary values better represent the studied relations (e.g. alliance/conflict between countries in
+international relationships [21]), or the authors prefer to use such values for practical reasons (e.g.
+limited or unreliable information [22]).
+We conduct our experiments on two datasets of random signed networks. All these data as well as
+the corresponding optimal solutions are publicly available online1.
+Dataset1: We generate these networks through the random signed network generator proposed in
+our previous work [6], which is publicly available online2. For complete unweighted signed networks, this
+model relies on only three parameters: n (number of vertices), ℓ0 (initial number of modules) and qm
+(proportion of misplaced edges, i.e. edges meant to be frustrated by construction). Moreover, we make
+the assumption that the proportion of misplaced edges is the same inside and between the modules.
+When it comes to incomplete unweighted signed networks, we introduce two more parameters, which
+are the density d of the graph and the proportion qneg of the negative edges. The last parameter qneg
+is defined as |E−|/|E| and allows controlling the ratio of positive to negative edges. For complete
+unweighted signed networks with d = 1, we generate 20 replications for parameter values ℓ0 = 3,
+n ∈ {32, 36, 40, 45, 50} and qm = {0.1, 0.2, 0.3, 0.4, 0.5, 0.6}. In these networks, the value of qneg
+with the considered parameters is approximately 0.7. For incomplete unweighted signed networks with
+d = {0.25, 0.50}, we generate 20 replications for parameter values ℓ0 = 3, n ∈ {32, 36, 40}, qm =
+{0.1, 0.2, 0.3, 0.4, 0.5, 0.6} and qneg = {0.3, 0.5, 0.7}. In total, we produce 600 and 1,080 instances for
+complete and incomplete networks, respectively, which makes a total of 1,680 instances. We use this
+dataset in Section 8.2.
+Dataset2: This dataset is different than Dataset1 in that the optimal solution for a generated
+network is known by construction. This allows to consider relatively large graph orders n without being
+limited by long running time of an exact partitioning method. For given n, d and ℓ0, we first create
+a perfectly structurally balanced (i.e., internally positive and externally negative) signed network with
+a built-in module structure. Clearly, the underlying module structure constitutes the optimal partition.
+Then, in order to take into account different positive to negative ratio values for internal and external
+edges we generate several signed networks by perturbing the initial signed network without affecting
+its underlying optimal partition, thanks to its definition of stability range [41]. We generate signed
+networks with parameter values n ∈ {30, 40, 50, 60, 70, 90}, d ∈ {0.25, 1.00} and ℓ0 = {2, 4, 6} through
+our random signed network generator, which is publicly available online3. In total, we produce 214 and
+184 instances for complete and incomplete networks, respectively, which makes a total of 398 instances.
+In each generated network, the associated optimal solution corresponds to the planted partition defined
+for the corresponding network without perturbation. We use this dataset only for Section 8.1.
+Dataset3: In addition to artificial graphs, we also considered 8 sparse real-world networks from
+two sources: four networks from Correlated of War (CoW) [43] and four biological networks retrieved
+from Samin Aref’s Figshare repository4, see [3] for more details. For each real-world signed network
+G, we apply a preprocessing step by retrieving the largest connected component defined in G+, for
+computational purposes.
+8
+Results
+We now assess the performance of our enumeration method EnumCC(rmax) when generating the space
+of optimal solutions to the CC problem. We first investigate the efficiency of the pruning conditions used
+in Algorithms 4 and 5 based on the MVMO property (Section 8.1). Then, we proceed with the evaluation
+of EnumCC(rmax), which includes all the pruning strategies used in the recurrent neighborhood search.
+We compare our method with OneTreeCC(), the best enumeration method available in the literature
+(Section 8.2).
+1DOI: 10.6084/m9.figshare.15043911
+2https://github.com/CompNet/SignedBenchmark.
+3https://github.com/CompNet/SignedStabilityBenchmark.
+4DOI: 10.6084/m9.figshare.5700832.v5
+18 / 30
+
+Arınık et al. – Efficient Enumeration of the Optimal Solutions to the Correlation Clustering problem
+8.1
+Evaluation of the MVMO-based Pruning Strategies
+The MVMO property has an important role in EnumCC, due to its ability to prune unfeasible edit
+operations in several parts of CoNS(r). To be able to consider relatively large graph orders n without
+being limited by the long running time of an exact partitioning method, we evaluate its performance
+based on Dataset2. The complete results1 as well as our source code3 are publicly available online.
+We apply CoNS(r) with and without the MVMO property onto all generated networks with r ∈
+{3, 4}. The version with MVMO property refers to the method described in Section 4, whereas the
+version without it is obtained by removing this property from Algorithms 4 and 5.
+Due to space
+considerations, Figure 8 illustrates only the results related to 4-edit distance. The results obtained for
+d = 0.25 and d = 1.00 are shown in separated subfigures. In each subfigure, the x-axis represents the
+graph order n, and execution times (in seconds) are shown on the log-scaled y-axis. The solid (resp.
+dashed) plot lines correspond to CoNS(r) with (resp. without) the MVMO property. Each plot line
+corresponds to a specific value of ℓ0 and is represented with a specific color. Each shaded region depicts
+a range of execution times around the plot line of the same color, based on the corresponding initial
+signed network and its perturbed versions.
+The first thing to notice is how graph density affects the execution times with increasing values of n.
+Indeed, the computational costs with and without the MVMO property are much larger with d = 1.00
+than those with d = 0.25. Second, substantially increasing n affects the execution times without the
+MVMO property, whereas including the MVMO property allows to better handle this effect. Indeed,
+considering d = 0.25 and the 4-edit distance, we observe that average execution times without the
+MVMO property are approximately ℓ0 times larger than with the MVMO property. Finally, including
+the MVMO property also allows to better handle increasing values of ℓ0. Indeed, when n = 70 and
+d = 0.25, the difference of average execution times between ℓ0 = 2 and ℓ0 = 6 without (resp. with)
+the MVMO property is 3.8s (resp. 0.6s) with the 3-edit distance, and 595s (resp. 63s) with the 4-edit
+distance.
+To conclude this part, the MVMO property makes a substantial improvement on CoNS(r). This
+improvement is apparent even with small values of n. Note that, even a small improvement (1s or 2s)
+can make a clear difference in terms of execution time for a solution space with 100 or more solutions,
+since CoNS(r) is repeated for each solution. Nevertheless, our results suggest that CoNS(4) should not
+be used for computational purposes, even with the MVMO property. In the following, we investigate if
+the improvements brought by the MVMO property and other pruning strategies allow EnumCC(rmax)
+to compete with OneTreeCC().
+8.2
+Evaluation of EnumCC
+We now compare the results of EnumCC(rmax) against OneTreeCC() based on the synthetic signed
+networks from Dataset1 and the real-world signed networks from Dataset3. As shown in Section 8.1,
+since CoNS(4) takes a substantial time, we rather apply EnumCC(3). We run both methods with a
+time limit of 12h and a limit of 50, 000 on the number of optimal solutions found. We first evaluate
+the results from Dataset1 for several values of density d with a fixed value of n (Section 8.2.1) and
+several values of n with a fixed value of d (Section 8.2.2), then pass to the evaluation of the results
+from Dataset3 (Section 8.2.3).
+Regarding the synthetic graphs, we present a selection of the most relevant results in Figures 9a–10c,
+for d ∈ {0.25, 1.00}. The complete results1 as well as our source code5,6 are available online, though.
+We first describe these plots generically here, before interpreting them. In these figures, there are 3
+subfigures. Each subfigure corresponds to a specific value of qm (hence, a specific parameter set),
+and displays the difference of execution times between EnumCC(3) and OneTreeCC() (i.e., EnumCC(3)
+minus OneTreeCC()), represented on the log-scaled y-axis of the plots. When such difference takes a
+negative value, this means that our proposed method runs faster than OneTreeCC(). The set of 20
+graphs generated for each parameter set are indexed and shown on the x-axis. Finally, to guide our
+discussion, we show for each graph the maximal number of solutions found by the method(s) within a
+time limit and the number of jumps related to EnumCC(3), i.e. njump(EnumCC(3)), where the latter
+is shown in parentheses.
+8.2.1
+Evaluation of EnumCC by Density
+In this section, the results are for n = 36 and d ∈ {0.25, 0.5, 1.0}. They are representative enough of
+the results obtained with other values of n. We first consider the d = 0.25 results, shown in Figure 9.
+5https://github.com/CompNet/Sosocc
+6https://github.com/CompNet/EnumCC
+19 / 30
+
+Arınık et al. – Efficient Enumeration of the Optimal Solutions to the Correlation Clustering problem
+0
+1
+10
+100
+1000
+5000
+30
+40
+50
+60
+70
+ℓ0
+2
+4
+6
+without
+MVMO
+with
+MVMO
+Execution time (s)
+Graph order (n)
+(a) Instances with d = 0.25.
+0
+1
+10
+100
+1000
+5000
+30
+40
+50
+60
+70
+Graph order (n)
+Execution time (s)
+ℓ0
+2
+4
+6
+without
+MVMO
+with
+MVMO
+(b) Instances with d = 1.00.
+Figure 8: Benchmark results for CoNS(r = 4) with vs. without MVMO-based pruning for d = 0.25 (8a)
+and d = 1.00 (8b). The x-axis represents graph order n, and execution times (in seconds) are shown in
+the log-scaled y-axis.
+20 / 30
+
+Arınık et al. – Efficient Enumeration of the Optimal Solutions to the Correlation Clustering problem
+The first thing that we notice is how the performance is affected by qneg, independently of qm. Indeed,
+we first see for qneg = 0.3 that EnumCC(3) runs faster than OneTreeCC() in the overwhelming majority
+of instances. Then, for qneg = 0.5, OneTreeCC() increases the number of instances, where it runs
+faster, but the dominance of EnumCC(3) is still preserved to a lesser extent. Finally, for qneg = 0.7,
+OneTreeCC() dominates EnumCC(3) on almost all instances.
+We observe that increasing qneg essentially results in the emergence of three features, which ad-
+vantages OneTreeCC() over EnumCC(3). The first one is large values of njump(EnumCC(3)). Since
+a branch-and-bound tree needs to be built from scratch, each additional jump has an extra cost for
+EnumCC(3) in terms of execution time. We observe that the values of njump(EnumCC(3)) are rel-
+atively larger for mainly qneg = 0.5 and qm = {0.4, 0.6}. This shows that increasing qm is likely to
+increase the dissimilarity between solutions, a fact that we have already pointed out in [6], for complete
+signed networks.
+The second feature is a very large size of the solution space. The results show that OneTreeCC() does
+a much better job in this case. Indeed, when qneg = 0.7, OneTreeCC() can solve instances associated
+with a very large number of solutions (e.g. 50, 000) within 5 minutes, whereas the same process often
+takes several hours for EnumCC(3). Nevertheless, such an extreme case happens only with some specific
+graph topology, as in qneg = 0.7. Indeed, in the random network generation, increasing qneg with a
+low graph density is more likely to produce instances having a large number of vertices with only few
+positive edges. Then, these vertices are often placed at the periphery of modules in the solutions, i.e.
+they can easily change their module from one solution to another. OneTreeCC() seems to handle well
+these vertices in the enumeration process, thanks to the mathematical modeling.
+Finally, the third feature, complementary to the previous one, is that instances having vertices with
+only few positive edges are often easier to solve, so that an initial optimal solution is often found at
+root relaxation, before passing to branch-and-bound. This usually results in a branch-and-bound tree
+with fewer branches for enumerating alternative optimal solutions in OneTreeCC(). It seems that this
+substantially advantages OneTreeCC() over EnumCC(3), when there are many optimal solutions to
+enumerate.
+For space considerations, we do not present the results for d ∈ {0.5, 1.0}. Note that for these values,
+we do not observe any extreme case with a very large number of solutions (as happens with d = 0.25 and
+qneg = 0.7). Indeed, the maximum number of solutions for those instances is 2, 455. This is probably
+because it is not as easy to break the underlying partition structure when graph density is large. This
+fact seems to advantage EnumCC(3) over OneTreeCC(). Indeed, the dominance of EnumCC(3) persists
+for qneg ∈ {0.3, 0.5} and even better than those for d = 0.25 (see Table 2). This holds for qneg = 0.7,
+too. OneTreeCC() outperforms EnumCC(3) only for qm = 0.1, whereas this was true for all values of qm
+for d = 0.25. These results confirm our previous observation: OneTreeCC() outperforms EnumCC(3)
+on instances with specific graph topology, where low density and large proportion of negative edges
+produce a very large number of solutions. In the other cases, EnumCC(3) performs much better. In the
+following, we study whether increasing graph order n affects these observations.
+8.2.2
+Evaluation of EnumCC by Graph Order
+In this section, we analyze the effect of increasing the graph order, for n ∈ {40, 45, 50}. To do so,
+we focus only on instances with d = 1.0 in order to reduce the total number of instances, hence the
+processing time of our analysis.
+Note that qneg approximately equals 0.7 in these instances.
+The
+corresponding results are shown in Figure 10. Overall, we see that the observations we made for n = 36
+and d ∈ {0.5, 1.0} in Section 8.2.1 are still valid when increasing n: EnumCC(3) performs much better.
+Furthermore, unlike the results obtained with d = 0.25, EnumCC(3) handles instances with a large value
+of njump(EnumCC(3)) much better (when we exclude some exceptional instances with more than 10
+jumps), and runs faster than OneTreeCC() in most of the instances. This point is even more valid when
+increasing n further.
+Table 1: Number of optimal solutions found by the considered methods within the time limit of 12h for
+unsolved instances with n = 50.
+methods
+instance no
+qm = 0.4
+qm = 0.5
+qm = 0.6
+1
+5
+6
+11
+16
+19
+2
+11
+12
+13
+15
+18
+20
+2
+6
+10
+12
+13
+17
+19
+20
+OneTreeCC()
+4
+4
+1
+1
+13
+2
+7
+21
+3
+1
+10
+2
+2
+3
+1
+3
+4
+5
+3
+1
+3
+EnumCC(3)
+10
+13
+6
+5
+255
+16
+217
+44
+8
+1
+115
+46
+13
+45
+10
+32
+6
+60
+7
+4
+65
+Another interesting point is the hardness of instances, when increasing n. We say that an instance
+is hard to solve when the resolution process takes too long, which amounts to exceeding a time limit.
+21 / 30
+
+Arınık et al. – Efficient Enumeration of the Optimal Solutions to the Correlation Clustering problem
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
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+2
+(1)
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+(1)
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+(1) 1
+(1)
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+(2)
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+(1) 1
+(1)
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+(1)
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+(1)
+3
+(1)
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+●
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+●
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+(1)
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+3
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+(1)
+8
+(1)
+3
+(1)
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+(1)
+79
+(3)
+25
+(3)
+8
+(2)
+1
+(1)
+1
+(1)
+37
+(2) 20
+(1)
+20
+(1)
+90
+(2)
+qm=0.2
+qm=0.4
+qm=0.6
+5
+10
+15
+20
+5
+10
+15
+20
+5
+10
+15
+20
+−100
+−10
+−1
+0
+1
+10
+100
+1000
+Instance number
+EnumCC(3) - OneTreeCC()
+(a) Instances with d = 0.25 and qneg = 0.3.
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+116
+(2)
+2
+(1)
+11
+(1)
+32
+(1)
+95
+(1)
+4
+(1)
+10
+(2)
+61
+(2)
+45
+(2)
+307
+(1)
+7
+(1)
+27
+(1)
+5
+(1)
+1304
+(1)
+21
+(1)
+6
+(2) 4
+(2)
+54
+(1)
+14
+(1)
+3
+(1)
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+66
+(1)
+2
+(1)
+2070
+(3)
+30
+(1) 14
+(1)
+96
+(2)
+112
+(3)
+142
+(2)
+183
+(4)
+12
+(2)
+653
+(4)
+35
+(1)
+62
+(1)
+57
+(1)
+203
+(6)
+580
+(8)
+41
+(2)
+1058
+(1)
+8
+(1)
+11
+(1)
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+169
+(3)
+12
+(1)
+18
+(1)
+129
+(1)
+228
+(4)
+231
+(3)
+4
+(1)
+72
+(1)
+13
+(1)
+9
+(1)
+55
+(3)
+15
+(2)
+56
+(1)
+37
+(1)
+84
+(1)
+308
+(2)
+8
+(1)
+13
+(1)
+8
+(3)
+45
+(3)
+qm=0.2
+qm=0.4
+qm=0.6
+5
+10
+15
+20
+5
+10
+15
+20
+5
+10
+15
+20
+−100
+−10
+−1
+0
+1
+10
+100
+1000
+Instance number
+EnumCC(3) - OneTreeCC()
+(b) Instances with d = 0.25 and qneg = 0.5.
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+7418
+(1)
+10934
+(1)
+465
+(1)
+94
+(1)
+9
+(1)
+11888
+(1)
+535
+(2)
+13
+(1)
+179
+(2)
+3918
+(1)
+4929
+(1)
+734
+(1)
+450
+(1)
+484
+(1)
+44506
+(1)
+50000
+(1)
+50000
+(1)
+50000
+(1)
+54
+(1)
+5008
+(1)
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+238
+(1)
+50000
+(1) 50000
+(1)
+32468
+(1)
+408
+(1)
+574
+(1)
+887
+(1)
+108
+(1)
+50000
+(1)
+9
+(1)
+5420
+(3)
+50000
+(1)
+40
+(1)
+26683
+(1)
+4
+(1)
+7564
+(1)
+1488
+(1)
+317
+(1)
+1042
+(1)
+4058
+(4)
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+50000
+(1)
+100
+(1)
+50000
+(1)
+467
+(1)
+50000
+(1) 50000
+(1)
+79
+(1)
+50000
+(1) 6999
+(2)
+3701
+(2)
+50000
+(1)
+50000
+(3)
+1916
+(1)
+50000
+(1)
+1596
+(1) 654
+(1)
+50000
+(1)
+50000
+(1)
+30772
+(1)
+1280
+(1)
+qm=0.2
+qm=0.4
+qm=0.6
+5
+10
+15
+20
+5
+10
+15
+20
+5
+10
+15
+20
+−100
+−10
+−101
+10
+100
+1000
+5000
+Instance number
+EnumCC(3) - OneTreeCC()
+(c) Instances with d = 0.25 and qneg = 0.7.
+Figure 9: Results for n = 36, d = 0.25 and qneg ∈ {0.3, 0.5, 0.7}. The log-scaled y-axis indicates the dif-
+ference of execution times between EnumCC(3) and OneTreeCC() (i.e., EnumCC(3) minus OneTreeCC()),
+in seconds. We show for each graph the maximal number of solutions found by EnumCC(3) within a 12h
+time limit, as well as the corresponding number of jumps, i.e. njump(EnumCC(3)), between parentheses.
+22 / 30
+
+Arınık et al. – Efficient Enumeration of the Optimal Solutions to the Correlation Clustering problem
+It is very important to analyze such cases in terms of the number of optimal solutions found by both
+methods.
+Moreover, the existence of such instances also indicates the maximum graph order, for
+which the enumeration methods can produce full enumeration of all alternate optimal solutions. In our
+experiments, a total of 21 such instances with n = 50 and qm > 0.3 are not solved within the time
+limit of 12h by both methods. These instances can be identified as the ones with y = 0 in Figure 10c,
+and they are summarized in Table 1. Each row corresponds to one of these methods, and each column
+represents an instance, identified by its id and its associated qm value.
+We see from Table 1 that
+EnumCC(3) handles these time-consuming instances better than OneTreeCC() does. Indeed, among
+the 21 instances not solved by any method, EnumCC(3) finds more solutions than OneTreeCC() for 20
+instances. Moreover, EnumCC(3) could solve 7 instances that OneTreeCC() could not, within the limit
+of 12h.
+To summarize the evaluation of EnumCC(3) with synthetic signed networks, there is not a single
+method which always gives the best results. On the one hand, OneTreeCC() handles very well the
+instances with a very large solution space.
+This extreme case is associated with a specific graph
+topology, based on low graph density and large qneg. On the other hand, EnumCC(3) performs better
+in the rest of the instances, which constitutes the overwhelming majority. This point is illustrated by
+Table 2, which summarizes our results by showing the proportion of cases where EnumCC(3) runs faster.
+Table 2: Summary regarding the proportion of the cases where EnumCC(3) is faster than OneTreeCC.
+We obtain these results by aggregating the instances by graph order n (excluding the instances, when the
+difference of execution times between EnumCC(3) and OneTreeCC() is in the margin of 5 seconds). N/A
+indicates that there is no available entry due to the exclusion of the instances, when the time difference is
+in the margin of 5 seconds.
+qm = 0.1
+qm = 0.2
+qm = 0.3
+qm = 0.4
+qm = 0.5
+qm = 0.6
+d = 0.25
+qneg = 0.3
+1.00
+0.92
+1.00
+0.87
+0.96
+0.92
+qneg = 0.5
+0.00
+0.52
+0.77
+0.61
+0.61
+0.86
+qneg = 0.7
+0.00
+0.08
+0.08
+0.05
+0.08
+0.03
+d = 0.50
+qneg = 0.3
+N/A
+1.00
+1.00
+1.00
+1.00
+1.00
+qneg = 0.5
+N/A
+1.00
+1.00
+0.92
+0.84
+0.75
+qneg = 0.7
+0.11
+0.80
+0.95
+0.90
+0.89
+0.87
+d = 1.00
+qneg ≈ 0.7
+1.00
+0.98
+0.98
+0.94
+0.95
+0.98
+8.2.3
+Evaluation Based on Real-World Instances
+Finally, we compare the results of EnumCC(3) against OneTreeCC() obtained on the real-world networks
+of Dataset3. We run both methods with a time limit of 12h and a limit of 50, 000 on the number of
+optimal solutions found. The results are summarized in Table 3. Each column corresponds to a real-
+world network, and they are sorted by increasing graph order n. We consider the first five networks as
+medium-sized and the last three as large-sized. The rows of Table 3 are organized in two parts. In the
+first part, we detail the characteristics of the considered networks, as well as the graph imbalance, for
+the sake of completeness. The second part corresponds to the evaluation results of OneTreeCC() and
+EnumCC(3). Therein, we indicate the execution time of the enumeration process, in seconds, and the
+number of optimal solutions found by these two methods within a time limit of 12h.
+We can summarize these results in three points. First, we notice that although the medium-sized net-
+works are larger than the synthetic graphs from Dataset2, in practice both OneTreeCC() and EnumCC(3)
+can handle real-world medium-sized networks in a very reasonable time. Second, we observe for medium-
+sized networks that EnumCC(3) is faster than OneTreeCC() for the first one, whereas OneTreeCC()
+dominates EnumCC(3) for the last four ones. EnumCC(3) takes longer because, in average, it spends
+86% of its execution time to prove that there is no alternate optimal solution in the end. Finally, as
+expected, both methods could not solve all three large-sized networks within the time limit of 12h. The
+particularity of these networks is that they are very sparse (d ≈ 1%) and their respective solution spaces
+contain at least 50, 000 optimal solutions. For all these networks, EnumCC(3) always finds more optimal
+solutions than OneTreeCC() does within the time limit, and it quickly explores a set of 50, 000 optimal
+solutions thanks to its RNS component. Since OneTreeCC() struggles to find more alternate solutions
+23 / 30
+
+Arınık et al. – Efficient Enumeration of the Optimal Solutions to the Correlation Clustering problem
+●
+●
+●
+●
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+(1)
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+(1)
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+(1)
+2
+(1)12
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+(1)6
+(1)
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+(1)
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+(2)37
+(1)
+2
+(2)
+9
+(1)
+4
+(2)
+●
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+●
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+(1)1
+(1)
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+(4)
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+(5)
+329
+(3)
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+(4)2
+(1)
+1
+(1)
+1
+(1)
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+(1)5
+(2)
+4
+(1)
+117
+(3)
+3
+(1)
+1
+(1)33
+(1)
+22
+(3)
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+78
+(13)
+9
+(2)
+30
+(1)
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+(2)
+1
+(1)14
+(3)
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+(2)
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+(3)
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+(1)
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+26
+(3)
+74
+(5)
+367
+(9)
+51
+(6)
+1
+(1)150
+(9)
+9
+(5)
+200
+(2)
+qm=0.2
+qm=0.4
+qm=0.6
+5
+10
+15
+20
+5
+10
+15
+20
+5
+10
+15
+20
+−5000
+−1000
+−100
+−10
+−101
+10
+100
+1000
+5000
+Instance number
+EnumCC(3) - OneTreeCC()
+(a) Instances with n = 40 and d = 1.00.
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
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+37
+(4)
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+(1)
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+(1)9
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+(1)
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+(1)
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+(1)1
+(1)16
+(1)
+1
+(1)
+27
+(2)
+14
+(2)
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+94
+(3)113
+(1)
+9
+(4)
+1
+(1)
+52
+(5)
+1
+(1)
+23
+(10)
+6
+(1)
+18
+(2)8
+(2)16
+(2)
+194
+(7)
+7
+(4)
+13
+(2)
+21
+(4)11
+(2)
+3
+(1)
+23
+(5)10
+(2)
+1
+(1)
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+8
+(2)
+10
+(6)
+17
+(1)
+6
+(3)
+42
+(6)
+6
+(3)
+4
+(1)
+8
+(1)
+16
+(2)69
+(10)
+4
+(2)
+5
+(2)5
+(1)
+1
+(1)
+3
+(2)
+13
+(6)
+3
+(1)
+4
+(1)
+24
+(4)
+7
+(1)
+qm=0.2
+qm=0.4
+qm=0.6
+5
+10
+15
+20
+5
+10
+15
+20
+5
+10
+15
+20
+−5000
+−1000
+−100
+−10
+−101
+10
+100
+1000
+Instance number
+EnumCC(3) - OneTreeCC()
+(b) Instances with n = 45 and d = 1.00.
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+3
+(2)
+36
+(1)
+2
+(1)36
+(2)
+14
+(1)6
+(1)
+2
+(1)1
+(1)
+6
+(1)8
+(1)
+1
+(1)
+1
+(1)
+2
+(1)
+24
+(1)
+2
+(1)
+5
+(1)6
+(1)6
+(2)
+1
+(1)
+8
+(1)
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+10
+(4)
+2
+(2)
+89
+(1)
+21
+(2)
+13
+(2)
+6
+(1)
+9
+(2)
+2
+(1)
+6
+(3)
+3
+(1)
+5
+(1)
+14
+(1)
+15
+(3)
+3
+(2)
+1
+(1)
+255
+(5)
+40
+(2)
+102
+(7)
+16
+(4)
+6
+(1)
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+1
+(1)
+45
+(2)
+4
+(1)8
+(1)
+5
+(2)
+10
+(4)
+1
+(1)
+1
+(1)
+35
+(3)
+32
+(3)
+1
+(1)
+6
+(4)
+60
+(8)
+9
+(2)
+8
+(3)
+3
+(1)
+7
+(3)
+24
+(2)
+4
+(1)
+65
+(3)
+qm=0.2
+qm=0.4
+qm=0.6
+5
+10
+15
+20
+5
+10
+15
+20
+5
+10
+15
+20
+−5000
+−1000
+−100
+−10
+−101
+10
+100
+1000
+Instance number
+EnumCC(3) - OneTreeCC()
+(c) Instances with n = 50 and d = 1.00.
+Figure 10: Results for n ∈ {40, 45, 50} and d = 1.00. The log-scaled y-axis indicates the difference of
+execution times between EnumCC(3) and OneTreeCC() (i.e., EnumCC(3) minus OneTreeCC()), in seconds.
+We show for each graph the maximal number of solutions found by EnumCC(3) within a time limit, as well
+as the corresponding number of jumps, i.e. njump(EnumCC(3)), between parentheses.
+24 / 30
+
+Arınık et al. – Efficient Enumeration of the Optimal Solutions to the Correlation Clustering problem
+Table 3: Evaluation results for sparse real-world signed networks from Dataset3. Each column corresponds
+to a real-world network and they are sorted by graph order n. The rows are organized in two parts. The
+first six rows detail the characteristics of the considered networks, as well as the graph imbalance. The last
+four rows indicate the execution time of the enumeration process in seconds and the number of optimal
+solutions found by OneTreeCC() and EnumCC(3) within a time limit of 12h.
+Medium-sized networks
+Large-sized networks
+CoW
+51-54
+CoW
+54-57
+CoW
+55-58
+CoW
+61-64
+Senate
+108th
+EGFR
+Yeast
+Macro-
+phage
+Characteristics
+graph order n = |V |
+61
+67
+72
+75
+99
+313
+575
+660
+graph size m = |E|
+413
+461
+508
+544
+2,413
+755
+910
+1,397
+density d
+0.23
+0.21
+0.20
+0.19
+0.50
+0.015
+0.005
+0.006
+prop. of neg. edges qneg
+0.08
+0.09
+0.08
+0.08
+0.41
+0.34
+0.09
+0.33
+# frustrated edges I(P)
+15
+27
+29
+34
+1157
+181
+35
+309
+prop. of frustrated edges I(P)/|E|
+0.04
+0.06
+0.06
+0.06
+0.48
+0.24
+0.04
+0.22
+Results
+# opt. solutions for OneTreeCC()
+46
+34
+41
+201
+22
+112
+50,000
+251
+# opt. solutions for EnumCC(3)
+46
+34
+41
+201
+22
+50,000
+50,000
+50,000
+exec. time for OneTreeCC()
+87.70s
+14.67s 15.62s 17.75s 858s
+43,200s 24,600s 43,200s
+exec. time for EnumCC(3)
+66.05s 67.38s
+134.87s 755.81s 2,424s
+3,751s 1,871s 2,280s
+within the time limit in two networks (i.e. EGFR and Macrophage), RNS appears to be useful in these
+cases.
+To conclude this part, all these results also support our conclusion from Section 8.2.2, in that
+there is not a single method which always gives the best results and both methods are complementary.
+Moreover, these results allow us to make our conclusions from Section 8.2.2 more precise, in two ways.
+First, it is more appropriate to use EnumCC(3), rather than OneTreeCC(), to explore a large number
+of optimal solutions within a time limit, for large signed networks. Nevertheless, it is more suitable to
+favor OneTreeCC() over EnumCC(3) for sparse signed networks with 65 < n < 100.
+9
+Conclusion
+For most clustering problems, due to their NP-hard nature, exact approaches do not scale well even
+when looking for a single optimal solution. In this work, we proposed an efficient enumeration method
+to identify all optimal solutions of the CC problem for a given signed graph. It combines an exhaustive
+enumeration strategy with neighborhoods of different sizes, designed for our problem. In our experiments
+based on synthetic and real-world signed networks, we first confirmed the findings of our previous work [6]
+about the existence of multiple optimal solutions for incomplete signed networks. We showed that such
+networks can have a very large number of optimal solutions for the CC problem, e.g., 50,000 and more
+for graphs containing only tens of vertices. Further investigation indicates that this extreme case of a
+very large number of optimal solutions is associated with two specific graph topologies, corresponding to
+1) low graph density and large proportion of negative edges for small- and medium-sized networks (i.e.
+n ≤ 100), and 2) very low graph density for large-sized networks (i.e. n > 100). Otherwise, multiple
+solutions can still exist, mostly for the graphs with considerably more imbalance.
+Finally, we also
+showed that our method EnumCC(3) performs better than CPLEX’s OneTreeCC() in the overwhelming
+majority of the considered networks. Nevertheless, there is not a single method which always gives the
+best results. Indeed, OneTreeCC() handles very well sparse medium-sized signed networks in general.
+We conclude that it is more appropriate to use OneTreeCC() in this case, whereas EnumCC(3) is more
+suitable for all the remaining cases.
+We believe that this work opens new directions for future research. First, the most straightforward
+perspective is to consider weighted signed graphs.
+This would require dealing with the generation
+of the edge weights in our random graph model, and to extend the pruning strategies proposed in
+this work to weighted signed graphs. Second, as we see in our experiments, the process of complete
+enumeration can be very costly, particularly because of the process of finding an undiscovered solution.
+To accelerate this process, it could be very beneficial to use a heuristic. Existing heuristics from the
+25 / 30
+
+Arınık et al. – Efficient Enumeration of the Optimal Solutions to the Correlation Clustering problem
+literature are designed to find a single high-quality solution though, so a better approach would be to
+redesign them by considering some tabu search-based operations allowing to escape from the discovered
+optimal solutions. Third, one could study how robust the solution spaces are, by slightly introducing
+some perturbations into the considered networks.
+This would also allow identifying critical vertices
+when the underlying space of optimal solutions is changed, akin to the concept of vitality [33]. This
+could be done through either repeating the process from scratch for each perturbed signed graph, or
+based on the definition of stability range [41]. Fourth, exploring the space of optimal solutions for other
+clustering problems with unsigned networks would be another interesting research line. Indeed, the work
+of Good et al. [26] showed the need of identifying multiple quasi-optimal solutions for the Modularity
+Maximization problem, which consists in detecting a community structure in an unsigned graph. From
+this perspective, an efficient enumeration method similar to ours could be implemented to see whether
+their empirical findings apply to optimal solutions too.
+Acknowledgments.
+This research benefited from the support of Agorantic research federation (FR
+3621), as well as the FMJH (Jacques Hadamard Mathematics Foundation) through PGMO (Gaspard
+Monge Program for Optimisation and operational research), and from the support to this program from
+EDF, Thales, Orange and Criteo.
+A
+2-partition and 2-chorded cycle inequalities
+For the sake of completeness, we detail below the 2-partition and 2-chorded cycle inequalities:
+• Let S, T ⊆ V be two nonempty disjoint subsets of V . Then, the 2-partition inequality, illustrated
+in Figure 11a, is defined as
+�
+u∈S
+�
+v∈T
+xuv −
+�
+(u,v)∈S
+u̸=v
+xuv −
+�
+(u,v)∈T
+u̸=v
+xuv ≤ min{|S|, |T|}.
+(20)
+• Let C ⊆ E be a cycle of length at least 5, and C = {vivi+2|i = 1, .., |C|−2}∪{v1v|C|−1, v2v|C|}
+be a 2-chorded cycle of C. Then, the 2-chorded cycle inequality, illustrated in Figure 11b, is
+defined as
+�
+(u,v)∈C
+xuv −
+�
+(u,v)∈C
+xuv ≤ ⌊|C|
+2 ⌋.
+(21)
+v1
+v2
+v3
+v4
+v5
+v1
+S
+v2
+v3
+v4
+T
+v5
+(a) 2-partition inequality.
+v1
+v2
+v3
+v4
+v5
+v6
+v7
+(b) 2-chorded cycle inequality.
+Figure 11: Illustrations of the 2-partition (left) and 2-chorded cycle (right) inequalities.
+B
+Edit Distance Between Two Membership Vectors
+Before calculating the edit distance between two membership vectors, we need to select one of them as
+the reference vector, in order to adapt the module assignments of the other vector based accordingly.
+Hence, the edit distance is calculated between the reference vector and this newly changed one, that
+we call relative vector.
+The task of adapting the relative vector w.r.t the reference one can be expressed as an assignment
+problem, also known as maximum weighted bipartite matching problem, as already done in the litera-
+ture [36]. Let πs and πt be two membership vectors of length n, associated with the partitions P s and
+26 / 30
+
+Arınık et al. – Efficient Enumeration of the Optimal Solutions to the Correlation Clustering problem
+a)
+b)
+c)
+d)
+f)
+g)
+h)
+i)
+j)
+k)
+l)
+m)
+n)
+o)
+p)
+r)
+e)
+z
+u
+u
+u
+u
+u
+u
+u
+u
+u
+u
+u
+u
+u
+u
+u
+u
+u
+v
+v
+v
+v
+v
+v
+v
+v
+v
+v
+v
+v
+v
+v
+v
+v
+z
+z
+z
+z
+z
+z
+z
+z
+z
+z
+z
+z
+z
+z
+z
+v
+z
+Figure 12: All atomic 3-edit operations.
+P t, which contain ℓs and ℓt modules, respectively. Also, since the edit distance is symmetric, without
+loss of generality, let ℓs ≤ ℓt. Moreover, let CM be the ℓs × ℓt confusion matrix of πs and πt. The
+term CMij, with 1 ≤ i ≤ ℓs and 1 ≤ j ≤ ℓt, represents the number of vertices in the intersection of
+modules M s
+i and M t
+j, i.e. |M s
+i ∩ M t
+j|. Then, we look for a bijection f : {1, 2, ..., ℓs} → {1, 2, ..., ℓt}
+that maximizes the number of vertices common to pairs of modules from both membership vectors, i.e.
+max
+ℓs
+�
+i=1
+CMi,f(i).
+(22)
+Since this problem can be modelled as an assignment or a maximum weighted bipartite matching
+problem, it can be solved in various ways. One of them is through the well-known Hungarian algorithm,
+whose complexity is O(n3) [34]. Nevertheless, the best polynomial time algorithm is currently based on
+the network simplex method, and it runs in O(|V ||E| + |V |2log(|V |)) time using a Fibonacci heap data
+structure [25]. One final remark is about the case of ℓs < ℓt, in which there are |ℓt − ℓs| unassigned
+module labels in πt. In this case, one can arbitrarily renumber these labels, starting from ℓs + 1.
+Finally, the edit distance between two membership vectors is calculated by simply counting the
+number of cases where the module labels of the vertices in the reference and relative vectors are
+different.
+C
+Proofs
+C.1
+All Proofs Related to the MVMO Property for 3-edit Operations on Com-
+plete Unweighted Signed Graphs
+We complete the proof of Lemma 12 by verifying below the conditions of all atomic 3-edit oper-
+ations for unweighted complete signed networks (illustrated in Figure 12), where
+s#»t
+V
+= {u, v, z}
+and (u, v), (u, z), (v, z) ∈ �E. Recall that �E = {(u, v) | (u, v) ∈ E and u, v ∈
+s#»t
+V
+and (πs(u) =
+πs(v) ∨ πt(u) = πs(v) ∨ πs(u) = πt(v) ∨ πt(u) = πt(v))}.
+a) We have (γleft
+u
+= auv + auz) > (γright
+u
+= −auv − auz), (γleft
+v
+= auv + avz) > (γright
+v
+=
+−auv − avz) and (γleft
+z
+= auz + avz) > (γright
+z
+= −auz − avz). We see that auv, auz and avz
+cannot be negative.
+b) We have (γleft
+u
+= auv + auz) > (γright
+u
+= 0), (γleft
+v
+= auv + avz) > (γright
+v
+= −avz) and
+(γleft
+z
+= auz + avz) > (γright
+z
+= −avz). We see that auv, auz and avz cannot be negative.
+c) We have (γleft
+u
+= auv + auz) > (γright
+u
+= 0), (γleft
+v
+= auv + avz) > (γright
+v
+= 0) and (γleft
+z
+=
+auz + avz) > (γright
+z
+= 0). We see that auv, auz and avz cannot be negative.
+27 / 30
+
+Arınık et al. – Efficient Enumeration of the Optimal Solutions to the Correlation Clustering problem
+d) We have (γleft
+u
+= auv) > (γright
+u
+= −auv − auz), (γleft
+v
+= auv) > (γright
+v
+= −auv − avz) and
+(γleft
+z
+= 0) > (γright
+z
+= −auz − avz). We see that auv, auz and avz must be positive.
+e) We have (γleft
+u
+= auv − auz) > (γright
+u
+= −auv + auz), (γleft
+v
+= auv − avz) > (γright
+v
+=
+−auv + avz) and (γleft
+z
+= −auz − avz) > (γright
+z
+= auz + avz). We see that auv (resp. auz and
+avz) cannot be negative (resp. positive).
+f) We have (γleft
+u
+= auv − auz) > (γright
+u
+= −auv), (γleft
+v
+= auv − avz) > (γright
+v
+= −auv) and
+(γleft
+z
+= 0) > (γright
+z
+= auz + avz). We see that auv, auz and avz cannot be negative.
+g) We have (γleft
+u
+= auv) > (γright
+u
+= auz − auv), (γleft
+v
+= auv) > (γright
+v
+= avz − auv) and
+(γleft
+z
+= −auz − avz) > (γright
+z
+= 0). We see that auv, auz and avz cannot be negative.
+h) We have (γleft
+u
+= auv) > (γright
+u
+= −auv), (γleft
+v
+= auv) > (γright
+v
+= −auv) and (γleft
+z
+=
+−auz − avz) > (γright
+z
+= 0). We see that auv, auz and avz cannot be negative.
+i) We have (γleft
+u
+= auv) > (γright
+u
+= auz), (γleft
+v
+= auv − avz) > (γright
+v
+= avz) and (γleft
+z
+=
+−auz − avz) > (γright
+z
+= +avz). We see that auv (resp. auz and avz) cannot be negative (resp.
+positive).
+j) We have (γleft
+u
+= auv) > (γright
+u
+= −auz), (γleft
+v
+= auv) > (γright
+v
+= −avz) and (γleft
+z
+= 0) >
+(γright
+z
+= −auz + avz). We see that auv and avz (resp. auz) must be positive (resp. negative).
+k) We have (γleft
+u
+= auv) > (γright
+u
+= 0), (γleft
+v
+= auv − avz) > (γright
+v
+= 0) and (γleft
+z
+= 0) >
+(γright
+z
+= avz). We see that auv and avz (resp. auz) must be positive (resp. negative).
+l) We have (γleft
+u
+= −auv) > (γright
+u
+= auz), (γleft
+v
+= −avz) > (γright
+v
+= auv) and (γleft
+z
+=
+−auz) > (γright
+z
+= avz). We see that auv and avz (resp. auz) must be positive (resp. negative).
+m) We have (γleft
+u
+= −auv) > (γright
+u
+= 0), (γleft
+v
+= −avz) > (γright
+v
+= auv) and (γleft
+z
+= 0) >
+(γright
+z
+= avz). We see that auv and avz (resp. auz) must be positive (resp. negative).
+n) We have (γleft
+u
+= −auv) > (γright
+u
+= auv − auz), (γleft
+v
+= −auv) > (γright
+v
+= auv + avz) and
+(γleft
+z
+= −avz) > (γright
+z
+= −auz). We see that auv (resp. auz and avz) cannot be negative
+(resp. positive).
+o) We have (γleft
+u
+= −auv) > (γright
+u
+= 0), (γleft
+v
+= 0) > (γright
+v
+= auv − avz) and (γleft
+z
+= 0) >
+(γright
+z
+= −avz). We see that auv (resp. auz and avz) cannot be negative (resp. positive).
+p) We have (γleft
+u
+= −auv) > (γright
+u
+= −auz), (γleft
+v
+= 0) > (γright
+v
+= auv + avz) and (γleft
+z
+=
+−avz) > (γright
+z
+= −auz). We see that auv (resp. auz and avz) cannot be negative (resp.
+positive).
+r) We have (γleft
+u
+= 0) > (γright
+u
+= −auv − auz), (γleft
+v
+= 0) > (γright
+v
+= −auv − avz) and
+(γleft
+z
+= 0) > (γright
+z
+= −auz − avz). We see that auv, auz and avz must be positive.
+References
+[1]
+R. K. Ahuja, Ö. Ergun, J. B. Orlin, and A. P. Punnen. “A survey of very large-scale neighborhood
+search techniques”. In: Discrete Applied Mathematics 123.1-3 (2002), pp. 75–102. doi: 10.1016/s0166-
+218x(01)00338-9.
+[2]
+Z. Ales, A. Knippel, and A. Pauchet. “Polyhedral Combinatorics of the K-partitioning Problem with
+Representative Variables”. In: Discrete Applied Mathematics 211 (2016), pp. 1–14. doi: 10.1016/j.
+dam.2016.04.002.
+[3]
+S. Aref and M. C. Wilson. “Balance and frustration in signed networks”. In: Journal of Complex Networks
+7.2 (2018), pp. 163–189. doi: 10.1093/comnet/cny015.
+[4]
+J. L. Arthur, M. Hachey, Sahr K., M. Huso, and A. R. Kiester. “Finding all optimal solutions to the
+reserve site selection problem”. In: Environmental and Ecological Statistics 4.2 (1997), pp. 153–165. doi:
+10.1023/a:1018570311399.
+[5]
+N. Arınık, R. Figueiredo, and V. Labatut. “Multiple partitioning of multiplex signed networks: Application
+to European parliament votes”. In: Social Networks 60 (2020), pp. 83 –102. doi: https://doi.org/10.
+1016/j.socnet.2019.02.001.
+[6]
+N. Arınık, R. Figueiredo, and V. Labatut. “Multiplicity and Diversity: Analyzing the Optimal Solution Space
+of the Correlation Clustering Problem on Complete Signed Graphs”. In: Journal of Complex Networks
+(2020). doi: 10.1093/comnet/cnaa025.
+[7]
+N. Arınık, R. Figueiredo, and V. Labatut. “Signed Graph Analysis for the Interpretation of Voting Behav-
+ior”. In: International Conference on Knowledge Technologies and Data-driven Business - International
+Workshop on Social Network Analysis and Digital Humanities. 2017.
+[8]
+N. Bansal, A. Blum, and S. Chawla. “Correlation Clustering”. In: 43rd Annual IEEE Symposium on
+Foundations of Computer Science. 2002, pp. 238–247. doi: 10.1109/SFCS.2002.1181947.
+[9]
+C. Blum and A. Roli. “Metaheuristics in combinatorial optimization”. In: ACM Computing Surveys 35.3
+(2003), pp. 268–308. doi: 10.1145/937503.937505.
+28 / 30
+
+Arınık et al. – Efficient Enumeration of the Optimal Solutions to the Correlation Clustering problem
+[10]
+Michael Brusco and Douglas Steinley. “K-balance partitioning: An exact method with applications to
+generalized structural balance and other psychological contexts”. In: Psychological Methods 15.2 (2010),
+pp. 145–157. doi: 10.1037/a0017738.
+[11]
+J. D. Camm, S. Polasky, A. Solow, and B. Csuti. “A note on optimal algorithms for reserve site selection”.
+In: Biological Conservation 78.3 (1996), pp. 353–355. doi: 10.1016/0006-3207(95)00132-8.
+[12]
+D. Cartwright and F. Harary. “Structural balance: A generalization of Heider’s theory”. In: Psychological
+Review 63 (1956), pp. 277–293. doi: 10.1037/h0046049.
+[13]
+Moses Charikar, Neha Gupta, and Roy Schwartz. “Local Guarantees in Graph Cuts and Clustering”. In:
+Integer Programming and Combinatorial Optimization. Springer International Publishing, 2017, pp. 136–
+147. doi: 10.1007/978-3-319-59250-3\_12.
+[14]
+P. Damaschke. “Fixed-Parameter Enumerability of Cluster Editing and Related Problems”. In: Theory of
+Computing Systems 46.2 (2010), pp. 261–283. doi: 10.1007/s00224-008-9130-1.
+[15]
+E. Danna, M. Fenelon, Z. Gu, and R. Wunderling. “Generating Multiple Solutions for Mixed Integer
+Programming Problems”. In: International Conference on Integer Programming and Combinatorial Opti-
+mization. Springer, 2007, pp. 280–294. doi: 10.1007/978-3-540-72792-7\_22.
+[16]
+B. DasGupta, G. A. Enciso, E. Sontag, and Y. Zhang. “Algorithmic and complexity results for decompo-
+sitions of biological networks into monotone subsystems”. In: Biosystems 9.1 (2007), pp. 161–178. doi:
+10.1016/j.biosystems.2006.08.001.
+[17]
+J. A. Davis. “Clustering and structural balance in graphs”. In: Human Relations 20.2 (1967), pp. 181–187.
+doi: 10.1177/001872676702000207.
+[18]
+Erik D. Demaine, Dotan Emanuel, Amos Fiat, and Nicole Immorlica. “Correlation clustering in general
+weighted graphs”. In: Theoretical Computer Science 361.2-3 (2006), pp. 172–187. doi: 10.1016/j.tcs.
+2006.05.008.
+[19]
+P. Doreian, P. Lloyd, and A. Mrvar. “Partitioning large signed two-mode networks: Problems and
+prospects”. In: Social Networks 35.2 (2013), pp. 178–203. doi: 10.1016/j.socnet.2012.01.002.
+[20]
+P. Doreian and A. Mrvar. “A partitioning approach to structural balance”. In: Social Networks 18.2 (1996),
+pp. 149–168. doi: 10.1016/0378-8733(95)00259-6.
+[21]
+P. Doreian and A. Mrvar. “Structural balance and signed international relations”. In: Journal of Social
+Structure 16 (2015), p. 1. doi: 10.21307/joss-2019-012.
+[22]
+J. Esteban, L. Mayoral, and D. Ray. “Ethnicity and Conflict: An Empirical Study”. In: American Economic
+Review 102.4 (2012), pp. 1310–1342. doi: 10.1257/aer.102.4.1310.
+[23]
+R. Figueiredo and G. Moura. “Mixed Integer Programming Formulations for Clustering Problems Related
+to Structural Balance”. In: Social Networks 35.4 (2013), pp. 639–651. doi: 10.1016/j.socnet.2013.
+09.002.
+[24]
+Matteo Fischetti, Fred Glover, and Andrea Lodi. “The feasibility pump”. In: Mathematical Programming
+104.1 (2005), pp. 91–104. doi: 10.1007/s10107-004-0570-3.
+[25]
+Michael L. Fredman and Robert Endre Tarjan. “Fibonacci heaps and their uses in improved network
+optimization algorithms”. In: Journal of the ACM (JACM) 34.3 (1987), pp. 596–615. doi: 10.1145/
+28869.28874.
+[26]
+B. H. Good, Y.-A. de Montjoye, and A. Clauset. “Performance of modularity maximization in practical
+contexts”. In: Physical Review E 81.4 (2010). doi: 10.1103/physreve.81.046106.
+[27]
+M. Grötschel and Y. Wakabayashi. “A cutting plane algorithm for a clustering problem”. In: Mathematical
+Programming 45.1-3 (1989), pp. 59–96. doi: 10.1007/bf01589097.
+[28]
+M. Grötschel and Y. Wakabayashi. “Facets of the clique partitioning polytope”. In: Mathematical Pro-
+gramming 47.1-3 (1990), pp. 367–387. doi: 10.1007/bf01580870.
+[29]
+F. Gürsoy and B. Badur. “Extracting the signed backbone of intrinsically dense weighted networks”. In:
+arXiv e-prints, arXiv:2012.05216 (2020), arXiv:2012.05216. arXiv: 2012.05216.
+[30]
+IBM. IBM ILOG CPLEX 12.8 User Manual IBM Corporation. 2018.
+[31]
+D. Jurafsky and J. H. Martin. Speech and Language Processing: An Introduction to Natural Language
+Processing,Computational Linguistics, and Speech Recognition. 2nd edition. Prentice-Hall, Inc., 2000.
+[32]
+M. Keuper, J. Lukasik, M. Singh, and J. Yarkony. “A Benders Decomposition Approach to Correlation
+Clustering”. In: The International Conference for High Performance Computing, Networking, Storage, and
+Analysis. 2020.
+[33]
+D. Koschützki, K. A. Lehmann, L. Peeters, S. Richter, D. Tenfelde-Podehl, and O. Zlotowski. “Central-
+ity Indices”. In: Network Analysis: Methodological Foundations. Vol. 3418. Lecture Notes in Computer
+Science. 2005, pp. 16–61. doi: 10.1007/978-3-540-31955-9_3.
+29 / 30
+
+Arınık et al. – Efficient Enumeration of the Optimal Solutions to the Correlation Clustering problem
+[34]
+H. W. Kuhn. “The Hungarian method for the assignment problem”. In: Naval Research Logistics Quarterly
+2.1-2 (1955), pp. 83–97. doi: 10.1002/nav.3800020109.
+[35]
+J. Kunegis, A. Lommatzsch, and C. Bauckhage. “The slashdot zoo”. In: Proceedings of the 18th interna-
+tional conference on World wide web. ACM Press, 2009, pp. 741–750. doi: 10.1145/1526709.1526809.
+[36]
+Xin Liu, Hui-Min Cheng, and Zhong-Yuan Zhang. “Evaluation of Community Detection Methods”. In:
+IEEE Transactions on Knowledge and Data Engineering (2019), pp. 1–1. doi: 10.1109/tkde.2019.
+2911943.
+[37]
+F. Ma and J.-K. Hao. “A multiple search operator heuristic for the max-k-cut problem”. In: Annals of
+Operations Research 248.1 (2016), pp. 365–403. doi: 10.1007/s10479-016-2234-0.
+[38]
+P. Massa and P. Avesani. “Controversial users demand local trust metrics: an experimental study on
+Epinions.com community”. In: Proceedings of the 20th national conference on Artificial intelligence.
+Vol. 1. 2005, pp. 121–126.
+[39]
+A. Mehrotra and M. A. Trick. “Cliques and clustering: A combinatorial approach”. In: Operations Research
+Letters 22.1 (1998), pp. 1–12. doi: 10.1016/s0167-6377(98)00006-6.
+[40]
+George Nemhauser and Laurence Wolsey. Integer and Combinatorial Optimization. Wiley, 1999.
+[41]
+S. Nowozin and S. Jegelka. “Solution stability in linear programming relaxations”. In: Proceedings of the
+26th Annual International Conference on Machine Learning - ICML. ACM Press, 2009. doi: 10.1145/
+1553374.1553473.
+[42]
+M. Oosten, J. H. G. C. Rutten, and F. C. R. Spieksma. “The clique partitioning problem: Facets and
+patching facets”. In: Networks 38.4 (2001), pp. 209–226. doi: 10.1002/net.10004.
+[43]
+J. Pevehouse, T. Nordstrom, and K. Warnke. “The Correlates of War 2 International Governmental
+Organizations Data Version 2.0”. In: Conflict Management and Peace Science 21.2 (2004), pp. 101–119.
+doi: 10.1080/07388940490463933.
+[44]
+E. Queiroga, A. Subramanian, R. Figueiredo, and Y. Frota. “Integer programming formulations and ef-
+ficient local search for relaxed correlation clustering”. In: Journal of Global Optimization (2021). doi:
+10.1007/s10898-020-00989-7.
+[45]
+S. Tan and J. Lü. “An evolutionary game approach for determination of the structural conflicts in signed
+networks”. In: Scientific Reports 6.1 (2016), p. 22022. doi: 10.1038/srep22022.
+[46]
+T. Zasĺavsky. “Balanced decompositions of a signed graph”. In: Journal of Combinatorial Theory, Series
+B 43.1 (1987), pp. 1–13. doi: 10.1016/0095-8956(87)90026-8.
+30 / 30
+
diff --git a/v9E5T4oBgHgl3EQfMQ4D/content/tmp_files/load_file.txt b/v9E5T4oBgHgl3EQfMQ4D/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf,len=2178
+page_content='Efficient Enumeration of the Optimal Solutions to the Correlation  Clustering problem  Nejat Arınık, Rosa Figueiredo & Vincent Labatut  January 16, 2023  Abstract  According to the structural balance theory, a signed graph is considered structurally balanced when  it can be partitioned into a number of modules such that positive and negative edges are respectively  located inside and between the modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In practice, real-world networks are rarely structurally balanced,  though.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In this case, one may want to measure the magnitude of their imbalance, and to identify the set  of edges causing this imbalance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' The correlation clustering (CC) problem precisely consists in looking  for the signed graph partition having the least imbalance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Recently, it has been shown that the space of  the optimal solutions of the CC problem can be constituted of numerous and diverse optimal solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Yet, this space is difficult to explore, as the CC problem is NP-hard, and exact approaches do not scale  well even when looking for a single optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' To alleviate this issue, in this work we propose  an efficient enumeration method allowing to retrieve the complete space of optimal solutions of the  CC problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' It combines an exhaustive enumeration strategy with neighborhoods of varying sizes, to  achieve computational effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Results obtained for middle-sized networks confirm the usefulness  of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Keywords: Correlation Clustering, Enumeration Strategy, Local Search, Optimal Solution Space,  Signed Graph, Structural Balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Cite as: N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Arınık, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Figueiredo & V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Labatut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Efficient Enumeration of the Optimal Solutions to  the Correlation Clustering problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Journal of Global Optimization, 2023 (forthcoming)  1  Introduction  Many real-world social systems involve antagonistic relations, a feature that can be directly modeled  through signed networks, which contain both positive and negative edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Alternatively, in some cases,  signed edges can be extracted from originally unsigned relations [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' The exact semantic of these edges  depends on the considered system: friendship/foe [35] and trust/distrust [38] in social networks, inhi-  bition/activation in biological networks [16, 45], agreement/disagreement in political vote networks [5,  7, 19] and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Determining the structural balance of a signed network is a key aspect in the investigation of the  structure of polarized relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' According to the structural balance theory, a signed graph is considered  to be balanced if it can be partitioned into two [12] or more [17] modules (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' clusters), such that all  positive edges are located inside these modules, whereas negative ones lie between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  However, it is very rare for a real-world network to be perfectly balanced, in which case one wants  to assess the magnitude and the causes of the imbalance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  For a given partition, this imbalance is  traditionally measured by counting the number of frustrated edges [12, 46], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' positive edges located  in-between modules and negative ones located inside them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Computing the graph imbalance amounts  to identifying the partition corresponding to the lowest imbalance measure over the space of all possible  partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' The optimal solution found exhibits the source of the imbalance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=', the frustrated edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  This minimization problem is known as the correlation clustering (CC) problem, proven to be NP-  hard [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  When solving an instance of the CC problem, the standard approach in the literature is to find a  single solution and focus the rest of the analysis on it, as if it were the only optimal solution [23, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Yet, as shown empirically, it is possible that several, and even many, alternative optimal solutions exist  for the considered instance [6, 10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' All these alternate solutions are equally relevant in terms of the  objective function of the CC problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Yet, they can be very different, in terms of how they partition  the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' One can then wonder how many solutions shape the space of optimal solutions, and if many,  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='05479v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='OC]  13 Jan 2023  Arınık et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' – Efficient Enumeration of the Optimal Solutions to the Correlation Clustering problem  how different/diverse they are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' From an application point of view, these are important questions to  answer in order to identify how many of these solutions are more suitable to the situation at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  In order to answer these questions, we recently conducted an empirical study in our previous work [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Therein, we first enumerated the optimal solutions of a collection of instances of the CC problem through  Integer Linear Programming (ILP) modeling [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We applied the best state-of-the-art optimal solution  enumeration method, as proposed by Danna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' [15] and incorporated in the commercial solver  CPLEX [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Then, we studied the obtained spaces of optimal solutions through complex network  analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In that work, we found out that slightly imbalanced networks tend to have fewer and structurally  more similar optimal solutions forming a single solution class, whereas a higher imbalance leads to many  diverse solutions possibly forming multiple classes of solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Consequently, we concluded that it is  of great importance that the decision-maker generates the full set, or as many optimal solutions as  possible, for a given instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  In such an analysis, guaranteeing the completeness of the space of optimal solutions requires em-  ploying exact approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' However, it is well known that, due to their NP-hard nature, generic exact  approaches able to solve most clustering problems (including the CC problem) are very costly in terms  of execution time and do not scale well, even when looking for a single optimal solution [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In that  respect, the enumeration method by Danna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' is the best state-of-the-art enumeration method able  to handle the CC problem so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' However, as it is a generic method designed to handle any combinato-  rial problem formulated in ILP, it cannot take full advantage of the problem knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Consequently,  the maximum number of vertices (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' graph order) that we could handle with their method in our  previous work was only 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Moreover, the execution time of the enumeration process for such small  graphs was very long (typically more than one day).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' This highlights the need for an efficient optimal  space enumeration method for the CC problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  In this work, we aim to obtain this efficiency by putting the problem knowledge into the enumeration  process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Based on the high similarity observed empirically between optimal solutions belonging to the  same class of solutions, it integrates into an exact approach a local search mechanism, which relies on  the use of neighborhoods of different sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In order to accelerate the proposed enumeration method,  we develop pruning strategies based on optimal conditions satisfied by partial solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Our main  contribution is the combination of local search and pruning strategies, which gives rise to an efficient  enumeration method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We present extensive computational experiments established for the proposed  method, relying on sparse and dense signed networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  The rest of the article is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In Section 2, we review the literature related to  the exact enumeration of all optimal solutions for the CC problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In Section 3, we give the formal  definition of the CC problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Next, we introduce our enumeration method in Section 4, and detail  our complete neighborhood search and pruning strategies in Section 5 and 6, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We put  the proposed method into practice on a selection of partially and fully connected signed networks in  Section 7 and discuss our results in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Finally, we review our main findings in Section 9, and  identify some perspectives for our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  2  Related Work  There are several methods proposed in the literature to solve the CC problem exactly [18, 23, 32],  however very few were designed to enumerate the whole space of its optimal solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In fact, very  few methods were proposed to enumerate all the optimal solutions of any combinatorial problem (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=',  [4] for maximal covering problem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In this section, we review the existing optimal solution enumeration  methods for the CC problem and for related problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  The literature provides two main approaches to enumerate all optimal solutions of a combinatorial  problem: Branch-and-Bound and Fixed-Parameter Enumeration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In the case of the CC problem, most  works focus on the former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Branch-and-Bound Approach  The enumeration of all optimal solutions of the CC problem (like  for any combinatorial problem) through a branch-and-bound method can be performed through two  algorithmic methods: Integer Linear Programming (ILP) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' ad hoc branch-and-bound programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Their main difference is that the former can tackle any problem translated in the language of mathe-  matical programming through any industrial optimization solver, whereas in the latter the construction  of the branch-and-bound tree relies on problem-specific idiosyncrasies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Based on how the branch-and-bound tree is used, the ILP approach can be implemented in two  different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' The first one relies on a simple sequential process: a slightly modified version of the  original problem is solved as many times as there are solutions in the optimal solution space (like in [4]  for the maximal covering problem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' At each iteration, already-found optimal solutions are sequentially  2 / 30  Arınık et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' – Efficient Enumeration of the Optimal Solutions to the Correlation Clustering problem  added as constraints into the mathematical model, in order to exclude them during future iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  However, the drawback of this approach is that a branch-and-bound tree needs to be built from scratch  for each new optimal solution found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' The second type of ILP implementation is called OneTree, and  was proposed by Danna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Instead of the previous simple sequential approach, it relies on a  more efficient two-step method that allows reducing the computational effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' As its name suggests, it  constructs a single branch-and-bound tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' It first builds and explores the search tree in order to find  efficiently the first optimal solution, and then enumerate all the alternative optimal solutions based on  the same tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' This method is available in the industrial optimization solver CPLEX [30], and we explain  in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='3 how we use it for the CC problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In our previous work [6], we used this approach for  generating the optimal solution space of complete random graphs up to 36 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  An ad hoc B&B programming method constructs the branch-and-bound tree differently from what  we described before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' For a given clustering problem, these methods systematically construct partial  solutions by assigning the vertices to one of the existing modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' This produces a search tree, whose  branches correspond to assignments of vertices to modules, and whose nodes correspond to partial  assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Similar to Danna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' [15], Brusco and Steinly [10] propose a two-step method for  generating all optimal CC solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' They first identify the optimal objective function value by finding  an optimal solution, then use the optimal value as input when building another branch-and-bound tree  from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' As shown by Figueiredo & Moura [23], this method does not deal well with finding a  single optimal solution for graphs with more than 20 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Fixed-Parameter Enumeration Approach  This is also an ad hoc method which requires to  transform a part of the considered problem into a new input parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  This in turn guarantees  to bound the overall running time as a function of an input parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Assuming that this input  parameter is small, the parameterized enumeration is efficient in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Damaschke [14] proposes  an FPT (Fixed-Parameter Tractable) algorithm to enumerate all optimal solutions in a given graph for  the Cluster Editing problem, which is equivalent to the CC problem when the input graph is complete  and unweighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Concretely, this FPT algorithm makes the number of frustrated edges in structural  balance parameterized to solve the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' However, as shown in our previous work [6], a drawback  of this approach is that the amount of imbalance, the input parameter, can be very large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Furthermore,  Damaschke does not provide any computational results in his theoretical work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  The computational results presented in the works mentioned in this section identify the ILP branch-  and-bound as the more efficient method for exactly solving the CC problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' This is explained by a  tightened initial linear relaxation of the ILP formulation and the quality of the cuts used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In this work,  we apply an ILP branch-and-bound method for the complete enumeration of the optimal CC solutions,  and propose a compromise strategy between the sequential approach and the OneTree method from  Danna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  3  Correlation Clustering Problem  Let us now introduce the notations necessary for defining the correlation clustering problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Let G =  (V, E) be an undirected graph, where V and E are the sets of vertices and edges, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We note  n = |V | and m = |E| the numbers of vertices (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' graph order) and edges, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We assume  that the graph contains no loops, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' edges connecting a vertex to itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Moreover, G[S] denotes the  subgraph induced by vertex subset S ⊂ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Consider a function s : E → {+, −} that assigns a sign to each edge in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' An undirected graph G  together with a function s is called a signed graph, denoted by G = (V, E, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' An edge (u, v) is called  negative if s(u, v) = − and positive if s(u, v) = +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We note E− and E+ the sets of negative and  positive edges in the signed graph, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Let also define the positive graph G+ of a given signed  graph G as the subgraph (V, E+, {+}) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' same vertices, but only the positive edges).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' The entries  auv of the signed adjacency matrix A associated with a signed graph G are defined in Equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  auv =  �  �  �  �  �  1,  if (u, v) ∈ E+,  −1,  if (u, v) ∈ E−,  0,  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  (1)  Let P = {M1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=', Mℓ} (1 ≤ ℓ ≤ n) be an ℓ-partition of V , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' a division of V into ℓ non-overlapping  and non-empty subsets Mi (1 ≤ i ≤ ℓ) called modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' The partition P is called a solution of the CC  problem for the given graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Given a solution P, an edge (u, v) is called internal if it is located inside a  module, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=', u and v belong to the same module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Likewise, an edge (u, v) is called external if it is located  3 / 30  Arınık et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' – Efficient Enumeration of the Optimal Solutions to the Correlation Clustering problem  between any two modules, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=', u and v belong to two different modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Given σ ∈ {+, −}, we define the  set of positive/negative (depending on σ) edges connecting two modules Mi, Mj ∈ P as Eσ(Mi, Mj) =  {(u, v) | (u, v) ∈ Eσ, u ∈ Mi and v ∈ Mj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We also define E(Mi, Mj) = E−(Mi, Mj) ∪ E+(Mi, Mj)  as the set of all edges connecting these modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Similarly, we note Ωσ(Mi, Mj) =  �  (u,v)∈Eσ(Mi,Mj)  auv  the signed number of positive/negative edges connecting these modules: note that this value can be  negative if σ = −.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We also define Ω(Mi, Mj) = Ω−(Mi, Mj) + Ω+(Mi, Mj) as the signed sum of the  edges connecting the same modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' It can also be negative, if there are more negative than positive  edges between Mi and Mj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  The Imbalance I(P) of a partition P is defined as the total number of frustrated edges, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' the  numbers of positive edges located between modules and of negative edges located inside them:  I(P) =  �  1≤i<j≤ℓ  Ω+(Mi, Mj) −  �  1≤i≤ℓ  Ω−(Mi, Mi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  (2)  The objective of the CC problem is to minimize the number of frustrated edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' This problem can  be formally described as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Problem 1 (CC problem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Given a signed graph G = (V, E, s), the correlation clustering problem  consists in finding a partition P of V such that the imbalance I(P) is minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  To the best of our knowledge, this NP-hard minimization problem appears under this name for the  first time in Bansal’s paper [8], although it is addressed before in the literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  4  Enumeration of Optimal CC Solutions  Any method that enumerates optimal solutions needs to find an initial optimal solution by optimally  solving the problem at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' This is why we first describe an efficient method for finding an optimal  solution for the CC problem (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1), before presenting the methods for identifying an alternative  optimal solution (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='2) and enumerating all optimal solutions (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1  Finding an Initial Optimal Solution  As largely illustrated in the literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' [18], [23]), the CC problem can be modeled by means of an  ILP proposed for the Uncapacitated Graph Clustering problem [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Note that the model can handle  weighted graphs, but we use it on unweighted ones in the context of this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  For each pair of vertices u, v ∈ V : u < v, a binary variable is first defined to describe the relative  module membership of pairs of vertices, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=',  xuv =  �  1,  if u and v are in a same module,  0,  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  (3)  Then, the ILP formulation of the CC problem for unweighted signed graphs is written as follows:  Min  �  u,v∈V :uv∈E−  xuv +  �  u,v∈V :uv∈E+  (1 − xuv)  (4)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  xuv + xvr − xur ≤ 1,  ∀u < v < r ∈ V  (5)  xuv − xvr + xur ≤ 1,  ∀u < v < r ∈ V  (6)  − xuv + xvr + xur ≤ 1,  ∀u < v < r ∈ V  (7)  xuv ∈ {0, 1},  ∀u, v ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  (8)  The objective function in (4) looks for a feasible solution minimizing the imbalance defined by  Equation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' A feasible solution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=', decision variables corresponding to a partition, must satisfy the  triangle inequalities (5), (6) and (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Finally, (8) describes the domain constraints for the variables in  the formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  This ILP model is sufficient to find an optimal solution through an optimization solver, but it  can be very time-consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' One way to deal with this issue is to strengthen it through a cutting  plane approach [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We use the 2-partition and the 2-chorded cycle valid inequalities as described by  Grötschel and Wakabayashi in [28] (see Appendix A for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In the cutting plane approach,  we add these two tight valid inequalities (only) during the root relaxation phase as described by Ales et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' [2], before proceeding to the construction of the branch-and-bound search tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' As reported in [2]  4 / 30  Arınık et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' – Efficient Enumeration of the Optimal Solutions to the Correlation Clustering problem  for a similar clustering problem, we also verified that adopting this cut strategy improves the overall  processing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' There are other inequalities in the literature that can further tighten the LP relaxation  for the CC problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' These are the 2-chorded even wheel [28] inequality and those defined in [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Nevertheless, in practice, together the 2-partition and the 2-chorded cycle inequalities describe a tight  linear relaxation [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Throughout this work, we denote ILP(G) as the formulation defined by Equations (4)–(8) and  B&C(ILP(G)) as the branch-and-cut procedure described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Also, we denote ILP+(G) as the  strengthened ILP obtained after executing B&C(ILP(G)), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=', the formulation obtained by adding to  ILP(G) all the cuts generated by B&C(ILP(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Finally, let B&B(ILP+(G)) be the branch-and-  bound procedure based on ILP+(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In the following, for the sake of simplicity, the term solution  refers to a graph partition, as well as a feasible solution to the formulation ILP+(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='2  Finding an Alternative Optimal Solution  An enumeration method needs to constantly find an optimal solution that would be different from those  identified before, and stored in a set S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' For this purpose, we need to define an extended model of  ILP+(G), that we denote ILP+jump(G, S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Consider a partition P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Let Xp ∈ {0, 1}n×n be the representative matrix of P defined as: xp  ij = 1  if i and j belong to the same module in P;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' and xp  ij = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' ILP+jump(G, S) includes the set  of constraints defined in Equation (9) on top of ILP+(G):  �  u,v∈V :u<v  |xp  uv − xuv| > 0, ∀P ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  (9)  We refer the reader to [24] for the implementation details of these constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  We see that this extended formulation allows us to find an alternative optimal solution other than  the ones in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Furthermore, given an optimal solution P ∈ S, to ensure that this alternative solution  has the same objective value as I(P), we add the objective function in (4) as a constraint, as shown in  Equation (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  �  u,v∈V :uv∈E−  xuv +  �  u,v∈V :uv∈E+  (1 − xuv) ≤ I(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  (10)  Equation (9) along with Equation (10) ensures that each feasible solution to ILP+jump(S) is  an optimal solution to the original problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Finally, we denote the corresponding branch-and-bound  procedure based on formulation ILP+jump(G, S) by B&B(ILP+jump(G, S)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='3  Enumerating All Optimal Solutions  As we have mentioned before, the CC problem can have multiple optimal solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In this section, we  are interested in methods enumerating all optimal solutions of the CC problem for a given signed graph  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In the following, we first present OneTreeCC [15], which is the best general method to enumerate  solutions available in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' It is incorporated in CPLEX [30], and we apply it to the formulation  ILP+(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Then, we introduce our ad hoc method EnumCC, in the aim of obtaining a faster method  able to handle relatively larger graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  We start with OneTreeCC(G), which takes as input a signed graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' The sketch of this two-step  method is shown in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In the first step (line 2), it finds an initial optimal solution based on  ILP+(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We note TB&B the branch-and-bound search tree constructed during this step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' The search  tree as well as the nodes considered at this first step are stored for further examination during the second  step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Moreover, Danna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' completely turn off dual tightening in the branch-and-bound procedure  in order to guarantee exhaustive enumeration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' According to the authors, this fact can have a negative  impact on performance, in terms of computational time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In the second step (line 3), OneTreeCC(G)  enumerates the set S of all alternative optimal solutions by expanding the search tree TB&B until  obtaining the complete set of all optimal solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  To compete with OneTreeCC(G), we propose a new method for the CC problem in order to com-  pletely enumerate the optimal partitions of a given signed graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We call it EnumCC(G, rmax), and  it takes as input a signed graph G and a maximum distance parameter rmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' As shown in Algorithm 2,  it can be viewed as an improved version of the sequential approach proposed by Arthur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' [4] for  the Maximal Covering problem (described in Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In the first step (line 2), EnumCC(G, rmax)  obtains an initial optimal solution thanks to B&B(ILP+(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  In the second step, instead of di-  rectly "jumping" onto undiscovered optimal solutions one by one through ILP+jump(G, S) (like in  the sequential approach), we slightly change this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Our method first discovers the recurrent  5 / 30  Arınık et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' – Efficient Enumeration of the Optimal Solutions to the Correlation Clustering problem  Algorithm 1: OneTreeCC(G)  Result: The set of all optimal solutions S  1 S = ∅  /* 1st step  /  2 Solve B&B(ILP+(G)) to obtain an optimal solution P, and let TB&B be the constructed  branch-and-bound search tree  /* 2nd step  /  3 Expand fathomed nodes in TB&B until enumerating the set S of all alternative optimal solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  neighborhood RN≤rmax(P) of the current optimal solution P (line 4), with the hope of discovering new  optimal solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' The recurrent neighborhood RN≤rmax(P) of an optimal solution P, is the set of  optimal solutions reached directly or indirectly from P, depending on the maximum distance parameter  rmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' The complete description of the Recurrent Neighborhood Search (RNS) is given in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Whether a new solution is found or not through RNS(G, P, rmax), the method then moves on to the  next step, consisting in jumping onto a new solution P (line 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' These RNS and jumping phases are  repeated, until all optimal solutions are discovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Clearly, EnumCC(G, rmax) can only improve the sequential approach as used in [4], provided  that RNS discovers at least one new solution and that its execution time is shorter than that of  B&B(ILP+jump(G, S)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Furthermore, the choice of rmax is sensitive: it does not contribute much  to the method when it is small, whereas the method becomes slower than the sequential approach when  rmax is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Algorithm 2: EnumCC(G, rmax)  Result: The set of all optimal solutions S  1 S = ∅  /* 1st step  /  2 Solve B&B(ILP+(G)) to obtain an optimal solution P  /* 2nd step  /  3 while P ̸= null do  /* step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1  /  4  Apply RNS(G, P, rmax) to generate the recurrent neighborhood RN≤rmax(P) of P  5  S = S � RN≤rmax(P)  /* step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='2  /  6  Jump onto an undiscovered optimal solution P, with P /∈ S, through  B&B(ILP+jump(G, S)) // if not possible, then P = null  7 end  The jumping phase in step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='2 of EnumCC(G, rmax) can be time-consuming for two reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' First,  even finding an alternative solution can be costly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Second, the number of jumps, denoted by the notation  njump(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=') (here, njump(EnumCC(G, rmax))), can be very large, despite the use of the neighborhood  RN≤rmax(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Nonetheless, we expect to save time and compensate the time spent in the jumping  phase thanks to RN≤rmax(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  In the rest of this work, we leave out the common parameter G from the notations of the men-  tioned methods and ILP models, for the sake of convenience: OneTreeCC(), EnumCC(rmax) and  ILP+jump(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  In the next section, we detail the recurrent neighborhood search RNS used in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  5  Recurrent Neighborhood Search (RNS)  Recurrent Neighborhood Search (RNS) is the common and undoubtedly the most important part of  our method EnumCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' As mentioned in the previous section, if we want this method to compete with  OneTreeCC(G), RNS needs to be efficient and fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Before defining the neighborhood of a solution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' of a partition, we need to select a distance  function between partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In the following, we first present the edit distance (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1), then  6 / 30  Arınık et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' – Efficient Enumeration of the Optimal Solutions to the Correlation Clustering problem  introduce the algorithmic details of the Complete Neighborhood Search (CoNS) used in our Recurrent  Neighborhood Search (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Finally, we explain the main procedure for Recurrent Neighborhood  Search (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Let us first introduce some definitions used in the rest of the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' A partition P with ℓ modules  can be associated with one or more membership vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Such a vector of size n, denoted by π, defines  a function over V → {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=', ℓ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' For a given vertex u, π(u) denotes the label of the module of P  that contains u, while Mπ(u) denotes the module itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Finally, given a partition P, we define an  r-neighborhood structure, denoted by Nr(P), as the family of partitions obtained by moving r vertices  into different modules of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We also define Nr(π) as the family of corresponding membership vectors  obtained by moving r vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1  Edit Distance  In Natural Language Processing, and more generally in Computer Science, the edit distance [31] is  defined as the minimum number of edit operations required to transform one string into another (or  more precisely, their total cost).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In the context of graph partitioning, with fixed vertices and edges,  edit operations are used as neighborhood search operators in local search heuristics developed for graph  problems [1, 9, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Such an operation consists in moving one or more vertices (called moving vertices)  from their current modules (called source modules) to other ones (called target modules).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' From the  perspective of the membership vectors corresponding to the concerned partitions, this transforms a  source vector into a target one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  The number of moving vertices constitutes the cost of the edit  operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In this work, we are interested in edit operations whose cost is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Definition 2 (Min-edit operation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Consider an edit operation transforming a source membership vector  into a target membership one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  If the set of moving vertices is the minimum set required for this  transformation, then it is a min-edit operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Otherwise, we call it non-min-edit operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Notice that the cost of an edit operation is not always minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' This is because two membership  vectors, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' the module labels of two partitions, can be very different, but essentially suggest very similar  module assignments for the vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' The distinction between min-edit and non-min-edit operations is  illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Importantly, the edit distance between two membership vectors is the cost of the min-edit operation  allowing to turn one vector into the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' The calculation of such distance between two membership  vectors can be done in polynomial time by solving an assignment problem (see Appendix B for more  details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Let us now introduce our notations related to the edit distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Let πs and πt represent the  source and target membership vectors for an edit operation, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' They are associated with the  source and target partitions P s = {M s  1, M s  2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=', M s  ℓs} and P t = {M t  1, M t  2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=', M t  ℓt}, containing ℓs and  ℓt modules, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Since the edit distance is symmetric, without loss of generality, let ℓs ≤ ℓt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Moreover, we note πs → πt an edit operation applied onto P s to obtain P t, whose set of moving vertices  is defined as  s#»t  V  = {u | πs(u) ̸= πt(u)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' This means that the remaining vertices, called non-moving  vertices, do not change modules, hence their module labels are the same in πs and πt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' The cost of this  edit operation is simply the number of vertices whose module has changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We denote this cost with  cost(πs → πt) and compute it by taking the cardinality of  s#»t  V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In the rest of the text, we use r for  short whenever we are interested only in the value of cost(πs → πt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Also, a r-edit operation (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  min-r-edit operation) is any edit operation (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' min-edit operation) whose cost equals r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Consider  an edit operation πs → πt whose moving vertices are in set  s#»t  V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' The distinct labels of the source and  target modules of a subset V ′ ⊂ V of vertices are denoted, respectively, by πs(V ′) = �  u∈V ′ πs(u) and  πt(V ′) = �  u∈V ′ πt(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Finally, we sometimes need to refer to several modules in an edit operation  πs → πt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' For this reason, we also define M s  L = �  l∈L M s  l for any L ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='., ℓs} in the source partition  P s and M t  L = �  l∈L M t  l for any L ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='., ℓt} in the target partition P t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  To illustrate the aforementioned concepts, let us consider the example of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' As shown in  Figure 1a, πs = (1, 1, 2, 2, 2), where M s  1 = {v1, v2} and M s  2 = {v3, v4, v5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Let us consider an edit  operation πs → πt with the moving vertices  s#»t  V  = {v2, v4} (Figure 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' The edit operation consists  in moving v2 into M s  2 and v4 into M s  1, which produces πt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Hence, we have M t  1 =  �  M s  1 \\ {v2}  �  ∪ {v4}  and M t  2 =  �  M s  2 \\ {v4}  �  ∪ {v2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  The labels of the source and target distinct modules of  s#»t  V  are  πs(  s#»t  V ) = {1, 2} and πt(  s#»t  V ) = {1, 2}, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Also, we have M s  πs(s #»t  V ) = M s  1 ∪ M s  2 and  M t  πt(s #»t  V ) = M t  1 ∪ M t  2 for this example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' The sets M s  πs(s #»t  V ) and M t  πt(s #»t  V ) could be different, if some of  7 / 30  Arınık et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' – Efficient Enumeration of the Optimal Solutions to the Correlation Clustering problem  the moving vertices move into a module M s  i other than M s  πs(s #»t  V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  M s  1  M s  2  v1  v2  v3  v4  v5  (a) Source member-  ship  vector  πs  =  (1, 1, 2, 2, 2),  where  M s  1 = {v1, v2} and  M s  2 = {v3, v4, v5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  M t  1  M t  2  v1  v4  v3  v2  v5  (b) Target membership vector πt =  (1, 2, 2, 1, 2), where M t  1 = {v1, v4}  and M t  2 = {v2, v3, v5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' The mov-  ing vertices are shown with a purple  border.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' The operation πs → πt is  min-2-edit, where  s#»t  V  = {v2, v4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  M t′  1 = M t  2  M t′  2 = M t  1  v2  v3  v5  v4  v1  (c) Target membership vector πt′  =  (2, 1, 1, 2, 1), where M t′  1  = {v2, v3, v5}  and M t′  2 = {v1, v4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' The moving vertices  are shown with a purple border.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Since  M t′  1 = M t  2 and M t′  2 = M t  1, πs → πt′ is  not a min-3-edit operation, where  s#»t′  V  =  {v1, v3, v5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Figure 1: Illustrative example for two edit operations applied to the same source membership vector πs,  whose associated partition is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' The first operation,  s#»t  V , is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' 1b, and the second,  s#»t′  V , in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' The underlying partition in P t and P t′ is the same, but the corresponding membership  vectors πt and πt′ are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Consequently,  s#»t  V  is a min-edit operation, whereas  s#»t′  V  is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='2  Complete Neighborhood Search (CoNS)  Our neighborhood search method CoNS(G, πs, r) takes as input a signed graph G, a membership  vector πs associated with an optimal partition P s of ℓ modules and an edit distance r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' It returns the set  Nr(πs) of membership vectors associated with all optimal partitions, obtained by applying min-r-edit  operations to a given membership vector πs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' To ensure the completeness of set Nr(πs), CoNS(G, πs, r)  analyses all combinations of moving vertices and their target modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We use two pruning strategies in  order to eliminate some combinations that do not lead to an optimal partition when an edit operation is  applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Both of them are described in this section, whereas the properties upon which they are based  are detailed in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In simple terms, they are based on the relations between a set of moving  vertices and their target modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In the following, we present CoNS(G, πs, r) as a pipeline procedure  consisting of four parts, as shown in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  In the first part, CoNS(G, πs, r) starts by selecting a candidate set of r moving vertices  s#»t  V  among all vertices in V (line 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' The set of all possible r moving vertices is obtained by generating all  combinations  �V  r  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We know that probably not every  s#»t  V  leads to a min-r-edit operation and an optimal  partition P t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' To detect such cases, we apply the first pruning procedure (line 3), called internalPruning  and depicted in Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' This procedure checks, without the knowledge of target modules, if  s#»t  V  satisfies some connectivity conditions related to the atomicity property (defined in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='2) and the  MVMO property up to three moving vertices (defined in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' As we will see in Section 6,  whenever one of these two conditions is not satisfied, the candidate set  s#»t  V  can be pruned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  At this point, the set of moving vertices  s#»t  V  is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Next, the algorithm defines step by step the  target membership vector πt by setting the target module of each vertex in  s#»t  V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We handle the process  of generating the target membership vector πt in three steps, which constitutes the second, third and  fourth parts of Algorithm 3, in order to take advantage of our pruning strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' The pruning strategy  used in these parts is slightly different from that of the first part: it is based on external connections  from  s#»t  V  to  s#»t  V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' The concerned procedure is called externalPruning and depicted in Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' It  verifies, based on external relations among  s#»t  V , if  s#»t  V  satisfies the min-edit (Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1), atomicity  8 / 30  Arınık et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' – Efficient Enumeration of the Optimal Solutions to the Correlation Clustering problem  Algorithm 3: CoNS(G, πs, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Input  : signed graph G(V, E, s), source membership vector πs with ℓ module labels, edit distance r  Output: Nr(πs)  1 T = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='., ℓ} // module labels in πs  /* 1st part  /  2 for each  s#»t  V  ∈ �V  r  �  do  3  if internalPruning(G, πs,  s#»t  V ) then  4  go to line 2  5  end  /* 2nd part  /  6  T0 = π(  s#»t  V ) �  {Unknown} // initial target module labels  7  Trem = T \\ T0 // remaining module labels  8  for each T ′  0 ∈ P(�T0  r  �  ), s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' πs(u) /∈ T ′  0, ∀u ∈  s#»t  V  do // P(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' ):  permutations  9  Let πt be the partition after assigning T ′  0 to  s#»t  V  10  if externalPruning(G, πs, πt,  s#»t  V ) then  11  go to line 8  12  end  /* 3rd part  /  13  Let B be the the number of Unknown labels in πt  14  if B ̸= 0 then  15  Let  s#»t  V  rem be the moving vertices, whose πt(  s#»t  V  rem) = Unknown  16  for b ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='., B} do  17  Let I be the set of all vectors of size B with elements belonging to {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='., b}, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' each element for  each vector appears once  18  for each I ∈ I do  19  Update πt with the assignment of I to πt(  s#»t  V  rem)  20  if externalPruning(G, πs, πt,  s#»t  V ) then  21  go to line 18  22  end  /* 4th part  /  23  T +  rem = Trem ∪ {ℓ + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='., ℓ + b} // expand for empty modules  24  for each T ′  r ∈ P(�T +  r  b  �  ) do  25  Replace the assignment of I to  s#»t  V  rem by Trem in πt  26  if I(πs) = I(πt) and ¬externalPruning(G, πs, πt,  s#»t  V ) then  27  Nr(πs) = Nr(πs) �  πt  28  end  29  end  30  end  31  end  32  else  33  if I(πs) = I(πt) then  34  Nr(πs) = Nr(πs) �  πt  35  end  36  end  37  end  38 end  9 / 30  Arınık et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' – Efficient Enumeration of the Optimal Solutions to the Correlation Clustering problem  Algorithm 4: internalPruning(G, πs,  s#»t  V )  Input  : signed graph G(V, E, s), source membership vector πs, moving vertices  s#»t  V  Output: Boolean variable  /* 1st part  /  1 if ¬intAtomic(G, πs,  s#»t  V ) then // Properties 6 and 7a  2  return true  3 end  /* 2nd part  /  4 if 2 ≤ |  s#»t  V | ≤ 3 and ¬intMVMO(G, πs,  s#»t  V ) then // Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1 and Lemma 12  5  return true;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  6 end  7 return false;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  (Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='2) and MVMO (Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='3) properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' As discussed in Section 6, whenever one of these  three conditions is not satisfied, the possibility of target modules for candidate set  s#»t  V  is pruned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' These  pruning strategies can be applied even when some target modules are not already decided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Therefore,  they are invoked, whenever the target modules of  s#»t  V  are updated (lines 10, 20 and 26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  In the second part, a moving vertex is allowed to move into a module, whose label is in T0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' The set  T0, defined at line 6, contains the labels of all unique source modules of  s#»t  V , as well as a label Unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  We use this label Unknown to indicate that the target module of a vertex is not already assigned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We  also define Trem as the remaining modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Notice that, from this point, a target membership vector πt  can contain multiple vertices with target modules labeled as Unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In the third part, this Unknown  label allows us to handle the other modules in Trem as target possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' At line 8, we consider all  possible permutations in P(  �T0  r  �  ) such that, for each vertex in  s#»t  V , the target and source modules are  different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  At this point, the externalPruning procedure can be applied for vertices whose target modules are  already assigned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' If the pruning conditions are not satisfied, the algorithm proceeds with the third part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Line 14 checks if no target module in πt is unknown (B = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' If so, then a new optimal partition  has been found, and the algorithm goes on with the final test (line 32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' However, if it is not the case,  the algorithm enters the fourth part: all the combinations in Trem (more precisely, T +  r ) are considered  as a possible assignment of unknown target modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' One can expect this task to be computationally  expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Thus, instead of directly determining the target modules of each vertex in  s#»t  V  rem, we first  generate all the coupling scenarios of moving vertices going into the same Unknown target module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  To handle this, let b be the number of distinct Unknown target modules (line 16), for each value  in the range of {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='., B}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We build all possible coupling scenarios I for the value b (line 17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We  couple a pair of remaining moving vertices, if they move into the same Unknown target module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' For  instance, {Unknown1, Unknown1, Unknown2} indicates that the last moving vertex moves into a  different Unknown target module than the first two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Once the target membership vector πt is enriched  with a coupling scenario I, the external pruning is repeated in line 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' If the pruning conditions are  not satisfied, the algorithm finally determines the target modules of  s#»t  V  rem, by respecting the actual  coupling scenario I (line 25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In the end, we add the current πs → πt into Nr(πs), if the optimality  condition is met (line 26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Without the problem-specific pruning strategies (internal and the external verification of properties  from Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='3), Algorithm 3 amounts to a brute force method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Although CoNS(G, πs, r) can be  very costly (even for small values of r), the properties described in Section 6 allow us to develop a  method whose cost is relatively lower than the brute force method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Indeed, given a membership vector  of size n with ℓ module labels, the number of non-repetitive permutations of all the combinations of  r moving vertices is nr × r!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' and the number of non-repetitive permutations of all the combinations  of target modules is ℓr × r!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='. In the end, the brute force method is Θ(nr × r!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' × ℓr × r!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' ), where we  assume that ℓ > r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Computational results in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1 show that the pruning strategies reduce the  computational cost of this procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  10 / 30  Arınık et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' – Efficient Enumeration of the Optimal Solutions to the Correlation Clustering problem  Algorithm 5: externalPruning(G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' πs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' πt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  s#»t  V )  Input  : signed graph G(V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' source membership vector πs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' target membership vector πt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  moving vertices  s#»t  V  Output: Boolean variable  /* 1st part  /  1 if ¬minEdit(πs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' πt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  s#»t  V ) or ¬extAtomic(G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' πs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' πt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  s#»t  V ) then // Properties 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' 7b and 8  2  return true  3 end  /* 2nd part  /  4 if 2 ≤ |  s#»t  V | and ¬extMVMO(G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' πs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' πt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  s#»t  V ) then // Lemmas 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='2, 12 and 13  5  return true  6 end  7 return false;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  P0  P1  1-edit  kmax-edit  2-edit  1-edit  2-edit  P6  P7  P8 P9  P10  P11  P12  P2 P3  P4  P5  kmax-edit  (a) RNS Tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  P1  P2  P3  P4  V  1  2  ={v1,v2}  V  3  4  ={v1,v2}  V  2  4  ={v3}  V  1  3  ={v3}  (b) Example of duplication in RNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Both P 2 and P 3 generate P 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Figure 2: Illustrations for the Recurrent Neighborhood Search (RNS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In both figures, circles represent  membership vectors associated with optimal partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='3  Main Procedure  The main procedure for Recurrent Neighborhood Search (RNS) consists in applying CoNS from Sec-  tion 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='2 in a recursive manner, in order to build a search tree called the RNS tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' This recursive  procedure is depicted in Figure 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' The root node of RNS tree corresponds to the initial partition P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  All other nodes of the RNS tree constitute the set of optimal solutions reached directly or indirectly  from P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' A solution associated with a node at some level h of the RNS tree is obtained after h − 1  applications of the CoNS procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' The height of the RNS tree is unknown but necessarily finite, as  we will see next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Method RNS(G, P, rmax), which takes as input a signed graph G, an optimal partition P and a  maximum edit distance rmax, is detailed in Algorithm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' It starts from the initial partition P (line 1),  and enumerates a set of its neighbor optimal partitions up to edit distance rmax (line 4), denoted  by RN≤rmax(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Each partition P t associated with πt ∈ Nr(πs) is considered as a potential initial  partition for seeking new neighbor partitions (line 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Despite the pruning techniques which avoid some  repetitions, it is possible that P t has been already discovered and belongs to RN≤rmax(P) (see such an  example in Figure 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Line 9 checks this case, and discards P t, if required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' This process is repeated in  a recursive manner, until there is no new partition obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  In the rest of this work, for the sake of convenience, we leave out the common parameter G and P s  (or P) when referring to the mentioned methods: CoNS(r) and RNS(rmax).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  6  Pruning Strategies  In this section, we present the pruning strategies incorporated in the method CoNS(G, πs, r), described  in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  But before this, we need to introduce some additional definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Consider an edit  operation πs → πt with moving vertices in set  s#»t  V , where membership vector πs (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' πt) is associated  11 / 30  Arınık et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' – Efficient Enumeration of the Optimal Solutions to the Correlation Clustering problem  Algorithm 6: RNS(G, P, rmax)  Input  : signed graph G(V, E, s), partition P, maximum edit distance rmax  Output: RN≤rmax(P)  1 U = {P} // a set of unprocessed partitions  2 RN≤rmax(P) = ∅ // a set of discovered partitions  3 while U ̸= ∅ do  4  for each r ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='., rmax} do  5  Let P s be an element of U, where πs is an associated membership vector of P s  6  RN≤rmax(P) = RN≤rmax(P) � P s  7  Nr(πs) = CoNS(G, πs, r) // Complete Neighborhood Search  8  for πt ∈ Nr(πs) do // P t is the associated partition of πt  /* memory-based duplication removal  /  9  if P t /∈ RN≤rmax(P) then  10  U = U � P t  11  end  12  end  13  end  14 end  with source partition P s (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  target partition P t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Let us define #»  V s  πs(u) =  s#»t  V  ∩ M s  πs(u) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  #»  V t  πs(u) =  s#»t  V  ∩ M t  πs(u)) as the set of moving vertices being in the source module of vertex u in  P s (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  in P t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Also, let #»  V s  πt(u) =  s#»t  V  ∩ M s  πt(u) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  #»  V t  πt(u) =  s#»t  V  ∩ M t  πt(u)) be the set  of moving vertices being in the target module of u in P s (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' in P t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Finally, we define  s#»t  V u =  �#»  V s  πs(u) ∪ #»  V t  πs(u) ∪ #»  V t  πt(u) ∪ #»  V s  πt(u)  �  \\ {u} as the set of moving vertices connected to vertex u in  �G[  s#»t  V ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' To illustrate the aforementioned notations related to  s#»t  V , we consider again the same example  from Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' For v1 ∈  s#»t  V , we have πs(v1) = 1 and πt(v1) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Then, #»  V s  πs(v1) = {v1, v2} is the  set of vertices in M s  1 and #»  V t  πt(v1) = {v1} since v1 is the only moving vertex belonging to M t  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Also,  #»  V s  πt(v1) = #»  V s  {3} = ∅ since there is no moving vertex in M s  3, and #»  V t  πs(v1) = {v3} since v3 ∈ M t  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Finally,  we get  s#»t  V v1 = {v2, v3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1  Non-Minimum Edit Operation Pruning  When generating the set Nr(πs) in RNS, only considering min-edit operations is sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Next, we  define a property allowing to detect, in some cases, that an edit operation is not minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' The idea  behind it is to define two sufficient conditions on the set of moving vertices implying that there exist  an operation of smaller cost leading to the same partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In the following, let πs → πt be an edit  operation with moving vertices in  s#»t  V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Property 3 (Non-min-edit operation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Let # »  Va ⊆  s#»t  V  be a subset of  s#»t  V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Assume that the vertices in  # »  Va constitute the majority part of module M s  a (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' |# »  Va| > |M s  a \\ # »  Va|) and that they are in the same  target module M t  b in P t (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' |πt(# »  Va)| = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Moreover, suppose there exists a subset of moving vertices  #»  Vb ⊆  s#»t  V , such that ∅ ⊆ #»  Vb ⊆ M s  b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Let us define V a (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' V b) as M s  a \\  s#»t  V  (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' M s  b \\  s#»t  V ) in P s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Each condition in the following is a sufficient condition for πs → πt to be a non-min-edit operation:  (a) If #»  Vb does not move into M s  a and |# »  Va| > |V a| + |V b|, then πs → πt is not a min-edit operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  (b) If #»  Vb moves into M s  a and |# »  Va| + |#»  Vb| > |V a| + |V b|, then πs → πt is not a min-edit operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Let us first handle the condition (3a), which is illustrated in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' When this condition  is satisfied, keeping # »  Va in M s  a and rather moving V a (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' V b) into M s  b (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' M s  a) results in an  r′-edit operation with r′ < r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In the case where condition (3b) is satisfied, keeping #»  Vb in M s  b while  moving V a and V b results in an r′-edit operation with r′ < r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Let us take the following example to  illustrate Property 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Two modules exchange all of their vertices with each other, such that # »  Va = M s  a,  12 / 30  Arınık et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' – Efficient Enumeration of the Optimal Solutions to the Correlation Clustering problem  M s  a  M s  b  # »  Va  V a  #»  Vb  V b  (a) Source partition P s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  M t  a  M t  b  # »  Va  V a  V b  (b) Target partition P t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Figure 3: Illustrative example for Property 3a, where vertices in # »  Va (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  #»  Vb) move from their source  module Ms  a (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Ms  b ) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' 3a) into target module Ms  b (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Ms  c , which is purposely not shown)  (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' V a and V b represent the non-moving vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  M1  M2  M3  M4  v1  v2  v3  v4  v1  v2  v3  v4  (a) An illustration of an r-edit operation πs → πt with  s#»t  V  = {v1, v2, v3, v4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' The mem-  bership vector πs associated with partition P s is defined by πs = (1, 1, 3, 4), whereas  πt = (2, 2, 2, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Positive (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  negative) edges between moving vertices are drawn in  green (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  v1  v2  v3  v4  (b) The corresponding interaction graph �G[  s#»t  V ] of the r-  edit operation illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' 4a  Figure 4: Illustrative example regarding the definition of an interaction subgraph, here �G[  s#»t  V ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  V a = ∅, #»  Vb = M s  b and V b = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In the end, we obtain the same two modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Since the inequality  |# »  Va| + |#»  Vb| > |V a| + |V b| holds in this particular scenario, this is not a min-edit operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='2  Decomposable Edit Operation  The next set of properties is related to the decomposability of an r-edit operation πs → πt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We assume  that I(P s) = I(P t), which is in line with our objective of enumerating all optimal solutions of a given  graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Definition 4 (Decomposable edit operation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We consider an r-edit operation πs → πt as decomposable  if there is an intermediate membership vector πt′, associated with a partition P t′, between πs and πt  such that:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' I(P s) = I(P t) = I(P t′);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  s#»t′  V  ⊂  s#»t  V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' and  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' πt′(u) = πt(u) for each u ∈  s#»t′  V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Property 5 (Atomic edit operation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We consider that an r-edit operation πs → πt is atomic if the  condition I(P s) < I(P t′) is satisfied for any r′-edit operation P s → P t′ with r′ < r satisfying (2) and  (3) in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We assume that there is an r′-edit operation such that I(P s) = I(P t′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Then, we can construct  an r′′-edit operation πt′ → πt satisfying (1) in Definition 4 and πt′(u) = πt(u) for each u ∈  s#»t  V  \\  s#»t′  V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Atomicity requires three types of connectivity conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We state the first one in the following  property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  13 / 30  Arınık et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' – Efficient Enumeration of the Optimal Solutions to the Correlation Clustering problem  Property 6 (Edge connectivity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In an atomic edit operation πs → πt, G[  s#»t  V ] is a single connected  component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Let  s#»t′  V  and  t′ #»t  V  be any two proper subsets of  s#»t  V , such that  s#»t′  V ∩  t′ #»t  V  = ∅ and E(  s#»t′  V ,  t′ #»t  V ) =  ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Then, we can construct with  s#»t′  V  an r′-edit operation satisfying Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Nevertheless, G[  s#»t  V ] being connected is not a sufficient condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' The next property states that  the signs of edges between moving vertices play a key role with respect to atomicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In the following,  we assume that G[  s#»t  V ] is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Property 7 (Spurious edge connectivity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Each condition in the following is a sufficient condition for  πs → πt to be a decomposable edit operation:  (a) Let S ⊆  s#»t  V  be a subset such that |πs(S)| = 1 and E(S, #»  V s  πs(S)) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Let G′ = (  s#»t  V , E′, w),  where E′ = E \\ E(S, #»  V s  πs(S)) and w(e) = 1 for each e ∈ E′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' If Ω(S, #»  V s  πs(S)) = 0 and G′ is not  connected, then πs → πt is decomposable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  (b) Let S ⊆  s#»t  V  be a subset such that |πt(S)| = 1 and E(S, #»  V t  πt(S)) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Let G′ = (  s#»t  V , E′, w),  where E′ = E \\ E(S, #»  V t  πt(S)) and w(e) = 1 for each e ∈ E′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' If Ω(S, #»  V t  πt(S)) = 0 and G′ is not  connected, then πs → πt is decomposable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Straightforwardly, we can imagine S as a contracted vertex v in both cases, such that the  vertices in S and the edges between them are replaced by a single vertex v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Since Ω(S, #»  V s  πs(S)) = 0  (Ω(S, #»  V t  πt(S)) = 0), the edges between v and #»  V s  πs(v) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' #»  V t  πt(v)) do not play a role (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=', as if they  did not exist).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Therefore, the proof is the same as for Property 6, with subsets S and  s#»t  V  \\ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  The last connectivity condition allows to verify if a moving vertex depends on the simultaneous  movement of a subset of other vertices in  s#»t  V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' It can be checked through the concept of interaction  subgraph �G[  s#»t  V ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  We define the interaction subgraph defined for the edit operation πs → πt as  �G[  s#»t  V ] = (  s#»t  V , �E), where �E = {(u, v) | u, v ∈  s#»t  V and (u, v) ∈ E  and (πs(u) = πs(v) ∨ πt(u) =  πs(v) ∨ πs(u) = πt(v) ∨ πt(u) = πt(v))}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Its extraction from its original graph G is illustrated in  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Property 8 (Interaction connectivity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Consider an atomic edit operation πs → πt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' The interaction  subgraph �G[  s#»t  V ] is a single connected component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Assume that �G[  s#»t  V ] is not connected (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  This implies that there is a subset  s#»t′  V  ⊆  s#»t  V , such that �E(  s#»t′  V ,  s#»t  V  \\  s#»t′  V ) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Let πs → πt′ be an edit operation with moving vertex  set  s#»t′  V , which satisfies (2)-(3) in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' When moving a vertex u ∈  s#»t′  V  from M s  πs(u) to M s  πt′(u)  the contribution of vertex u to the imbalance is  Ω({u}, #»  V s  πs(u)) − Ω({u}, #»  V s  πt′(u)) + Ω({u}, #»  V t′  πt′(u)) − Ω({u}, #»  V t′  πs(u))  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  (11)  From the definition of �E and from �E(  s#»t′  V ,  s#»t  V \\  s#»t′  V ) = ∅, there is no vertex v ∈  s#»t  V \\  s#»t′  V  which  contributes to the contribution of vertex u, as shown in Equation (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Therefore, Equation (1) in  Definition 4 is also satisfied for πs → πt′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  For the sake of simplicity, in Algorithm 4, we define the function intAtomic(G, πs,  s#»t  V ) which is  used to indicate whether πs → πt satisfies Properties 6 and 7a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Notice that we do so when the target  modules of  s#»t  V  are not known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Likewise, in Algorithm 5, we define the function extAtomic(G, πs, πt,  s#»t  V ), which indicates whether πs → πt satisfies Properties 3, 5, 7b and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Notice that we do so when  the target modules of  s#»t  V  are partially or completely known, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=', in lines 10, 20 and 26 of Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='3  Multiple Vertex Moves between Optima (MVMO) Property  In this section, we introduce our last family of properties, which rely on the assumption that both  partitions associated to an edit operation are optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  14 / 30  Arınık et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' – Efficient Enumeration of the Optimal Solutions to the Correlation Clustering problem  Property 9 (MVMO on weighted signed networks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Consider an atomic edit operation πs → πt with  r > 1, where P s and P t are optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' The following property holds for each u ∈  s#»t  V :  γleft  u  > γright  u  ,  (12)  where  γleft  u  = Ω({u}, #»  V s  πs(u)) − Ω({u}, #»  V s  πt(u)),  (13)  γright  u  = −Ω({u}, #»  V t  πt(u)) + Ω({u}, #»  V t  πs(u)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  (14)  (15)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Let πs → πt be an atomic min-r-edit operation applied on Ps to obtain Pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' First, we recall a  simple condition satisfied by any optimal partition for the CC problem [13] when moving a single vertex,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' when r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Given an optimal partition Ps, the placement of any vertex u in its module M s  πs(u)  is also optimal with respect to any other modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' This means that any vertex u maximizes (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  minimizes) its number of positive (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' negative) edges in module M s  πs(u) of Ps, so that moving u to  any other module does not improve the objective function value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We can write this condition as  Ω({u}, M s  πs(u)) ≥ Ω({u}, M s  πt(u)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  (16)  Now, when we move more than one vertex in πs → πt, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' if r > 1, this condition becomes  Ω({u}, M s  πs(u)) > Ω({u}, M s  πt(u)),  (17)  for each u ∈  s#»t  V , due to the atomicity assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Next, in order to take into account the existence of  other moving vertices in the source and target modules of vertex u, Equation (17) can be rewritten as  γleft  u  >  s#»t  ∆u, where  s#»t  ∆u = Ω({u}, M s  πt(u) \\ #»  V s  πt(u)) − Ω({u}, M s  πs(u) \\ #»  V s  πs(u)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Similarly, considering  the atomic min-r-edit operation P t → P s, we have also −  t#»s  ∆u > γright  u  , where  t#»s  ∆u = Ω({u}, M t  πs(u)\\  #»  V t  πs(u)) − Ω({u}, M t  πt(u) \\ #»  V t  πt(u)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Since both  s#»t  ∆u and  t#»s  ∆u are defined on relations of vertex u with  non-moving vertices in P s and P t, we have  s#»t  ∆u = −  t#»s  ∆u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Finally, by rewriting the previous equations  we obtain γleft  u  >  s#»t  ∆u = −  t#»s  ∆u > γright  u  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Corollary 10 (MVMO on unweighted signed networks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' The inequality γleft  u  − γright  u  ≥ 2 holds for  each vertex u, on top of Property 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' The proof is straightforward from the proof of Property 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Figure 5 can be used to illustrate Property 9 and Corollary 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' For instance, for vertex v1, the  equation in Property 9 becomes  a12 − a13 > a15 − a12 − a14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  (18)  Similarly, for vertex v3, the equation in Property 9 becomes  −a34 − a35 > a13 + a23 + a34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  (19)  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='4  Tractable Cases of the MVMO Property  Notice that the MVMO property requires πs and πt to be known, which makes this property not very  useful for computational purposes (in Algorithm 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In this section, we analyze the MVMO property  when only partial information of πt is known (lines 10 and 20 in Algorithm 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  We start by analysing some special relational patterns for 2-edit and 3-edit operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Lemma 11  focuses on 2-edit operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Recall that �E = {(u, v) ∈ E | u, v ∈  s#»t  V  ∧ (πs(u) = πs(v) ∨ πt(u) =  πs(v) ∨ πs(u) = πt(v) ∨ πt(u) = πt(v))}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Lemma 11 (MVMO 2-edit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Consider an atomic 2-edit operation πs → πt, where  s#»t  V  = {u, v}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Since  it is an atomic edit operation, we have (u, v) ∈ �E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Then  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' If (πs(u) = πs(v)) ∨ (πt(u) = πt(v)), then auv > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  15 / 30  Arınık et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' – Efficient Enumeration of the Optimal Solutions to the Correlation Clustering problem  M1  M2  M3  v1  v2  v3  v4  v5  v1  v2  v4  v3  v5  a12  a15  a34  Figure 5: Illustrative example, where five vertices v1, v2, v3, v4 and v5 move from their source modules  to their target ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Dashed circles indicate the moving vertices, when they are in their target modules  after the edit operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Note that we do not show non-moving vertices for the sake of simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Positive  (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' negative) edges between moving vertices are drawn in green (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  a)  d)  e)  b)  c)  u  v  u  v  u  v  u  u  v  v  Figure 6: All 2-edit operation scenarios, where (u, v) ∈ �E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Note that only scenarios (6a) and (6b) are  atomic 2-edit operations, with auv > 0 and auv < 0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' If (πs(u) = πt(v)) ∨ (πt(u) = πs(v)), then auv < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' From Property 6, we have (u, v) ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In this proof, we use an exhausting strategy: moving  vertices u and v, with (u, v) ∈ �E, can be in one of the five scenarios presented in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Since  Corollary 10 holds for each vertex u ∈  s#»t  V , only scenarios (6a) and (6b) are atomic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Therefore, auv > 0  for scenario (a) whereas auv < 0 for scenario (b):  a) We have (γleft  u  = auv) > (γright  u  = −auv) and (γleft  v  = auv) > (γright  v  = −auv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We see that  auv must be positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Since Corollary 10 is satisfied with auv > 0, it is atomic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  b) We have (γleft  u  = −auv) > (γright  u  = auv) and (γleft  v  = −auv) > (γright  v  = auv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We see that  auv must be negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Since Corollary 10 is satisfied with auv < 0, it is atomic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  c) We have (γleft  u  = 0) > (γright  u  = −auv) and (γleft  v  = 0) > (γright  v  = −auv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We see that auv  must be positive and this satisfies Property 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' However, Corollary 10 is not satisfied, which makes  this edit operation decomposable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  d) We have (γleft  u  = auv) > (γright  u  = 0) and (γleft  v  = auv) > (γright  v  = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We see that auv must  be positive and this satisfies Property 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' However, Corollary 10 is not satisfied, which makes this  edit operation decomposable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  e) We have (γleft  u  = −auv) > (γright  u  = 0) and (γleft  v  = 0) > (γright  v  = −auv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We see that auv  must be negative and this satisfies Property 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' However, Corollary 10 is not satisfied, which makes  this edit operation decomposable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Next, we focus on 3-edit operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We verify some cases in which Lemma 11 is also valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Lemma 12 (MVMO 3-edit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Consider an atomic r-edit operation πs → πt with r = 3, where  s#»t  V  =  {u, v, z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Since it is an atomic edit operation, we have (u, v), (u, z), (v, z) ∈ �E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Lemma 11 holds true  for each pair (u, v) of vertices, with two exceptions:  16 / 30  Arınık et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' – Efficient Enumeration of the Optimal Solutions to the Correlation Clustering problem  b)  c)  a)  d)  e)  u  u  u  v  v  v  z  z  z  u  v  z  u  v  z  Figure 7: Five of the possible 3-edit operation scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  The full list can be found in the Appendix  (Figure 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In this figure, all edit operations are atomic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' all vertices in  s#»t  V  are in the same source module and are moved into the same target module  (Figure 7a),  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' only two of  s#»t  V  are in the same source module and are moved into the source module of the third  moving vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Reciprocally, the third moving vertex moves into the source module of the others’  source module (Figure 7b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Without these two exceptions, G[  s#»t  V ] forms a triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Nevertheless, in these two exceptions,  G[  s#»t  V ] can form a path, and, as will see next, this path ensures that moving vertices from the same  source module are positively connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Similar to Lemma 11, the proof is based on all possible 17  scenarios of three moving vertices with (u, v), (u, z), (v, z) ∈ �E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Some of those scenarios are shown in  Figure 7, and we provide the full list in the Appendix (Figure 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' The proof is straightforward when  one adapts Corollary 10 to those scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We verify below only those in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' The full list of  verifications is found in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  a) We have (γleft  u  = auv + auz) > (γright  u  = −auv − auz), (γleft  v  = auv + avz) > (γright  v  =  −auv − avz) and (γleft  z  = auz + avz) > (γright  z  = −auz − avz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We see that auv, auz and avz  cannot be negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Since Corollary 10 is satisfied with a positive path formed in G[  s#»t  V ] for each  vertex u ∈  s#»t  V , it is atomic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  b) We have (γleft  u  = auv − auz) > (γright  u  = −auv + auz), (γleft  v  = auv − avz) > (γright  v  =  −auv + avz) and (γleft  z  = −auz − avz) > (γright  z  = auz + avz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We see that auv (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' auz  and avz) cannot be negative (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' positive).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Since Corollary 10 is satisfied with a path formed  in G[  s#»t  V ], it is atomic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  c) We have (γleft  u  = 0) > (γright  u  = −auv − auz), (γleft  v  = 0) > (γright  v  = −auv − avz) and  (γleft  z  = 0) > (γright  z  = −auz − avz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We see that auv, auz and avz must be positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Since  Corollary 10 is satisfied with a positive triangle formed in G[  s#»t  V ], it is atomic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  d) We have (γleft  u  = auv) > (γright  u  = −auv − auz), (γleft  v  = auv) > (γright  v  = −auv − avz) and  (γleft  z  = 0) > (γright  z  = −auz − avz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We see that auv, auz and avz must be positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Since  Corollary 10 is satisfied with a positive triangle formed in G[  s#»t  V ], it is atomic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  e) We have (γleft  u  = auv −auz) > (γright  u  = 0), (γleft  v  = auv) > (γright  v  = −avz) and (γleft  z  = 0) >  (γright  z  = auz − avz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We see that auv and avz (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' auz) must be positive (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' negative).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Since Corollary 10 is satisfied with a triangle formed in G[  s#»t  V ], it is atomic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Unlike Lemma 12, we cannot completely generalize Lemma 11 for more than three moving vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Nevertheless, there are some circumstances, where Lemma 11 is still valid for a subset of  s#»t  V , and this  is formalized in Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Lemma 13 (MVMO r-edit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Consider an atomic r-edit operation πs → πt with r ≥ 4 and a vertex  u ∈  s#»t  V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' If 2 ≤ |u ∪  s#»t  V u| ≤ 3, Lemma 11 holds true for each pair (u, v) with v ∈  s#»t  V u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' The condition of 2 ≤ |u∪  s#»t  V u| ≤ 3 ensures that subset u∪  s#»t  V u represents one of the scenarios in  2- or 3-edit operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We note that two exceptional scenarios mentioned in Lemma 12 are not of interest  here, because they necessarily satisfy the definition of atomic edit operation, hence |u ∪  s#»t  V u| > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  For the sake of simplicity, in Algorithm 4, we define the function intMVMO(G, πs,  s#»t  V ) which  indicates whether πs → πt satisfies Lemmas 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1 and 12, when the target modules of  s#»t  V  are not  known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Likewise, in Algorithm 5 we define the function extMVMO(G, πs, πt,  s#»t  V ), which indicates  17 / 30  Arınık et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' – Efficient Enumeration of the Optimal Solutions to the Correlation Clustering problem  whether πs → πt satisfies Lemmas 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='2, 12 and 13, when the target modules of  s#»t  V  are partially or  completely known (10, 20 and 26 of Algorithm 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  7  Dataset  This section is dedicated to the description of the dataset used in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' As mentioned in the  introduction, in this article we focus on unweighted graphs as we have done in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Application-wise,  unweighted signed networks fit certain modeling situations and methodological choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Indeed, in some  works, binary values better represent the studied relations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' alliance/conflict between countries in  international relationships [21]), or the authors prefer to use such values for practical reasons (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  limited or unreliable information [22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  We conduct our experiments on two datasets of random signed networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' All these data as well as  the corresponding optimal solutions are publicly available online1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Dataset1: We generate these networks through the random signed network generator proposed in  our previous work [6], which is publicly available online2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' For complete unweighted signed networks, this  model relies on only three parameters: n (number of vertices), ℓ0 (initial number of modules) and qm  (proportion of misplaced edges, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' edges meant to be frustrated by construction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Moreover, we make  the assumption that the proportion of misplaced edges is the same inside and between the modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  When it comes to incomplete unweighted signed networks, we introduce two more parameters, which  are the density d of the graph and the proportion qneg of the negative edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' The last parameter qneg  is defined as |E−|/|E| and allows controlling the ratio of positive to negative edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' For complete  unweighted signed networks with d = 1, we generate 20 replications for parameter values ℓ0 = 3,  n ∈ {32, 36, 40, 45, 50} and qm = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='6}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In these networks, the value of qneg  with the considered parameters is approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' For incomplete unweighted signed networks with  d = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='50}, we generate 20 replications for parameter values ℓ0 = 3, n ∈ {32, 36, 40}, qm =  {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='6} and qneg = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='7}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In total, we produce 600 and 1,080 instances for  complete and incomplete networks, respectively, which makes a total of 1,680 instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We use this  dataset in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Dataset2: This dataset is different than Dataset1 in that the optimal solution for a generated  network is known by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' This allows to consider relatively large graph orders n without being  limited by long running time of an exact partitioning method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' For given n, d and ℓ0, we first create  a perfectly structurally balanced (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=', internally positive and externally negative) signed network with  a built-in module structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Clearly, the underlying module structure constitutes the optimal partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Then, in order to take into account different positive to negative ratio values for internal and external  edges we generate several signed networks by perturbing the initial signed network without affecting  its underlying optimal partition, thanks to its definition of stability range [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We generate signed  networks with parameter values n ∈ {30, 40, 50, 60, 70, 90}, d ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='25, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='00} and ℓ0 = {2, 4, 6} through  our random signed network generator, which is publicly available online3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In total, we produce 214 and  184 instances for complete and incomplete networks, respectively, which makes a total of 398 instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  In each generated network, the associated optimal solution corresponds to the planted partition defined  for the corresponding network without perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We use this dataset only for Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Dataset3: In addition to artificial graphs, we also considered 8 sparse real-world networks from  two sources: four networks from Correlated of War (CoW) [43] and four biological networks retrieved  from Samin Aref’s Figshare repository4, see [3] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' For each real-world signed network  G, we apply a preprocessing step by retrieving the largest connected component defined in G+, for  computational purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  8  Results  We now assess the performance of our enumeration method EnumCC(rmax) when generating the space  of optimal solutions to the CC problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We first investigate the efficiency of the pruning conditions used  in Algorithms 4 and 5 based on the MVMO property (Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Then, we proceed with the evaluation  of EnumCC(rmax), which includes all the pruning strategies used in the recurrent neighborhood search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  We compare our method with OneTreeCC(), the best enumeration method available in the literature  (Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  1DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='6084/m9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='figshare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='15043911  2https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='com/CompNet/SignedBenchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  3https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='com/CompNet/SignedStabilityBenchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  4DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='6084/m9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='figshare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='5700832.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='v5  18 / 30  Arınık et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' – Efficient Enumeration of the Optimal Solutions to the Correlation Clustering problem  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1  Evaluation of the MVMO-based Pruning Strategies  The MVMO property has an important role in EnumCC, due to its ability to prune unfeasible edit  operations in several parts of CoNS(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' To be able to consider relatively large graph orders n without  being limited by the long running time of an exact partitioning method, we evaluate its performance  based on Dataset2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' The complete results1 as well as our source code3 are publicly available online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  We apply CoNS(r) with and without the MVMO property onto all generated networks with r ∈  {3, 4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' The version with MVMO property refers to the method described in Section 4, whereas the  version without it is obtained by removing this property from Algorithms 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Due to space  considerations, Figure 8 illustrates only the results related to 4-edit distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' The results obtained for  d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='25 and d = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='00 are shown in separated subfigures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In each subfigure, the x-axis represents the  graph order n, and execution times (in seconds) are shown on the log-scaled y-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' The solid (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  dashed) plot lines correspond to CoNS(r) with (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' without) the MVMO property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Each plot line  corresponds to a specific value of ℓ0 and is represented with a specific color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Each shaded region depicts  a range of execution times around the plot line of the same color, based on the corresponding initial  signed network and its perturbed versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  The first thing to notice is how graph density affects the execution times with increasing values of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Indeed, the computational costs with and without the MVMO property are much larger with d = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='00  than those with d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Second, substantially increasing n affects the execution times without the  MVMO property, whereas including the MVMO property allows to better handle this effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Indeed,  considering d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='25 and the 4-edit distance, we observe that average execution times without the  MVMO property are approximately ℓ0 times larger than with the MVMO property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Finally, including  the MVMO property also allows to better handle increasing values of ℓ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Indeed, when n = 70 and  d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='25, the difference of average execution times between ℓ0 = 2 and ℓ0 = 6 without (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' with)  the MVMO property is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='8s (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='6s) with the 3-edit distance, and 595s (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' 63s) with the 4-edit  distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  To conclude this part, the MVMO property makes a substantial improvement on CoNS(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' This  improvement is apparent even with small values of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Note that, even a small improvement (1s or 2s)  can make a clear difference in terms of execution time for a solution space with 100 or more solutions,  since CoNS(r) is repeated for each solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Nevertheless, our results suggest that CoNS(4) should not  be used for computational purposes, even with the MVMO property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In the following, we investigate if  the improvements brought by the MVMO property and other pruning strategies allow EnumCC(rmax)  to compete with OneTreeCC().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='2  Evaluation of EnumCC  We now compare the results of EnumCC(rmax) against OneTreeCC() based on the synthetic signed  networks from Dataset1 and the real-world signed networks from Dataset3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' As shown in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1,  since CoNS(4) takes a substantial time, we rather apply EnumCC(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We run both methods with a  time limit of 12h and a limit of 50, 000 on the number of optimal solutions found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We first evaluate  the results from Dataset1 for several values of density d with a fixed value of n (Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1) and  several values of n with a fixed value of d (Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='2), then pass to the evaluation of the results  from Dataset3 (Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Regarding the synthetic graphs, we present a selection of the most relevant results in Figures 9a–10c,  for d ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='25, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='00}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' The complete results1 as well as our source code5,6 are available online, though.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  We first describe these plots generically here, before interpreting them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In these figures, there are 3  subfigures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Each subfigure corresponds to a specific value of qm (hence, a specific parameter set),  and displays the difference of execution times between EnumCC(3) and OneTreeCC() (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=', EnumCC(3)  minus OneTreeCC()), represented on the log-scaled y-axis of the plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' When such difference takes a  negative value, this means that our proposed method runs faster than OneTreeCC().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' The set of 20  graphs generated for each parameter set are indexed and shown on the x-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Finally, to guide our  discussion, we show for each graph the maximal number of solutions found by the method(s) within a  time limit and the number of jumps related to EnumCC(3), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' njump(EnumCC(3)), where the latter  is shown in parentheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1  Evaluation of EnumCC by Density  In this section, the results are for n = 36 and d ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' They are representative enough of  the results obtained with other values of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We first consider the d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='25 results, shown in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  5https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='com/CompNet/Sosocc  6https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='com/CompNet/EnumCC  19 / 30  Arınık et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' – Efficient Enumeration of the Optimal Solutions to the Correlation Clustering problem  0  1  10  100  1000  5000  30  40  50  60  70  ℓ0  2  4  6  without  MVMO  with  MVMO  Execution time (s)  Graph order (n)  (a) Instances with d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  0  1  10  100  1000  5000  30  40  50  60  70  Graph order (n)  Execution time (s)  ℓ0  2  4  6  without  MVMO  with  MVMO  (b) Instances with d = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Figure 8: Benchmark results for CoNS(r = 4) with vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' without MVMO-based pruning for d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='25 (8a)  and d = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='00 (8b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' The x-axis represents graph order n, and execution times (in seconds) are shown in  the log-scaled y-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  20 / 30  Arınık et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' – Efficient Enumeration of the Optimal Solutions to the Correlation Clustering problem  The first thing that we notice is how the performance is affected by qneg, independently of qm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Indeed,  we first see for qneg = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='3 that EnumCC(3) runs faster than OneTreeCC() in the overwhelming majority  of instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Then, for qneg = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='5, OneTreeCC() increases the number of instances, where it runs  faster, but the dominance of EnumCC(3) is still preserved to a lesser extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Finally, for qneg = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='7,  OneTreeCC() dominates EnumCC(3) on almost all instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  We observe that increasing qneg essentially results in the emergence of three features, which ad-  vantages OneTreeCC() over EnumCC(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' The first one is large values of njump(EnumCC(3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Since  a branch-and-bound tree needs to be built from scratch, each additional jump has an extra cost for  EnumCC(3) in terms of execution time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We observe that the values of njump(EnumCC(3)) are rel-  atively larger for mainly qneg = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='5 and qm = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='6}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' This shows that increasing qm is likely to  increase the dissimilarity between solutions, a fact that we have already pointed out in [6], for complete  signed networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  The second feature is a very large size of the solution space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' The results show that OneTreeCC() does  a much better job in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Indeed, when qneg = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='7, OneTreeCC() can solve instances associated  with a very large number of solutions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' 50, 000) within 5 minutes, whereas the same process often  takes several hours for EnumCC(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Nevertheless, such an extreme case happens only with some specific  graph topology, as in qneg = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Indeed, in the random network generation, increasing qneg with a  low graph density is more likely to produce instances having a large number of vertices with only few  positive edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Then, these vertices are often placed at the periphery of modules in the solutions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  they can easily change their module from one solution to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' OneTreeCC() seems to handle well  these vertices in the enumeration process, thanks to the mathematical modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Finally, the third feature, complementary to the previous one, is that instances having vertices with  only few positive edges are often easier to solve, so that an initial optimal solution is often found at  root relaxation, before passing to branch-and-bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' This usually results in a branch-and-bound tree  with fewer branches for enumerating alternative optimal solutions in OneTreeCC().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' It seems that this  substantially advantages OneTreeCC() over EnumCC(3), when there are many optimal solutions to  enumerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  For space considerations, we do not present the results for d ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Note that for these values,  we do not observe any extreme case with a very large number of solutions (as happens with d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='25 and  qneg = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Indeed, the maximum number of solutions for those instances is 2, 455.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' This is probably  because it is not as easy to break the underlying partition structure when graph density is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' This  fact seems to advantage EnumCC(3) over OneTreeCC().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Indeed, the dominance of EnumCC(3) persists  for qneg ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='5} and even better than those for d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='25 (see Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' This holds for qneg = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='7,  too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' OneTreeCC() outperforms EnumCC(3) only for qm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1, whereas this was true for all values of qm  for d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' These results confirm our previous observation: OneTreeCC() outperforms EnumCC(3)  on instances with specific graph topology, where low density and large proportion of negative edges  produce a very large number of solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In the other cases, EnumCC(3) performs much better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In the  following, we study whether increasing graph order n affects these observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='2  Evaluation of EnumCC by Graph Order  In this section, we analyze the effect of increasing the graph order, for n ∈ {40, 45, 50}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' To do so,  we focus only on instances with d = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='0 in order to reduce the total number of instances, hence the  processing time of our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Note that qneg approximately equals 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='7 in these instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  The  corresponding results are shown in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Overall, we see that the observations we made for n = 36  and d ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='0} in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1 are still valid when increasing n: EnumCC(3) performs much better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Furthermore, unlike the results obtained with d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='25, EnumCC(3) handles instances with a large value  of njump(EnumCC(3)) much better (when we exclude some exceptional instances with more than 10  jumps), and runs faster than OneTreeCC() in most of the instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' This point is even more valid when  increasing n further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Table 1: Number of optimal solutions found by the considered methods within the time limit of 12h for  unsolved instances with n = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  methods  instance no  qm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='4  qm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='5  qm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='6  1  5  6  11  16  19  2  11  12  13  15  18  20  2  6  10  12  13  17  19  20  OneTreeCC()  4  4  1  1  13  2  7  21  3  1  10  2  2  3  1  3  4  5  3  1  3  EnumCC(3)  10  13  6  5  255  16  217  44  8  1  115  46  13  45  10  32  6  60  7  4  65  Another interesting point is the hardness of instances, when increasing n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We say that an instance  is hard to solve when the resolution process takes too long, which amounts to exceeding a time limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  21 / 30  Arınık et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' – Efficient Enumeration of the Optimal Solutions to the Correlation Clustering problem 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
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+page_content='1280  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='(1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='qm=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='2  qm=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='4  qm=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='6  5  10  15  20  5  10  15  20  5  10  15  20  −100  −10  −101  10  100  1000  5000  Instance number  EnumCC(3) - OneTreeCC()  (c) Instances with d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='25 and qneg = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Figure 9: Results for n = 36, d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='25 and qneg ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='7}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' The log-scaled y-axis indicates the dif-  ference of execution times between EnumCC(3) and OneTreeCC() (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=', EnumCC(3) minus OneTreeCC()),  in seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We show for each graph the maximal number of solutions found by EnumCC(3) within a 12h  time limit, as well as the corresponding number of jumps, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' njump(EnumCC(3)), between parentheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  22 / 30  Arınık et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' – Efficient Enumeration of the Optimal Solutions to the Correlation Clustering problem  It is very important to analyze such cases in terms of the number of optimal solutions found by both  methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Moreover, the existence of such instances also indicates the maximum graph order, for  which the enumeration methods can produce full enumeration of all alternate optimal solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In our  experiments, a total of 21 such instances with n = 50 and qm > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='3 are not solved within the time  limit of 12h by both methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' These instances can be identified as the ones with y = 0 in Figure 10c,  and they are summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Each row corresponds to one of these methods, and each column  represents an instance, identified by its id and its associated qm value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  We see from Table 1 that  EnumCC(3) handles these time-consuming instances better than OneTreeCC() does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Indeed, among  the 21 instances not solved by any method, EnumCC(3) finds more solutions than OneTreeCC() for 20  instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Moreover, EnumCC(3) could solve 7 instances that OneTreeCC() could not, within the limit  of 12h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  To summarize the evaluation of EnumCC(3) with synthetic signed networks, there is not a single  method which always gives the best results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' On the one hand, OneTreeCC() handles very well the  instances with a very large solution space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  This extreme case is associated with a specific graph  topology, based on low graph density and large qneg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' On the other hand, EnumCC(3) performs better  in the rest of the instances, which constitutes the overwhelming majority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' This point is illustrated by  Table 2, which summarizes our results by showing the proportion of cases where EnumCC(3) runs faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Table 2: Summary regarding the proportion of the cases where EnumCC(3) is faster than OneTreeCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  We obtain these results by aggregating the instances by graph order n (excluding the instances, when the  difference of execution times between EnumCC(3) and OneTreeCC() is in the margin of 5 seconds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' N/A  indicates that there is no available entry due to the exclusion of the instances, when the time difference is  in the margin of 5 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  qm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1  qm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='2  qm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='3  qm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='4  qm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='5  qm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='6  d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='25  qneg = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='92  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='87  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='96  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='92  qneg = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='52  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='77  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='61  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='61  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='86  qneg = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='03  d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='50  qneg = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='3  N/A  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='00  qneg = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='5  N/A  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
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+page_content='75  qneg = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
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+page_content='89  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='87  d = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='00  qneg ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='98  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
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+page_content='94  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='95  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='98  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='3  Evaluation Based on Real-World Instances  Finally, we compare the results of EnumCC(3) against OneTreeCC() obtained on the real-world networks  of Dataset3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We run both methods with a time limit of 12h and a limit of 50, 000 on the number of  optimal solutions found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' The results are summarized in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Each column corresponds to a real-  world network, and they are sorted by increasing graph order n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We consider the first five networks as  medium-sized and the last three as large-sized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' The rows of Table 3 are organized in two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In the  first part, we detail the characteristics of the considered networks, as well as the graph imbalance, for  the sake of completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' The second part corresponds to the evaluation results of OneTreeCC() and  EnumCC(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Therein, we indicate the execution time of the enumeration process, in seconds, and the  number of optimal solutions found by these two methods within a time limit of 12h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  We can summarize these results in three points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' First, we notice that although the medium-sized net-  works are larger than the synthetic graphs from Dataset2, in practice both OneTreeCC() and EnumCC(3)  can handle real-world medium-sized networks in a very reasonable time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Second, we observe for medium-  sized networks that EnumCC(3) is faster than OneTreeCC() for the first one, whereas OneTreeCC()  dominates EnumCC(3) for the last four ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' EnumCC(3) takes longer because, in average, it spends  86% of its execution time to prove that there is no alternate optimal solution in the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Finally, as  expected, both methods could not solve all three large-sized networks within the time limit of 12h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' The  particularity of these networks is that they are very sparse (d ≈ 1%) and their respective solution spaces  contain at least 50, 000 optimal solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' For all these networks, EnumCC(3) always finds more optimal  solutions than OneTreeCC() does within the time limit, and it quickly explores a set of 50, 000 optimal  solutions thanks to its RNS component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Since OneTreeCC() struggles to find more alternate solutions  23 / 30  Arınık et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' – Efficient Enumeration of the Optimal Solutions to the Correlation Clustering problem 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
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+page_content='qm=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='2  qm=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='4  qm=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='6  5  10  15  20  5  10  15  20  5  10  15  20  −5000  −1000  −100  −10  −101  10  100  1000  5000  Instance number  EnumCC(3) - OneTreeCC()  (a) Instances with n = 40 and d = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' 37  (4)  4  (1)  2  (1)  4  (1)  10  (1)  6  (2)  8  (1)  2  (1)  4  (1)  6  (1)  2  (1)9  (2)  9  (1)  3  (1)  6  (1)1  (1)16  (1)  1  (1)  27  (2)  14  (2) 94  (3)113  (1)  9  (4)  1  (1)  52  (5)  1  (1)  23  (10)  6  (1)  18  (2)8  (2)16  (2)  194  (7)  7  (4)  13  (2)  21  (4)11  (2)  3  (1)  23  (5)10  (2)  1  (1) 8  (2)  10  (6)  17  (1)  6  (3)  42  (6)  6  (3)  4  (1)  8  (1)  16  (2)69  (10)  4  (2)  5  (2)5  (1)  1  (1)  3  (2)  13  (6)  3  (1)  4  (1)  24  (4)  7  (1)  qm=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='2  qm=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='4  qm=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='6  5  10  15  20  5  10  15  20  5  10  15  20  −5000  −1000  −100  −10  −101  10  100  1000  Instance number  EnumCC(3) - OneTreeCC()  (b) Instances with n = 45 and d = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' 3  (2)  36  (1)  2  (1)36  (2)  14  (1)6  (1)  2  (1)1  (1)  6  (1)8  (1)  1  (1)  1  (1)  2  (1)  24  (1)  2  (1)  5  (1)6  (1)6  (2)  1  (1)  8  (1) 10  (4)  2  (2)  89  (1)  21  (2)  13  (2)  6  (1)  9  (2)  2  (1)  6  (3)  3  (1)  5  (1)  14  (1)  15  (3)  3  (2)  1  (1)  255  (5)  40  (2)  102  (7)  16  (4)  6  (1) 1  (1)  45  (2)  4  (1)8  (1)  5  (2)  10  (4)  1  (1)  1  (1)  35  (3)  32  (3)  1  (1)  6  (4)  60  (8)  9  (2)  8  (3)  3  (1)  7  (3)  24  (2)  4  (1)  65  (3)  qm=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='2  qm=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='4  qm=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='6  5  10  15  20  5  10  15  20  5  10  15  20  −5000  −1000  −100  −10  −101  10  100  1000  Instance number  EnumCC(3) - OneTreeCC()  (c) Instances with n = 50 and d = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Figure 10: Results for n ∈ {40, 45, 50} and d = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' The log-scaled y-axis indicates the difference of  execution times between EnumCC(3) and OneTreeCC() (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=', EnumCC(3) minus OneTreeCC()), in seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  We show for each graph the maximal number of solutions found by EnumCC(3) within a time limit, as well  as the corresponding number of jumps, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' njump(EnumCC(3)), between parentheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  24 / 30  Arınık et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' – Efficient Enumeration of the Optimal Solutions to the Correlation Clustering problem  Table 3: Evaluation results for sparse real-world signed networks from Dataset3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Each column corresponds  to a real-world network and they are sorted by graph order n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' The rows are organized in two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' The  first six rows detail the characteristics of the considered networks, as well as the graph imbalance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' The last  four rows indicate the execution time of the enumeration process in seconds and the number of optimal  solutions found by OneTreeCC() and EnumCC(3) within a time limit of 12h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Medium-sized networks  Large-sized networks  CoW  51-54  CoW  54-57  CoW  55-58  CoW  61-64  Senate  108th  EGFR  Yeast  Macro-  phage  Characteristics  graph order n = |V |  61  67  72  75  99  313  575  660  graph size m = |E|  413  461  508  544  2,413  755  910  1,397  density d  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='23  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='21  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='19  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='006  prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' of neg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' edges qneg  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='09  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='41  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='34  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='09  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='33  # frustrated edges I(P)  15  27  29  34  1157  181  35  309  prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' of frustrated edges I(P)/|E|  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='48  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='24  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='22  Results  # opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' solutions for OneTreeCC()  46  34  41  201  22  112  50,000  251  # opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' solutions for EnumCC(3)  46  34  41  201  22  50,000  50,000  50,000  exec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' time for OneTreeCC()  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='70s  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='67s 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='62s 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='75s 858s  43,200s 24,600s 43,200s  exec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' time for EnumCC(3)  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='05s 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='38s  134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='87s 755.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='81s 2,424s  3,751s 1,871s 2,280s  within the time limit in two networks (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' EGFR and Macrophage), RNS appears to be useful in these  cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  To conclude this part, all these results also support our conclusion from Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='2, in that  there is not a single method which always gives the best results and both methods are complementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Moreover, these results allow us to make our conclusions from Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='2 more precise, in two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  First, it is more appropriate to use EnumCC(3), rather than OneTreeCC(), to explore a large number  of optimal solutions within a time limit, for large signed networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Nevertheless, it is more suitable to  favor OneTreeCC() over EnumCC(3) for sparse signed networks with 65 < n < 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  9  Conclusion  For most clustering problems, due to their NP-hard nature, exact approaches do not scale well even  when looking for a single optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In this work, we proposed an efficient enumeration method  to identify all optimal solutions of the CC problem for a given signed graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' It combines an exhaustive  enumeration strategy with neighborhoods of different sizes, designed for our problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In our experiments  based on synthetic and real-world signed networks, we first confirmed the findings of our previous work [6]  about the existence of multiple optimal solutions for incomplete signed networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We showed that such  networks can have a very large number of optimal solutions for the CC problem, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=', 50,000 and more  for graphs containing only tens of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Further investigation indicates that this extreme case of a  very large number of optimal solutions is associated with two specific graph topologies, corresponding to  1) low graph density and large proportion of negative edges for small- and medium-sized networks (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  n ≤ 100), and 2) very low graph density for large-sized networks (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' n > 100).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Otherwise, multiple  solutions can still exist, mostly for the graphs with considerably more imbalance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Finally, we also  showed that our method EnumCC(3) performs better than CPLEX’s OneTreeCC() in the overwhelming  majority of the considered networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Nevertheless, there is not a single method which always gives the  best results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Indeed, OneTreeCC() handles very well sparse medium-sized signed networks in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  We conclude that it is more appropriate to use OneTreeCC() in this case, whereas EnumCC(3) is more  suitable for all the remaining cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  We believe that this work opens new directions for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' First, the most straightforward  perspective is to consider weighted signed graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  This would require dealing with the generation  of the edge weights in our random graph model, and to extend the pruning strategies proposed in  this work to weighted signed graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Second, as we see in our experiments, the process of complete  enumeration can be very costly, particularly because of the process of finding an undiscovered solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  To accelerate this process, it could be very beneficial to use a heuristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Existing heuristics from the  25 / 30  Arınık et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' – Efficient Enumeration of the Optimal Solutions to the Correlation Clustering problem  literature are designed to find a single high-quality solution though, so a better approach would be to  redesign them by considering some tabu search-based operations allowing to escape from the discovered  optimal solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Third, one could study how robust the solution spaces are, by slightly introducing  some perturbations into the considered networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  This would also allow identifying critical vertices  when the underlying space of optimal solutions is changed, akin to the concept of vitality [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' This  could be done through either repeating the process from scratch for each perturbed signed graph, or  based on the definition of stability range [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Fourth, exploring the space of optimal solutions for other  clustering problems with unsigned networks would be another interesting research line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Indeed, the work  of Good et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' [26] showed the need of identifying multiple quasi-optimal solutions for the Modularity  Maximization problem, which consists in detecting a community structure in an unsigned graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' From  this perspective, an efficient enumeration method similar to ours could be implemented to see whether  their empirical findings apply to optimal solutions too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  This research benefited from the support of Agorantic research federation (FR  3621), as well as the FMJH (Jacques Hadamard Mathematics Foundation) through PGMO (Gaspard  Monge Program for Optimisation and operational research), and from the support to this program from  EDF, Thales, Orange and Criteo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  A  2-partition and 2-chorded cycle inequalities  For the sake of completeness, we detail below the 2-partition and 2-chorded cycle inequalities:  Let S, T ⊆ V be two nonempty disjoint subsets of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Then, the 2-partition inequality, illustrated  in Figure 11a, is defined as  �  u∈S  �  v∈T  xuv −  �  (u,v)∈S  u̸=v  xuv −  �  (u,v)∈T  u̸=v  xuv ≤ min{|S|, |T|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  (20)  Let C ⊆ E be a cycle of length at least 5, and C = {vivi+2|i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='., |C|−2}∪{v1v|C|−1, v2v|C|}  be a 2-chorded cycle of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Then, the 2-chorded cycle inequality, illustrated in Figure 11b, is  defined as  �  (u,v)∈C  xuv −  �  (u,v)∈C  xuv ≤ ⌊|C|  2 ⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  (21)  v1  v2  v3  v4  v5  v1  S  v2  v3  v4  T  v5  (a) 2-partition inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  v1  v2  v3  v4  v5  v6  v7  (b) 2-chorded cycle inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Figure 11: Illustrations of the 2-partition (left) and 2-chorded cycle (right) inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  B  Edit Distance Between Two Membership Vectors  Before calculating the edit distance between two membership vectors, we need to select one of them as  the reference vector, in order to adapt the module assignments of the other vector based accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Hence, the edit distance is calculated between the reference vector and this newly changed one, that  we call relative vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  The task of adapting the relative vector w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='t the reference one can be expressed as an assignment  problem, also known as maximum weighted bipartite matching problem, as already done in the litera-  ture [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Let πs and πt be two membership vectors of length n, associated with the partitions P s and  26 / 30  Arınık et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' – Efficient Enumeration of the Optimal Solutions to the Correlation Clustering problem  a)  b)  c)  d)  f)  g)  h)  i)  j)  k)  l)  m)  n)  o)  p)  r)  e)  z  u  u  u  u  u  u  u  u  u  u  u  u  u  u  u  u  u  v  v  v  v  v  v  v  v  v  v  v  v  v  v  v  v  z  z  z  z  z  z  z  z  z  z  z  z  z  z  z  v  z  Figure 12: All atomic 3-edit operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  P t, which contain ℓs and ℓt modules, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Also, since the edit distance is symmetric, without  loss of generality, let ℓs ≤ ℓt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Moreover, let CM be the ℓs × ℓt confusion matrix of πs and πt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' The  term CMij, with 1 ≤ i ≤ ℓs and 1 ≤ j ≤ ℓt, represents the number of vertices in the intersection of  modules M s  i and M t  j, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' |M s  i ∩ M t  j|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Then, we look for a bijection f : {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=', ℓs} → {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=', ℓt}  that maximizes the number of vertices common to pairs of modules from both membership vectors, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  max  ℓs  �  i=1  CMi,f(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  (22)  Since this problem can be modelled as an assignment or a maximum weighted bipartite matching  problem, it can be solved in various ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' One of them is through the well-known Hungarian algorithm,  whose complexity is O(n3) [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Nevertheless, the best polynomial time algorithm is currently based on  the network simplex method, and it runs in O(|V ||E| + |V |2log(|V |)) time using a Fibonacci heap data  structure [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' One final remark is about the case of ℓs < ℓt, in which there are |ℓt − ℓs| unassigned  module labels in πt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In this case, one can arbitrarily renumber these labels, starting from ℓs + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Finally, the edit distance between two membership vectors is calculated by simply counting the  number of cases where the module labels of the vertices in the reference and relative vectors are  different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  C  Proofs  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1  All Proofs Related to the MVMO Property for 3-edit Operations on Com-  plete Unweighted Signed Graphs  We complete the proof of Lemma 12 by verifying below the conditions of all atomic 3-edit oper-  ations for unweighted complete signed networks (illustrated in Figure 12), where  s#»t  V  = {u, v, z}  and (u, v), (u, z), (v, z) ∈ �E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Recall that �E = {(u, v) | (u, v) ∈ E and u, v ∈  s#»t  V  and (πs(u) =  πs(v) ∨ πt(u) = πs(v) ∨ πs(u) = πt(v) ∨ πt(u) = πt(v))}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  a) We have (γleft  u  = auv + auz) > (γright  u  = −auv − auz), (γleft  v  = auv + avz) > (γright  v  =  −auv − avz) and (γleft  z  = auz + avz) > (γright  z  = −auz − avz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We see that auv, auz and avz  cannot be negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  b) We have (γleft  u  = auv + auz) > (γright  u  = 0), (γleft  v  = auv + avz) > (γright  v  = −avz) and  (γleft  z  = auz + avz) > (γright  z  = −avz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We see that auv, auz and avz cannot be negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  c) We have (γleft  u  = auv + auz) > (γright  u  = 0), (γleft  v  = auv + avz) > (γright  v  = 0) and (γleft  z  =  auz + avz) > (γright  z  = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We see that auv, auz and avz cannot be negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  27 / 30  Arınık et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' – Efficient Enumeration of the Optimal Solutions to the Correlation Clustering problem  d) We have (γleft  u  = auv) > (γright  u  = −auv − auz), (γleft  v  = auv) > (γright  v  = −auv − avz) and  (γleft  z  = 0) > (γright  z  = −auz − avz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We see that auv, auz and avz must be positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  e) We have (γleft  u  = auv − auz) > (γright  u  = −auv + auz), (γleft  v  = auv − avz) > (γright  v  =  −auv + avz) and (γleft  z  = −auz − avz) > (γright  z  = auz + avz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We see that auv (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' auz and  avz) cannot be negative (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' positive).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  f) We have (γleft  u  = auv − auz) > (γright  u  = −auv), (γleft  v  = auv − avz) > (γright  v  = −auv) and  (γleft  z  = 0) > (γright  z  = auz + avz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We see that auv, auz and avz cannot be negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  g) We have (γleft  u  = auv) > (γright  u  = auz − auv), (γleft  v  = auv) > (γright  v  = avz − auv) and  (γleft  z  = −auz − avz) > (γright  z  = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We see that auv, auz and avz cannot be negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  h) We have (γleft  u  = auv) > (γright  u  = −auv), (γleft  v  = auv) > (γright  v  = −auv) and (γleft  z  =  −auz − avz) > (γright  z  = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We see that auv, auz and avz cannot be negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  i) We have (γleft  u  = auv) > (γright  u  = auz), (γleft  v  = auv − avz) > (γright  v  = avz) and (γleft  z  =  −auz − avz) > (γright  z  = +avz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We see that auv (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' auz and avz) cannot be negative (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  positive).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  j) We have (γleft  u  = auv) > (γright  u  = −auz), (γleft  v  = auv) > (γright  v  = −avz) and (γleft  z  = 0) >  (γright  z  = −auz + avz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We see that auv and avz (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' auz) must be positive (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' negative).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  k) We have (γleft  u  = auv) > (γright  u  = 0), (γleft  v  = auv − avz) > (γright  v  = 0) and (γleft  z  = 0) >  (γright  z  = avz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We see that auv and avz (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' auz) must be positive (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' negative).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  l) We have (γleft  u  = −auv) > (γright  u  = auz), (γleft  v  = −avz) > (γright  v  = auv) and (γleft  z  =  −auz) > (γright  z  = avz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We see that auv and avz (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' auz) must be positive (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' negative).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  m) We have (γleft  u  = −auv) > (γright  u  = 0), (γleft  v  = −avz) > (γright  v  = auv) and (γleft  z  = 0) >  (γright  z  = avz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We see that auv and avz (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' auz) must be positive (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' negative).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  n) We have (γleft  u  = −auv) > (γright  u  = auv − auz), (γleft  v  = −auv) > (γright  v  = auv + avz) and  (γleft  z  = −avz) > (γright  z  = −auz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We see that auv (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' auz and avz) cannot be negative  (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' positive).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  o) We have (γleft  u  = −auv) > (γright  u  = 0), (γleft  v  = 0) > (γright  v  = auv − avz) and (γleft  z  = 0) >  (γright  z  = −avz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We see that auv (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' auz and avz) cannot be negative (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' positive).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  p) We have (γleft  u  = −auv) > (γright  u  = −auz), (γleft  v  = 0) > (γright  v  = auv + avz) and (γleft  z  =  −avz) > (γright  z  = −auz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We see that auv (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' auz and avz) cannot be negative (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  positive).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  r) We have (γleft  u  = 0) > (γright  u  = −auv − auz), (γleft  v  = 0) > (γright  v  = −auv − avz) and  (γleft  z  = 0) > (γright  z  = −auz − avz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' We see that auv, auz and avz must be positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  References  [1]  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Ahuja, Ö.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Ergun, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Orlin, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Punnen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' “A survey of very large-scale neighborhood  search techniques”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In: Discrete Applied Mathematics 123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1-3 (2002), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' 75–102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1016/s0166-  218x(01)00338-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  [2]  Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Ales, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Knippel, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Pauchet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' “Polyhedral Combinatorics of the K-partitioning Problem with  Representative Variables”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In: Discrete Applied Mathematics 211 (2016), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' 1–14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  dam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  [3]  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Aref and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Wilson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' “Balance and frustration in signed networks”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In: Journal of Complex Networks  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='2 (2018), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' 163–189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1093/comnet/cny015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  [4]  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Arthur, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Hachey, Sahr K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Huso, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Kiester.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' “Finding all optimal solutions to the  reserve site selection problem”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In: Environmental and Ecological Statistics 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='2 (1997), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' 153–165.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' doi:  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1023/a:1018570311399.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  [5]  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Arınık, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Figueiredo, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Labatut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' “Multiple partitioning of multiplex signed networks: Application  to European parliament votes”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In: Social Networks 60 (2020), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' 83 –102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' doi: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='socnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  [6]  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Arınık, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Figueiredo, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Labatut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' “Multiplicity and Diversity: Analyzing the Optimal Solution Space  of the Correlation Clustering Problem on Complete Signed Graphs”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In: Journal of Complex Networks  (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1093/comnet/cnaa025.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  [7]  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Arınık, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Figueiredo, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Labatut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' “Signed Graph Analysis for the Interpretation of Voting Behav-  ior”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In: International Conference on Knowledge Technologies and Data-driven Business - International  Workshop on Social Network Analysis and Digital Humanities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  [8]  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Bansal, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Blum, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Chawla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' “Correlation Clustering”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In: 43rd Annual IEEE Symposium on  Foundations of Computer Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' 2002, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' 238–247.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1109/SFCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1181947.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  [9]  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Blum and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Roli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' “Metaheuristics in combinatorial optimization”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In: ACM Computing Surveys 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='3  (2003), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' 268–308.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1145/937503.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='937505.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  28 / 30  Arınık et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' – Efficient Enumeration of the Optimal Solutions to the Correlation Clustering problem  [10]  Michael Brusco and Douglas Steinley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' “K-balance partitioning: An exact method with applications to  generalized structural balance and other psychological contexts”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In: Psychological Methods 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='2 (2010),  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' 145–157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1037/a0017738.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  [11]  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Camm, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Polasky, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Solow, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Csuti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' “A note on optimal algorithms for reserve site selection”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  In: Biological Conservation 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='3 (1996), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' 353–355.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1016/0006-3207(95)00132-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  [12]  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Cartwright and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Harary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' “Structural balance: A generalization of Heider’s theory”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In: Psychological  Review 63 (1956), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' 277–293.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1037/h0046049.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  [13]  Moses Charikar, Neha Gupta, and Roy Schwartz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' “Local Guarantees in Graph Cuts and Clustering”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In:  Integer Programming and Combinatorial Optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Springer International Publishing, 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' 136–  147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1007/978-3-319-59250-3\\_12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  [14]  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Damaschke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' “Fixed-Parameter Enumerability of Cluster Editing and Related Problems”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In: Theory of  Computing Systems 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='2 (2010), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' 261–283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1007/s00224-008-9130-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  [15]  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Danna, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Fenelon, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Gu, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Wunderling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' “Generating Multiple Solutions for Mixed Integer  Programming Problems”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In: International Conference on Integer Programming and Combinatorial Opti-  mization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Springer, 2007, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' 280–294.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1007/978-3-540-72792-7\\_22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  [16]  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' DasGupta, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Enciso, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Sontag, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' “Algorithmic and complexity results for decompo-  sitions of biological networks into monotone subsystems”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In: Biosystems 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1 (2007), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' 161–178.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' doi:  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='biosystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  [17]  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Davis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' “Clustering and structural balance in graphs”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In: Human Relations 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='2 (1967), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' 181–187.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1177/001872676702000207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  [18]  Erik D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Demaine, Dotan Emanuel, Amos Fiat, and Nicole Immorlica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' “Correlation clustering in general  weighted graphs”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In: Theoretical Computer Science 361.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='2-3 (2006), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' 172–187.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='tcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  [19]  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Doreian, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Lloyd, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Mrvar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' “Partitioning large signed two-mode networks: Problems and  prospects”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In: Social Networks 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='2 (2013), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' 178–203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='socnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  [20]  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Doreian and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Mrvar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' “A partitioning approach to structural balance”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In: Social Networks 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='2 (1996),  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' 149–168.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1016/0378-8733(95)00259-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  [21]  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Doreian and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Mrvar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' “Structural balance and signed international relations”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In: Journal of Social  Structure 16 (2015), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='21307/joss-2019-012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  [22]  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Esteban, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Mayoral, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Ray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' “Ethnicity and Conflict: An Empirical Study”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In: American Economic  Review 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='4 (2012), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' 1310–1342.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1257/aer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1310.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  [23]  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Figueiredo and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Moura.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' “Mixed Integer Programming Formulations for Clustering Problems Related  to Structural Balance”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In: Social Networks 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='4 (2013), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' 639–651.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='socnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  [24]  Matteo Fischetti, Fred Glover, and Andrea Lodi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' “The feasibility pump”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In: Mathematical Programming  104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1 (2005), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' 91–104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1007/s10107-004-0570-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  [25]  Michael L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Fredman and Robert Endre Tarjan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' “Fibonacci heaps and their uses in improved network  optimization algorithms”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In: Journal of the ACM (JACM) 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='3 (1987), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' 596–615.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1145/  28869.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='28874.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  [26]  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Good, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' de Montjoye, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Clauset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' “Performance of modularity maximization in practical  contexts”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In: Physical Review E 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='4 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1103/physreve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='046106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  [27]  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Grötschel and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Wakabayashi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' “A cutting plane algorithm for a clustering problem”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In: Mathematical  Programming 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1-3 (1989), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' 59–96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1007/bf01589097.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  [28]  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Grötschel and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Wakabayashi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' “Facets of the clique partitioning polytope”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In: Mathematical Pro-  gramming 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1-3 (1990), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' 367–387.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1007/bf01580870.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  [29]  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Gürsoy and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Badur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' “Extracting the signed backbone of intrinsically dense weighted networks”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In:  arXiv e-prints, arXiv:2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='05216 (2020), arXiv:2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='05216.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' arXiv: 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='05216.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  [30]  IBM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' IBM ILOG CPLEX 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='8 User Manual IBM Corporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  [31]  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Jurafsky and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Martin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Speech and Language Processing: An Introduction to Natural Language  Processing,Computational Linguistics, and Speech Recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' 2nd edition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Prentice-Hall, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=', 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  [32]  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Keuper, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Lukasik, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Singh, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Yarkony.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' “A Benders Decomposition Approach to Correlation  Clustering”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In: The International Conference for High Performance Computing, Networking, Storage, and  Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  [33]  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Koschützki, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Lehmann, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Peeters, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Richter, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Tenfelde-Podehl, and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Zlotowski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' “Central-  ity Indices”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In: Network Analysis: Methodological Foundations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' 3418.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Lecture Notes in Computer  Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' 2005, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' 16–61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1007/978-3-540-31955-9_3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  29 / 30  Arınık et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' – Efficient Enumeration of the Optimal Solutions to the Correlation Clustering problem  [34]  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Kuhn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' “The Hungarian method for the assignment problem”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In: Naval Research Logistics Quarterly  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1-2 (1955), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' 83–97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1002/nav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='3800020109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  [35]  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Kunegis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Lommatzsch, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Bauckhage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' “The slashdot zoo”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In: Proceedings of the 18th interna-  tional conference on World wide web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' ACM Press, 2009, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' 741–750.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1145/1526709.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1526809.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  [36]  Xin Liu, Hui-Min Cheng, and Zhong-Yuan Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' “Evaluation of Community Detection Methods”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In:  IEEE Transactions on Knowledge and Data Engineering (2019), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' 1–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1109/tkde.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  2911943.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  [37]  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Ma and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Hao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' “A multiple search operator heuristic for the max-k-cut problem”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In: Annals of  Operations Research 248.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1 (2016), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' 365–403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1007/s10479-016-2234-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  [38]  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Massa and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Avesani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' “Controversial users demand local trust metrics: an experimental study on  Epinions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='com community”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In: Proceedings of the 20th national conference on Artificial intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' 2005, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' 121–126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  [39]  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Mehrotra and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Trick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' “Cliques and clustering: A combinatorial approach”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In: Operations Research  Letters 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1 (1998), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' 1–12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1016/s0167-6377(98)00006-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  [40]  George Nemhauser and Laurence Wolsey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Integer and Combinatorial Optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Wiley, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  [41]  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Nowozin and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Jegelka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' “Solution stability in linear programming relaxations”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In: Proceedings of the  26th Annual International Conference on Machine Learning - ICML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' ACM Press, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1145/  1553374.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1553473.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  [42]  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Oosten, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Rutten, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Spieksma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' “The clique partitioning problem: Facets and  patching facets”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In: Networks 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='4 (2001), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' 209–226.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1002/net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='10004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  [43]  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Pevehouse, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Nordstrom, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Warnke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' “The Correlates of War 2 International Governmental  Organizations Data Version 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='0”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In: Conflict Management and Peace Science 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='2 (2004), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' 101–119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1080/07388940490463933.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  [44]  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Queiroga, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Subramanian, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Figueiredo, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Frota.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' “Integer programming formulations and ef-  ficient local search for relaxed correlation clustering”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In: Journal of Global Optimization (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' doi:  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1007/s10898-020-00989-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  [45]  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Tan and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Lü.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' “An evolutionary game approach for determination of the structural conflicts in signed  networks”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In: Scientific Reports 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1 (2016), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' 22022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1038/srep22022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  [46]  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' Zasĺavsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' “Balanced decompositions of a signed graph”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' In: Journal of Combinatorial Theory, Series  B 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1 (1987), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' 1–13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='1016/0095-8956(87)90026-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
+page_content='  30 / 30' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E5T4oBgHgl3EQfMQ4D/content/2301.05479v1.pdf'}
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+Point Production of a Nonrelativistic Unparticle
+Recoiling Against a Particle
+Eric Braaten
+Department of Physics, The Ohio State University, Columbus, OH 43210, USA
+Hans-Werner Hammer
+Technische Universit¨at Darmstadt, Department of Physics,
+Institut f¨ur Kernphysik, 64289 Darmstadt, Germany and
+ExtreMe Matter Institute EMMI and Helmholtz
+Forschungsakademie Hessen f¨ur FAIR (HFHF),
+GSI Helmholtzzentrum f¨ur Schwerionenforschung GmbH, 64291 Darmstadt, Germany
+(Dated: January 12, 2023)
+Abstract
+A nonrelativistic unparticle can be defined as an excitation created by an operator with a definite
+scaling dimension in a nonrelativistic field theory with an approximate conformal symmetry. The
+point production rate of an unparticle has power-law dependence on its total energy with an
+exponent determined by its scaling dimension. We use the exact result for the 3-point function
+of primary operators in a nonrelativistic conformal field theory to derive the contribution to the
+point production rate of the unparticle from its decay into another unparticle recoiling against a
+particle. In the case where the conformal symmetry is broken by a large positive scattering length,
+we deduce the exponent of the energy in the point production rate of the loosely bound two-particle
+state recoiling against a particle with large relative momentum.
+1
+arXiv:2301.04399v1  [hep-th]  11 Jan 2023
+
+I.
+INTRODUCTION
+A definition of an elementary particle that is popular in group-theoretic circles is an irre-
+ducible representation of the Poincar´e group. The concept of an unparticle was introduced
+by Georgi [1]. An unparticle is a system created by a local operator with a definite scaling
+dimension in a conformal field theory (CFT). It can therefore be defined as an irreducible rep-
+resentation of the conformal symmetry group. The conformal group on a 3+1-dimensional
+Minkowski space-time is a 15-dimensional group that includes the Poincar´e group and scale
+transformations as subgroups. A relativistic unparticle is characterized by a single number:
+the scaling dimension of the operator. If the CFT belongs to a hidden sector beyond the
+Standard Model of particle physics, the unparticle cannot be observed directly. However it
+can be observed indirectly through the momentum distribution of Standard Model particles
+produced in association with the unparticle [1]. If the unparticle is produced in association
+with a single Standard Model particle, the invariant mass distribution of the unparticle can
+be determined by measuring the recoil-momentum distribution of the Standard Model par-
+ticle. It has power-law behavior with an exponent determined by the scaling dimension of
+the unparticle. The existence of unparticles in a hidden sector would produce novel signals
+in high energy colliders [2–4]. The CMS collaboration has searched for signals of unparticles
+in pp collisions at the Large Hadron Collider [5–7].
+Hammer and Son recently pointed out that unparticles can also arise in nonrelativistic
+physics [8]. The nonrelativistic conformal symmetry group on a 3+1-dimensional Galilean
+space-time is a 13-dimensional group that includes the Galilean group and scale transfor-
+mations as subgroups.1 It is also called the Schr¨odinger group, because it is the symmetry
+group of the free Schr¨odinger equation. A nonrelativistic conformal field theory (NRCFT)
+is a field theory with nonrelativistic conformal symmetry [10]. A nonrelativistic unparticle
+is a system created by a local operator with a definite scaling dimension in such a theory. In
+contrast to the relativistic case, a nonrelativistic unparticle is characterized by two numbers:
+its kinetic mass M and the scaling dimension ∆ of the operator [8].
+Hammer and Son pointed out that systems of low-energy neutrons produced by a short-
+distance reaction provide physical examples of nonrelativistic unparticles [8]. Neutrons have
+a negative scattering length a that is much larger than their effective range r0. As a con-
+sequence, the behavior of a system of neutrons whose kinetic energies in their center-of-
+momentum (CM) frame are all in the scaling region between 1/(mna2) and 1/(mnr2
+0), where
+mn is the neutron mass, is approximately scale invariant. In the unitary limit 1/a → 0,
+r0 → 0, a nonrelativistic effective field theory describing the neutrons has nonrelativistic
+conformal symmetry. A convenient basis for local operators in the effective field theory are
+those with definite scaling behavior under the conformal symmetry. A system of N neutrons
+created by a local operator with scaling dimension ∆N can be interpreted as an unparticle
+with mass Nmn. For the 2-neutron unparticle, the lowest scaling dimension is ∆2 = 2.
+1 For a discussion of the relation to the relativistic conformal group, see e.g. Ref. [9].
+2
+
+0.1
+1
+10
+E/εnn
+dR/dE [arbitrary units]
+  2n
+  3n
+  4n 
+-0.5
+E
+E
+E
+E
+E
+E
+0.5
+1.77
+2.57
+3
+5.5
+FIG. 1.
+Scaling prediction of Ref. [8] for the invariant energy distribution dR/dE of 2, 3, and
+4 neutrons in their CM frame. The transition from the free phase-space behavior to the scaling
+region around E ∼ εnn is illustrated by the dotted lines.
+For unparticles with larger N, the scaling dimensions are transcendental numbers. For the
+3-neutron unparticle, the lowest scaling dimension is ∆3 = 4.27272. The lowest scaling
+dimensions for 4 and 5 neutrons are 5.07 and 7.6, respectively. A summary of the lowest
+scaling dimensions for up to 6 neutrons is given in Table I of [10].
+An N-neutron unparticle can be created by a short-distance nuclear reaction of the form
+A1 +A2 −→ B +(nn . . .) [8]. The invariant energy E of the N neutrons, which is their total
+kinetic energy in their CM frame, can be determined by measuring the momentum of the
+recoiling nucleus B or by detecting the neutrons directly. There is a scaling region of E in
+which the differential cross section has the scaling behavior E∆N−5/2dE. The corresponding
+power-law behavior for N noninteracting particles as E approaches the threshold is governed
+by the N-particle phase space, E(3N−5)/2dE. In the case of fermions, there is an additional
+Pauli suppression factor E for each pair of identical particles. For E of the order of the energy
+scale set by the neutron-neutron scattering length εnn = 1/(mna2), there is a transition from
+the phase-space behavior for non-interacting particles to the power-law behavior governed
+by ∆N in the scaling region. This nontrivial scaling behavior is the smoking gun for an
+unparticle. The resulting predictions for the invariant energy distributions of 2, 3, and 4
+neutrons are shown in Fig. 1. In the case of 2 neutrons, there is a maximum in the region
+E ∼ εnn. For larger neutron numbers, there is a change in the slope of the log-log plot in
+the region E ∼ εnn.
+The predicted invariant energy distribution, dR/dE, for four neutrons in Fig. 1 is not
+consistent with a resonance-like structure at E/εnn ≈ 20 in the point production of four
+3
+
+neutrons.
+Such a structure was recently observed in the 4n spectrum measured in the
+knock-out reaction 8He(p, pα)4n by Duer et al. [11]. However, the neutron distribution of
+the initial 8He nucleus clearly plays a role in this process and the applicability of the point-
+production approximation is questionable as the neutrons are emitted from a 8He source.
+A recent theoretical study of the reaction [12] attributes the structure to the final state
+interaction among the four neutrons and the presence of preexisting four neutrons in the
+periphery of 8He nucleus projectile. While the former effect is captured in the the invariant
+4n energy distribution of Fig. 1, the latter is not.
+In Ref. [13], we pointed out that a low-energy system of neutral charm mesons created by
+a short-distance reaction is an unparticle because of the X(3872) resonance in the D∗0 ¯D0 +
+D0 ¯D∗0 channel [14]. The scattering length a in that channel is large and negative. Because
+the resonant channel is a superposition of charm mesons, there is no Efimov effect [15, 16].
+The behavior of a system of neutral charm mesons whose kinetic energies in their CM frame
+are all in the scaling region between 1/(2µa2) and 1/(2µr2
+0), where µ is the reduced mass
+of D∗0 and ¯D0, is therefore approximately scale invariant. In the unitary limit 1/a → 0,
+r0 → 0, a nonrelativistic effective field theory describing the neutral charm mesons has
+nonrelativistic conformal symmetry. A convenient basis for local operators in the effective
+field theory are those with definite scaling behavior under the conformal symmetry. A system
+of N neutral charm mesons created by a local operator with scaling dimension ∆N can be
+interpreted as an unparticle. The scaling dimensions for 2-charm-meson unparticles are the
+same as for two neutrons. The lowest scaling dimensions for 3-charm-meson unparticles
+are ∆3 = 3.10119 for D0 ¯D∗0D0 and ¯D0D∗0 ¯D0 as well as ∆3 = 3.08697 for D0 ¯D∗0 ¯D∗0 and
+¯D0D∗0D∗0. In each case, there are two pairs of particles with a large scattering length a.
+The slight difference in the two values of ∆3 is due to the different masses of the D0 and D∗0.
+The 2-charm-meson unparticle can be observed through the recoil-momentum spectrum of
+the kaon in the inclusive decay B± → K± + (anything). A 3-charm-meson unparticle can
+be observed through the prompt production of X(3872) D0 at the Large Hadron Collider.
+Because the scattering length a in the case of neutral charm mesons is large and posi-
+tive, the unparticles have components that include bound states. There are reactions for
+producing final states that include bound states whose rates have power-law behavior with
+exponents determined by conformal symmetry. The simplest such reaction is the production
+of X(3872) D0 from the creation of a 3-charm-meson unparticle at a point, whose rate scales
+as a power of the unparticle energy E [13].
+In this paper, we consider the production of a pair of unparticles from the creation of a
+single unparticle at a point. We use the analytic result for the 3-point function of primary
+operators in a NRCFT to derive the contribution to the production rate from another
+unparticle recoiling against a single particle. In the case where the conformal symmetry is
+broken by a large positive scattering length, we deduce the exponent of the energy in the
+point production rate of the loosely bound two-particle state recoiling against a particle with
+large relative momentum.
+4
+
+�
+n
+FIG. 2.
+Space-time propagator for unparticle n with kinetic mass Mn and scaling dimension ∆n.
+II.
+THREE-POINT FUNCTION FOR PRIMARY OPERATORS
+We start by calculating the Fourier transform of the 3-point function for primary operators
+in a NRCFT. We take the dimension of space to be D, and we denote a space-time position
+by x = (x, t). An operator φ†
+3(x) with scaling dimension ∆3 creates an unparticle with
+mass M3.
+An operator φ2(x) with scaling dimension ∆2 annihilates an unparticle with
+mass M2. An operator φ1(x) with scaling dimension ∆1 annihilates an unparticle with mass
+M1 = M. The masses satisfy the constraint M1 + M2 = M3 from Galilean symmetry. We
+are particularly interested in the case where φ1(x) annihilates a single particle with mass
+M, in which case its scaling dimension is ∆1 = D/2.
+A.
+Propagators
+The space-time propagator for a primary operator φn(x) with mass Mn and scaling di-
+mension ∆n is
+�
+φn(x1) φ†
+n(x2)
+�
+= Cn (t12)−∆nθ(t12) exp
+�
+iMn x2
+12
+2 t12
+�
+,
+(1)
+where tij = ti − tj, x2
+ij = (xi − xj)2, and Cn is a constant. We represent the space-time
+propagator by the diagram in Fig. 2.
+The energy-momentum propagator is obtained by Fourier-transforming in both space-
+time positions and then factoring out an energy-momentum delta function. It is advan-
+tageous to take the variables (x1 + x2)/2 and x1 − x2 for the Fourier transform.
+After
+performing the trivial Fourier transform in (x1 + x2)/2, the remaining Fourier transform in
+space is a Gaussian integral that depends on the momentum p:
+�
+dDx12 exp
+�
+− i p · x12 + iMn x2
+12
+2t12
+�
+=
+�
+2πi t12
+Mn
+�D/2
+exp
+�
+− i t12
+2Mn
+p2
+�
+.
+(2)
+The subsequent Fourier transform in time gives a simple function of the energy E:
+� ∞
+0
+dt12 (it12)D/2−∆n exp
+�
+i E t12 − i p2
+2Mn
+t12
+�
+= −i Γ
+�D
+2 + 1 − ∆n
+� � p2
+2Mn
+− E
+�∆n−D/2−1
+.
+(3)
+With an appropriate choice for the constant Cn in Eq. (1), our final result for the energy-
+momentum propagator is
+Dn(E, p) = −i C′
+n
+� p2
+2Mn
+− E
+�∆n−D/2−1
+,
+(4)
+5
+
+�
+3
+1
+2
+FIG. 3.
+Space-time 3-point function for primary operators associated with unparticles 1, 2, and
+3.
+where C′
+n is a constant.
+In the case where φ1(x) annihilates a single particle, its scaling dimension is ∆1 = D/2
+and the propagator in Eq. (4) has a simple pole in the energy E at p2/(2M). With the
+conventional choice C′
+1 = 1, the discontinuity in the energy is
+D1(E + iϵ, p) − D1(E − iϵ, p) = 2π δ
+�
+E − p2
+2M
+�
+.
+(5)
+For other unparticles, the discontinuity in the energy is
+Dn(E+iϵ, p)−Dn(E−iϵ, p) = 2C′
+n sin
+�
+π(∆n−D/2)
+� �
+E− p2
+2Mn
+�∆n−D/2−1
+θ
+�
+E− p2
+2Mn
+�
+. (6)
+B.
+Space-time three-point function
+The 3-point function
+�
+φ1(x1) φ2(x2) φ†
+3(x3)
+�
+for primary operators in a NRCFT was first
+considered by Henkel in 1993 [17]. The 3-point function can be represented by the diagram
+in Fig. 3. He used the Schr¨odinger symmetry to determine the 3-point function analytically
+up to a scaling function of a single variable:
+�
+φ1(x1) φ2(x2) φ†
+3(x3)
+�
+= (t13)−∆13,2/2θ(t13) exp
+�
+iM1 x2
+13
+2 t13
+�
+×(t23)−∆23,1/2θ(t23) exp
+�
+iM2 x2
+23
+2 t23
+�
+(t12)−∆12,3/2 Φ(w),
+(7)
+where ∆ij,k = ∆i + ∆j − ∆k and the argument of the scaling function Φ is
+w =
+�
+t23(x1 − x3) − t13(x2 − x3)
+�2
+2 t13 t23 t12
+= 1
+2
+�x2
+12
+t12
+− x2
+13
+t13
++ x2
+23
+t23
+�
+.
+(8)
+The first expression shows that the sign of w is the same as the sign of t12. The scaling
+function Φ(w) depends also on the masses M1 and M2 and on the scaling dimensions ∆1,
+6
+
+�
+3
+2
+1
+FIG. 4.
+Fourier-transformed 3-point function for primary operators associated with unparticles
+1, 2, and 3.
+∆2, and ∆3. The 3-point function in Eq. (7) is symmetric under the interchange of the
+subscripts 1 and 2.
+An integral representation for the scaling function Φ(w) was first obtained by Henkel
+and Unterberger from the 3-point function for a CFT in two higher dimensions [18]. It has
+also been obtained by Fuertes and Moroz and by Volovich and Wen using holography and
+the AdS/CFT correspondence [19, 20]. In the Anti-de Sitter space (AdS) formulation, the
+external points in Fig. 3 are on the boundary of AdS and the blob is replaced by a point
+interaction inside the bulk of AdS. In Refs. [18, 19], the 3-point function is expressed as
+a function of Euclidean times. The expression in Ref. [20] in terms of real times is more
+directly useful for our purposes. The integral representation for the scaling function is
+Φ(w) = C12,3
+� +∞
+−∞
+du (u + iϵ)−∆13,2/2 e−iM1u
+� +∞
+−∞
+dv (v + iϵ)−∆23,1/2 e−iM2v
+×
+�
+u − v + (1 + iϵ)w
+�−∆12,3/2,
+(9)
+where C12,3 is a constant. Because the signs of t12 and w are identical (cf. Eq.(8)), the
+product of (t12)−∆12,3/2 and the last factor in Eq. (9) has a well-defined imaginary part:
+(t12)−∆12,3/2 �
+u − v + (1 + iϵ)w
+�−∆12,3/2 =
+�
+t12 (u − v + w) + iϵ
+�−∆12,3/2.
+(10)
+The scaling function can also be expressed analytically in terms of a confluent hypergeometric
+function [20].
+C.
+Fourier transform in space
+The Fourier transform of the 3-point function in position space can be represented by
+the diagram in Fig. 3. To evaluate the Fourier transforms in the three spatial positions, it
+7
+
+is convenient to express the last factor in Eq. (10) as the integral of an exponential:
+�
+t12(u − v + w) + iϵ
+�−∆12,3/2 =
+�
+eiπ/2 |t12|
+�−∆12,3/2
+1
+Γ(∆12,3/2)
+×
+� ∞
+0
+dm m∆12,3/2−1 exp
+�
+− m[ϵ − i sign(t12)(u − v + w)]
+�
+.
+(11)
+The Fourier transform can be expressed as the product of the momentum-conserving delta
+function δD(p1 + p2 − p3) and a Gaussian integral in x13 and x23. In the case t12 > 0, the
+Gaussian integral is
+�
+dDx13 exp
+�
+− i p1 · x13 + iM1 x2
+13
+2t13
+� �
+dDx23 exp
+�
+− i p2 · x23 + iM2 x2
+23
+2t23
+�
+e+i m w
+=
+�
+2πi
+t12t13
+t12M1 + t23m
+�D/2 �
+2πi
+t23(t12M1 + t23m)
+t12M1M2 + t13M1m + t23M2m
+�D/2
+× exp
+�
+− it12t13(M2 + m)p2
+1 + t12t23(M1 − m)p2
+2 + t13t23mp2
+3
+2(t12M1M2 + t13M1m + t23M2m)
+�
+.
+(12)
+We can go to the center-of-momentum (CM) frame by setting p3 = 0 and |p1| = |p2| = p.
+In the case t12 > 0, the 3-point function with the momentum delta function factored out
+reduces to
+C′
+12,3 (t13)(D−∆13,2)/2θ(t13) (t23)(D−∆23,1)/2θ(t23) |t12|−∆12,3/2
+� ∞
+0
+dm m∆12,3/2−1 e−mϵ
+×
+� +∞
+−∞
+du (u + iϵ)−∆13,2/2 e−i(M1−m)u
+� +∞
+−∞
+dv (v + iϵ)−∆23,1/2 e−i(M2+m)v
+×
+�
+t12M1M2
+t12M1M2 + t13M1m + t23M2m
+�D/2
+exp
+�
+− i
+t12(t23M1 + t13M2 + t12m)
+2(t12M1M2 + t13M1m + t23M2m)p2
+�
+,
+(13)
+where C′
+12,3 is a constant. The integrals over u and v can be evaluated analytically by closing
+the integration contour in the lower half-plane:
+� +∞
+−∞
+du (u + iϵ)−∆/2 e−iMu = e−iπ∆/4
+2π
+Γ(∆/2)M ∆/2−1θ(M).
+(14)
+After these integrals have been evaluated, the limit ϵ → 0 can be taken in the remainder.
+The choice t12 > 0 breaks the symmetry under interchange of the subscripts 1 and 2. Given
+this choice, it is convenient to change to a dimensionless integration variable x = m/M1.
+Our final result for the spatial Fourier transform of the 3-point function in the CM frame
+with t12 > 0 is
+C′′
+12,3 (t13)(D−∆13,2)/2θ(t13) (t23)(D−∆23,1)/2θ(t23) |t12|−∆12,3/2
+×
+� 1
+0
+dx x∆12,3/2−1 (1 − x)∆13,2/2−1 �
+1 + (M1/M2)x
+�∆23,1/2−1
+×
+�
+t12
+t12 + (M1/M2)xt13 + xt23
+�D/2
+exp
+�
+− i t12[xt12 + (M2/M1)t13 + t23]
+2M2[t12 + (M1/M2)xt13 + xt23]p2
+�
+, (15)
+where C′′
+12,3 is a constant.
+8
+
+D.
+Fourier transform in time
+The Fourier transforms in the three times can be expressed as the product of the energy-
+conserving delta function δ(E1 + E2 − E3), a Fourier transform in t13 with energy E1, and a
+Fourier transform in t23 with energy E2. We first isolate the contribution from the Fourier
+transform in t13 from the region t13 ≫ t23 and then evaluate the Fourier transform in t23.
+In the limit t13 ≫ t23, the exponential in Eq. (15), reduces to
+exp
+�
+− i t12
+�
+xt12 + (M2/M1)t13 + t23
+�
+2M2
+�
+t12 + (M1/M2)xt13 + xt23
+�p2
+�
+−→ exp
+�
+− i p2
+2M1
+t13
+�
+exp
+�
+− i[1 − 2x − (M2/M1)x] p2
+2M2[1 + (M1/M2)x]
+t23
+�
+,
+(16)
+where the identity t12 = t13 − t23 has been used. The contribution to the Fourier transform
+in t13 from the region t13 ≫ t23 can be reduced to the integral
+� ∞
+0
+dt13 (it13)(D−∆13,2−∆12,3)/2 exp
+�
+iE1t13 − i p2
+2M1
+t13
+�
+= −i Γ
+�
+D/2 + 1 − ∆1
+� � p2
+2M1
+− E1
+�∆1−D/2−1
+.
+(17)
+This integral is the propagator for φ1 multiplied by a constant. The subsequent Fourier
+transform in t23 can be reduced to the integral
+� ∞
+0
+dt23 (it23)(D−∆23,1)/2 exp
+�
+iE2t23 − i[1 − 2x − (M2/M1)x] p2
+2M2[1 + (M1/M2)x]
+t23
+�
+= −i Γ
+�
+[D − ∆23,1]/2 + 1
+� �[1 − 2x − (M2/M1)x] p2
+2M2[1 + (M1/M2)x]
+− E2
+�(∆23,1−D)/2−1
+.
+(18)
+We now specialize to the case of φ1(x) being the operator that annihilates a single particle.
+Its scaling dimension is ∆1 = D/2, so the integral in Eq. (17) has a simple pole at E1 =
+p2/(2M1). The residue of the pole in the Fourier-transformed three-point function is
+Gpole(E2, p) = C′′′
+12,3
+� 1
+0
+dx x∆12,3/2−1 (1 − x)∆13,2/2−1 �
+1 + (M1/M2)x
+�∆23,1/2−1−D/2
+×
+� p2
+2M2
+− E2 −
+(M3/M2)x
+1 + (M1/M2)x
+p2
+2M12
+�(∆23,1−D)/2−1
+,
+(19)
+where M12 = M1M2/M3 is a reduced mass and C′′′
+12,3 is a constant. The integrand has been
+expressed as a function of the energy E2−p2/(2M2) of the unparticle relative to its threshold
+and the total energy p2/(2M12) of the unparticle at its threshold and the particle.
+In the limit E2 → p2/(2M2) with p2 fixed, the integral over x in Eq. (19) has a divergent
+term that comes from the region near the lower endpoint.
+The divergent factor is the
+propagator of the unparticle. The limiting behavior of the 3-point function in Eq. (19) as
+E2 → p2/(2M2) is
+Gpole(E2, p) −→ C′′′′
+12,3 D2(E2, p)
+� p2
+2M12
+�−∆12,3/2
+,
+(20)
+9
+
+where C′′′′
+12,3 is a constant.
+This is our result for the scaling behavior of the Fourier-
+transformed 3-point function at large momentum p.
+The exponent −∆12,3/2 in Eq. (20) can also be deduced from the analytic result for an
+integral whose integrand has the same dependence on x near the lower endpoint as the
+integrand in Eq. (19):
+� 1
+0
+dx x∆12,3/2−1 (1 − x)∆13,2/2−1
+� p2
+2M2
+− E2 − x(M3/M2) p2
+2M12
+�(∆23,1−D)/2−1
+= B
+�∆12,3
+2
+, ∆13,2
+2
+� � p2
+2M2
+− E2
+�(∆23,1−D)/2−1
+× 2F1
+�∆12,3
+2
+, D + 2 − ∆23,1
+2
+; ∆1; (M3/M2) (p2/2M12)
+p2/(2M2) − E2
+�
+,
+(21)
+where B(x, y) is the beta function and 2F1(a, b; c; z) is the ordinary hypergeometric function
+(cf. Ref.[21]). Using the transformation formula that changes the argument of the hyperge-
+ometric function from z to 1/z, we find that its leading divergence as z → ∞ has the factor
+z−∆12,3/2 provided D > 2.
+III.
+POINT PRODUCTION RATES
+We are now in the position to determine the energy dependence of the production rate
+of an unparticle recoiling against a particle with large relative momentum from the creation
+of an unparticle at short distances. We then determine the exponent of the energy in the
+production rate of a loosely bound 2-particle state recoiling against a particle with large
+relative momentum. For definiteness, we focus on the case D = 3.
+A.
+Unparticle recoiling against a particle
+We first consider the production rate in the NRCFT of a particle with mass M1 plus the
+unparticle with mass M2 and scaling dimension ∆2 from the creation at short distances of the
+unparticle with mass M3 = M2+M1 and scaling dimension ∆3. The inclusive production rate
+is proportional to the imaginary part of the Green function
+�
+φ3(x1) φ†
+3(x2)
+�
+. The contribution
+to the production rate from intermediate states consisting of a particle and an unparticle
+can be obtained from the Fourier transform of the 3-point function
+�
+φ1(x1) φ2(x2) φ†
+3(x3)
+�
+in the CM frame, which we denote as G(E1, E2, p). The Green function G(E1, E2, p) with
+the particle propagator D1(E1, p) amputated and evaluated on shell at E1 = p2/(2M) is
+Gpole(E2, p) in Eq. (19). That amplitude can be represented by the diagram in Fig. 5. By
+energy conservation, the total energy is E3 = E1 + E2. Using the optical theorem, the
+differential rate dR as a function of the total energy E3 can be obtained from Gpole(E2, p)
+in several steps:
+10
+
+�
+3
+2
+FIG. 5.
+Amplitude for the creation of unparticle 3 at a point and its evolution to unparticle 2
+and a single particle with large relative momentum. The particle is represented by a single line.
+• complete the amputation of the final-state propagators by dividing Gpole(E2, p) by the
+unparticle propagator D2(E2, p),
+• multiply the amputated amplitude Gamp(E2, p) = Gpole(E2, p)/D2(E2, p) by its com-
+plex conjugate,
+• multiply |Gamp(E2, p)|2 by the discontinuities in the particle propagator D1(E1, p) and
+in the unparticle propagator D2(E2, p), which are given in Eqs. (5) and (6),
+• integrate over the energy E1 of the particle and its momentum p in the CM frame
+with the measure dE1d3p/(2π)4.
+The resulting expression for the differential production rate in the CM frame as a function
+of E3 is
+dR = C
+� dE1
+2π
+d3p
+(2π)3
+��Gamp(E3 − E1, p)
+��2 2π δ
+�
+E1 − p2
+2M1
+�
+×
+�
+E3 − E1 − p2
+2M2
+�∆2−5/2
+θ
+�
+E3 − E1 − p2
+2M2
+�
+,
+(22)
+where C is a constant. The integral over E1 can be evaluated using the delta function, which
+sets E1 = p2/(2M1). The discontinuity in D2(E2, p) provides the threshold E3 > p2/(2M12),
+where M12 = M1M2/M3 is a reduced mass. The differential rate for E3 above the threshold
+reduces to
+dR = C
+�
+E3 −
+p2
+2M12
+�∆2−5/2 ��Gamp(E3 − p2/(2M1), p)
+��2 d3p
+(2π)3.
+(23)
+We can use Eq. (20) to deduce the rate for energy E3 close to the threshold p2/(2M12)
+for fixed p. The limiting behavior as E2 → p2/(2M2) of the amputated 3-point function is
+Gamp(E2, p) −→ C′′′′
+12,3
+� p2
+2M12
+�−∆12,3/2
+.
+(24)
+11
+
+The limiting behavior of the differential rate as E3 → p2/(2M12) is therefore
+dR −→ C′
+�
+E3 −
+p2
+2M12
+�∆2−5/2 � p2
+2M12
+�∆3−∆2−3/2
+d3p
+(2π)3,
+(25)
+where C′ = C′′′′
+12,3C and we have used ∆1 = 3/2.
+The above results for the point production rate in Eq. (25) can be applied to multi-
+neutron systems. In this case, ∆3 and ∆2 are the scaling dimensions of (N + 1)-neutron
+and N-neutron operators, respectively. The contribution to the point production rate of the
+(N + 1)-neutron unparticle from its decay into an N-neutron unparticle recoiling against
+a single neutron is given by Eq. (23). In the region near the threshold for the N-neutron
+unparticle, that production rate reduces at large relative momentum to Eq. (25). The lowest
+scaling dimensions for 3, 4, and 5 neutrons are 4.27, 5.07, 7.6, respectively. (See [10] for
+further discussion and references.)
+B.
+Bound state recoiling against a particle
+Next we consider the case where a bound state arises from a deformation of a nonrelativis-
+tic conformal field theory that produces a large positive scattering length a in the channel
+associated with the unparticle with mass M2 and scaling dimension ∆2 = 2. The bound
+state has constituents with masses M1 and M2 − M1. Its binding energy is |εX| = 1/(2µa2),
+where µ = M1(M2 −M1)/M2 is the reduced mass of its constituents. The field φ†
+2(x) creates
+a particle of mass M1 and a particle of mass M2 − M1. The propagator for the field φ2(x) is
+D2(E, p) = −i C′
+2
+�� p2
+2M2
+− E
+�1/2
+−
+1
+�
+2µa2
+�−1
+.
+(26)
+This propagator has a pole at the energy
+Epole = −
+1
+2µa2 + p2
+2M2
+.
+(27)
+The pole is associated with the bound state, which we will refer to as X. The discontinuity
+of the propagator is
+D2(E + iϵ, p) − D2(E − iϵ, p) = 2C′
+2
+�
+�
+E − p2/(2M2)
+E − p2/(2M2) + |εX| θ
+�
+E −
+p2
+2Mn
+�
++2π
+�
+|εX| δ
+�
+E + |εX| − p2
+2M2
+��
+.
+(28)
+The differential rate dR for the production of either three particles or a single particle
+plus X can be obtained from the differential rate in Eq. (22) by taking into account the
+dependence of Gamp(E3 − E1, p) on a and replacing the discontinuity of the propagator
+12
+
+�
+3
+FIG. 6.
+Amplitude for the creation of unparticle 3 at a point and its evolution to a loosely bound
+state of two particles and a single particle with large relative momentum. The bound state is
+represented by a double line.
+D2(E3 − E1, p) by the discontinuity given in Eq. (28). The amplitude for producing the
+bound state X plus a single particle can be represented by the diagram in Fig. 6. The
+integrated rate RX for producing a single particle plus X can be obtained by using the delta
+function in Eq. (28) to evaluate the integral over p in Eq. (23):
+RX = CX
+�
+|εX|
+�
+E3 + |εX|
+�1/2
+����Gamp
+�M1E3 − M2|εX|
+M3
+,
+�
+2M12
+�
+E3 + |εX|
+��1/2
+�����
+2
+,
+(29)
+where CX = CC′
+2(2M12)3/2/π. The amputated 3-point function depends on a explicitly
+through εX in its two arguments and also implicitly through the interactions between the
+particles. In the limit E3 ≫ |εX|, its dependence on E3 is given in Eq. (20). The dependence
+of RX on E3 in that limit therefore has the power-law behavior
+RX −→ C′
+X
+�
+|εX| E
+−∆12,3+1/2
+3
+,
+(30)
+where C′
+X is a constant. The exponent is ∆3 − ∆2 − ∆1 + 1
+2 = ∆3 − 3.
+This case can be realized with neutral D and D∗ mesons and the X(3872) [13]. The
+X(3872) is a resonance extremely close to the threshold in the JPC = 1++ channel of D∗0 ¯D0
+and D0 ¯D∗0. The most precise measurements of the mass by the LHCb collaboration give
+an energy relative to the D∗0 ¯D0 threshold of εX = −0.07 ± 0.12 MeV [22, 23], which implies
+|εX| < 0.22 MeV at the 90% confidence level. This tiny binding energy corresponds to a
+large positive scattering length a = 1/
+�
+2µ|εX|, where µ is the D∗0 ¯D0 reduced mass.
+The production rate dR/dE of an X and a neutral D or D∗ meson near the threshold
+is determined by the energy scale εDX = 1/(2µDXa2
+DX), where µDX is the reduced mass
+of the XD (or XD∗) system and aDX is the corresponding S-wave scattering length. To
+leading order in the contact effective field theory for the neutral D and D∗ mesons, the
+scattering lengths for D0X and D∗0X are universal und equal to a multiplied by a large
+negative coefficient: aD0X = −9.7 a, aD∗0X = −16.6 a [24]. Thus they show an additional
+enhancement beyond the already large D∗0 ¯D0 scattering length a. Inserting explicit values
+this leads to the tiny energy scales εD0X = 0.82 keV or εD∗0X = 0.26 keV.
+13
+
+0.01
+1
+100
+E/|εX|
+dR/dE [arbitrary units]
+  XD
+  XD*
+E
+-1/2
+0.10119
+0.08697
+E
+E
+E
+-1/2
+FIG. 7.
+Production rates dR/dE for D0X (solid curve) and D∗0X (dashed curve) from the
+creation of neutral charm mesons at short distances as functions of the invariant energy E. The
+dotted lines show the transition between different scaling regions.
+As shown in Fig. 7, dR/dE increases from zero at the threshold to a peak near εDX/|εX| ≈
+0.01 for D0X and 0.004 for D∗0X, respectively. Then it decreases to a local minimum at an
+energy of order |εX|. In this region, the interior structure of the X is not resolved and the
+scaling exponent −1/2 for two structureless particles in the final state appears [8]. Beyond
+the minimum, there is a scaling region where dR/dE increases with a power-law determined
+by the three-body scaling dimensions ∆3 = 3.10119 for D0X and ∆3 = 3.08697 for D∗0X.
+The power-law behavior of the amplitudes Γ for D0X and D∗0X is predicted as E0.10119 and
+E0.08697, respectively [13]. It has been verified by explicit calculations of the point production
+rate in a contact effective field theory for the neutral D and D∗ mesons. There is a crossover
+to a more rapidly increasing production rate at even higher energies when non-universal
+effects kick in, which is not shown in Fig. 7.
+IV.
+CONCLUSION
+Nonrelativistic unparticles arise naturally in any system that can be described by a non-
+relativistic field theory close to a conformally invariant limit. Systems of neutrons with
+small invariant energy created by point reactions are examples of nonrelativistic unparticles
+in nuclear physics [8]. Systems of neutral charm mesons with small invariant energy created
+by point reactions are examples of nonrelativistic unparticles in particle physics [13]. The
+concept of nonrelativistic unparticles is useful, because it allows scaling regions to be identi-
+fied in which reaction rates have power-law behavior characterized by nontrivial exponents.
+14
+
+It would be interesting to find other physical realizations of nonrelativistic unparticles in
+nature. It would also be interesting to exploit the remarkable control of interactions that is
+possible with ultracold atoms to create new systems with unparticles.
+We have used the analytic results for the three-point function for primary operators in a
+nonrelativistic conformal field theory in Refs. [18–20] to derive the contribution to the point
+production rate of an unparticle from its decay into another unparticle recoiling against a
+particle. We also used it to derive a scaling law for the decay into a loosely bound 2-particle
+state recoiling against a particle with large momentum. If analytic results for the 4-point
+function of primary operators in a nonrelativistic conformal field theory were available,
+they could be applied to the elastic scattering of unparticles or the elastic scattering of an
+unparticle with a particle. They could also be used to derive scaling laws for the elastic
+scattering of a loosely bound 2-particle state and a particle at large momentum transfer. It
+would be interesting to compare the analytic exponents with numerical results for the elastic
+scattering of D0 and D∗0 with X(3872) [13].
+ACKNOWLEDGMENTS
+H.-W.H. acknowleges the hospitality of KITP, Santa Barbara where part of this work
+was carried out. The research of E.B. was supported in part by the U.S. Department of
+Energy under grant DE-SC0011726. H.-W.H. was supported in part by the National Science
+Foundation under Grant No. NSF PHY-1748958, by the Deutsche Forschungsgemeinschaft
+(DFG, German Research Foundation) - Project-ID 279384907 - SFB 1245 and by the German
+Federal Ministry of Education and Research (BMBF) (Grant no. 05P21RDFNB).
+15
+
+[1] H. Georgi, Unparticle physics, Phys. Rev. Lett. 98, 221601 (2007) [arXiv:hep-ph/0703260].
+[2] K. Cheung, W.Y. Keung and T.C. Yuan, Collider signals of unparticle physics, Phys. Rev.
+Lett. 99, 051803 (2007) [arXiv:0704.2588].
+[3] H. Georgi, Another odd thing about unparticle physics, Phys. Lett. B 650, 275-278 (2007)
+[arXiv:0704.2457].
+[4] K. Cheung, W.Y. Keung and T.C. Yuan, Collider Phenomenology of Unparticle Physics, Phys.
+Rev. D 76, 055003 (2007) [arXiv:0706.3155].
+[5] V. Khachatryan et al. [CMS], Search for dark matter, extra dimensions, and unparticles in
+monojet events in proton–proton collisions at √s = 8 TeV, Eur. Phys. J. C 75, 235 (2015)
+[arXiv:1408.3583].
+[6] V. Khachatryan et al. [CMS], Search for dark matter and unparticles produced in association
+with a Z boson in proton-proton collisions at √s = 8 TeV, Phys. Rev. D 93, 052011 (2016)
+[erratum: Phys. Rev. D 97, 099903 (2018)] [arXiv:1511.09375].
+[7] A. M. Sirunyan et al. [CMS], Search for dark matter and unparticles in events with a Z boson
+and missing transverse momentum in proton-proton collisions at √s = 13 TeV, JHEP 03, 061
+(2017) [erratum: JHEP 09, 106 (2017)] [arXiv:1701.02042].
+[8] H.W. Hammer and D.T. Son, Unnuclear physics, Proc. Nat. Acad. Sci. 118, e2108716118
+(2021) [arXiv:2103.12610].
+[9] G. K. Karananas and A. Monin, More on the operator-state map in nonrelativistic CFTs,
+Phys. Rev. D 105, 065008 (2022) [arXiv:2109.03836 [hep-th]].
+[10] Y. Nishida and D.T. Son, Nonrelativistic conformal field theories, Phys. Rev. D 76, 086004
+(2007) [arXiv:0706.3746].
+[11] M. Duer et al., Observation of a correlated free four-neutron system, Nature 606, 678 (2022).
+[12] R. Lazauskas, E. Hiyama and J. Carbonell, Low energy structures in nuclear reactions with
+4n in the final state, [arXiv:2207.07575 [nucl-th]].
+[13] E. Braaten and H. W. Hammer, Interpretation of Neutral Charm Mesons near Threshold as
+Unparticles, Phys. Rev. Lett. 128, 032002 (2022) [arXiv:2107.02831].
+[14] S.K. Choi et al. [Belle Collaboration], Observation of a narrow charmonium-like state in ex-
+clusive B± → K±π+π−J/ψ decays, Phys. Rev. Lett. 91, 262001 (2003) [hep-ex/0309032].
+[15] E. Braaten and M. Kusunoki, Low-energy universality and the new charmonium resonance at
+3870-MeV, Phys. Rev. D 69, 074005 (2004) [arXiv:hep-ph/0311147 [hep-ph]].
+[16] E. Braaten and H.W. Hammer, Universality in few-body systems with large scattering length,
+Phys. Rept. 428, 259-390 (2006) [arXiv:cond-mat/0410417].
+[17] M. Henkel, Schrodinger invariance in strongly anisotropic critical systems, J. Statist. Phys.
+75, 1023 (1994) [arXiv:hep-th/9310081].
+[18] M. Henkel and J. Unterberger, Schrodinger invariance and space-time symmetries, Nucl. Phys.
+B 660, 407 (2003) [arXiv:hep-th/0302187].
+16
+
+[19] C.A. Fuertes and S. Moroz, Correlation functions in the non-relativistic AdS/CFT correspon-
+dence, Phys. Rev. D 79, 106004 (2009) [arXiv:0903.1844].
+[20] A. Volovich and C. Wen, Correlation Functions in Non-Relativistic Holography, JHEP 05,
+087 (2009) [arXiv:0903.2455].
+[21] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic Press,
+New York, 1965).
+[22] R. Aaij et al. [LHCb], Study of the lineshape of the χc1(3872) state, Phys. Rev. D 102, 092005
+(2020) [arXiv:2005.13419].
+[23] R. Aaij et al. [LHCb], Study of the ψ2(3823) and χc1(3872) states in B+ → (J/ψπ+π−) K+
+decays, JHEP 08, 123 (2020) [arXiv:2005.13422].
+[24] D.L. Canham, H.-W. Hammer and R.P. Springer, On the scattering of D and D∗ mesons off
+the X(3872), Phys. Rev. D 80, 014009 (2009) [arXiv:0906.1263].
+17
+
diff --git a/x9E3T4oBgHgl3EQfPAm3/content/tmp_files/load_file.txt b/x9E3T4oBgHgl3EQfPAm3/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..5c25b94912852306edeb1b3f4c60d77ceff6aeff
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+page_content='Point Production of a Nonrelativistic Unparticle  Recoiling Against a Particle  Eric Braaten  Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' The Ohio State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Columbus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' OH 43210,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' USA  Hans-Werner Hammer  Technische Universit¨at Darmstadt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  Institut f¨ur Kernphysik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' 64289 Darmstadt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Germany and  ExtreMe Matter Institute EMMI and Helmholtz  Forschungsakademie Hessen f¨ur FAIR (HFHF),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  GSI Helmholtzzentrum f¨ur Schwerionenforschung GmbH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' 64291 Darmstadt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Germany  (Dated: January 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' 2023)  Abstract  A nonrelativistic unparticle can be defined as an excitation created by an operator with a definite  scaling dimension in a nonrelativistic field theory with an approximate conformal symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' The  point production rate of an unparticle has power-law dependence on its total energy with an  exponent determined by its scaling dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' We use the exact result for the 3-point function  of primary operators in a nonrelativistic conformal field theory to derive the contribution to the  point production rate of the unparticle from its decay into another unparticle recoiling against a  particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' In the case where the conformal symmetry is broken by a large positive scattering length,  we deduce the exponent of the energy in the point production rate of the loosely bound two-particle  state recoiling against a particle with large relative momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='04399v1  [hep-th]  11 Jan 2023  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  INTRODUCTION  A definition of an elementary particle that is popular in group-theoretic circles is an irre-  ducible representation of the Poincar´e group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' The concept of an unparticle was introduced  by Georgi [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' An unparticle is a system created by a local operator with a definite scaling  dimension in a conformal field theory (CFT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' It can therefore be defined as an irreducible rep-  resentation of the conformal symmetry group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' The conformal group on a 3+1-dimensional  Minkowski space-time is a 15-dimensional group that includes the Poincar´e group and scale  transformations as subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' A relativistic unparticle is characterized by a single number:  the scaling dimension of the operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' If the CFT belongs to a hidden sector beyond the  Standard Model of particle physics, the unparticle cannot be observed directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' However it  can be observed indirectly through the momentum distribution of Standard Model particles  produced in association with the unparticle [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' If the unparticle is produced in association  with a single Standard Model particle, the invariant mass distribution of the unparticle can  be determined by measuring the recoil-momentum distribution of the Standard Model par-  ticle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' It has power-law behavior with an exponent determined by the scaling dimension of  the unparticle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' The existence of unparticles in a hidden sector would produce novel signals  in high energy colliders [2–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' The CMS collaboration has searched for signals of unparticles  in pp collisions at the Large Hadron Collider [5–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  Hammer and Son recently pointed out that unparticles can also arise in nonrelativistic  physics [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' The nonrelativistic conformal symmetry group on a 3+1-dimensional Galilean  space-time is a 13-dimensional group that includes the Galilean group and scale transfor-  mations as subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='1 It is also called the Schr¨odinger group, because it is the symmetry  group of the free Schr¨odinger equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' A nonrelativistic conformal field theory (NRCFT)  is a field theory with nonrelativistic conformal symmetry [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' A nonrelativistic unparticle  is a system created by a local operator with a definite scaling dimension in such a theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' In  contrast to the relativistic case, a nonrelativistic unparticle is characterized by two numbers:  its kinetic mass M and the scaling dimension ∆ of the operator [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  Hammer and Son pointed out that systems of low-energy neutrons produced by a short-  distance reaction provide physical examples of nonrelativistic unparticles [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Neutrons have  a negative scattering length a that is much larger than their effective range r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' As a con-  sequence, the behavior of a system of neutrons whose kinetic energies in their center-of-  momentum (CM) frame are all in the scaling region between 1/(mna2) and 1/(mnr2  0), where  mn is the neutron mass, is approximately scale invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' In the unitary limit 1/a → 0,  r0 → 0, a nonrelativistic effective field theory describing the neutrons has nonrelativistic  conformal symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' A convenient basis for local operators in the effective field theory are  those with definite scaling behavior under the conformal symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' A system of N neutrons  created by a local operator with scaling dimension ∆N can be interpreted as an unparticle  with mass Nmn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' For the 2-neutron unparticle, the lowest scaling dimension is ∆2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  1 For a discussion of the relation to the relativistic conformal group, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='1  1  10  E/εnn  dR/dE [arbitrary units]  2n  3n  4n  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='5  E  E  E  E  E  E  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='77  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='57  3  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='5  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  Scaling prediction of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' [8] for the invariant energy distribution dR/dE of 2, 3, and  4 neutrons in their CM frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' The transition from the free phase-space behavior to the scaling  region around E ∼ εnn is illustrated by the dotted lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  For unparticles with larger N, the scaling dimensions are transcendental numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' For the  3-neutron unparticle, the lowest scaling dimension is ∆3 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='27272.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' The lowest scaling  dimensions for 4 and 5 neutrons are 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='07 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='6, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' A summary of the lowest  scaling dimensions for up to 6 neutrons is given in Table I of [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  An N-neutron unparticle can be created by a short-distance nuclear reaction of the form  A1 +A2 −→ B +(nn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=') [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' The invariant energy E of the N neutrons, which is their total  kinetic energy in their CM frame, can be determined by measuring the momentum of the  recoiling nucleus B or by detecting the neutrons directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' There is a scaling region of E in  which the differential cross section has the scaling behavior E∆N−5/2dE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' The corresponding  power-law behavior for N noninteracting particles as E approaches the threshold is governed  by the N-particle phase space, E(3N−5)/2dE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' In the case of fermions, there is an additional  Pauli suppression factor E for each pair of identical particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' For E of the order of the energy  scale set by the neutron-neutron scattering length εnn = 1/(mna2), there is a transition from  the phase-space behavior for non-interacting particles to the power-law behavior governed  by ∆N in the scaling region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' This nontrivial scaling behavior is the smoking gun for an  unparticle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' The resulting predictions for the invariant energy distributions of 2, 3, and 4  neutrons are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' In the case of 2 neutrons, there is a maximum in the region  E ∼ εnn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' For larger neutron numbers, there is a change in the slope of the log-log plot in  the region E ∼ εnn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  The predicted invariant energy distribution, dR/dE, for four neutrons in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' 1 is not  consistent with a resonance-like structure at E/εnn ≈ 20 in the point production of four  3  neutrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  Such a structure was recently observed in the 4n spectrum measured in the  knock-out reaction 8He(p, pα)4n by Duer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' However, the neutron distribution of  the initial 8He nucleus clearly plays a role in this process and the applicability of the point-  production approximation is questionable as the neutrons are emitted from a 8He source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  A recent theoretical study of the reaction [12] attributes the structure to the final state  interaction among the four neutrons and the presence of preexisting four neutrons in the  periphery of 8He nucleus projectile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' While the former effect is captured in the the invariant  4n energy distribution of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' 1, the latter is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' [13], we pointed out that a low-energy system of neutral charm mesons created by  a short-distance reaction is an unparticle because of the X(3872) resonance in the D∗0 ¯D0 +  D0 ¯D∗0 channel [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' The scattering length a in that channel is large and negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Because  the resonant channel is a superposition of charm mesons, there is no Efimov effect [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  The behavior of a system of neutral charm mesons whose kinetic energies in their CM frame  are all in the scaling region between 1/(2µa2) and 1/(2µr2  0), where µ is the reduced mass  of D∗0 and ¯D0, is therefore approximately scale invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' In the unitary limit 1/a → 0,  r0 → 0, a nonrelativistic effective field theory describing the neutral charm mesons has  nonrelativistic conformal symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' A convenient basis for local operators in the effective  field theory are those with definite scaling behavior under the conformal symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' A system  of N neutral charm mesons created by a local operator with scaling dimension ∆N can be  interpreted as an unparticle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' The scaling dimensions for 2-charm-meson unparticles are the  same as for two neutrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' The lowest scaling dimensions for 3-charm-meson unparticles  are ∆3 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='10119 for D0 ¯D∗0D0 and ¯D0D∗0 ¯D0 as well as ∆3 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='08697 for D0 ¯D∗0 ¯D∗0 and  ¯D0D∗0D∗0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' In each case, there are two pairs of particles with a large scattering length a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  The slight difference in the two values of ∆3 is due to the different masses of the D0 and D∗0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  The 2-charm-meson unparticle can be observed through the recoil-momentum spectrum of  the kaon in the inclusive decay B± → K± + (anything).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' A 3-charm-meson unparticle can  be observed through the prompt production of X(3872) D0 at the Large Hadron Collider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  Because the scattering length a in the case of neutral charm mesons is large and posi-  tive, the unparticles have components that include bound states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' There are reactions for  producing final states that include bound states whose rates have power-law behavior with  exponents determined by conformal symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' The simplest such reaction is the production  of X(3872) D0 from the creation of a 3-charm-meson unparticle at a point, whose rate scales  as a power of the unparticle energy E [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  In this paper, we consider the production of a pair of unparticles from the creation of a  single unparticle at a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' We use the analytic result for the 3-point function of primary  operators in a NRCFT to derive the contribution to the production rate from another  unparticle recoiling against a single particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' In the case where the conformal symmetry is  broken by a large positive scattering length, we deduce the exponent of the energy in the  point production rate of the loosely bound two-particle state recoiling against a particle with  large relative momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  4  �  n  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  Space-time propagator for unparticle n with kinetic mass Mn and scaling dimension ∆n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  THREE-POINT FUNCTION FOR PRIMARY OPERATORS  We start by calculating the Fourier transform of the 3-point function for primary operators  in a NRCFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' We take the dimension of space to be D, and we denote a space-time position  by x = (x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' An operator φ†  3(x) with scaling dimension ∆3 creates an unparticle with  mass M3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  An operator φ2(x) with scaling dimension ∆2 annihilates an unparticle with  mass M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' An operator φ1(x) with scaling dimension ∆1 annihilates an unparticle with mass  M1 = M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' The masses satisfy the constraint M1 + M2 = M3 from Galilean symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' We  are particularly interested in the case where φ1(x) annihilates a single particle with mass  M, in which case its scaling dimension is ∆1 = D/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  Propagators  The space-time propagator for a primary operator φn(x) with mass Mn and scaling di-  mension ∆n is  �  φn(x1) φ†  n(x2)  �  = Cn (t12)−∆nθ(t12) exp  �  iMn x2  12  2 t12  �  ,  (1)  where tij = ti − tj, x2  ij = (xi − xj)2, and Cn is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' We represent the space-time  propagator by the diagram in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  The energy-momentum propagator is obtained by Fourier-transforming in both space-  time positions and then factoring out an energy-momentum delta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' It is advan-  tageous to take the variables (x1 + x2)/2 and x1 − x2 for the Fourier transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  After  performing the trivial Fourier transform in (x1 + x2)/2, the remaining Fourier transform in  space is a Gaussian integral that depends on the momentum p:  �  dDx12 exp  �  − i p · x12 + iMn x2  12  2t12  �  =  �  2πi t12  Mn  �D/2  exp  �  − i t12  2Mn  p2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  (2)  The subsequent Fourier transform in time gives a simple function of the energy E:  � ∞  0  dt12 (it12)D/2−∆n exp  �  i E t12 − i p2  2Mn  t12  �  = −i Γ  �D  2 + 1 − ∆n  � � p2  2Mn  − E  �∆n−D/2−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  (3)  With an appropriate choice for the constant Cn in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' (1), our final result for the energy-  momentum propagator is  Dn(E, p) = −i C′  n  � p2  2Mn  − E  �∆n−D/2−1  ,  (4)  5  �  3  1  2  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  Space-time 3-point function for primary operators associated with unparticles 1, 2, and  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  where C′  n is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  In the case where φ1(x) annihilates a single particle, its scaling dimension is ∆1 = D/2  and the propagator in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' (4) has a simple pole in the energy E at p2/(2M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' With the  conventional choice C′  1 = 1, the discontinuity in the energy is  D1(E + iϵ, p) − D1(E − iϵ, p) = 2π δ  �  E − p2  2M  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  (5)  For other unparticles, the discontinuity in the energy is  Dn(E+iϵ, p)−Dn(E−iϵ, p) = 2C′  n sin  �  π(∆n−D/2)  � �  E− p2  2Mn  �∆n−D/2−1  θ  �  E− p2  2Mn  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' (6)  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  Space-time three-point function  The 3-point function  �  φ1(x1) φ2(x2) φ†  3(x3)  �  for primary operators in a NRCFT was first  considered by Henkel in 1993 [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' The 3-point function can be represented by the diagram  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' He used the Schr¨odinger symmetry to determine the 3-point function analytically  up to a scaling function of a single variable:  �  φ1(x1) φ2(x2) φ†  3(x3)  �  = (t13)−∆13,2/2θ(t13) exp  �  iM1 x2  13  2 t13  �  ×(t23)−∆23,1/2θ(t23) exp  �  iM2 x2  23  2 t23  �  (t12)−∆12,3/2 Φ(w),  (7)  where ∆ij,k = ∆i + ∆j − ∆k and the argument of the scaling function Φ is  w =  �  t23(x1 − x3) − t13(x2 − x3)  �2  2 t13 t23 t12  = 1  2  �x2  12  t12  − x2  13  t13  + x2  23  t23  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  (8)  The first expression shows that the sign of w is the same as the sign of t12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' The scaling  function Φ(w) depends also on the masses M1 and M2 and on the scaling dimensions ∆1,  6  �  3  2  1  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  Fourier-transformed 3-point function for primary operators associated with unparticles  1, 2, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  ∆2, and ∆3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' The 3-point function in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' (7) is symmetric under the interchange of the  subscripts 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  An integral representation for the scaling function Φ(w) was first obtained by Henkel  and Unterberger from the 3-point function for a CFT in two higher dimensions [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' It has  also been obtained by Fuertes and Moroz and by Volovich and Wen using holography and  the AdS/CFT correspondence [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' In the Anti-de Sitter space (AdS) formulation, the  external points in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' 3 are on the boundary of AdS and the blob is replaced by a point  interaction inside the bulk of AdS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' In Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' [18, 19], the 3-point function is expressed as  a function of Euclidean times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' The expression in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' [20] in terms of real times is more  directly useful for our purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' The integral representation for the scaling function is  Φ(w) = C12,3  � +∞  −∞  du (u + iϵ)−∆13,2/2 e−iM1u  � +∞  −∞  dv (v + iϵ)−∆23,1/2 e−iM2v  ×  �  u − v + (1 + iϵ)w  �−∆12,3/2,  (9)  where C12,3 is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Because the signs of t12 and w are identical (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' (8)), the  product of (t12)−∆12,3/2 and the last factor in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' (9) has a well-defined imaginary part:  (t12)−∆12,3/2 �  u − v + (1 + iϵ)w  �−∆12,3/2 =  �  t12 (u − v + w) + iϵ  �−∆12,3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  (10)  The scaling function can also be expressed analytically in terms of a confluent hypergeometric  function [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  Fourier transform in space  The Fourier transform of the 3-point function in position space can be represented by  the diagram in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' To evaluate the Fourier transforms in the three spatial positions, it  7  is convenient to express the last factor in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' (10) as the integral of an exponential:  �  t12(u − v + w) + iϵ  �−∆12,3/2 =  �  eiπ/2 |t12|  �−∆12,3/2  1  Γ(∆12,3/2)  ×  � ∞  0  dm m∆12,3/2−1 exp  �  − m[ϵ − i sign(t12)(u − v + w)]  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  (11)  The Fourier transform can be expressed as the product of the momentum-conserving delta  function δD(p1 + p2 − p3) and a Gaussian integral in x13 and x23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' In the case t12 > 0, the  Gaussian integral is  �  dDx13 exp  �  − i p1 · x13 + iM1 x2  13  2t13  � �  dDx23 exp  �  − i p2 · x23 + iM2 x2  23  2t23  �  e+i m w  =  �  2πi  t12t13  t12M1 + t23m  �D/2 �  2πi  t23(t12M1 + t23m)  t12M1M2 + t13M1m + t23M2m  �D/2  × exp  �  − it12t13(M2 + m)p2  1 + t12t23(M1 − m)p2  2 + t13t23mp2  3  2(t12M1M2 + t13M1m + t23M2m)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  (12)  We can go to the center-of-momentum (CM) frame by setting p3 = 0 and |p1| = |p2| = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  In the case t12 > 0, the 3-point function with the momentum delta function factored out  reduces to  C′  12,3 (t13)(D−∆13,2)/2θ(t13) (t23)(D−∆23,1)/2θ(t23) |t12|−∆12,3/2  � ∞  0  dm m∆12,3/2−1 e−mϵ  ×  � +∞  −∞  du (u + iϵ)−∆13,2/2 e−i(M1−m)u  � +∞  −∞  dv (v + iϵ)−∆23,1/2 e−i(M2+m)v  ×  �  t12M1M2  t12M1M2 + t13M1m + t23M2m  �D/2  exp  �  − i  t12(t23M1 + t13M2 + t12m)  2(t12M1M2 + t13M1m + t23M2m)p2  �  ,  (13)  where C′  12,3 is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' The integrals over u and v can be evaluated analytically by closing  the integration contour in the lower half-plane:  � +∞  −∞  du (u + iϵ)−∆/2 e−iMu = e−iπ∆/4  2π  Γ(∆/2)M ∆/2−1θ(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  (14)  After these integrals have been evaluated, the limit ϵ → 0 can be taken in the remainder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  The choice t12 > 0 breaks the symmetry under interchange of the subscripts 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Given  this choice, it is convenient to change to a dimensionless integration variable x = m/M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  Our final result for the spatial Fourier transform of the 3-point function in the CM frame  with t12 > 0 is  C′′  12,3 (t13)(D−∆13,2)/2θ(t13) (t23)(D−∆23,1)/2θ(t23) |t12|−∆12,3/2  ×  � 1  0  dx x∆12,3/2−1 (1 − x)∆13,2/2−1 �  1 + (M1/M2)x  �∆23,1/2−1  ×  �  t12  t12 + (M1/M2)xt13 + xt23  �D/2  exp  �  − i t12[xt12 + (M2/M1)t13 + t23]  2M2[t12 + (M1/M2)xt13 + xt23]p2  �  , (15)  where C′′  12,3 is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  8  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  Fourier transform in time  The Fourier transforms in the three times can be expressed as the product of the energy-  conserving delta function δ(E1 + E2 − E3), a Fourier transform in t13 with energy E1, and a  Fourier transform in t23 with energy E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' We first isolate the contribution from the Fourier  transform in t13 from the region t13 ≫ t23 and then evaluate the Fourier transform in t23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  In the limit t13 ≫ t23, the exponential in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' (15), reduces to  exp  �  − i t12  �  xt12 + (M2/M1)t13 + t23  �  2M2  �  t12 + (M1/M2)xt13 + xt23  �p2  �  −→ exp  �  − i p2  2M1  t13  �  exp  �  − i[1 − 2x − (M2/M1)x] p2  2M2[1 + (M1/M2)x]  t23  �  ,  (16)  where the identity t12 = t13 − t23 has been used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' The contribution to the Fourier transform  in t13 from the region t13 ≫ t23 can be reduced to the integral  � ∞  0  dt13 (it13)(D−∆13,2−∆12,3)/2 exp  �  iE1t13 − i p2  2M1  t13  �  = −i Γ  �  D/2 + 1 − ∆1  � � p2  2M1  − E1  �∆1−D/2−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  (17)  This integral is the propagator for φ1 multiplied by a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' The subsequent Fourier  transform in t23 can be reduced to the integral  � ∞  0  dt23 (it23)(D−∆23,1)/2 exp  �  iE2t23 − i[1 − 2x − (M2/M1)x] p2  2M2[1 + (M1/M2)x]  t23  �  = −i Γ  �  [D − ∆23,1]/2 + 1  � �[1 − 2x − (M2/M1)x] p2  2M2[1 + (M1/M2)x]  − E2  �(∆23,1−D)/2−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  (18)  We now specialize to the case of φ1(x) being the operator that annihilates a single particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  Its scaling dimension is ∆1 = D/2, so the integral in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' (17) has a simple pole at E1 =  p2/(2M1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' The residue of the pole in the Fourier-transformed three-point function is  Gpole(E2, p) = C′′′  12,3  � 1  0  dx x∆12,3/2−1 (1 − x)∆13,2/2−1 �  1 + (M1/M2)x  �∆23,1/2−1−D/2  ×  � p2  2M2  − E2 −  (M3/M2)x  1 + (M1/M2)x  p2  2M12  �(∆23,1−D)/2−1  ,  (19)  where M12 = M1M2/M3 is a reduced mass and C′′′  12,3 is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' The integrand has been  expressed as a function of the energy E2−p2/(2M2) of the unparticle relative to its threshold  and the total energy p2/(2M12) of the unparticle at its threshold and the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  In the limit E2 → p2/(2M2) with p2 fixed, the integral over x in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' (19) has a divergent  term that comes from the region near the lower endpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  The divergent factor is the  propagator of the unparticle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' The limiting behavior of the 3-point function in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' (19) as  E2 → p2/(2M2) is  Gpole(E2, p) −→ C′′′′  12,3 D2(E2, p)  � p2  2M12  �−∆12,3/2  ,  (20)  9  where C′′′′  12,3 is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  This is our result for the scaling behavior of the Fourier-  transformed 3-point function at large momentum p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  The exponent −∆12,3/2 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' (20) can also be deduced from the analytic result for an  integral whose integrand has the same dependence on x near the lower endpoint as the  integrand in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' (19):  � 1  0  dx x∆12,3/2−1 (1 − x)∆13,2/2−1  � p2  2M2  − E2 − x(M3/M2) p2  2M12  �(∆23,1−D)/2−1  = B  �∆12,3  2  , ∆13,2  2  � � p2  2M2  − E2  �(∆23,1−D)/2−1  × 2F1  �∆12,3  2  , D + 2 − ∆23,1  2  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' ∆1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' (M3/M2) (p2/2M12)  p2/(2M2) − E2  �  ,  (21)  where B(x, y) is the beta function and 2F1(a, b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' z) is the ordinary hypergeometric function  (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='[21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Using the transformation formula that changes the argument of the hyperge-  ometric function from z to 1/z, we find that its leading divergence as z → ∞ has the factor  z−∆12,3/2 provided D > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  POINT PRODUCTION RATES  We are now in the position to determine the energy dependence of the production rate  of an unparticle recoiling against a particle with large relative momentum from the creation  of an unparticle at short distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' We then determine the exponent of the energy in the  production rate of a loosely bound 2-particle state recoiling against a particle with large  relative momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' For definiteness, we focus on the case D = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  Unparticle recoiling against a particle  We first consider the production rate in the NRCFT of a particle with mass M1 plus the  unparticle with mass M2 and scaling dimension ∆2 from the creation at short distances of the  unparticle with mass M3 = M2+M1 and scaling dimension ∆3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' The inclusive production rate  is proportional to the imaginary part of the Green function  �  φ3(x1) φ†  3(x2)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' The contribution  to the production rate from intermediate states consisting of a particle and an unparticle  can be obtained from the Fourier transform of the 3-point function  �  φ1(x1) φ2(x2) φ†  3(x3)  �  in the CM frame, which we denote as G(E1, E2, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' The Green function G(E1, E2, p) with  the particle propagator D1(E1, p) amputated and evaluated on shell at E1 = p2/(2M) is  Gpole(E2, p) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' That amplitude can be represented by the diagram in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' By  energy conservation, the total energy is E3 = E1 + E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Using the optical theorem, the  differential rate dR as a function of the total energy E3 can be obtained from Gpole(E2, p)  in several steps:  10  �  3  2  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  Amplitude for the creation of unparticle 3 at a point and its evolution to unparticle 2  and a single particle with large relative momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' The particle is represented by a single line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  complete the amputation of the final-state propagators by dividing Gpole(E2, p) by the  unparticle propagator D2(E2, p),  multiply the amputated amplitude Gamp(E2, p) = Gpole(E2, p)/D2(E2, p) by its com-  plex conjugate,  multiply |Gamp(E2, p)|2 by the discontinuities in the particle propagator D1(E1, p) and  in the unparticle propagator D2(E2, p), which are given in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' (5) and (6),  integrate over the energy E1 of the particle and its momentum p in the CM frame  with the measure dE1d3p/(2π)4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  The resulting expression for the differential production rate in the CM frame as a function  of E3 is  dR = C  � dE1  2π  d3p  (2π)3  ��Gamp(E3 − E1, p)  ��2 2π δ  �  E1 − p2  2M1  �  ×  �  E3 − E1 − p2  2M2  �∆2−5/2  θ  �  E3 − E1 − p2  2M2  �  ,  (22)  where C is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' The integral over E1 can be evaluated using the delta function, which  sets E1 = p2/(2M1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' The discontinuity in D2(E2, p) provides the threshold E3 > p2/(2M12),  where M12 = M1M2/M3 is a reduced mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' The differential rate for E3 above the threshold  reduces to  dR = C  �  E3 −  p2  2M12  �∆2−5/2 ��Gamp(E3 − p2/(2M1), p)  ��2 d3p  (2π)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  (23)  We can use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' (20) to deduce the rate for energy E3 close to the threshold p2/(2M12)  for fixed p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' The limiting behavior as E2 → p2/(2M2) of the amputated 3-point function is  Gamp(E2, p) −→ C′′′′  12,3  � p2  2M12  �−∆12,3/2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  (24)  11  The limiting behavior of the differential rate as E3 → p2/(2M12) is therefore  dR −→ C′  �  E3 −  p2  2M12  �∆2−5/2 � p2  2M12  �∆3−∆2−3/2  d3p  (2π)3,  (25)  where C′ = C′′′′  12,3C and we have used ∆1 = 3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  The above results for the point production rate in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' (25) can be applied to multi-  neutron systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' In this case, ∆3 and ∆2 are the scaling dimensions of (N + 1)-neutron  and N-neutron operators, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' The contribution to the point production rate of the  (N + 1)-neutron unparticle from its decay into an N-neutron unparticle recoiling against  a single neutron is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' In the region near the threshold for the N-neutron  unparticle, that production rate reduces at large relative momentum to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' The lowest  scaling dimensions for 3, 4, and 5 neutrons are 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='27, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='07, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='6, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' (See [10] for  further discussion and references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=')  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  Bound state recoiling against a particle  Next we consider the case where a bound state arises from a deformation of a nonrelativis-  tic conformal field theory that produces a large positive scattering length a in the channel  associated with the unparticle with mass M2 and scaling dimension ∆2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' The bound  state has constituents with masses M1 and M2 − M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Its binding energy is |εX| = 1/(2µa2),  where µ = M1(M2 −M1)/M2 is the reduced mass of its constituents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' The field φ†  2(x) creates  a particle of mass M1 and a particle of mass M2 − M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' The propagator for the field φ2(x) is  D2(E, p) = −i C′  2  �� p2  2M2  − E  �1/2  −  1  �  2µa2  �−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  (26)  This propagator has a pole at the energy  Epole = −  1  2µa2 + p2  2M2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  (27)  The pole is associated with the bound state, which we will refer to as X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' The discontinuity  of the propagator is  D2(E + iϵ, p) − D2(E − iϵ, p) = 2C′  2  �  �  E − p2/(2M2)  E − p2/(2M2) + |εX| θ  �  E −  p2  2Mn  �  +2π  �  |εX| δ  �  E + |εX| − p2  2M2  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  (28)  The differential rate dR for the production of either three particles or a single particle  plus X can be obtained from the differential rate in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' (22) by taking into account the  dependence of Gamp(E3 − E1, p) on a and replacing the discontinuity of the propagator  12  �  3  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  Amplitude for the creation of unparticle 3 at a point and its evolution to a loosely bound  state of two particles and a single particle with large relative momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' The bound state is  represented by a double line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  D2(E3 − E1, p) by the discontinuity given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' The amplitude for producing the  bound state X plus a single particle can be represented by the diagram in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' The  integrated rate RX for producing a single particle plus X can be obtained by using the delta  function in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' (28) to evaluate the integral over p in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' (23):  RX = CX  �  |εX|  �  E3 + |εX|  �1/2  ����Gamp  �M1E3 − M2|εX|  M3  ,  �  2M12  �  E3 + |εX|  ��1/2  �����  2  ,  (29)  where CX = CC′  2(2M12)3/2/π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' The amputated 3-point function depends on a explicitly  through εX in its two arguments and also implicitly through the interactions between the  particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' In the limit E3 ≫ |εX|, its dependence on E3 is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' The dependence  of RX on E3 in that limit therefore has the power-law behavior  RX −→ C′  X  �  |εX| E  −∆12,3+1/2  3  ,  (30)  where C′  X is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' The exponent is ∆3 − ∆2 − ∆1 + 1  2 = ∆3 − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  This case can be realized with neutral D and D∗ mesons and the X(3872) [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' The  X(3872) is a resonance extremely close to the threshold in the JPC = 1++ channel of D∗0 ¯D0  and D0 ¯D∗0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' The most precise measurements of the mass by the LHCb collaboration give  an energy relative to the D∗0 ¯D0 threshold of εX = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='12 MeV [22, 23], which implies  |εX| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='22 MeV at the 90% confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' This tiny binding energy corresponds to a  large positive scattering length a = 1/  �  2µ|εX|, where µ is the D∗0 ¯D0 reduced mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  The production rate dR/dE of an X and a neutral D or D∗ meson near the threshold  is determined by the energy scale εDX = 1/(2µDXa2  DX), where µDX is the reduced mass  of the XD (or XD∗) system and aDX is the corresponding S-wave scattering length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' To  leading order in the contact effective field theory for the neutral D and D∗ mesons, the  scattering lengths for D0X and D∗0X are universal und equal to a multiplied by a large  negative coefficient: aD0X = −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='7 a, aD∗0X = −16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='6 a [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Thus they show an additional  enhancement beyond the already large D∗0 ¯D0 scattering length a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Inserting explicit values  this leads to the tiny energy scales εD0X = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='82 keV or εD∗0X = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='26 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  13  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='01  1  100  E/|εX|  dR/dE [arbitrary units]  XD  XD*  E  1/2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='10119  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='08697  E  E  E  1/2  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  Production rates dR/dE for D0X (solid curve) and D∗0X (dashed curve) from the  creation of neutral charm mesons at short distances as functions of the invariant energy E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' The  dotted lines show the transition between different scaling regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' 7, dR/dE increases from zero at the threshold to a peak near εDX/|εX| ≈  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='01 for D0X and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='004 for D∗0X, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Then it decreases to a local minimum at an  energy of order |εX|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' In this region, the interior structure of the X is not resolved and the  scaling exponent −1/2 for two structureless particles in the final state appears [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Beyond  the minimum, there is a scaling region where dR/dE increases with a power-law determined  by the three-body scaling dimensions ∆3 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='10119 for D0X and ∆3 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='08697 for D∗0X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  The power-law behavior of the amplitudes Γ for D0X and D∗0X is predicted as E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='10119 and  E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='08697, respectively [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' It has been verified by explicit calculations of the point production  rate in a contact effective field theory for the neutral D and D∗ mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' There is a crossover  to a more rapidly increasing production rate at even higher energies when non-universal  effects kick in, which is not shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  CONCLUSION  Nonrelativistic unparticles arise naturally in any system that can be described by a non-  relativistic field theory close to a conformally invariant limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Systems of neutrons with  small invariant energy created by point reactions are examples of nonrelativistic unparticles  in nuclear physics [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Systems of neutral charm mesons with small invariant energy created  by point reactions are examples of nonrelativistic unparticles in particle physics [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' The  concept of nonrelativistic unparticles is useful, because it allows scaling regions to be identi-  fied in which reaction rates have power-law behavior characterized by nontrivial exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  14  It would be interesting to find other physical realizations of nonrelativistic unparticles in  nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' It would also be interesting to exploit the remarkable control of interactions that is  possible with ultracold atoms to create new systems with unparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  We have used the analytic results for the three-point function for primary operators in a  nonrelativistic conformal field theory in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' [18–20] to derive the contribution to the point  production rate of an unparticle from its decay into another unparticle recoiling against a  particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' We also used it to derive a scaling law for the decay into a loosely bound 2-particle  state recoiling against a particle with large momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' If analytic results for the 4-point  function of primary operators in a nonrelativistic conformal field theory were available,  they could be applied to the elastic scattering of unparticles or the elastic scattering of an  unparticle with a particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' They could also be used to derive scaling laws for the elastic  scattering of a loosely bound 2-particle state and a particle at large momentum transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' It  would be interesting to compare the analytic exponents with numerical results for the elastic  scattering of D0 and D∗0 with X(3872) [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' acknowleges the hospitality of KITP, Santa Barbara where part of this work  was carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' The research of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' was supported in part by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Department of  Energy under grant DE-SC0011726.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' was supported in part by the National Science  Foundation under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' NSF PHY-1748958, by the Deutsche Forschungsgemeinschaft  (DFG, German Research Foundation) - Project-ID 279384907 - SFB 1245 and by the German  Federal Ministry of Education and Research (BMBF) (Grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' 05P21RDFNB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  15  [1] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Georgi, Unparticle physics, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' 98, 221601 (2007) [arXiv:hep-ph/0703260].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  [2] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Cheung, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Keung and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Yuan, Collider signals of unparticle physics, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' 99, 051803 (2007) [arXiv:0704.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='2588].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  [3] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Georgi, Another odd thing about unparticle physics, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' B 650, 275-278 (2007)  [arXiv:0704.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='2457].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  [4] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Cheung, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Keung and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Yuan, Collider Phenomenology of Unparticle Physics, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' D 76, 055003 (2007) [arXiv:0706.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='3155].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  [5] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Khachatryan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' [CMS], Search for dark matter, extra dimensions, and unparticles in  monojet events in proton–proton collisions at √s = 8 TeV, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' C 75, 235 (2015)  [arXiv:1408.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='3583].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  [6] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Khachatryan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' [CMS], Search for dark matter and unparticles produced in association  with a Z boson in proton-proton collisions at √s = 8 TeV, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' D 93, 052011 (2016)  [erratum: Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' D 97, 099903 (2018)] [arXiv:1511.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='09375].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  [7] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Sirunyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' [CMS], Search for dark matter and unparticles in events with a Z boson  and missing transverse momentum in proton-proton collisions at √s = 13 TeV, JHEP 03, 061  (2017) [erratum: JHEP 09, 106 (2017)] [arXiv:1701.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='02042].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  [8] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Hammer and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Son, Unnuclear physics, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' 118, e2108716118  (2021) [arXiv:2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='12610].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  [9] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Karananas and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Monin, More on the operator-state map in nonrelativistic CFTs,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' D 105, 065008 (2022) [arXiv:2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='03836 [hep-th]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  [10] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Nishida and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Son, Nonrelativistic conformal field theories, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' D 76, 086004  (2007) [arXiv:0706.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='3746].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  [11] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Duer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=', Observation of a correlated free four-neutron system, Nature 606, 678 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  [12] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Lazauskas, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Hiyama and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Carbonell, Low energy structures in nuclear reactions with  4n in the final state, [arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='07575 [nucl-th]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  [13] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Braaten and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Hammer, Interpretation of Neutral Charm Mesons near Threshold as  Unparticles, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' 128, 032002 (2022) [arXiv:2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='02831].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  [14] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Choi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' [Belle Collaboration], Observation of a narrow charmonium-like state in ex-  clusive B± → K±π+π−J/ψ decays, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' 91, 262001 (2003) [hep-ex/0309032].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  [15] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Braaten and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Kusunoki, Low-energy universality and the new charmonium resonance at  3870-MeV, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' D 69, 074005 (2004) [arXiv:hep-ph/0311147 [hep-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  [16] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Braaten and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Hammer, Universality in few-body systems with large scattering length,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Rept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' 428, 259-390 (2006) [arXiv:cond-mat/0410417].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  [17] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Henkel, Schrodinger invariance in strongly anisotropic critical systems, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Statist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  75, 1023 (1994) [arXiv:hep-th/9310081].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  [18] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Henkel and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Unterberger, Schrodinger invariance and space-time symmetries, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  B 660, 407 (2003) [arXiv:hep-th/0302187].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  16  [19] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Fuertes and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Moroz, Correlation functions in the non-relativistic AdS/CFT correspon-  dence, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' D 79, 106004 (2009) [arXiv:0903.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='1844].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  [20] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Volovich and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Wen, Correlation Functions in Non-Relativistic Holography, JHEP 05,  087 (2009) [arXiv:0903.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='2455].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  [21] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Gradshteyn and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Ryzhik, Table of Integrals, Series, and Products (Academic Press,  New York, 1965).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  [22] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Aaij et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' [LHCb], Study of the lineshape of the χc1(3872) state, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' D 102, 092005  (2020) [arXiv:2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='13419].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  [23] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Aaij et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' [LHCb], Study of the ψ2(3823) and χc1(3872) states in B+ → (J/ψπ+π−) K+  decays, JHEP 08, 123 (2020) [arXiv:2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='13422].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  [24] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Canham, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Hammer and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Springer, On the scattering of D and D∗ mesons off  the X(3872), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content=' D 80, 014009 (2009) [arXiv:0906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='1263].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
+page_content='  17' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQfPAm3/content/2301.04399v1.pdf'}
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+Near horizon approximation and beyond for a two-level
+atom falling into a Kerr-Newman black hole
+Soham Sena∗, Rituparna Mandala† and Sunandan Gangopadhyaya‡
+a Department of Astrophysics and High Energy Physics
+S.N. Bose National Centre for Basic Sciences
+JD Block, Sector III, Salt Lake, Kolkata 700 106, India
+Abstract
+In this work we investigate the phenomena of acceleration radiation for a two-level atom falling into the event horizon
+of a Kerr-Newman black hole. In (Phys. Rev. D 104 (2021) 065006), it has been shown that conformal quantum
+mechanics has a connection to the generated Planck-like spectrum due to acceleration radiation. In this particular
+aspect, the near horizon approximation played a significant role. In Phys. Rev. D 106 (2022) 025004, we have used the
+beyond near horizon approximation to show that the excitation probability attains a Planck-like spectrum irrespective
+of the non-existence of an underlying conformal symmetry for a general class of static spherically symmetric black
+holes. In our analysis we have gone beyond the near horizon approximation for the rotating and charged case and
+even without the consideration of the conformal symmetry we observe a similar Planck-like spectrum. However, the
+coefficient of the spectrum is significantly different from the near horizon case. We have then considered a different
+scenario where a two-level atom emits multiple photons while freely falling into the event horizon of the Kerr-Newman
+black hole. It is observed that the Planck factor in the excitation probability is significantly small than that of the
+case of single-photon emission (for large number of simultaneously emitted photons from the two-level atom). Finally,
+we have computed the von-Neumann entropy which is also known as the horizon brightened acceleration radiation
+entropy or the HBAR entropy. We have carried out our analysis for a scalar field only to see the effect of the charge
+and rotation of the black hole in this particular scenario.
+1
+Introduction
+The greatest endeavour of modern theoretical physics is
+to unify multiple mainstream descriptions of the under-
+lying theories of nature under a more fundamental theo-
+retical description. General relativity [1, 2] and quantum
+mechanics, being the two most intrigued theoretical de-
+scriptions of the universe, are the hardest to unify in a
+single stream. Although there have been several successful
+attempts to unify other fundamental theories like quan-
+tum field theory, electrodynamics, and the weak and the
+strong nuclear forces, none have been able to unify quan-
+tum mechanics and general relativity successfully.
+The
+efforts to unify gravity, thermodynamics, and geometry
+led to several ground-breaking discoveries like black hole
+thermodynamics [5–9], Hawking radiation [7–15], parti-
+cle emission from a black hole [16–18], acceleration ra-
+diation [19–28, 31–34, 36, 37], and the Unruh effect [38].
+Hawking radiation is observed from a black hole if one
+considers quantum effects in curved space-time. Another
+phenomenon describing the spontaneous creation of par-
+ticles is the Schwinger mechanism [39].
+In this case, a
+very strong electric field is applied which results in the
+production of pairs of particles and antiparticles. In the
+case of the Unruh-Fulling effect it has been observed that
+if an observer is uniformly accelerated (with acceleration
+‘a’) through an inertial vacuum, the observer can detect
+a thermal bath of particles (quanta of the fields) with a
+∗sensohomhary@gmail.com, soham.sen@bose.res.in
+†drimit.ritu@gmail.com
+‡sunandan.gangopadhyay@gmail.com
+temperature
+T =
+ℏa
+2πkBc
+(1)
+where kB denotes the Boltzmann constant.
+In contrast
+to the accelerated observer, the inertial observer does not
+detect any such swarm of excited particles. In this pro-
+cess the two-level atom jumps to the excited state with the
+simultaneous emission of a virtual photon. By interrupt-
+ing this virtual process one can observe real photons. The
+same particle creation scenario from a different perspec-
+tive was observed in [28] for a two-level atom freely falling
+into the event horizon of a Schwarzschild black hole. In
+this process the two-level atom emits real photons and the
+excitation probability is similar to that of the excitation
+probability for the case of a mirror accelerating with re-
+spect to a fixed two-level atom in flat space-time. It was
+also observed that in this process that the freely falling
+atom feels the effect of thermal radiation similar to that
+of Hawking radiation. The von-Neumann entropy consid-
+ering this radiation was termed as the horizon brightened
+acceleration radiation entropy or the HBAR entropy [28].
+Very few works also considered the effects of the conformal
+symmetry as a contributing factor towards near horizon
+analysis of a black hole [26, 27, 31–34]. In [31], it was ob-
+served that a near horizon analysis in principle harbours
+a conformal symmetry leading to a Planck-like distribu-
+tion for the simultaneously emitted photons from two-level
+atoms freely falling into static spherically symmetric black
+holes. Later in [32], the same analysis was carried out for
+a Kerr black hole and similar Planck factor was observed
+if viewed from the perspective of a co-rotating observer
+near the event horizon of the rotating black hole and the
+dominating cause for the Planck-like distribution was con-
+1
+arXiv:2301.04834v1  [gr-qc]  12 Jan 2023
+
+sidered to be the underlying conformal symmetry arising
+due to near horizon approximation. Recently, in [37], we
+considered two-level atoms freely falling into a general class
+of static spherically symmetric black holes with the simul-
+taneous emission of scalar photons. Our analysis revolved
+around the consideration of the “beyond near horizon anal-
+ysis” which in principle breaks the conformal symmetry
+present in the earlier analysis. We found that apart from
+some changes in the coefficient, the Planck-like factor re-
+tained its integrity concluding that the conformal symme-
+try does not play a crucial role in case of the existence of a
+Planck like factor in the analytical form of the excitation
+probabilities.
+Inspired by the existing works in these fields, we consider a
+two-level uncharged atom freely falling into the event hori-
+zon of a Kerr-Newman black hole with a perfectly reflecting
+mirror at the event horizon of the black hole in this work.
+We have also implemented the “beyond near horizon” in
+our current analysis. We have also considered a much gen-
+eralized scenario, where the two level atom emits multiple
+scalar photons instead of a single atom, to investigate the
+effect of the coupling between multiple scalar fields with
+the two-level atoms on the excitation probability.
+The organization of the paper goes as follows.
+In sec-
+tion(2), we compute the scalar field solution from the co-
+variant Klein-Gordon equation in the Kerr-Newman back-
+ground. In section(3), we write down the trajectories of the
+freely falling atom in the event horizon of Kerr-Newman
+black hole. In section(4), we compute the excitation and
+absorption probabilities and in the following section we
+compute the excitation probability for the case of simul-
+taneous emission of N number of photons from a single
+two-level atom.
+In section(6), we calculate the horizon
+brightened acceleration radiation (HBAR) entropy for such
+a process and then finally conclude this work.
+2
+The Kerr Newman geometry and the
+beyond near horizon approximation
+In this section, we shall compute the solution of the Klein-
+Gordon equation in the Kerr-Newman black hole back-
+ground. The exact solution of the Einstein’s field equations
+for a charged and rotating black of mass M, charge Q, and
+angular momentum J in the Boyer-Lindquist coordinates
+({t, r, θ, φ}) is given by [40–42]
+ds2 = − ∆Q
+ρ2 (dt − a sin2 θdφ)2 + ρ2
+∆Q
+dr2 + ρ2dθ2
++ sin2 θ
+ρ2
+[(r2 + a2)dφ − adt]2
+(2)
+where a =
+J
+M , ∆Q = r2 − 2Mr + a2 + Q2, and ρ = r2 +
+a2 cos2 θ. In this case we are considering geometrized units
+(c = G = 1).
+By setting ∆Q = 0, one can obtain the
+outer and inner horizons of the Kerr black hole as r± =
+M ±
+�
+M 2 − a2 − Q2. The line element in eq.(2), can be
+recast in the following form
+ds2 = −ρ2∆Q
+Σ2
+dt2 + ρ2
+∆Q
+dr2 +ρ2dθ2 + Σ2
+ρ2 sin2 θ(dφ−ϖdt)2
+(3)
+where Σ2 = (r2 + a2)2 − ∆Qa2 sin2 θ and ϖ = − gtφ
+gφφ =
+−
+a(Q2−2Mr)
+(r2+a2)2−∆Qa2 sin2 θ .
+The position dependent angular
+velocity (ϖ) with respect to an external reference frame
+can be reduced to the angular velocity (˜Ω) of the black
+hole while approaching the external horizon of the Kerr-
+Newman black hole. The form ˜Ω reads
+˜Ω = lim
+r→r+ ϖ =
+a
+r2
++ + a2
+(4)
+where we have made use of the identity r2
++ +a2 = 2Mr+ −
+Q2. In order to proceed further, we need to compute the
+determinant and the inverse components of the metric ten-
+sor for the Kerr-Newman geometry. The determinant g of
+the metric tensor reads
+g = gttgφφ − g2
+tφ = −ρ4 sin2(θ) ,
+(5)
+and the non-vanishing metric components of gµν reads
+gtt = −(r2 + a2)2 − ∆Qa2 sin2 θ
+ρ2∆2
+Q
+, grr = ∆Q
+ρ2 , gθθ = 1
+ρ2 ,
+gtφ = gφt = a(∆Q − (r2 + a2))
+ρ2∆Q
+, gφφ = ∆Q − a2 sin2 θ
+ρ2∆Q sin2 θ
+.
+(6)
+In our analysis we are considering the case of acceleration
+radiation of a scalar field of mass σ0 from an atom of mass
+σa near the outer horizon of a Kerr-Newman black hole.
+The covariant Klein-Gordon equation for a scalar field Ψ
+reads
+1
+√−g ∂µ(√−ggµν∂νΨ) − σ2
+0Ψ = 0 .
+(7)
+Using the form of g from eq.(5) and the non-vanishing com-
+ponents of the inverse of the metric tensor from eq.(6), we
+can recast the above equation in the following form
+− Σ2
+∆Q
+∂2Ψ
+∂t2 + 2a(Q2 − 2Mr)
+∆Q
+∂2Ψ
+∂t∂φ + ∂
+∂r
+�
+∆Q
+∂Ψ
+∂r
+�
++
+1
+sin θ
+∂
+∂θ
+�
+sin θ∂Ψ
+∂θ
+�
++
+�
+1
+sin2 θ − a2
+∆Q
+� ∂2Ψ
+∂φ2 = σ2
+0ρ2Ψ.
+(8)
+In order to solve eq.(8), we denote Ψ = ψνlm in terms of the
+frequency (ν) and the quantum numbers l, m and consider
+a separation of variables as follows
+ψνlm(t, r, θ, φ) = e−iνtRνlm(r)Θνlm(θ)eimφ
+(9)
+where we will use R(r) and Θ(θ) instead of Rνlm(r) and
+Θνlm(θ) respectively throughout our analysis. We can now
+define a new frame corotating with the black hole with an
+angular velocity ˜Ω by introducing the following coordinate
+changes
+˜t = t , ˜φ = φ − ˜Ωt ,
+(10)
+and the form of the shifted frequency is given as follows
+˜ν = ν − m˜Ω .
+(11)
+Using the new definitions in eq.(s)(10,11) we can rewrite
+the form of the scalar field in eq.(9), as
+ψ˜νlm(t, r, θ, ˜φ) = e−i˜νtR(r)Θ(θ)eim ˜φ .
+(12)
+Using eq.(9) back in eq.(8), we can separate the equations
+involving the radial part of the solution and the θ depen-
+dent part of the solution. The radial equation reads
+d
+dr
+�
+∆Q
+dR
+dr
+�
++
+�(r2 + a2)2ν2 + 2aνm(Q2 − 2Mr) + a2m2
+∆Q
+− a2ν2 − σ2
+0r2 − β
+�
+R = 0
+(13)
+2
+
+with β being the separation constant and the differential
+equation involving Θ reads
+1
+sin θ
+d
+dθ
+�
+sin θdΘ
+dθ
+�
++
+�
+a2ν2 cos2 θ −
+m2
+sin2 θ + β
+− σ2
+0a2 cos2 θ
+�
+Θ = 0 .
+(14)
+In order to solve the radial equation, we use the beyond
+near horizon approximation. The form of ∆ and its higher
+order derivatives in the beyond near horizon approximation
+takes the following form
+∆Q(r) ≃ (r − r+)∆′
+Q(r+) + 1
+2(r − r+)2∆′′
+Q(r+)
+(15)
+= y∆′
+Q+ + 1
+2y2∆′′
+Q+ ,
+∆′
+Q(r) ≃ ∆′
+Q+ + y∆′′
+Q+ ,
+(16)
+∆′′
+Q(r) ≃ ∆′′
+Q+ = 2
+(17)
+where ∆′
+Q(r+) = ∆′
+Q+, ∆′′
+Q(r+) = ∆′′
+Q+, r − r+ = y, and
+y is much smaller with respect to the outer horizon ra-
+dius. When sufficiently close to the outer horizon radius
+the Killing vector is time-like and one can define the asso-
+ciated surface gravity as
+κ = −1
+2(∇µξν)(∇µξν)
+=
+r+ − r−
+2(r2
++ + a2)
+=
+∆′
+Q+
+2(r2
++ + a2) .
+(18)
+Using the beyond near horizon approximation, we can re-
+cast the radial equation in eq.(13) given as
+1
+y
+�
+d
+dx
+�
+y
+�
+1 +
+y
+∆′
+Q+
+�
+dR(y)
+dy
+��
++
+� ˜ν2
+4κ2
+1
+y2++
+�
+4ν
+� ˜νr+
+2κ
++ 1
+2am
+�
+− ˜ν2
+4κ2 − (a2ν2 + σ2
+0r2
++ + β)
+�
+1
+∆′
+Q+y
+�
+R(y) = 0 .
+(19)
+In order to solve the above radial equation, we consider the
+following relation
+R(y) =
+�
+y +
+y2
+∆′
+Q+
+�− 1
+2
+u(y) .
+(20)
+Substituting the above equation back in eq.(19), we can
+recast the radial equation in terms of u(y) as follows
+y2u′′(y) +
+�� ˜ν
+2κ
+�2
++ 1
+4
+�
+u(y) − Byu(y) = 0
+(21)
+where ′ denotes differentiation with respect to the variable
+y and
+B =
+2
+∆′
+Q+
+�� ˜ν
+2κ
+�2
++ 1
+4
+�
++ a2ν2 + β + σ2
+0r2
++
+∆′
+Q+
+−
+4ν
+∆′
+Q+
+� ˜ν
+2κr+ + 1
+2am
+�
+.
+(22)
+It is very important to observe that the B dependent lin-
+ear term in eq.(21) arises due to the consideration of the
+beyond near horizon approximation only.
+If we now set
+B = 0, we obtain the usual scaling invariant equation
+[31,32] exhibiting the asymptotic conformal symmetry. To
+solve eq.(22), we consider a trial solution of the form
+u(y) =
+∞
+�
+k=0
+Akyp+k
+(23)
+with p being a constant. Substituting the above equation
+back in eq.(21), we obtain the following relation
+∞
+�
+k=0
+Akyp+k
+�
+(p + k)(p + k − 1) − By + ˜ν2
+4κ2 + 1
+4
+�
+= 0.
+(24)
+In order to compute the individual unknown coefficients
+Ak for each k, we shall compare each equation containing
+different powers of the variable y from eq.(24) to zero. The
+equation from the pth power of order y as follows
+p2 − p + ˜ν2
+4κ2 + 1
+4 = 0 .
+(25)
+The solution of eq.(25) is given as follows
+p = 1
+2 ± i ˜ν
+2κ .
+(26)
+From the higher order equations in y, one can obtain the
+recursion relation involving the coefficients Ak. Hence, we
+can represent every coefficient Ak in terms of the coefficient
+A0 using the recursion relation as follows
+Ak = BkA0Γ
+�
+1 + i ˜ν
+κ
+�
+k!Γ
+�
+k + 1 + i ˜ν
+κ
+� .
+(27)
+As A0 is an arbitrary constant, we set its value equals to
+1. Using the form of the coefficients in the above equation,
+we obtain the formal solution of eq.(21) given as
+u(y) = y
+1
+2 + i˜ν
+2κ
+∞
+�
+k=0
+ykBkΓ
+�
+1 + i˜ν
+κ
+�
+k!Γ
+�
+k + 1 + i˜ν
+κ
+� .
+(28)
+As we are considering a very small value of the variable y,
+we shall keep upto the linear order term in y in the above
+equation. Using eq.(28) back in eq.(20), we obtain the form
+of R(y) upto O(y) to be
+R(y) =
+�
+y +
+y2∆′′
+Q+
+2∆′
+Q+
+�− 1
+2
+y
+1
+2 + i˜ν
+2κ
+∞
+�
+k=0
+ykBkΓ
+�
+1 + i˜ν
+κ
+�
+k!Γ
+�
+k + 1 + i˜ν
+κ
+�
+≃y− i˜ν
+2κ
+�
+1 +
+�
+κ2B2
+κ2 + ˜ν2 −
+1
+2∆′
+Q+
+�
+y + i˜νκB
+κ2 + ˜ν2 y
+�
+.
+(29)
+With the radial part of the solution in hand, we shall now
+proceed to obtain the θ part of the solution. We consider
+that the angle θ will attain a value θ+ when the particle
+reaches near the outer horizon radius r+. To proceed fur-
+ther, we define a new variable as follows
+ϕ = θ − θ+
+(30)
+where ϕ ≪ θ+. We can now Taylor expand any function
+f(θ) about θ+ up to O(ϕ) as f(θ) ≃ f(θ+) + ϕf ′(θ+). Up
+3
+
+to O(ϕ), we can recast eq.(14) as
+d
+dϕ
+�
+[1 + ϕ cot θ+]dΘ(ϕ)
+dϕ
+�
++
+�
+a2 cos2 θ+[ν2 − σ2
+0] −
+m2
+sin2 θ+
++ β
+�
+Θ(ϕ) + ϕ
+�
+a2 cos2 θ+(cotθ+ − 2 tan θ+)(ν2 − σ2
+0)
++ β cot θ+ + m2 cot θ+
+sin2 θ+
+�
+Θ(ϕ) = 0 .
+(31)
+In order to solve the above equation we consider the fol-
+lowing form of Θ(ϕ)
+Θ(ϕ) = (1 + ϕ cot θ+)− 1
+2 S(ϕ) .
+(32)
+By using the above relation, we can recast eq.(31) in the
+following form
+d2S(ϕ)
+dϕ2
++ p1S(ϕ) − p2ϕS(ϕ) = 0
+(33)
+where the analytical forms of p1 and p2 are given as follows
+p1 = cot2 θ+
+4
++ a2 cos2 θ+(ν2 − σ2
+0) + β −
+m2
+sin2 θ+
+, (34)
+p2 = a2 sin 2θ+(ν2 − σ2
+0) − 2m2 cot θ+
+sin2 θ+
++ cot3 θ+
+4
+.
+(35)
+Now eq.(33) has an exact solution of the form
+S(ϕ) = C1Ai
+�
+−p1 + ϕp2
+p
+2
+3
+2
+�
++C2Bi
+�
+−p1 + ϕp2
+p
+2
+3
+2
+�
+(36)
+with Ai and Bi denoting the ‘AiryAi’ and ‘AiryBi’ func-
+tions respectively, and C1 and C2 being the integration
+constants. It is very important to observe that we need a
+perturbative solution and we are doing a perturbative cal-
+culation. Although when the particle approaches the event
+horizon of the Kerr-Newman black hole, according to our
+earlier consideration, θ → θ+ and ϕ → 0 and therefore we
+do not explicitly write the form of the solution upto O(ϕ).
+Therefore, in this limit, eq.(32) reduces to the following
+form
+lim
+ϕ→0 Θ(ϕ) = Θ+ = C1Ai
+�
+− p1
+p
+2
+3
+2
+�
++C2Bi
+�
+−p1
+p
+2
+3
+2
+�
+. (37)
+We can now rewrite the solution of the Klein-Gordon equa-
+tion as follows
+ψ˜νlm = e−i˜νty− i˜ν
+2κ
+�
+1+
+� κ2B2
+κ2 + ˜ν2 −
+1
+2∆′
+Q+
+�
+y + i˜νκB
+κ2 + ˜ν2
+�
+× Θ+eim ˜φ .
+(38)
+To compute the excitation probability we need to obtain
+the equations governing the trajectory of the two level atom
+freely falling into the event horizon of the black hole.
+3
+Trajectories of the atom freely falling
+into the black hole
+In terms of the time-like and azimuthal Killing vectors ϑ
+and ς, we can define two conserved quantities as follows
+E = −ϑ.p = −gtt
+dt
+dλ − gtφ
+dφ
+dλ ,
+(39)
+Lz = ς.p = gtφ
+dt
+dλ + gφφσ0
+dφ
+dλ
+(40)
+where E denotes the energy, Lz denotes the angular mo-
+mentum, p denotes the four momentum of the atom, and λ
+denotes the affine parameter. One can make an alternative
+choice of the affine parameter by denoting τ = λσ0 and
+define the new quantities as follows
+e = E
+σ0
+= −gtt
+dt
+dτ − gtφ
+dφ
+dτ ,
+l = Lz
+σ0
+= gtφ
+dt
+dτ + gφφ
+dφ
+dτ
+(41)
+where e denotes the energy per unit mass and l denotes
+the angular momentum per unit mass. Using the above
+two relations, one can obtain the following two trajectory
+equations
+ρ2 dt
+dτ = a(l − ae sin2 θ) + r2 + a2
+∆
+K(r) ,
+(42)
+ρ2 dφ
+dτ = −
+�
+ae −
+l
+sin2 θ
+�
++
+a
+∆Q
+K(r)
+(43)
+where K(r) = (r2 + a2)e − al. One can also define one
+other conserved quantity as follows
+q = Q
+σ2
+0
+=
+� pθ
+σ0
+�2
++ cos2 θ
+�
+a2(1 − e2) +
+� l
+sin θ
+�2�
+.
+(44)
+Following the standard procedure, one can obtain the fol-
+lowing two trajectory equations as well
+ρ2 dr
+dτ = −
+�
+R(r) ,
+(45)
+ρ2 dφ
+dτ = ±
+�
+U(θ)
+(46)
+where
+R(r) = K(r)2 − ∆Q
+�
+r2 + (l − ae)2 + q
+�
+,
+(47)
+U(θ) = q − cos2 θ
+�
+a2(1 − e2) +
+l2
+sin2 θ
+�
+.
+(48)
+From eq.(45), we can obtain the radial trajectory equation
+in terms of the new modified variable y upto O(y) as follows
+(r2
++ + a2 cos2 θ+ + 2r+y)dy
+dτ = −
+�
+P2
+0 − P1x + O(y2)
+dτ
+dy ≃ −ρ2
++
+�
+1 + P1
+2P2
+0
++ 2r+x
+ρ2
++
+�
+(49)
+where ρ2
++ = r2
++ + a2 cos2 θ+, P0 = (r2
++ + a2)(e − ˜Ωl), and
+P1 = −4r+eP0 + ∆′
+Q+(r2
++ + (l − ae)2 + q). It is easy
+to solve eq.(49) and obtain the form of τ in terms of the
+spatial variable y as follows (up to O(y))
+τ(y) ≃ −ρ2
++y
+P0
+.
+(50)
+Now the energy measured from the viewpoint of the coro-
+tating frame (dragged with an angular velocity ˜Ω) reads
+˜e = (e − ˜Ωl) implying that the constant P0 as well as
+the leading order term is eq.(50) is proportional to ˜e. Now
+using eq.(42) along with eq.(45), we obtain the form of the
+time t up to O(y) as
+t(y) ≃ − 1
+2κ ln y − δy
+(51)
+4
+
+where the constant δ is given as
+δ =
+�
+P1
+2P2
+0
+−
+1
+2κ(r2
++ + a2) +
+4r+
+r2
++ + a2
+�
+1 +
+˜Ωl
+2˜e
+��
+1
+2κ
+−
+˜Ω
+˜e (aes2
++ − l)
+(52)
+with s+ = sin θ+. We can now obtain the form of φ up to
+O(y) using eq.(s)(43,45) as follows
+φ ≃ −
+˜Ω
+2κ ln y +
+�
+ae − l
+s2
++
+� y
+P0
+−
+˜Ωy
+2κ
+�
+P1
+2P2
+0
+−
+1
+2(r+2 + a2)κ +
+2r+
+r2
++ + a2
+˜e + ˜Ωl
+˜e
+�
+.
+(53)
+Using eq.(10), we can redefine the form of ˜φ up to O(y) as
+follows
+˜φ = φ − ˜Ωt = λy
+(54)
+where the constant λ is given by
+λ = −
+1
+(r2
++ + a2) ˜e(aes2
++ − l)
+�
+˜Ωa − 1
+s2
++
+�
++
+˜Ωr+
+κ(r2
++ + a2) .
+(55)
+We are now in a position to compute the analytical form
+of the excitation probability.
+4
+The transition probability
+In this case we are considering a two-level atom freely
+falling into the outer horizon of a Kerr-Newman black hole
+and its emitting photons as observed from the viewpoint of
+an observer corotating with an angular velocity ˜Ω. If the
+atomic transition frequency is ω and the ground state and
+excited states of the two-level atom are defined as |g⟩ and
+|e⟩ then we can write the atom field interaction Hamilto-
+nian considering only the coupling with a single scalar field
+as
+ˆHI(τ) = ℏG
+�
+ˆb˜νψ˜νlm(r(τ), t(τ)) + h.c.
+�
+(|g⟩⟨e|e−iωτ +h.c.)
+(56)
+where G is the atom-filed coupling constant, ˆb˜ν is the anni-
+hilation operator of the scalar field mode. Here the scalar
+photon is being emitted simultaneously with the excita-
+tion of the two level atom. Hence, we can write down the
+excitation probability given as
+Pexc = 1
+ℏ2
+����
+ˆ
+dτ ⟨1˜ν, e| ˆHI(τ) |0˜ν, g⟩
+����
+2
+= 1
+ℏ2
+����
+ˆ ∞
+0
+dy dτ
+dy ⟨1˜ν, e| ˆHI[τ(y)] |0˜ν, g⟩
+����
+2
+≃G2ρ4
++
+P2
+0
+����
+ˆ ∞
+0
+dy
+�
+1 + 2r+y
+ρ2
++
++ P1y
+2P2
+0
+�
+ψ∗
+˜νlm(y)eiωτ
+����
+2
+.
+(57)
+In the last line of eq.(57), we observe that the only con-
+tributions in the form of the excitation probability comes
+due to the existence of the counter-rotating terms in the
+interaction Hamiltonian in eq.(56). We shall now substi-
+tute the form of τ from eq.(49) and ψ∗
+˜νlm(r(y), t(y)) using
+eq.(s)(38,50,51) in the above equation. Keeping terms upto
+O(y), we can rewrite the form of the excitation probability
+as follows
+Pexc = G2ρ4
++|Θ+|2
+P2
+0
+����
+ˆ ∞
+0
+dy
+�
+1+
+�2r+
+ρ2
++
++ P1
+2P2
+0
++
+� κ2B
+κ2 + ˜ν2
+−
+1
+2∆′
+Q+
+��
+y + i˜νBκ
+κ2 + ˜ν2 y
+�
+y− i˜ν
+2κ e− i˜ν
+2κ ln ye
+−i
+�
+mλ+˜νδ+
+ρ2
++ω
+P0
+�
+y .
+(58)
+We can rewrite the above form of the excitation probability
+in a much simpler form as follows
+Pexc = G2ξ2
+����
+ˆ ∞
+0
+dy (1 + Dy + iEy) y− i˜ν
+κ e−iχy
+����
+2
+(59)
+where we have defined ξ ≡
+ρ2
++|Θ+|
+P0
+, χ ≡ mλ + ˜νδ +
+ρ2
++ω
+P0 ,
+D ≡ 2r+
+ρ2
++ +
+P1
+2P2
+0 +
+κ2B
+κ2+˜ν2 −
+1
+2∆′
+Q+ , and E ≡
+˜νBκ
+κ2+˜ν2 . Let us
+now define a new variable of the form χy = α. In terms of
+this new variable, we can rewrite the form of the excitation
+probability in eq.(59) as
+Pexc = G2ξ2
+χ2
+����
+ˆ ∞
+0
+dα
+�
+1 + D
+χ α + iE
+χ α
+�
+α− i˜ν
+κ e−iα
+����
+2
+= 2πG2ξ2˜ν
+κ3χ4
+(κ2(D2 + (E + χ)2) + (D2 + E2)˜ν2
+− 2κDχ˜ν)
+1
+e
+2π˜ν
+κ − 1
+.
+(60)
+Eq.(60) is one of the main results in our paper. We shall
+now compute the probability for the de-excitation of the
+two-level atom from the excited state to the ground state
+along with a simultaneous absorption of a photon with fre-
+quency ˜ν. The initial and the final states for such a process
+are given as
+|initial⟩ = |0˜ν, e⟩ ,
+|final⟩ = |1˜ν, g⟩ .
+(61)
+With the redefinition of a constant χ′ ≡
+ρ2
++ω
+P0 − mλ − ˜νδ
+and using the forms of the initial and final states from
+eq.(s)(61,62), we can calculate the absorption probability
+as follows
+Pabs = 2πG2ξ2˜ν
+κ3χ′4
+(κ2(D2 + (E − χ′)2) + (D2 + E2)˜ν2
++ 2κDχ′˜ν)
+1
+1 − e− 2π˜ν
+κ
+.
+(62)
+We can now obtain the form of the excitation probability
+for a simultaneous emission of a photon due to a two-level
+atom freely falling into the event horizon of A Kerr black
+hole by setting Q → 0 in eq.(60). In order to compare the
+difference between the two cases, we plot the Planck-factor
+for the two black holes with change in the frequency of
+the emitted photon in Fig.(1). As we are in geometrized
+units, we take the values of the parameters as a = 0.1a0,
+Q = 0.4a0 (only in case of the Kerr-Neumann black hole),
+and M = 0.5a0 with respect to some arbitrary length scale
+a0. From the figure we find that for the Kerr-Newman
+black hole the Planck-factor has lower values compared to
+the Planck factor in the Kerr black hole case. In the next
+5
+
+Figure 1: Planck-factor vs frequency plot for the Kerr and
+the Kerr-Newman black hole
+section we shall compute the excitation probability of a
+two-level atom with the simultaneous emission of multiple
+photons.
+5
+Simultaneous emission of N number of
+photons
+We now consider a more of a special case where during the
+excitation of the two-level atom, there are simultaneous
+emission of N number of photons with frequency ˜νj with
+the index j running from 1 to N (as observed from the
+frame corotating with an angular velocity ˜Ω). We consider
+the atomic transition frequency to be ϖ. In this scenario
+of virtual transition, the final and the initial states have
+the following form
+|initial⟩ = |0˜ν1, 0˜ν2, · · · , 0˜νN , g⟩ ,
+(63)
+|final⟩ = |1˜ν1, 1˜ν2, · · · , 1˜νN , e⟩ .
+(64)
+It is important to note that this initial and final state repre-
+sents a very special case where excitation consists of emis-
+sion of N number of virtual photons. There can be much
+complicated scenarios where when the two-level atom gets
+excited to its higher energy state by N − j number of vir-
+tual excitation processes and j number of virtual absorp-
+tion processes. The atom-field interaction Hamiltonian in
+this particular scenario is given as follows
+ˆ˜HI(τ) =ℏ ˜G[ˆb˜ν1ψ˜ν1 + h.c.] ⊗ [ˆb˜ν2ψ˜ν2 + h.c.] · · · ⊗ [ˆb˜νN ψ˜νN
++ h.c.] ⊗ (|g⟩ ⟨e| e−iϖτ + h.c)
+(65)
+where ˜G denotes the new coupling constant within the
+atom and the scalar fields and ˆb˜νj denotes the annihilation
+operator corresponding to the j-th photon with j ranging
+from 1 to N. The form of ψ˜νk, for k ∈ {1, · · · , N} is given
+as (upto O(y))
+ψ˜νklm ≃ e−i˜νkty−
+i˜νk
+2κ
+�
+1+
+� κ2B2
+k
+κ2 + ˜ν2
+k
+−
+1
+2∆′
+Q+
+�
+y + i˜νkκBk
+κ2 + ˜ν2
+k
+�
+× Θ+eim ˜φ
+(66)
+where the form of Bk is given to be
+Bk =
+2
+∆′
+Q+
+�� ˜νk
+2κ
+�2
++ 1
+4
+�
++ a2ν2
+k + β + σ2
+0r2
++
+∆′
+Q+
+− 4νk
+∆′
+Q+
+� ˜νk
+2κr+ + 1
+2am
+�
+(67)
+with ˜νk = νk − m˜Ω.
+Following the previous analysis it
+is easy to understand that only the counter-rotating terms
+will contribute to the excitation probability. The excitation
+probability for this particular scenario is given by
+PN =
+˜G2ρ4
++
+P2
+0
+����
+ˆ ∞
+0
+dy
+�
+1 + 2r+y
+ρ2
++
++ P1y
+2P2
+0
+�
+ψ∗
+˜ν1 · · · ψ∗
+˜νN eiϖτ
+����
+2
+(68)
+where we have not explicitly written the other two quan-
+tum numbers defining each mode of the scalar field. For a
+rather simpler approach, we shall now use ˜ν1 = ˜ν2 = · · · =
+˜νN = ˜ν. For this scenario, we also get B1 = B2 = · · · =
+BN = B. Hence, the excitation probability for the simul-
+taneous emission of N photons with identical frequencies
+is given as
+P ′
+N = 1
+ℏ2
+����
+ˆ
+dτ ⟨final| ˆ˜HI(τ) |initial⟩
+����
+2
+≃
+˜G2ρ4
++|Θ+|2N
+P2
+0
+����
+ˆ ∞
+0
+dy
+�
+1 + ˜Dy + i˜Ey
+�
+e−i˜χyy− iN ˜ν
+κ
+����
+2
+(69)
+where the analytical forms of the constants ˜D, ˜E, and ˜χ
+are given by
+˜D = 2r+
+ρ2
++
++ P1
+2P2
+0
++ N
+�
+κ2B
+κ2 + ˜ν2 −
+1
+2∆′
+Q+
+�
+,
+(70)
+˜E = ˜νBκN
+κ2 + ˜ν2 ,
+(71)
+˜χ = (mλ + ˜νδ)N + ρ2
++ϖ
+P0
+.
+(72)
+We now define a new constant ˜ξ ≡
+ρ2
++|Θ+|N
+P0
+and a new
+variable ˜α ≡ ˜χy. In terms of ˜α and ˜ξ, we can rewrite the
+excitation probability in eq.(69) as
+P ′
+N =
+˜G2 ˜ξ2
+˜χ2
+����
+ˆ ∞
+0
+d˜α
+�
+1 +
+˜D
+˜χ ˜α + i
+˜E
+˜χ ˜α
+�
+e−i˜α ˜α− iN ˜ν
+κ
+����
+2
+= 2Nπ ˜G2 ˜ξ2˜ν
+κ3 ˜χ4
+(κ2( ˜D2 + (˜E + ˜χ)2) + ( ˜D2 + ˜E2)N 2˜ν2
+− 2Nκ ˜D˜χ˜ν)
+1
+e
+2Nπ˜ν
+κ
+− 1
+.
+(73)
+Comparing eq.(60) along with eq.(73), we observe that the
+Planck factor in the N-photon emission case has a signif-
+icantly smaller value than the single photon emission case
+6
+
+Planck-factor
+0.025
+0.020
+0.015
+Kerr
+0.010
+Kerr-Newman
+0.005
+vao
+0.4
+0.6
+0.8
+1.0leading to an exponential decrease in the case of multi-
+photon emission case for a higher value of N. In the next
+section, we shall compute the von-Neumann entropy for
+photon emission when a two-level atom falls freely into the
+event horizon of a black hole.
+6
+Horizon brightened acceleration radia-
+tion entropy
+We now consider that a stream of two-level atoms are
+falling into the event horizon of the black hole at a con-
+stant rate k. If ∆N number of particles are falling in a
+time interval ∆t, we can write
+∆N
+∆t = k.
+(74)
+We shall now revert to a density matrix approach.
+We
+consider that for the i-th atom the change in the density
+matrix is δϱi. Hence, for N number of atoms, the change
+in the density matrix takes the form
+∆ϱ =
+∆N
+�
+i=0
+δϱi = ∆Nδϱ
+(75)
+where we have considered that the change in the density
+matrix is constant for each atom. Using eq.(75) back in
+eq.(74), we get the following relation
+∆ϱ
+∆t = kδϱ.
+(76)
+The absorption and emission rates can be written as Γexc =
+kPexc and Γabs = kPabs. In terms of the absorption and
+emission rates, the Lindblad master equation reads [43]
+˙ϱ = − Γabs
+2
+�
+ϱb†b + b†bϱ − 2bϱb†�
+− Γexc
+2
+�
+ϱbb† + bb†ϱ − 2b†ϱb
+�
+(77)
+The rate of change of the {n, n}-th element of the density
+matrix is given as
+˙ϱn,n = − Γabs (nϱn,n − (n + 1)ϱn+1,n+1)
+− Γexc ((n + 1)ϱn,n − nϱn−1,n−1) .
+(78)
+The steady-state single-mode solution takes the form
+ϱS.S
+n,n =
+�Γexc
+Γabs
+�n �
+1 − Γexc
+Γabs
+�
+(79)
+In order to compute the Horizon Brightened acceleration
+radiation entropy, we consider ω ≫ ˜ν, mλ. Following this
+assumption we find χ = χ′ ≃
+ρ2
++ω
+P0
+and χ > E. As addi-
+tional assumptions, we consider χ > D and κ > ˜ν. Using
+the above assumptions, we can write
+Γexc
+Γabs
+= Pexc
+Pabs
+=
+�
+1 + 4E
+χ − 4D˜ν
+κχ
+�
+e− 2π˜ν
+κ
+≃
+�
+1 + 4B˜ν
+κχ − 4D˜ν
+κχ
+�
+e− 2π˜ν
+κ
+.
+(80)
+The von-Neumann entropy for the system reads
+Sϱ = −Tr[ϱ ln ϱ] = −
+�
+n,ν,l,m
+ϱn,n ln ϱ.
+(81)
+The rate of change of the von-Neumann entropy in eq.(81)
+can be computed as follows (with the Boltzmann constant
+set to 1)
+˙Sϱ = − d
+dt (Tr[ϱ ln ϱ]) ≃ −
+�
+n,ν,l,m
+˙ϱn,n ln ϱS.S.
+n,n .
+(82)
+Now for the n, n element of the density matrix with the
+set of quantum numbers s =
+˜
+ν, l, m, we get the following
+relation
+�
+n
+˙ϱn,nn = ˙¯ns .
+(83)
+If we now sum ˙¯ns˜ν over all the quantum numbers, we ob-
+tain
+�
+s={ν,l,m}
+˙¯ns˜ν =
+�
+s={ν,l,m}
+˙¯ns(ν − m˜Ω) = ˙Ep − ˜Ω ˙Jp .
+(84)
+In eq.(84) ˙Ep is the rate of change in the total energy of the
+black hole and ˙Jp is the rate of change of the axial angular
+momentum of the black hole (defined about the rotational
+axis of the black hole). The extra term in eq.(84) generates
+due to the angular momentum of the black hole. Using the
+definition of a =
+J
+M , we can rewrite eq.(84) as follows
+�
+s={ν,l,m}
+˙¯ns˜ν = ˙mp −
+a2
+r2
++ + a2 ˙mp =
+˙mpr2
++
+r2
++ + a2
+(85)
+with ˙mp denoting the rate of change of the mass of the
+black hole due to emitting photons. In order to express
+the rate of change in the von-Neumann entropy in terms
+of the area of the black hole, we compute the area of the
+Kerr-Newman black hole at r = r+ and some fixed time as
+A =
+ˆ √gθθgφφdθdφ = 4π(r2
++ + a2) .
+(86)
+The rate of change of the area of the black hole due to si-
+multaneous emission of the photons can be directly related
+to the rate of change of the mass of the black hole due to
+emitting photons via the following relation
+˙Ap = 16πr2
++
+∆′
+Q+
+˙mp .
+(87)
+Substituting eq.(79) into eq.(82) and using eq.(s)(83-87),
+we obtain the following form for the rate of change of the
+von-Neumann entropy
+˙Sϱ = −
+�
+n,ν,l,m
+n ˙ϱn,n
+�
+ln
+�
+1 − 4˜ν
+κχ (D − B)
+�
+− 2π˜ν
+κ
+�
+=
+˙mpr2
++
+r2
++ + a2
+2π
+κ +
+4 ˙mpr2
++
+κχ(r2
++ + a2) (D − B)
+=⇒
+˙Sϱ ≃
+˙Ap
+4 +
+˙ApP0
+2πρ2
++ω
+�
+2r+
+ρ2
++
++ P1
+2P2
+0
+−
+1
+2∆′
+Q+
+�
+.
+(88)
+Eq.(88) is one of the most important results in our work,
+the first term of the above result can be termed as the
+“ ˙A/4 law” in the context of HBAR entropy. It shows that
+because of the consideration of the beyond near horizon
+approximation the the rate of change of the HBAR entropy
+picks up an extra correction term apart from the usual
+total time derivative of the “area divided by four” form
+factor [34].
+7
+
+7
+Conclusion
+In this work we have considered a two-level atom freely
+falling into the event horizon of a rotating and charged
+black hole. In our analysis we have considered the event
+horizon of the black hole to be covered by a perfectly re-
+flecting mirror which shields the infalling atoms from inter-
+acting with the outgoing Hawking-radiation. This particu-
+lar setup is identical to that of a Boulware like vacuum [44].
+Our analysis captures the essence of going beyond the near
+horizon approximation.
+In case of a near horizon anal-
+ysis, there exists an underlying conformal symmetry. In
+the beyond near horizon analysis, the underlying confor-
+mal symmetry breaks down but the excitation probability
+still retains a Planck like factor. This observation implies
+that the occurrence of the Planck like factor is independent
+of the underlying conformal symmetry. Next we have plot-
+ted the Planck-factor with respect to frequency for a Kerr
+and a Kerr-Newman black hole.
+From Fig.(1), we have
+observed that the Planck factor has higher value in case
+of the Kerr black hole for different values of the frequency
+implying a lower amount in the emission of photons for a
+Kerr-Newman black hole with same mass and same angular
+momentum. In the next part of our analysis, we have con-
+sidered a much generalized case where with the excitation
+of a two-level atom from its ground state there are simulta-
+neous emission of multiple photons (N number of photons
+in our case). We observe that the excitation probability
+has identical form factor to that of a single-photon emission
+case. The only difference lies in the fact the exponential
+decay is much more amplified when a significantly higher
+number of photons are being emitted simultaneously. Fi-
+nally we have computed the von-Neumann entropy which
+is also termed as the horizon brightened acceleration radi-
+ation entropy [28]. We observe that in the analytical form
+of the rate of change of the von-Neumann entropy, apart
+from the total time derivative of the area divided by four
+term, there exists extra correction terms. These correction
+terms generate only due to the consideration of the beyond
+near horizon approximation.
+References
+[1] A. Einstein, Sitzungsber Preuss Akad Wiss (1915) 844.
+[2] A. Einstein, Ann. der Physik 49 (1916) 769.
+[3] P. Weiss, Sci. News 157 (2000) 86.
+[4] T. G. Philbin, et al., Science 319 (2008) 1367.
+[5] J.D. Bekenstein, Lett. Nuovo Cimento 4 (1972) 737.
+[6] J. D. Bekenstein, Phys. Rev. D 7 (1973) 2333.
+[7] S.W. Hawking, Nature 248 (1974) 30.
+[8] S.W. Hawking, Commun. Math. Phys 43 (1975) 199.
+[9] S.W. Hawking, Phys. Rev. D 13 (1976) 191.
+[10] S. Gangopadhyay and S. Kulkarni, Phys. Rev. D 77
+(2008) 024038.
+[11] S. Gangopadhyay, Phys. Rev. D 77 (2008) 064027.
+[12] S. Gangopadhyay, Phys. Rev. D 78 (2008) 044026.
+[13] R. Banerjee, S. Gangopadhyay, and S. Kulkarni, Gen.
+Relativ. Gravit. 42 (2010) 2865.
+[14] S. Gangopadhyay, Eur. Phys. Lett. 85 (2009) 10004.
+[15] S. Gangopadhyay, Gen. Relativ. Gravit. 42 (2010)
+1183.
+[16] D. N. Page, Phys. Rev. D 13 (1976) 198.
+[17] D. N. Page, Phys. Rev. D 14 (1976) 3260.
+[18] D. N. Page, Phys. Rev. D 16 (1977) 2402.
+[19] S. A. Fulling, Phys. Rev. D 7 (1973) 2850.
+[20] P. Davies, J. Phys. A 8 (1975) 609.
+[21] B. S. DeWitt, General Relativity: An Einstein Cente-
+nary Survey, eds S. W. Hawking, W. Israel (Cambridge
+Univ Press (1979), Cambridge, UK).
+[22] W. G. Unruh and R. M. Wald, Phys. Rev. D 29 (1984)
+1047.
+[23] R. M¨uller, Phys. Rev. D 56 (1997) 953.
+[24] D. A. T. Vanzella and G. E. A. Matsas, Phys. Rev.
+Lett. 87 (2001) 151301.
+[25] L. C. B. Crispino, A. Higuchi, G. E. A. Matsas, Rev.
+Mod. Phys. 80 (2008) 787.
+[26] A. Castro, A. Maloney and A. Strominger, Phys. Rev.
+D 82 (2010) 024008.
+[27] I. Agullo, J. Navarro-Salas, G. J. Olmo and L. Parker,
+Phys. Rev. Lett. 105 (2010) 211305.
+[28] M. O. Scully, S. A. Fulling, D. M. Lee, D. N. Page,
+W. P. Schleich and A. A. Svidzinsky, Proc. Natl. Acad.
+Sci. U.S.A. 115 (2018) 8131.
+[29] A. A. Svidzinsky, S. J. Ben-Benjamin, S. A. Fulling
+and D. N. Page, Phys. Rev. Lett 121 (2018) 071301.
+[30] S. A. Fulling and J. H. Wilson, Phys. Scr. 94 (2019)
+014004.
+[31] H. E. Camblong, A. Chakraborty and C. R. Ord´o˜nez,
+Phys. Rev. D 102 (2020) 085010.
+[32] A. Azizi, H. E. Camblong, A. Chakraborty, C. R.
+Ord´o˜nez and M. O. Scully, Phys. Rev. D 104 (2021)
+065006.
+[33] A. Azizi, H. E. Camblong, A. Chakraborty, C. R.
+Ord´o˜nez and M. O. Scully, Phys. Rev. D 104 (2021)
+084086.
+[34] A. Azizi, H. E. Camblong, A. Chakraborty, C. R.
+Ord´o˜nez and M. O. Scully, Phys. Rev. D 104 (2021)
+084085.
+[35] R. Chatterjee, S. Gangopadhyay and A. S. Majumdar,
+Phys. Rev. D 104 (2021) 124001.
+[36] S. Sen, R. Mandal and S. Gangopadhyay, Phys. Rev.
+D 105 (2022) 085007.
+[37] S. Sen, R. Mandal and S. Gangopadhyay, Phys. Rev.
+D 106 (2022) 025004.
+8
+
+[38] W. G. Unruh, Phys. Rev. D 14 (1976) 870.
+[39] J. Schwinger, Phys. Rev. 82 (1951) 664.
+[40] R. P. Kerr, Phys. Rev. Lett. 11 (1963) 237.
+[41] E. T. Newman, E. Couch, K. Chinnapared, A. Exton,
+A. Prakash, and R. Torrence, J. Math. Phys. 6 (1965)
+918.
+[42] R. H. Boyer and R. W. Lindquist, J. Math. Phys. 8
+(1957) 265.
+[43] G. Lindblad, Commun. Math. Phys. 48 (1976) 119.
+[44] D. G. Boulware, Phys. Rev. D 11 (1975) 1404.
+9
+
diff --git a/xdE3T4oBgHgl3EQf_gsD/content/tmp_files/load_file.txt b/xdE3T4oBgHgl3EQf_gsD/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..1d108b29281ae65301f961e4811444dee625c7ad
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@@ -0,0 +1,562 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf,len=561
+page_content='Near horizon approximation and beyond for a two-level  atom falling into a Kerr-Newman black hole  Soham Sena∗, Rituparna Mandala† and Sunandan Gangopadhyaya‡  a Department of Astrophysics and High Energy Physics  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Bose National Centre for Basic Sciences  JD Block, Sector III, Salt Lake, Kolkata 700 106, India  Abstract  In this work we investigate the phenomena of acceleration radiation for a two-level atom falling into the event horizon  of a Kerr-Newman black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' In (Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' D 104 (2021) 065006), it has been shown that conformal quantum  mechanics has a connection to the generated Planck-like spectrum due to acceleration radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' In this particular  aspect, the near horizon approximation played a significant role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' In Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' D 106 (2022) 025004, we have used the  beyond near horizon approximation to show that the excitation probability attains a Planck-like spectrum irrespective  of the non-existence of an underlying conformal symmetry for a general class of static spherically symmetric black  holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' In our analysis we have gone beyond the near horizon approximation for the rotating and charged case and  even without the consideration of the conformal symmetry we observe a similar Planck-like spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' However, the  coefficient of the spectrum is significantly different from the near horizon case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' We have then considered a different  scenario where a two-level atom emits multiple photons while freely falling into the event horizon of the Kerr-Newman  black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' It is observed that the Planck factor in the excitation probability is significantly small than that of the  case of single-photon emission (for large number of simultaneously emitted photons from the two-level atom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Finally,  we have computed the von-Neumann entropy which is also known as the horizon brightened acceleration radiation  entropy or the HBAR entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' We have carried out our analysis for a scalar field only to see the effect of the charge  and rotation of the black hole in this particular scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  1  Introduction  The greatest endeavour of modern theoretical physics is  to unify multiple mainstream descriptions of the under-  lying theories of nature under a more fundamental theo-  retical description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' General relativity [1, 2] and quantum  mechanics, being the two most intrigued theoretical de-  scriptions of the universe, are the hardest to unify in a  single stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Although there have been several successful  attempts to unify other fundamental theories like quan-  tum field theory, electrodynamics, and the weak and the  strong nuclear forces, none have been able to unify quan-  tum mechanics and general relativity successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  The  efforts to unify gravity, thermodynamics, and geometry  led to several ground-breaking discoveries like black hole  thermodynamics [5–9], Hawking radiation [7–15], parti-  cle emission from a black hole [16–18], acceleration ra-  diation [19–28, 31–34, 36, 37], and the Unruh effect [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  Hawking radiation is observed from a black hole if one  considers quantum effects in curved space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Another  phenomenon describing the spontaneous creation of par-  ticles is the Schwinger mechanism [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  In this case, a  very strong electric field is applied which results in the  production of pairs of particles and antiparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' In the  case of the Unruh-Fulling effect it has been observed that  if an observer is uniformly accelerated (with acceleration  ‘a’) through an inertial vacuum, the observer can detect  a thermal bath of particles (quanta of the fields) with a  ∗sensohomhary@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='com, soham.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='sen@bose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='in  †drimit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='ritu@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='com  ‡sunandan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='gangopadhyay@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='com  temperature  T =  ℏa  2πkBc  (1)  where kB denotes the Boltzmann constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  In contrast  to the accelerated observer, the inertial observer does not  detect any such swarm of excited particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' In this pro-  cess the two-level atom jumps to the excited state with the  simultaneous emission of a virtual photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' By interrupt-  ing this virtual process one can observe real photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' The  same particle creation scenario from a different perspec-  tive was observed in [28] for a two-level atom freely falling  into the event horizon of a Schwarzschild black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' In  this process the two-level atom emits real photons and the  excitation probability is similar to that of the excitation  probability for the case of a mirror accelerating with re-  spect to a fixed two-level atom in flat space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' It was  also observed that in this process that the freely falling  atom feels the effect of thermal radiation similar to that  of Hawking radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' The von-Neumann entropy consid-  ering this radiation was termed as the horizon brightened  acceleration radiation entropy or the HBAR entropy [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  Very few works also considered the effects of the conformal  symmetry as a contributing factor towards near horizon  analysis of a black hole [26, 27, 31–34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' In [31], it was ob-  served that a near horizon analysis in principle harbours  a conformal symmetry leading to a Planck-like distribu-  tion for the simultaneously emitted photons from two-level  atoms freely falling into static spherically symmetric black  holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Later in [32], the same analysis was carried out for  a Kerr black hole and similar Planck factor was observed  if viewed from the perspective of a co-rotating observer  near the event horizon of the rotating black hole and the  dominating cause for the Planck-like distribution was con-  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='04834v1  [gr-qc]  12 Jan 2023  sidered to be the underlying conformal symmetry arising  due to near horizon approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Recently, in [37], we  considered two-level atoms freely falling into a general class  of static spherically symmetric black holes with the simul-  taneous emission of scalar photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Our analysis revolved  around the consideration of the “beyond near horizon anal-  ysis” which in principle breaks the conformal symmetry  present in the earlier analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' We found that apart from  some changes in the coefficient, the Planck-like factor re-  tained its integrity concluding that the conformal symme-  try does not play a crucial role in case of the existence of a  Planck like factor in the analytical form of the excitation  probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  Inspired by the existing works in these fields, we consider a  two-level uncharged atom freely falling into the event hori-  zon of a Kerr-Newman black hole with a perfectly reflecting  mirror at the event horizon of the black hole in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  We have also implemented the “beyond near horizon” in  our current analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' We have also considered a much gen-  eralized scenario, where the two level atom emits multiple  scalar photons instead of a single atom, to investigate the  effect of the coupling between multiple scalar fields with  the two-level atoms on the excitation probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  The organization of the paper goes as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  In sec-  tion(2), we compute the scalar field solution from the co-  variant Klein-Gordon equation in the Kerr-Newman back-  ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' In section(3), we write down the trajectories of the  freely falling atom in the event horizon of Kerr-Newman  black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' In section(4), we compute the excitation and  absorption probabilities and in the following section we  compute the excitation probability for the case of simul-  taneous emission of N number of photons from a single  two-level atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  In section(6), we calculate the horizon  brightened acceleration radiation (HBAR) entropy for such  a process and then finally conclude this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  2  The Kerr Newman geometry and the  beyond near horizon approximation  In this section, we shall compute the solution of the Klein-  Gordon equation in the Kerr-Newman black hole back-  ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' The exact solution of the Einstein’s field equations  for a charged and rotating black of mass M, charge Q, and  angular momentum J in the Boyer-Lindquist coordinates  ({t, r, θ, φ}) is given by [40–42]  ds2 = − ∆Q  ρ2 (dt − a sin2 θdφ)2 + ρ2  ∆Q  dr2 + ρ2dθ2  + sin2 θ  ρ2  [(r2 + a2)dφ − adt]2  (2)  where a =  J  M , ∆Q = r2 − 2Mr + a2 + Q2, and ρ = r2 +  a2 cos2 θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' In this case we are considering geometrized units  (c = G = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  By setting ∆Q = 0, one can obtain the  outer and inner horizons of the Kerr black hole as r± =  M ±  �  M 2 − a2 − Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' The line element in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' (2), can be  recast in the following form  ds2 = −ρ2∆Q  Σ2  dt2 + ρ2  ∆Q  dr2 +ρ2dθ2 + Σ2  ρ2 sin2 θ(dφ−ϖdt)2  (3)  where Σ2 = (r2 + a2)2 − ∆Qa2 sin2 θ and ϖ = − gtφ  gφφ =  −  a(Q2−2Mr)  (r2+a2)2−∆Qa2 sin2 θ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  The position dependent angular  velocity (ϖ) with respect to an external reference frame  can be reduced to the angular velocity (˜Ω) of the black  hole while approaching the external horizon of the Kerr-  Newman black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' The form ˜Ω reads  ˜Ω = lim  r→r+ ϖ =  a  r2  + + a2  (4)  where we have made use of the identity r2  + +a2 = 2Mr+ −  Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' In order to proceed further, we need to compute the  determinant and the inverse components of the metric ten-  sor for the Kerr-Newman geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' The determinant g of  the metric tensor reads  g = gttgφφ − g2  tφ = −ρ4 sin2(θ) ,  (5)  and the non-vanishing metric components of gµν reads  gtt = −(r2 + a2)2 − ∆Qa2 sin2 θ  ρ2∆2  Q  , grr = ∆Q  ρ2 , gθθ = 1  ρ2 ,  gtφ = gφt = a(∆Q − (r2 + a2))  ρ2∆Q  , gφφ = ∆Q − a2 sin2 θ  ρ2∆Q sin2 θ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  (6)  In our analysis we are considering the case of acceleration  radiation of a scalar field of mass σ0 from an atom of mass  σa near the outer horizon of a Kerr-Newman black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  The covariant Klein-Gordon equation for a scalar field Ψ  reads  1  √−g ∂µ(√−ggµν∂νΨ) − σ2  0Ψ = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  (7)  Using the form of g from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' (5) and the non-vanishing com-  ponents of the inverse of the metric tensor from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' (6), we  can recast the above equation in the following form  − Σ2  ∆Q  ∂2Ψ  ∂t2 + 2a(Q2 − 2Mr)  ∆Q  ∂2Ψ  ∂t∂φ + ∂  ∂r  �  ∆Q  ∂Ψ  ∂r  �  +  1  sin θ  ∂  ∂θ  �  sin θ∂Ψ  ∂θ  �  +  �  1  sin2 θ − a2  ∆Q  � ∂2Ψ  ∂φ2 = σ2  0ρ2Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  (8)  In order to solve eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' (8), we denote Ψ = ψνlm in terms of the  frequency (ν) and the quantum numbers l, m and consider  a separation of variables as follows  ψνlm(t, r, θ, φ) = e−iνtRνlm(r)Θνlm(θ)eimφ  (9)  where we will use R(r) and Θ(θ) instead of Rνlm(r) and  Θνlm(θ) respectively throughout our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' We can now  define a new frame corotating with the black hole with an  angular velocity ˜Ω by introducing the following coordinate  changes  ˜t = t , ˜φ = φ − ˜Ωt ,  (10)  and the form of the shifted frequency is given as follows  ˜ν = ν − m˜Ω .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  (11)  Using the new definitions in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' (s)(10,11) we can rewrite  the form of the scalar field in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' (9), as  ψ˜νlm(t, r, θ, ˜φ) = e−i˜νtR(r)Θ(θ)eim ˜φ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  (12)  Using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' (9) back in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' (8), we can separate the equations  involving the radial part of the solution and the θ depen-  dent part of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' The radial equation reads  d  dr  �  ∆Q  dR  dr  �  +  �(r2 + a2)2ν2 + 2aνm(Q2 − 2Mr) + a2m2  ∆Q  − a2ν2 − σ2  0r2 − β  �  R = 0  (13)  2  with β being the separation constant and the differential  equation involving Θ reads  1  sin θ  d  dθ  �  sin θdΘ  dθ  �  +  �  a2ν2 cos2 θ −  m2  sin2 θ + β  − σ2  0a2 cos2 θ  �  Θ = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  (14)  In order to solve the radial equation, we use the beyond  near horizon approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' The form of ∆ and its higher  order derivatives in the beyond near horizon approximation  takes the following form  ∆Q(r) ≃ (r − r+)∆′  Q(r+) + 1  2(r − r+)2∆′′  Q(r+)  (15)  = y∆′  Q+ + 1  2y2∆′′  Q+ ,  ∆′  Q(r) ≃ ∆′  Q+ + y∆′′  Q+ ,  (16)  ∆′′  Q(r) ≃ ∆′′  Q+ = 2  (17)  where ∆′  Q(r+) = ∆′  Q+, ∆′′  Q(r+) = ∆′′  Q+, r − r+ = y, and  y is much smaller with respect to the outer horizon ra-  dius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' When sufficiently close to the outer horizon radius  the Killing vector is time-like and one can define the asso-  ciated surface gravity as  κ = −1  2(∇µξν)(∇µξν)  =  r+ − r−  2(r2  + + a2)  =  ∆′  Q+  2(r2  + + a2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  (18)  Using the beyond near horizon approximation, we can re-  cast the radial equation in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' (13) given as  1  y  �  d  dx  �  y  �  1 +  y  ∆′  Q+  �  dR(y)  dy  ��  +  � ˜ν2  4κ2  1  y2++  �  4ν  � ˜νr+  2κ  + 1  2am  �  − ˜ν2  4κ2 − (a2ν2 + σ2  0r2  + + β)  �  1  ∆′  Q+y  �  R(y) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  (19)  In order to solve the above radial equation, we consider the  following relation  R(y) =  �  y +  y2  ∆′  Q+  �− 1  2  u(y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  (20)  Substituting the above equation back in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' (19), we can  recast the radial equation in terms of u(y) as follows  y2u′′(y) +  �� ˜ν  2κ  �2  + 1  4  �  u(y) − Byu(y) = 0  (21)  where ′ denotes differentiation with respect to the variable  y and  B =  2  ∆′  Q+  �� ˜ν  2κ  �2  + 1  4  �  + a2ν2 + β + σ2  0r2  +  ∆′  Q+  −  4ν  ∆′  Q+  � ˜ν  2κr+ + 1  2am  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  (22)  It is very important to observe that the B dependent lin-  ear term in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' (21) arises due to the consideration of the  beyond near horizon approximation only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  If we now set  B = 0, we obtain the usual scaling invariant equation  [31,32] exhibiting the asymptotic conformal symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' To  solve eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' (22), we consider a trial solution of the form  u(y) =  ∞  �  k=0  Akyp+k  (23)  with p being a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Substituting the above equation  back in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' (21), we obtain the following relation  ∞  �  k=0  Akyp+k  �  (p + k)(p + k − 1) − By + ˜ν2  4κ2 + 1  4  �  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  (24)  In order to compute the individual unknown coefficients  Ak for each k, we shall compare each equation containing  different powers of the variable y from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' (24) to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' The  equation from the pth power of order y as follows  p2 − p + ˜ν2  4κ2 + 1  4 = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  (25)  The solution of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' (25) is given as follows  p = 1  2 ± i ˜ν  2κ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  (26)  From the higher order equations in y, one can obtain the  recursion relation involving the coefficients Ak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Hence, we  can represent every coefficient Ak in terms of the coefficient  A0 using the recursion relation as follows  Ak = BkA0Γ  �  1 + i ˜ν  κ  �  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='Γ  �  k + 1 + i ˜ν  κ  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  (27)  As A0 is an arbitrary constant, we set its value equals to  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Using the form of the coefficients in the above equation,  we obtain the formal solution of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' (21) given as  u(y) = y  1  2 + i˜ν  2κ  ∞  �  k=0  ykBkΓ  �  1 + i˜ν  κ  �  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='Γ  �  k + 1 + i˜ν  κ  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  (28)  As we are considering a very small value of the variable y,  we shall keep upto the linear order term in y in the above  equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' (28) back in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' (20), we obtain the form  of R(y) upto O(y) to be  R(y) =  �  y +  y2∆′′  Q+  2∆′  Q+  �− 1  2  y  1  2 + i˜ν  2κ  ∞  �  k=0  ykBkΓ  �  1 + i˜ν  κ  �  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='Γ  �  k + 1 + i˜ν  κ  �  ≃y− i˜ν  2κ  �  1 +  �  κ2B2  κ2 + ˜ν2 −  1  2∆′  Q+  �  y + i˜νκB  κ2 + ˜ν2 y  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  (29)  With the radial part of the solution in hand, we shall now  proceed to obtain the θ part of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' We consider  that the angle θ will attain a value θ+ when the particle  reaches near the outer horizon radius r+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' To proceed fur-  ther, we define a new variable as follows  ϕ = θ − θ+  (30)  where ϕ ≪ θ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' We can now Taylor expand any function  f(θ) about θ+ up to O(ϕ) as f(θ) ≃ f(θ+) + ϕf ′(θ+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Up  3  to O(ϕ), we can recast eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' (14) as  d  dϕ  �  [1 + ϕ cot θ+]dΘ(ϕ)  dϕ  �  +  �  a2 cos2 θ+[ν2 − σ2  0] −  m2  sin2 θ+  + β  �  Θ(ϕ) + ϕ  �  a2 cos2 θ+(cotθ+ − 2 tan θ+)(ν2 − σ2  0)  + β cot θ+ + m2 cot θ+  sin2 θ+  �  Θ(ϕ) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  (31)  In order to solve the above equation we consider the fol-  lowing form of Θ(ϕ)  Θ(ϕ) = (1 + ϕ cot θ+)− 1  2 S(ϕ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  (32)  By using the above relation, we can recast eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' (31) in the  following form  d2S(ϕ)  dϕ2  + p1S(ϕ) − p2ϕS(ϕ) = 0  (33)  where the analytical forms of p1 and p2 are given as follows  p1 = cot2 θ+  4  + a2 cos2 θ+(ν2 − σ2  0) + β −  m2  sin2 θ+  , (34)  p2 = a2 sin 2θ+(ν2 − σ2  0) − 2m2 cot θ+  sin2 θ+  + cot3 θ+  4  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  (35)  Now eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' (33) has an exact solution of the form  S(ϕ) = C1Ai  �  −p1 + ϕp2  p  2  3  2  �  +C2Bi  �  −p1 + ϕp2  p  2  3  2  �  (36)  with Ai and Bi denoting the ‘AiryAi’ and ‘AiryBi’ func-  tions respectively, and C1 and C2 being the integration  constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' It is very important to observe that we need a  perturbative solution and we are doing a perturbative cal-  culation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Although when the particle approaches the event  horizon of the Kerr-Newman black hole, according to our  earlier consideration, θ → θ+ and ϕ → 0 and therefore we  do not explicitly write the form of the solution upto O(ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  Therefore, in this limit, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' (32) reduces to the following  form  lim  ϕ→0 Θ(ϕ) = Θ+ = C1Ai  �  − p1  p  2  3  2  �  +C2Bi  �  −p1  p  2  3  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' (37)  We can now rewrite the solution of the Klein-Gordon equa-  tion as follows  ψ˜νlm = e−i˜νty− i˜ν  2κ  �  1+  � κ2B2  κ2 + ˜ν2 −  1  2∆′  Q+  �  y + i˜νκB  κ2 + ˜ν2  �  × Θ+eim ˜φ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  (38)  To compute the excitation probability we need to obtain  the equations governing the trajectory of the two level atom  freely falling into the event horizon of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  3  Trajectories of the atom freely falling  into the black hole  In terms of the time-like and azimuthal Killing vectors ϑ  and ς, we can define two conserved quantities as follows  E = −ϑ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='p = −gtt  dt  dλ − gtφ  dφ  dλ ,  (39)  Lz = ς.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='p = gtφ  dt  dλ + gφφσ0  dφ  dλ  (40)  where E denotes the energy, Lz denotes the angular mo-  mentum, p denotes the four momentum of the atom, and λ  denotes the affine parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' One can make an alternative  choice of the affine parameter by denoting τ = λσ0 and  define the new quantities as follows  e = E  σ0  = −gtt  dt  dτ − gtφ  dφ  dτ ,  l = Lz  σ0  = gtφ  dt  dτ + gφφ  dφ  dτ  (41)  where e denotes the energy per unit mass and l denotes  the angular momentum per unit mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Using the above  two relations, one can obtain the following two trajectory  equations  ρ2 dt  dτ = a(l − ae sin2 θ) + r2 + a2  ∆  K(r) ,  (42)  ρ2 dφ  dτ = −  �  ae −  l  sin2 θ  �  +  a  ∆Q  K(r)  (43)  where K(r) = (r2 + a2)e − al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' One can also define one  other conserved quantity as follows  q = Q  σ2  0  =  � pθ  σ0  �2  + cos2 θ  �  a2(1 − e2) +  � l  sin θ  �2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  (44)  Following the standard procedure, one can obtain the fol-  lowing two trajectory equations as well  ρ2 dr  dτ = −  �  R(r) ,  (45)  ρ2 dφ  dτ = ±  �  U(θ)  (46)  where  R(r) = K(r)2 − ∆Q  �  r2 + (l − ae)2 + q  �  ,  (47)  U(θ) = q − cos2 θ  �  a2(1 − e2) +  l2  sin2 θ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  (48)  From eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' (45), we can obtain the radial trajectory equation  in terms of the new modified variable y upto O(y) as follows  (r2  + + a2 cos2 θ+ + 2r+y)dy  dτ = −  �  P2  0 − P1x + O(y2)  dτ  dy ≃ −ρ2  +  �  1 + P1  2P2  0  + 2r+x  ρ2  +  �  (49)  where ρ2  + = r2  + + a2 cos2 θ+, P0 = (r2  + + a2)(e − ˜Ωl), and  P1 = −4r+eP0 + ∆′  Q+(r2  + + (l − ae)2 + q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' It is easy  to solve eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' (49) and obtain the form of τ in terms of the  spatial variable y as follows (up to O(y))  τ(y) ≃ −ρ2  +y  P0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  (50)  Now the energy measured from the viewpoint of the coro-  tating frame (dragged with an angular velocity ˜Ω) reads  ˜e = (e − ˜Ωl) implying that the constant P0 as well as  the leading order term is eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' (50) is proportional to ˜e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Now  using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' (42) along with eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' (45), we obtain the form of the  time t up to O(y) as  t(y) ≃ − 1  2κ ln y − δy  (51)  4  where the constant δ is given as  δ =  �  P1  2P2  0  −  1  2κ(r2  + + a2) +  4r+  r2  + + a2  �  1 +  ˜Ωl  2˜e  ��  1  2κ  −  ˜Ω  ˜e (aes2  + − l)  (52)  with s+ = sin θ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' We can now obtain the form of φ up to  O(y) using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' (s)(43,45) as follows  φ ≃ −  ˜Ω  2κ ln y +  �  ae − l  s2  +  � y  P0  −  ˜Ωy  2κ  �  P1  2P2  0  −  1  2(r+2 + a2)κ +  2r+  r2  + + a2  ˜e + ˜Ωl  ˜e  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  (53)  Using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' (10), we can redefine the form of ˜φ up to O(y) as  follows  ˜φ = φ − ˜Ωt = λy  (54)  where the constant λ is given by  λ = −  1  (r2  + + a2) ˜e(aes2  + − l)  �  ˜Ωa − 1  s2  +  �  +  ˜Ωr+  κ(r2  + + a2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  (55)  We are now in a position to compute the analytical form  of the excitation probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  4  The transition probability  In this case we are considering a two-level atom freely  falling into the outer horizon of a Kerr-Newman black hole  and its emitting photons as observed from the viewpoint of  an observer corotating with an angular velocity ˜Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' If the  atomic transition frequency is ω and the ground state and  excited states of the two-level atom are defined as |g⟩ and  |e⟩ then we can write the atom field interaction Hamilto-  nian considering only the coupling with a single scalar field  as  ˆHI(τ) = ℏG  �  ˆb˜νψ˜νlm(r(τ), t(τ)) + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  �  (|g⟩⟨e|e−iωτ +h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=')  (56)  where G is the atom-filed coupling constant, ˆb˜ν is the anni-  hilation operator of the scalar field mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Here the scalar  photon is being emitted simultaneously with the excita-  tion of the two level atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Hence, we can write down the  excitation probability given as  Pexc = 1  ℏ2  ����  ˆ  dτ ⟨1˜ν, e| ˆHI(τ) |0˜ν, g⟩  ����  2  = 1  ℏ2  ����  ˆ ∞  0  dy dτ  dy ⟨1˜ν, e| ˆHI[τ(y)] |0˜ν, g⟩  ����  2  ≃G2ρ4  +  P2  0  ����  ˆ ∞  0  dy  �  1 + 2r+y  ρ2  +  + P1y  2P2  0  �  ψ∗  ˜νlm(y)eiωτ  ����  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  (57)  In the last line of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' (57), we observe that the only con-  tributions in the form of the excitation probability comes  due to the existence of the counter-rotating terms in the  interaction Hamiltonian in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='(56).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' We shall now substi-  tute the form of τ from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' (49) and ψ∗  ˜νlm(r(y), t(y)) using  eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' (s)(38,50,51) in the above equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Keeping terms upto  O(y), we can rewrite the form of the excitation probability  as follows  Pexc = G2ρ4  +|Θ+|2  P2  0  ����  ˆ ∞  0  dy  �  1+  �2r+  ρ2  +  + P1  2P2  0  +  � κ2B  κ2 + ˜ν2  −  1  2∆′  Q+  ��  y + i˜νBκ  κ2 + ˜ν2 y  �  y− i˜ν  2κ e− i˜ν  2κ ln ye  −i  �  mλ+˜νδ+  ρ2  +ω  P0  �  y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  (58)  We can rewrite the above form of the excitation probability  in a much simpler form as follows  Pexc = G2ξ2  ����  ˆ ∞  0  dy (1 + Dy + iEy) y− i˜ν  κ e−iχy  ����  2  (59)  where we have defined ξ ≡  ρ2  +|Θ+|  P0  , χ ≡ mλ + ˜νδ +  ρ2  +ω  P0 ,  D ≡ 2r+  ρ2  + +  P1  2P2  0 +  κ2B  κ2+˜ν2 −  1  2∆′  Q+ , and E ≡  ˜νBκ  κ2+˜ν2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Let us  now define a new variable of the form χy = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' In terms of  this new variable, we can rewrite the form of the excitation  probability in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' (59) as  Pexc = G2ξ2  χ2  ����  ˆ ∞  0  dα  �  1 + D  χ α + iE  χ α  �  α− i˜ν  κ e−iα  ����  2  = 2πG2ξ2˜ν  κ3χ4  (κ2(D2 + (E + χ)2) + (D2 + E2)˜ν2  − 2κDχ˜ν)  1  e  2π˜ν  κ − 1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  (60)  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' (60) is one of the main results in our paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' We shall  now compute the probability for the de-excitation of the  two-level atom from the excited state to the ground state  along with a simultaneous absorption of a photon with fre-  quency ˜ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' The initial and the final states for such a process  are given as  |initial⟩ = |0˜ν, e⟩ ,  |final⟩ = |1˜ν, g⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  (61)  With the redefinition of a constant χ′ ≡  ρ2  +ω  P0 − mλ − ˜νδ  and using the forms of the initial and final states from  eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' (s)(61,62), we can calculate the absorption probability  as follows  Pabs = 2πG2ξ2˜ν  κ3χ′4  (κ2(D2 + (E − χ′)2) + (D2 + E2)˜ν2  + 2κDχ′˜ν)  1  1 − e− 2π˜ν  κ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  (62)  We can now obtain the form of the excitation probability  for a simultaneous emission of a photon due to a two-level  atom freely falling into the event horizon of A Kerr black  hole by setting Q → 0 in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='(60).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' In order to compare the  difference between the two cases, we plot the Planck-factor  for the two black holes with change in the frequency of  the emitted photon in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' As we are in geometrized  units, we take the values of the parameters as a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='1a0,  Q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='4a0 (only in case of the Kerr-Neumann black hole),  and M = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='5a0 with respect to some arbitrary length scale  a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' From the figure we find that for the Kerr-Newman  black hole the Planck-factor has lower values compared to  the Planck factor in the Kerr black hole case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' In the next  5  Figure 1: Planck-factor vs frequency plot for the Kerr and  the Kerr-Newman black hole  section we shall compute the excitation probability of a  two-level atom with the simultaneous emission of multiple  photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  5  Simultaneous emission of N number of  photons  We now consider a more of a special case where during the  excitation of the two-level atom, there are simultaneous  emission of N number of photons with frequency ˜νj with  the index j running from 1 to N (as observed from the  frame corotating with an angular velocity ˜Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' We consider  the atomic transition frequency to be ϖ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' In this scenario  of virtual transition, the final and the initial states have  the following form  |initial⟩ = |0˜ν1, 0˜ν2, · · · , 0˜νN , g⟩ ,  (63)  |final⟩ = |1˜ν1, 1˜ν2, · · · , 1˜νN , e⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  (64)  It is important to note that this initial and final state repre-  sents a very special case where excitation consists of emis-  sion of N number of virtual photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' There can be much  complicated scenarios where when the two-level atom gets  excited to its higher energy state by N − j number of vir-  tual excitation processes and j number of virtual absorp-  tion processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' The atom-field interaction Hamiltonian in  this particular scenario is given as follows  ˆ˜HI(τ) =ℏ ˜G[ˆb˜ν1ψ˜ν1 + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='] ⊗ [ˆb˜ν2ψ˜ν2 + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='] · · · ⊗ [ˆb˜νN ψ˜νN  + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='] ⊗ (|g⟩ ⟨e| e−iϖτ + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='c)  (65)  where ˜G denotes the new coupling constant within the  atom and the scalar fields and ˆb˜νj denotes the annihilation  operator corresponding to the j-th photon with j ranging  from 1 to N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' The form of ψ˜νk, for k ∈ {1, · · · , N} is given  as (upto O(y))  ψ˜νklm ≃ e−i˜νkty−  i˜νk  2κ  �  1+  � κ2B2  k  κ2 + ˜ν2  k  −  1  2∆′  Q+  �  y + i˜νkκBk  κ2 + ˜ν2  k  �  × Θ+eim ˜φ  (66)  where the form of Bk is given to be  Bk =  2  ∆′  Q+  �� ˜νk  2κ  �2  + 1  4  �  + a2ν2  k + β + σ2  0r2  +  ∆′  Q+  − 4νk  ∆′  Q+  � ˜νk  2κr+ + 1  2am  �  (67)  with ˜νk = νk − m˜Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  Following the previous analysis it  is easy to understand that only the counter-rotating terms  will contribute to the excitation probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' The excitation  probability for this particular scenario is given by  PN =  ˜G2ρ4  +  P2  0  ����  ˆ ∞  0  dy  �  1 + 2r+y  ρ2  +  + P1y  2P2  0  �  ψ∗  ˜ν1 · · · ψ∗  ˜νN eiϖτ  ����  2  (68)  where we have not explicitly written the other two quan-  tum numbers defining each mode of the scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' For a  rather simpler approach, we shall now use ˜ν1 = ˜ν2 = · · · =  ˜νN = ˜ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' For this scenario, we also get B1 = B2 = · · · =  BN = B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Hence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' the excitation probability for the simul-  taneous emission of N photons with identical frequencies  is given as  P ′  N = 1  ℏ2  ����  ˆ  dτ ⟨final| ˆ˜HI(τ) |initial⟩  ����  2  ≃  ˜G2ρ4  +|Θ+|2N  P2  0  ����  ˆ ∞  0  dy  �  1 + ˜Dy + i˜Ey  �  e−i˜χyy− iN ˜ν  κ  ����  2  (69)  where the analytical forms of the constants ˜D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' ˜E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' and ˜χ  are given by  ˜D = 2r+  ρ2  +  + P1  2P2  0  + N  �  κ2B  κ2 + ˜ν2 −  1  2∆′  Q+  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  (70)  ˜E = ˜νBκN  κ2 + ˜ν2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  (71)  ˜χ = (mλ + ˜νδ)N + ρ2  +ϖ  P0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  (72)  We now define a new constant ˜ξ ≡  ρ2  +|Θ+|N  P0  and a new  variable ˜α ≡ ˜χy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' In terms of ˜α and ˜ξ, we can rewrite the  excitation probability in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' (69) as  P ′  N =  ˜G2 ˜ξ2  ˜χ2  ����  ˆ ∞  0  d˜α  �  1 +  ˜D  ˜χ ˜α + i  ˜E  ˜χ ˜α  �  e−i˜α ˜α− iN ˜ν  κ  ����  2  = 2Nπ ˜G2 ˜ξ2˜ν  κ3 ˜χ4  (κ2( ˜D2 + (˜E + ˜χ)2) + ( ˜D2 + ˜E2)N 2˜ν2  − 2Nκ ˜D˜χ˜ν)  1  e  2Nπ˜ν  κ  − 1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  (73)  Comparing eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' (60) along with eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' (73), we observe that the  Planck factor in the N-photon emission case has a signif-  icantly smaller value than the single photon emission case  6  Planck-factor  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='020  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='015  Kerr  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='010  Kerr-Newman  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='005  vao  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='0leading to an exponential decrease in the case of multi-  photon emission case for a higher value of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' In the next  section, we shall compute the von-Neumann entropy for  photon emission when a two-level atom falls freely into the  event horizon of a black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  6  Horizon brightened acceleration radia-  tion entropy  We now consider that a stream of two-level atoms are  falling into the event horizon of the black hole at a con-  stant rate k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' If ∆N number of particles are falling in a  time interval ∆t, we can write  ∆N  ∆t = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  (74)  We shall now revert to a density matrix approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  We  consider that for the i-th atom the change in the density  matrix is δϱi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Hence, for N number of atoms, the change  in the density matrix takes the form  ∆ϱ =  ∆N  �  i=0  δϱi = ∆Nδϱ  (75)  where we have considered that the change in the density  matrix is constant for each atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' (75) back in  eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' (74), we get the following relation  ∆ϱ  ∆t = kδϱ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  (76)  The absorption and emission rates can be written as Γexc =  kPexc and Γabs = kPabs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' In terms of the absorption and  emission rates, the Lindblad master equation reads [43]  ˙ϱ = − Γabs  2  �  ϱb†b + b†bϱ − 2bϱb†�  − Γexc  2  �  ϱbb† + bb†ϱ − 2b†ϱb  �  (77)  The rate of change of the {n, n}-th element of the density  matrix is given as  ˙ϱn,n = − Γabs (nϱn,n − (n + 1)ϱn+1,n+1)  − Γexc ((n + 1)ϱn,n − nϱn−1,n−1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  (78)  The steady-state single-mode solution takes the form  ϱS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='S  n,n =  �Γexc  Γabs  �n �  1 − Γexc  Γabs  �  (79)  In order to compute the Horizon Brightened acceleration  radiation entropy, we consider ω ≫ ˜ν, mλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Following this  assumption we find χ = χ′ ≃  ρ2  +ω  P0  and χ > E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' As addi-  tional assumptions, we consider χ > D and κ > ˜ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Using  the above assumptions, we can write  Γexc  Γabs  = Pexc  Pabs  =  �  1 + 4E  χ − 4D˜ν  κχ  �  e− 2π˜ν  κ  ≃  �  1 + 4B˜ν  κχ − 4D˜ν  κχ  �  e− 2π˜ν  κ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  (80)  The von-Neumann entropy for the system reads  Sϱ = −Tr[ϱ ln ϱ] = −  �  n,ν,l,m  ϱn,n ln ϱ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  (81)  The rate of change of the von-Neumann entropy in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' (81)  can be computed as follows (with the Boltzmann constant  set to 1)  ˙Sϱ = − d  dt (Tr[ϱ ln ϱ]) ≃ −  �  n,ν,l,m  ˙ϱn,n ln ϱS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  n,n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  (82)  Now for the n, n element of the density matrix with the  set of quantum numbers s =  ˜  ν, l, m, we get the following  relation  �  n  ˙ϱn,nn = ˙¯ns .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  (83)  If we now sum ˙¯ns˜ν over all the quantum numbers, we ob-  tain  �  s={ν,l,m}  ˙¯ns˜ν =  �  s={ν,l,m}  ˙¯ns(ν − m˜Ω) = ˙Ep − ˜Ω ˙Jp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  (84)  In eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' (84) ˙Ep is the rate of change in the total energy of the  black hole and ˙Jp is the rate of change of the axial angular  momentum of the black hole (defined about the rotational  axis of the black hole).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' The extra term in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' (84) generates  due to the angular momentum of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Using the  definition of a =  J  M , we can rewrite eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' (84) as follows  �  s={ν,l,m}  ˙¯ns˜ν = ˙mp −  a2  r2  + + a2 ˙mp =  ˙mpr2  +  r2  + + a2  (85)  with ˙mp denoting the rate of change of the mass of the  black hole due to emitting photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' In order to express  the rate of change in the von-Neumann entropy in terms  of the area of the black hole, we compute the area of the  Kerr-Newman black hole at r = r+ and some fixed time as  A =  ˆ √gθθgφφdθdφ = 4π(r2  + + a2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  (86)  The rate of change of the area of the black hole due to si-  multaneous emission of the photons can be directly related  to the rate of change of the mass of the black hole due to  emitting photons via the following relation  ˙Ap = 16πr2  +  ∆′  Q+  ˙mp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  (87)  Substituting eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' (79) into eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' (82) and using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' (s)(83-87),  we obtain the following form for the rate of change of the  von-Neumann entropy  ˙Sϱ = −  �  n,ν,l,m  n ˙ϱn,n  �  ln  �  1 − 4˜ν  κχ (D − B)  �  − 2π˜ν  κ  �  =  ˙mpr2  +  r2  + + a2  2π  κ +  4 ˙mpr2  +  κχ(r2  + + a2) (D − B)  =⇒  ˙Sϱ ≃  ˙Ap  4 +  ˙ApP0  2πρ2  +ω  �  2r+  ρ2  +  + P1  2P2  0  −  1  2∆′  Q+  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  (88)  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' (88) is one of the most important results in our work,  the first term of the above result can be termed as the  “ ˙A/4 law” in the context of HBAR entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' It shows that  because of the consideration of the beyond near horizon  approximation the the rate of change of the HBAR entropy  picks up an extra correction term apart from the usual  total time derivative of the “area divided by four” form  factor [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  7  7  Conclusion  In this work we have considered a two-level atom freely  falling into the event horizon of a rotating and charged  black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' In our analysis we have considered the event  horizon of the black hole to be covered by a perfectly re-  flecting mirror which shields the infalling atoms from inter-  acting with the outgoing Hawking-radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' This particu-  lar setup is identical to that of a Boulware like vacuum [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  Our analysis captures the essence of going beyond the near  horizon approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  In case of a near horizon anal-  ysis, there exists an underlying conformal symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' In  the beyond near horizon analysis, the underlying confor-  mal symmetry breaks down but the excitation probability  still retains a Planck like factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' This observation implies  that the occurrence of the Planck like factor is independent  of the underlying conformal symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Next we have plot-  ted the Planck-factor with respect to frequency for a Kerr  and a Kerr-Newman black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' (1), we have  observed that the Planck factor has higher value in case  of the Kerr black hole for different values of the frequency  implying a lower amount in the emission of photons for a  Kerr-Newman black hole with same mass and same angular  momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' In the next part of our analysis, we have con-  sidered a much generalized case where with the excitation  of a two-level atom from its ground state there are simulta-  neous emission of multiple photons (N number of photons  in our case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' We observe that the excitation probability  has identical form factor to that of a single-photon emission  case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' The only difference lies in the fact the exponential  decay is much more amplified when a significantly higher  number of photons are being emitted simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Fi-  nally we have computed the von-Neumann entropy which  is also termed as the horizon brightened acceleration radi-  ation entropy [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' We observe that in the analytical form  of the rate of change of the von-Neumann entropy, apart  from the total time derivative of the area divided by four  term, there exists extra correction terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' These correction  terms generate only due to the consideration of the beyond  near horizon approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  References  [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Einstein, Sitzungsber Preuss Akad Wiss (1915) 844.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  [2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Einstein, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' der Physik 49 (1916) 769.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  [3] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Weiss, Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' News 157 (2000) 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  [4] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Philbin, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=', Science 319 (2008) 1367.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  [5] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Bekenstein, Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Nuovo Cimento 4 (1972) 737.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  [6] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Bekenstein, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' D 7 (1973) 2333.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  [7] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Hawking, Nature 248 (1974) 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  [8] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Hawking, Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Phys 43 (1975) 199.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  [9] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Hawking, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' D 13 (1976) 191.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  [10] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Gangopadhyay and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Kulkarni, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' D 77  (2008) 024038.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  [11] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Gangopadhyay, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' D 77 (2008) 064027.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  [12] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Gangopadhyay, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' D 78 (2008) 044026.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  [13] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Banerjee, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Gangopadhyay, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Kulkarni, Gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  Relativ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Gravit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' 42 (2010) 2865.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  [14] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Gangopadhyay, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' 85 (2009) 10004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  [15] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Gangopadhyay, Gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Relativ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Gravit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' 42 (2010)  1183.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  [16] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Page, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' D 13 (1976) 198.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  [17] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Page, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' D 14 (1976) 3260.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  [18] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Page, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' D 16 (1977) 2402.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  [19] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Fulling, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' D 7 (1973) 2850.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  [20] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Davies, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' A 8 (1975) 609.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  [21] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' DeWitt, General Relativity: An Einstein Cente-  nary Survey, eds S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Hawking, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Israel (Cambridge  Univ Press (1979), Cambridge, UK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  [22] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Unruh and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Wald, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' D 29 (1984)  1047.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  [23] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' M¨uller, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' D 56 (1997) 953.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  [24] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Vanzella and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Matsas, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' 87 (2001) 151301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  [25] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Crispino, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Higuchi, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Matsas, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' 80 (2008) 787.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  [26] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Castro, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Maloney and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Strominger, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  D 82 (2010) 024008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  [27] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Agullo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Navarro-Salas, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Olmo and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Parker,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' 105 (2010) 211305.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  [28] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Scully, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Fulling, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Lee, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Page,  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Schleich and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Svidzinsky, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Natl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' 115 (2018) 8131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  [29] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Svidzinsky, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Ben-Benjamin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Fulling  and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Page, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Lett 121 (2018) 071301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  [30] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Fulling and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Wilson, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Scr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' 94 (2019)  014004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  [31] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Camblong, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Chakraborty and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Ord´o˜nez,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' D 102 (2020) 085010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  [32] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Azizi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Camblong, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Chakraborty, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  Ord´o˜nez and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Scully, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' D 104 (2021)  065006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  [33] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Azizi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Camblong, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Chakraborty, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  Ord´o˜nez and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Scully, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' D 104 (2021)  084086.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  [34] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Azizi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Camblong, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Chakraborty, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  Ord´o˜nez and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Scully, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' D 104 (2021)  084085.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  [35] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Chatterjee, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Gangopadhyay and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Majumdar,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' D 104 (2021) 124001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  [36] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Sen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Mandal and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Gangopadhyay, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  D 105 (2022) 085007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  [37] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Sen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Mandal and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Gangopadhyay, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  D 106 (2022) 025004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  8  [38] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Unruh, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' D 14 (1976) 870.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  [39] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Schwinger, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' 82 (1951) 664.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  [40] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Kerr, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' 11 (1963) 237.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  [41] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Newman, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Couch, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Chinnapared, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Exton,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Prakash, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Torrence, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' 6 (1965)  918.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  [42] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Boyer and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Lindquist, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' 8  (1957) 265.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  [43] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Lindblad, Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' 48 (1976) 119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  [44] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Boulware, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content=' D 11 (1975) 1404.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
+page_content='  9' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQf_gsD/content/2301.04834v1.pdf'}
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+arXiv:2301.05485v1  [math.DS]  13 Jan 2023
+From Equilibrium Statistical Physics
+Under Experimental Constraints
+to Macroscopic Port-Hamiltonian Systems
+Judy Najnudel, Thomas Hélie, David Roze, Rémy Müller
+S3AM Team, Laboratory STMS (UMR 9912), IRCAM-CNRS-SU, 1 Place Igor Stravinsky, 75004
+Paris, France
+Abstract
+This paper proposes to build a bridge between microscopic descriptions of matter with
+internal energy, composed of many fast interacting particles inside an environment, and
+their port-Hamiltonian (PH) descriptions at macroscopic scale. The environment, as-
+sumed to be slow, is modeled through experimental constraints on macroscopic quantities
+(e.g. energy, particle number, etc), with a partitioning into two classes: non fluctuating
+and fluctuating values. The method to derive the PH macroscopic laws is detailed in sev-
+eral steps and illustrated on two standard cases (ideal gas, Ising ferromagnets). It revisits
+equilibrium statistical physics with a focus on this partitioning. First, the Boltzmann’s
+principle is used to provide the statistic law of the matter.
+It defines a macroscopic
+equilibrium characterized by a scalar value, the entropy, together with thermodynamic
+quantities emerging from each constraint. Then, the port-Hamiltonian system is derived.
+The Hamiltonian (macroscopic energy) is derived as a function of the macroscopic state
+(entropy and the macroscopic quantities associated with the fluctuating class). The ports
+(flows/efforts) are related to the time-derivative of the state and the Hamiltonian gra-
+dient in a conservative way. This open system defines the reversible laws that govern
+standard thermodynamic quantities. Lastly, this paper presents a strategy to extend this
+PH system to an irreversible conservative one, given a macroscopic dissipative law.
+Keywords:
+equilibrium statistical physics, macroscopic port-Hamiltonian Systems,
+statistical entropy, experimental conditions
+1. Introduction
+A macroscopic system (of size 10−2 m or bigger) is constituted of matter, that is,
+billions of microscopic particles (of size 10−9 m or smaller) which are collectively re-
+sponsible for the system’s behavior. However, studying a single particle tells nothing
+about the macroscopic system, just as following the trajectory of a single person is not
+sufficient to predict a crowd movement. Yet, solving exhaustive equations with billions
+of variables would be all at once much too complex and irrelevant: at a high enough
+scale, individual behaviors do not matter. Indeed, one is usually not interested in the
+particular trajectories of water molecules in one’s glass, but rather in the volume, on
+Preprint submitted to Physica D
+January 16, 2023
+
+average, that they take. Likewise, one is not only interested in the day’s weather report,
+but rather in the global tendency .
+Averages and tendencies belong to the domain of statistics, which aims to describe
+complex systems with a reduced number of variables. Thus, Statistical Physics (SP)
+computes averages on (fast) fluctuations of complex systems in order to derive (slower)
+macroscopic quantities, given some experimental conditions. Statistical arguments for
+the description of a system transitioning towards thermodynamic equilibrium were intro-
+duced by Ludwig Boltzmann in 1877 [1]. This framework allows the prediction of macro-
+scopic thermodynamic phenomena such as temperature, entropy creation, and phase
+transitions [2].
+Thermodynamics has been broadly studied in the context of port-Hamiltonian system
+(PHS), as well as their modeling and their control (see e.g. [3, 4, 5, 6, 7, 8, 9]). However,
+the proper derivation of macroscopic thermodynamic PHS from complex systems with
+numerous degrees of freedom is seldom addressed. As it happens, the choice of a system
+representation for this kind of model reduction is all but inconsequential, and must be
+handled with care [10, 11]. In this paper, we propose a series of systematical steps in
+order to construct a simplified yet physically-based structured macroscopic PHS from a
+system that can be described by SP.
+Note that in the scope of this work, we limit ourselves to equilibrium SP, in the
+sense that average quantities are determined for a system at thermodynamic equilibrium,
+given some experimental conditions. It is compatible with studying the system dynam-
+ics, assuming that thermodynamic relaxation (the process of reaching thermodynamic
+equilibrium) is infinitely faster than the rate of change of experimental conditions. Based
+on this assumption, a macroscopic trajectory is to be understood as a succession of ther-
+modynamic equilibrium states.
+This paper is structured as follows. In Section 2, we formalize the description of
+the microscopic configurations of a system through the choice of (i) an ad hoc parti-
+cle representation and (ii) a set of characterization functions that evaluate macroscopic
+quantities . Section 3 addresses the experimental conditions at the macroscopic level and
+their influence on the system configuration space. In section 4, we introduce a stochas-
+tic description for microscopic configurations . Then, in Section 5, we determine the
+conditional probability distribution according to the Boltzmann principle for a system at
+thermodynamic equilibrium. This allows the derivation of relevant macroscopic variables
+as expectations for this probability distribution. In Section 6, we introduce the ports and
+relates them to those macroscopic variables, leading to the macroscopic PHS model. Fi-
+nally, section 7 summarizes the practical sequence of these steps to derive the macroscopic
+PHS model from the microscopic description. In addition, it presents how to derive a
+conservative irreversible PHS model from an additional macroscopic dissipation law.
+All the steps are detailed in the following sections, as recapped in Fig. 1.
+2. Microstate of a system
+2.1. Particle representation (p ∈ P)
+In order to describe a system at a microscopic level, each of its particles must be
+described in a relevant way. Depending on the system under study, one may choose to
+represent a particle by its position, momentum, charge, magnetic moment, etc.
+2
+
+A. Microscopic description - Section 2
+1. P-valued particle representation
+2. Particle configuration m ∈ M = P⋆
+3. Set of characterizing functions F
+B. Experimental conditions - Section 3
+4. F = Ffixed ∪ Ffree
+5. Fixed values θfixed
+6. Set of accessible configurations Ma(θfixed)
+C. Stochastic setting - Section 4
+7. Probability distribution p
+8. Surprisal Sb
+p(m) (in base b)
+9. Statistical entropy Sk(p), k = 1/ln b
+D. Boltzmann principle - Section 5
+10. Ergodicity Ep
+�
+Fi ∈ Ffree�
+= Fi
+11. Maximum entropy given
+experimental conditions (B.)
+→ Thermodynamic entropy Sk(Fi)
+and Lagrange mult. λi
+E. Macroscopic PHS - Section 6
+12. Entropy to energy representation
+Sk(E, Fj) ↔ E(S, Fj
+� �� �
+x
+)
+13. Connection to ports (effort u, flow y)
+Fj ↔ ˙x ↔ u,
+λj ↔ ∇E(x) ↔ y
+Figure 1: from equilibrium statistical physics to macroscopic port-Hamiltonian systems (PHS): method
+recap with the labels of the main mathematical objects introduced in each step.
+Definition 1 (Particle set P). Given a chosen representation to encode a particle state,
+we denote P the set of all its possible values.
+Example 1 (Particle represented by its position and momentum). For a particle chosen
+to be represented by its position in space r ∈ R3 and momentum p ∈ R3, such as in a
+gas, the particle set is defined as P = R3 × R3.
+Example 2 (Particle represented by its magnetic moment). For a particle chosen to be
+represented by its magnetic moment s ∈ {−1, 1} such as in the Ising model [12, 13], the
+particle set is defined as P = {−1, 1}.
+2.2. Configuration space (m ∈ M)
+As an element of P represents the state of one particle, a natural way to represent
+a configuration of particles is to concatenate elements of P. By analogy with formal
+language theory [14], a particular configuration of particles is chosen to be encoded as a
+word over the alphabet P (see remark 1 for other choices) .
+Definition 2 (Encoding space W). We denote W := P⋆ the space of encodable config-
+urations (or, for short, the encoding space) , where ⋆ is the Kleene operator defined by
+P = {ǫ},
+Pi+1 =
+�
+p1 · p2 | (p1, p2) ∈ Pi × P
+�
+∀i ≥ 0,
+(1a)
+P⋆ =
+�
+i≥0
+Pi,
+(1b)
+with ǫ the empty configuration and · the concatenation operation.
+Property 1 (W is a monoid). By construction, the encoding space W is a monoid
+(see Def. (3)) with associative binary operation · (concatenation) and identity element ǫ
+(empty configuration).
+3
+
+•1
+•2
+•1
+•2
+•1
+•2
+•1
+•2
+(a) Examples of microstates for a system of two particles described by their spin s ∈ {−1 (blue) , 1 (red)}.
+(b) Examples of microstates for a system with three particles described by their position (circle) and momentum
+(arrow).
+Figure 2: Examples of microstates for different systems.
+Definition 3 (Monoid). A set S is a monoid if it is equipped with an associative binary
+operation · : S × S �→ S and identity element ǫ, such that for all (s1, s2, s3) ∈ S3, the
+following properties hold
+1. s1 · (s2 · s3) = (s1 · s2) · s3,
+2. ǫ · s1 = s1 · ǫ = s1.
+Based on the chosen representation, some configurations may not be physically ad-
+missible 1, therefore we introduce the set of microstates as follows.
+Definition 4 (Admissible configuration set M and microstate m ∈ M). We denote
+M ⊆ W the set of encodable configurations that are also physically admissible .
+An
+element m ∈ M is called a microstate of the system.
+In the following, we choose M = W (see remark 1(i) for an interpretation and (ii-iii)
+for examples with M ̸= W) . Figure 2a shows examples of microstates for a system of
+particles described by their spin, and Fig. 2b shows a system of particles described by
+their position and momentum.
+Property 2 (M is measurable). The pair
+�
+M, P(M)
+�
+where P(M) denotes the powerset
+of M is a measurable space, that is, it verifies
+1. M ∈ P(M),
+2. P(M) is closed under complements: M\P ∈ P(M),
+∀P ∈ P(M),
+3. P(M) is closed under countable unions: �∞
+i=1 Pi ∈ P(M)
+∀P1, P2, . . . ∈ P(M).
+As mentioned above, the set M could be defined on encoding spaces generated with
+operators other than the concatenation. Examples are outlined in the following remark 1.
+1For instance, if the chosen representation assigns a unique label to each particle, configurations in
+which several particles share the same label are not admissible (see remark 1(iii) for more more details) .
+4
+
+Remark 1 (Examples of combinatorial structures and interpretations). The monoid W
+provides a simple and natural way to encode microscopic configurations, which consists
+in choosing a prioritized and distinguishable representation of particles (see specification
+(i) below). But the configuration encoding can be addressed by using any appropriate
+combinatorics of particles, possibly choosing other specifications (see e.g. [15, § I.2] for
+details on operators Seq, Mset, Pset, etc., mentioned below):
+(i) Distinguishability with prioritization. A word pi · pi−1 · · · · · p1 ∈ Pi ⊂ W can be
+interpreted as describing the state values pn ∈ P of particles number n = 1, . . . , i.
+In this sense, definition 2 encodes a physics with distinguishable particles and with
+a priority ordering on involved particles (particle 1 can be encoded alone, particle 2
+only if 1 is involved, etc). From the combinatorics point of view, the encoding space
+W = P⋆ corresponds to the sequence construction, also denoted Seq(P) in [15].
+(ii) Undistinguishability. A natural encoding of a physics described with undistinguish-
+able particles is the multiset W(ii) =Mset(P) composed of all the finite sets of P
+(the order between elements does not count), in which arbitrary finite repetitions
+of elements are allowed. Examples of admissible configuration sets M(ii) are W(ii),
+or the powerset Pset(P) ⊂Mset(P) (no repetition allowed) if the physics under
+consideration forbids two particles of a microstate to be excited by the same state
+value.
+(iii) Distinguishability without priorization. A means other than (i) to encode configu-
+rations with distinguishable particles is to add a distinct label to each particle such
+as its number n ∈ L ⊆ N. In this case, the encoding space can be the powerset
+W(iii) =Pset(Plabeled), where the set of labeled particles Plabeled is the cartesian
+product L × P. An admissible configuration set that forbids the particle ubiquity or
+replication is a subset M(iii) of W(iii) for which all the elements make the number
+of occurrences of each label n ∈ L be 0 or 1.
+(iv) Complex structures. More generally, the encoding space can be composed of complex
+elements, with structures involving sequences, cycles (for e.g. aromatic molecules),
+trees (etc.) and their combination according to precise combinatorial specifications
+(see [15, I.2.3].
+2.3. Characterizing functions (Fi ∈ F)
+In order to characterize the system at a microscopic level, one may choose to equip
+M with a finite set of characterizing functions, labeled by i ∈ I, denoted Fi : M → Fi.
+Definition 5 (Extensivity). A function Fi is extensive if Fi is a R+-semimodule (see
+Def. 6) and if it verifies
+m3 = m1 · m2 ⇒ Fi(m3) = Fi(m1) + Fi(m2)
+∀(m1, m2, m3) ∈ M3.
+(2)
+Definition 6 (R+-semimodule). A set S is a R+-semimodule if for all (r1, r2) ∈ R+2
+and (s1, s2) ∈ S2, the following properties hold
+1. r1 (s1 + s2) = r1 s1 + r2 s2,
+5
+
+2. (r1 + r2) s1 = r1 s1 + r2 s1,
+3. (r1 r2) s1 = r1 (r2 s1),
+4. 1 s1 = s1,
+5. 0 s1 = 0.
+Example 1 (Continued). For a gas of N identical, non-interacting particles, the function
+Fe : M �→ R+ defined as
+Fe : m �−→ Fe(m) =
+N
+�
+i=1
+��pi(m)
+��2
+2 m
+(3)
+gives the energy of the system in microstate m, where pi(m) is the momentum of particle
+i, and m is the mass of a particle. It fulfills the extensivity property defined in Eq. (2).
+Example 2 (Continued). In the Ising model [16] , the function Fe : M �→ R defined as
+Fe : m �−→ Fe(m) = −1
+2 m⊺ Jex m
+(4)
+gives the energy of the system in microstate m, where each coefficient Jexi,j is the ex-
+change energy between atom i and atom j. It does not fulfill the extensivity property
+defined in Eq. (2).
+Example 3. The function Fn : M �→ N+ defined as
+Fn : m �−→ Fn(m)
+(5)
+where Fn(m) is the number of particles of the system in microstate m is extensive .
+Example 4. The function Fr defined as
+Fr(m) =
+�
+ri�
+1≤i≤Fn(m) ,
+(6)
+where ri ∈ R3 is the position of particle i, gives the set of all particle positions for the
+system in microstate m. It is not extensive.
+Example 5. The function Fv : M �→ R+ defined as
+Fv : m �−→ Fv(m)
+(7)
+where Fv(m) is the volume occupied by the system in microstate m. The choice of such
+function Fv is hardly unique (see remark 2 below). Here, we propose to define Fv(m)
+as the minimal bounding volume enclosing all particle positions of microstate m that
+accounts for the container geometry and its degrees of freedom.
+For instance, for a
+cylindrical container of fixed base A closed by a piston moving freely along axis z, we can
+define the volume as
+Fv(m) = A × h(m),
+with h(m) = max{ri
+z | ri ∈ Fr(m)}.
+(8)
+This function does not fulfill the extensivity property defined in Eq. (2).
+6
+
+Remark 2 (Volume). The mathematical conceptualization of the volume is a challenging
+issue2. In the context of SP, its physical conceptualization is also an issue:
+(i) a possible choice could be the volume occupied by the particles, e.g. that delimited
+by the 3D simplicial envelope of all the particle positions,
+(ii) an alternative is to consider the volume of a container, in which the particle are
+authorized to evolve.
+The case (i) allows the definition of a characterizing function Fv, the volume being intrin-
+sically related to the microstate. In (ii), the microstate does not encode the information
+of the container volume: its set of particles in contact with the container boundary can
+even be empty. This information is an "experimental constraint" (presented in section 3
+below). Note also that example 5 corresponds to a hybrid description in between (i) and
+(ii).
+In the following, we denote F = {Fi}i∈I the set of characterizing functions on M.
+3. Experimental conditions and accessible microstates (m ∈ Ma)
+Experimental conditions may constrain characterizing functions to take values that
+are compatible with these experimental conditions. Thus, under experimental conditions,
+the configuration space becomes restricted to a set of accessible microstates Ma ⊂ M.
+Definition 7 (Set of accessible microstates Ma). Denote Ifixed ⊆ I the set of labels of
+characterizing functions that are experimentally constrained. Due to the constraints, a
+function Fi, i ∈ Ifixed can only take admissible values in Ffixed
+i
+⊆ Fi.
+Denote θfixed :=
+�
+Ffixed
+i
+�
+i∈Ifixed. The set of accessible microstates Ma
+�
+θfixed�
+is
+Ma
+�
+θfixed�
+=
+�
+m ∈ M | Fi(m) ∈ Ffixed
+i
+∀i ∈ Ifixed�
+.
+(9)
+Remark that Ifixed = I defines an isolated system with respect to the chosen charac-
+terizing functions.
+Example 1 (Continued). Consider a gas of N particles in a closed tank. The system
+cannot exchange particles with the environment, therefore the number of particles Fn(m)
+is fixed to N.
+Denote Ffixed
+n
+= {N}. The set of accessible microstates is Ma (N) =
+�
+m ∈ M | Fn(m) ∈ Ffixed
+n
+�
+.
+Example 1 (Continued). Consider a gas of N particles in a closed tank occupying a
+space Π ⊂ R3. Denoting Ffixed
+n
+= {N} and Ffixed
+r
+= ΠN, the set of accessible microstates
+is Ma (N, Π) =
+�
+m ∈ M | Fn(m) ∈ Ffixed
+n
+, Fr(m) ∈ Ffixed
+r
+�
+. Note that the constraint on
+Ffixed
+r
+corresponds to the case (ii) in remark 2, the container being described by Π.
+Other examples of experimental conditions are shown on Fig. 3.
+2In [17, Chap. 3,p.3], Grothendieck mentions the absence (in most textbooks) of any "serious" defi-
+nition of the notion of length (of a curve), of area (of a surface), of volume (of a solid).
+7
+
+(a) Fixed number of particles
+and fixed volume.
+(b) Fixed number of particles.
+(c) Fixed volume.
+Figure 3: Examples of experimental conditions for a gas in a tank.
+Intermediate point. At this step, the physics is described through: M (microscopic rep-
+resentation of physical particles, i.e. microstates), Ma ⊆ M (microstates accessible under
+fixed macroscopic experimental constraints), F (links to quantities that can be observed
+at the macroscopic scale). This is not sufficient to derive an autonomous physical law at
+macroscopic scale. This issue can be solved by: (i) completing F with a single new func-
+tion, namely, the surprisal, which is elaborated from a stochastic description, giving rise
+to the entropy; and (ii) applying the fundamental principle introduced by Boltzmann [1]
+to derive a microstate probability that physically makes sense.
+The following sections revisit this approach, focusing on the propensity of macroscopic
+quantities to communicate with their peer in an external environment. To this end, we
+assume that all characterizing functions that are not explicitly fixed by experimental
+conditions can still depend on microstate m, and we denote Ifree := I\Ifixed the set of
+labels of characterizing functions not fixed by experimental conditions.
+4. Stochastic representation and measure of uncertainty
+4.1. Microstate stochastic description
+The system fluctuates from one accessible microstate to another. It is considered to
+be impossible to predict these fluctuations in a deterministic fashion at the macroscopic
+level: SP adopts a stochastic framework that model their random description.
+Indeed,
+from Prop. (2),
+�
+M, P(M)
+�
+is measurable, therefore so is
+�
+Ma, Tr
+�
+P(M)
+�
+Ma
+�
+, where
+Tr
+�
+P(M)
+�
+Ma denotes the trace of P(M) on Ma [18]. Assuming that Ma is countable and
+that the distribution p is discrete3, we can define a probability distribution p
+�
+. | θfixed� :
+Ma
+�
+θfixed�
+�→ [0, 1], denoted p for short below, which assigns to each microstate m ∈
+Ma
+�
+θfixed�
+a probability p(m) to be the actual microstate of the system (the Boltzmann
+principle in section 5 will provide a tool to determine this probability).
+The average of a random quantity F(m) is given by its expectation Ep[F], defined as
+Ep[F] =
+�
+m∈Ma(θfixed)
+p(m) F(m).
+(10)
+3Extensions to continuous measurable spaces are available and similar, replacing the sum by an
+integral with a Lebesgues measure and using distributions in (10).
+8
+
+4.2. Statistical entropy
+Given some basis of information units b > 1, a microstate m with probability p(m)
+has a surprisal Sb
+p(m) defined as
+Sb
+p(m) = logb
+1
+p(m)
+with logb =
+1
+ln(b) ln .
+(11)
+The surprisal, or information content, quantifies how much the occurrence of microstate
+m is surprising. For example, if some microstate m is the state of the system for certain,
+it has probability 1 and surprisal 0.
+As the probability of two independent events m1 and m2 verifies
+p (m1 · m2) = p(m1) p(m2),
+(12)
+the surprisal function Sb
+p verifies the extensivity property defined in Eq. (2).
+The surprisal allows the definition of a measure of lack of information on average for a
+probability distribution p and a basis b, namely, the statistical entropy Sb(p) [19] defined
+as
+Sb(p) = Ep[Sb
+p].
+(13)
+The statistical entropy can be interpreted of as “the average number of questions to ask
+with b possible answers per question” in order to know the actual microstate for certain.
+Example 6. Consider the outcomes of tossing a coin twice. The coin can come up heads
+or tails after each toss, hence 2 × 2 = 4 possible outcomes (Fig. 4). If all outcomes are
+equiprobable, one needs at least two questions with two possible answers each to know the
+exact outcome:
+1. Did the coin come up heads or tails after the first toss?
+2. Did the coin come up heads or tails after the second toss?
+As it happens, taking p1 : m �→ p1(m) =
+1
+4 and b = 2 in Eq. (13) yields Sb(p1) =
+− log2
+1
+4 = 2.
+However, if the probability distribution is not uniform, some outcomes are more prob-
+able than others, and the uncertainty is lower; ditto the entropy.
+For instance, with
+a probability distribution p2 assigning
+1
+2 to outcome (A), 1
+4 to outcome (B), and 1
+8 to
+outcomes (C) and (D), the entropy becomes Sb(p2) = 1.75 < Sb(p1) = 2.
+In information theory, statistical entropy relates to optimal encoding of information.
+Suppose you repeat the coin toss experiment of Ex. (6) for a long period of time, and wish
+to record every outcome on a computer. For a sequence of two tosses with distribution
+p1, an outcome cannot be encoded in less than two bits; while with distribution p2,
+outcome (A) can be encoded on one bit, outcome (B) on two, and outcomes (C) and (D)
+on three, that is, 1× 1
+2 +2× 1
+4 +6× 1
+8 = 1.75 bits on average. The most frequent outcome
+takes the least encoding space; conversely, the comparatively large encoding space taken
+by outcomes (C) and (D) is compensated by the rarity of their occurrence.
+On the
+whole, exploiting the knowledge underpinned by distribution p2 reduces the encoding
+cost. This principle underlies Morse code (and, more generally, lossless entropy encoding
+like Huffman coding [20]): very common letters such as “e” or “i” take much fewer dots
+than less common letters like “j” or “q”.
+9
+
+(a)
+(b)
+(c)
+(d)
+Figure 4: Possible outcomes for a coin tossed twice.
+For compacity, the statistical entropy becomes in the following
+Sk(p) := Sb=exp(1/k)(p) = −k
+�
+m∈Ma(θfixed)
+p(m) ln p(m),
+(14)
+for all p defined on Ma(θfixed).
+Remark 3 (Random structures). Following remark 1, the cases of microstates involving
+combinatorial structures based on elaborated specifications (such as molecules and chem-
+ical reaction processes) require elaborate tools addressing random structures (see e.g. [15,
+part. C]) that are out of the scope of this paper.
+5. Microstate probability distribution at equilibrium and partition function
+5.1. Thermodynamic equilibrium
+A system is at thermodynamic equilibrium when its statistics stops evolving. At this
+point, the ergodic hypothesis postulates that over a “sufficiently long” period of time t,
+the system explores all its accessible microstates. Assuming that a microstate m can
+be measured at a time τ through M : R+ �→ Ma, this means that at thermodynamic
+equilibrium, the temporal mean of a quantity Fi coincides with its expectation Ep[Fi]:
+Fi :=
+lim
+t→+∞
+1
+t
+� t
+0
+(Fi ◦ M) (τ) dτ = Ep[Fi].
+(15)
+Therefore, for a given set θfree of mean values (Fi)i∈Ifree, the ergodic hypothesis
+translates into a set of hypotheses H
+�
+θfree�
+defined as
+H
+�
+θfree�
+=
+�
+Ep[Fi] = Fi
+�
+i∈Ifree .
+(16)
+While still discussed [21] (especially regarding the definition of “sufficiently long”),
+this hypothesis is the foundation of equilibrium statistical physics, and we assume its
+validity in the following.
+5.2. Boltzmann principle: maximum entropy at thermodynamic equilibrium (reminder)
+By definition, a system at equilibrium does not evolve. Since change is new informa-
+tion, the information given by a system at equilibrium is minimal. As statistical entropy
+is a measure of lack of information, it follows that at equilibrium, the entropy is maximal:
+this is the Boltzmann principle.
+10
+
+It follows that the microstate probability distribution at thermodynamic equilibrium
+p⋆ is
+p⋆ = arg max
+p
+Sk(p) subject to
+�
+m∈Ma(θfixed)
+p(m) = 1
+= arg
+p
+max
+p,λ Sk(p) + λ
+
+
+
+�
+m∈Ma(θfixed)
+p(m) − 1
+
+
+ ,
+(17)
+where the second line specifies the constraint using a Lagrange multiplier λ.
+5.3. Boltzmann principle in a macroscopic external environment
+The macroscopic quantities (Fi∈Ifree(m)) whose fluctuation is experimentally allowed
+are prone to communicate with the external environment (for example, through exchanges
+of particle, energy, etc). Assuming ergodicity and an infinite ratio between macroscopic
+and microscopic time scales, this means that the expectation of these fluctuating quanti-
+ties (with value Fi∈Ifree) can change over time (at the slow macroscopic scale), while the
+thermodynamic equilibrium is satisfied (at the fast microscopic scale) and continuously
+updated (at the slow macroscopic scale).
+As a main step of this paper, this issue is addressed by deriving the conditional
+probability p⋆ �
+m | H
+�
+θfree��
+that maximizes entropy (Boltzmann principle) constrained
+by the given macroscopic values θfree = (Fi)i∈Ifree, namely,
+p⋆ =arg max
+p
+Sk(p) subject to
+
+
+
+
+
+
+
+
+
+�
+m∈Ma(θfixed)
+p(m) = 1,
+H
+�
+θfree�
+,
+=arg
+p
+max
+p,λ0,λi∈IfreeSk(p) + λ0
+
+
+
+�
+m∈Ma(θfixed)
+p(m) − 1
+
+
+
++
+�
+i∈Ifree
+λi
+�
+Ep[Fi] − Fi
+�
+,
+(18)
+where H
+�
+θfree�
+accounts for the experimental conditions (see Eq. (16)).
+Note that, by definition, this probability naturally restores that of (17), if the values
+Fi∈Ifree are the expectations of Fi∈Ifree computed for probability (17).
+Theorem 1. Let θfree :=
+�
+Fi
+�
+i∈Ifree ∈ ×
+i∈IfreeFi, where ×
+i∈IfreeFi denotes the Cartesian
+product of the (Fi)i∈Ifree.
+Then for all m ∈ Ma
+�
+θfixed�
+,
+p⋆
+�
+m | H
+�
+θfree��
+=
+exp
+� �
+i∈Ifree λi Fi(m)
+k
+�
+Z
+�
+λfree�
+,
+(19a)
+11
+
+where
+Z
+�
+λfree�
+:=
+�
+m∈Ma(θfixed)
+exp
+��
+i∈Ifree λi Fi(m)
+k
+�
+(19b)
+is the partition function of the system, and, for all i ∈ Ifree, λi verifies
+∂
+∂λi
+k ln Z
+�
+λfree�
+= Fi.
+(19c)
+The proof is in Appendix A.
+Definition 8 (Thermodynamic entropy). The thermodynamic entropy Sk �
+θfree�
+is de-
+fined as the statistical entropy for the probability distribution at equilibrium given θfree:
+Sk �
+θfree�
+= Sk
+�
+p⋆
+�
+· | H
+�
+θfree���
+.
+(20)
+Property 3. The thermodynamic entropy function Sk is a Legendre transform of k ln Z
+and we have
+Sk �
+θfree�
+= k ln Z
+�
+λfree�
+−
+�
+i∈Ifree
+λi Fi.
+(21)
+Proof.
+Sk �
+θfree� (a)
+= Sk
+�
+p⋆
+�
+· | H
+�
+θfree���
+(b)
+= −k
+�
+m∈Ma(θfixed)
+p⋆
+�
+m | H
+�
+θfree��
+ln p⋆
+�
+m | H
+�
+θfree��
+(c)
+= −k
+�
+m∈Ma(θfixed)
+p⋆
+�
+m | H
+�
+θfree��
+ln
+
+
+
+exp
+� �
+i∈Ifree λi Fi(m)
+k
+�
+Z
+�
+λfree�
+
+
+
+= −k
+�
+m∈Ma(θfixed)
+p⋆
+�
+m | H
+�
+θfree�� ��
+i∈Ifree λi Fi(m)
+k
+− ln Z
+�
+λfree��
+(d)
+= −
+�
+i∈Ifree
+λi Fi + k ln Z
+�
+λfree�
+,
+using (a) Eq. (20), (b) Eq. (14), (c) Eq. (19a), and (d) Eqs. (10)-(15).
+We deduce that Sk is a Legendre transform of k ln Z (see also [22]).
+Property 4. It follows from Prop. (3) that for all i ∈ Ifree, the Lagrange multiplier λi
+is the derivative of the thermodynamic entropy function with respect to average Fi
+λi = − ∂Sk
+∂Fi
+�
+θfree�
+.
+(22)
+12
+
+Example 7. In particular, this defines the system temperature T , chemical potential µ,
+and pressure P as
+1
+T := ∂Sk
+∂Fe
+�
+θfree�
+,
+µ
+T := − ∂Sk
+∂Fn
+�
+θfree�
+,
+P
+T := ∂Sk
+∂Fv
+�
+θfree�
+.
+(23)
+Adiabatic process. For a system going through an adiabatic process (no thermal exchange
+with the environment), the surprisal is independent of m so that
+Sb
+p⋆(m) = S
+∀m.
+(24)
+That implies that for such systems, all microstates have the same probability
+p⋆(m) = 1
+Ω,
+with Ω = card
+�
+Ma
+�
+θfixed��
+.
+(25)
+From Eq. (19a), it follows that for such a system, we have
+�
+i∈Ifree
+λi Fi(m) = C,
+(26)
+where C is independent of m.
+Example 8. In particular, a system that can only exchange volume with its environment
+verifies Fe(m) + P Fv(m) = C, where C is independent of m.
+5.4. Identification of Boltzmann constant
+To ensure that the statistical entropy does coincide with the thermodynamic entropy
+at equilibrium, the constant k must be chosen as the Boltzmann constant4 kB = 1.38 ×
+10−23 J.K−1.
+Indeed, consider an ideal gas of N non-interacting atoms in a box of
+volume V at temperature T , represented by their position and momentum. The partition
+function Z is given by
+Z(T | N, V ) = V N
+�2 π m k T
+h2
+�3 N/2
+,
+(27)
+where here m denotes the mass of an atom, and h is the Planck constant. From Prop. (3),
+the thermodynamic entropy Sk(Fe, N, V ) is given by
+Sk(F e, N, V ) = k ln Z(T | N, V ) + Fe
+T .
+(28)
+Moreover, from Eq. (23), the pressure P is given by
+P = T ∂Sk
+∂V (Fe, N, V ) = N k T
+V
+.
+(29)
+Therefore, k must be identified with kB so that the ideal gas law P V = N kB T is
+verified.
+4Note that, from definition (14), choosing the unit USI reference k0 = 1 J.K−1 as a unit information
+quantity, this value corresponds to the question number base bB = exp(k0/kB) = exp(1023/1.38) ≈
+103.147e+22 ≈ 29.4736e+21. This means that explaining +1 J.K−1 requires about 1022 bits for a gas.
+13
+
+6. Final PHS model
+The macrosopic description of open system can be achieved by using balanced equa-
+tions of variations of entropy, energy, mass, etc. Port-Hamiltonian systems provide an
+adapted framework for such physical descriptions. This section addresses this issue by
+using a standard formulation (recalled in section 6.1), in which the energy is expressed as
+a function of entropy and state variables: this requires to invert S : (E, . . . ) �→ S(E, . . . )
+w.r.t. E, to introduce the hamiltonian H = E : (S, . . . ) �→ H(S, . . . ) in a first step. Note
+that this inversion is a technical step that could be avoided (to still use the entropy func-
+tion) by considering contact forms as in [23] (see also [24] for the alternative GENERIC
+formulation).
+6.1. Reminder on port-Hamiltonian systems
+The PHS formalism provides a unified formalism for the modeling of multiphysical
+systems, in the sense that it recognizes energy as a universal currency. Indeed, any phys-
+ical system can be divided into parts that interact with each other via energy exchanges.
+Detailed presentations of PHS are available in [25, 26]. In this paper, we rely on a
+differential-algebraic formulation adapted to multiphysical systems [27, 28]. This formu-
+lation allows the representation of a dynamical system as a network of
+1. storage components of state x and energy E(x);
+2. passive memoryless components described by an effort law z : w �→ z(w), such as
+the dissipated power Pd = z (w)⊺ w is non-negative for all flows w;
+3. connection ports conveying the outgoing power Pext = u⊺y where u are inputs and
+y are outputs.
+The system flows f and efforts e are coupled through a (possibly dependent on x) skew-
+symmetric interconnection matrix S = −S⊺, so that
+
+
+˙x
+w
+y
+
+
+� �� �
+f
+= S
+
+
+∇E(x)
+z(w)
+u
+
+
+�
+��
+�
+e
+.
+(30)
+Such systems satisfy the power balance
+Ps + Pd + Pext = 0
+(31)
+where Ps = ∇E(x)⊺ ˙x denotes the stored power.
+Proof.
+Ps + Pd + Pext = ∇E(x)⊺ ˙x + z (w)⊺ w + u⊺ y = e⊺f = e⊺Se = (e⊺Se)⊺ = −e⊺Se = 0
+due to the skew-symmetry of S.
+Note that in this paper, we adopt the passive sign convention (also called receiver
+convention) for all components, including external sources. This means that a flow is
+defined positive when entering the component [29].
+14
+
+6.2. Macroscopic state and energy
+In the previous sections, we derived the thermodynamic entropy as a function of
+the energy and other macroscopic variables.
+In order to obtain a port-Hamiltonian
+formulation such as Eq. (30), we choose to express the energy as a function of the
+thermodynamic entropy and other macroscopic variables instead.
+Denote S := Sk �
+θfree�
+, and θx := θfree\Fe the set of macroscopic quantities that are
+not the energy, with corresponding set of labels Ix. We choose to define the (extensive)
+macroscopic state x as
+x =
+�
+S, θx�⊺
+,
+(32)
+so that the flow ˙x accounts for the time variation of extensive quantities. Assuming
+that the entropy function Sk is invertible with respect to Fe, we define the macroscopic
+energy function E as
+E : x �−→ E
+�
+Sk(Fe, θx), θx�
+= Fe,
+(33)
+so that the effort ∇E accounts for intensive quantities.
+Otherwise, the macroscopic
+energy function can be defined implicitly via Eq. (21) and contact forms [23, 8].
+Remark: the energy function E should be homogeneous of degree 1, so that it verifies
+for all γ
+E(γ x) = γ E(x).
+(34)
+6.3. Connection to ports
+Ext
+Sys
+(isolated) {Sys + ext}
+fsys
+fext
+esys = eext
+Figure 5:
+Flows f and efforts e of a system and its environment.
+The considered system and its
+environment as a whole form an isolated system.
+The environment acts on the system flow so that at thermodynamic equilibrium, the
+flows are balanced, and the effort is shared at the system interface (Fig 5):
+∂Esys
+∂F
+sys
+i
+= ∂Eext
+∂F
+ext
+i
+∀i ∈ Ix.
+(35)
+(see Appendix B for proof).
+Adopting the notations of Eq. (30), together with Eq. (22), we obtain the relations
+between flows, efforts and external ports in Table 1.
+15
+
+Table 1: Port variables and their relations.
+State x
+�
+S, Fi
+�
+u + ˙x = 0
+Effort ∇E(x)
+[T, T λi]
+y = ∇E(x)
+6.4. Conservative, reversible PHS
+Denoting σext the outgoing entropy flow, the conservative PHS interconnection matrix
+of an open system is found to be
+∇E(x)
+u
+T
+µ
+−P
+σext
+˙Next
+˙Vext
+
+
+
+
+˙S
+.
+.
+.
+−1
+.
+.
+˙x
+˙Fn
+.
+.
+.
+.
+−1
+.
+˙Fv
+.
+.
+.
+.
+.
+−1
+Text
+1
+.
+.
+.
+.
+.
+y
+µext
+.
+1
+.
+.
+.
+.
+−Pext
+.
+.
+1
+.
+.
+.
+.
+(36)
+7. Summary of the method and generalization route to irreversible systems
+This section first summarises the main steps to derive a reversible conservative macro-
+scopic PHS from a microscopic description of matter in an experimental context. Second,
+a process is proposed to complete this modelling under a macroscopic irreversible con-
+servative form, given a dissipation law.
+7.1. Reversible conservative macroscopic PHS
+In summary, the macroscopic PHS can be described from the microscopic description
+by completing the following steps:
+1. Microstate representation Define P, W = P⋆ and M ⊆ W equipped with char-
+acterizing functions F = {Fi : M �→ Fi}i∈I.
+2. Experimental conditions and accessible microstates
+(a) Partition F = Ffixed ∪ Ffree into the set Ffixed of functions the values of which
+are physically constrained by the experiment and its complement Ffree, with
+corresponding sets of indices Ifixed and Ifree.
+(b) Denote θfixed ⊂ ×i∈IfixedFi the set of experimentally admissible values for
+functions in Ffixed.
+(c) Denote Ma
+�
+θfixed�
+the corresponding set of admissible microstates.
+3. Stochastic description For all probability distributions p : Ma
+�
+θfixed�
+�−→ [0, 1],
+(a) Derive the surprisal Sb
+p : m ∈ Ma
+�
+θfixed�
+�−→ logb
+1
+p(m) ∈ R+.
+(b) Derive the statistical entropy function Sk : p �−→ Ep
+�
+Sb=exp(1/k)
+p
+�
+∈
+�
+0, 1
+Ω
+�
+.
+4. Boltzmann principle for ergodic systems at thermodynamic equilibrium
+16
+
+(a) Introduce θfree :=
+�
+Fi
+�
+i∈Ifree the values of functions in Ffree observed at a
+macroscopic scale.
+(b) Define p⋆ �
+m | H
+�
+θfree��
+according to Th. 1.
+(c) Define the thermodynamic entropy function Sk : θfree �→ Sk
+�
+p⋆ �
+. | H
+�
+θfree���
+.
+For common experimental constraints (i.e., constraints on F =
+�
+Fe, Fn, Fv, Sb
+p
+�
+), we
+obtain the results in Table 2 (see also [30]).
+Note that if there is no analytic solution for Fe �→ Sk(Fe, . . . ) and its inverse (note
+that ∂Sk
+∂Fe
+is monotonic), approximation strategies can be used (see [31] for an example).
+17
+
+Table 2: Statistical ensembles and associated constraints for usual experimen-
+tal conditions. Ω denotes the cardinal of Ma (set of accessible microstates).
+Ensemble
+θfixed
+θfree
+p⋆(m)
+Entropy
+Example
+Micro-canonical
+(E, N, V, S)
+1
+Ω
+kB ln Ω
+Gas in an isolated tank
+Isoenthalpic-isobaric
+(N, S)
+�
+Fe, Fv
+�
+1
+Ω
+kB ln Ω
+Gas in a closed tank
+No
+with a piston,
+thermally insulated
+thermal contact
+Unnamed
+(V, S)
+�
+Fe, Fn
+�
+1
+Ω
+kB ln Ω
+Gas in a porous tank,
+thermally insulated
+Unnamed
+S
+�
+Fe, Fn, Fv
+�
+1
+Ω
+kB ln Ω
+Gas in a porous tank
+with a piston,
+thermally insulated
+Thermal contact
+Canonical
+(N, V )
+Fe
+exp
+�
+− Fe(m)
+kB T
+�
+Z(T )
+kB ln Z(T ) + Fe
+T
+Gas in a closed tank,
+in contact with a thermostat
+Isothermal-isobaric
+N
+�
+Fe, Fv
+�
+exp
+�
+− Fe(m)+P Fv(m)
+kB T
+�
+Z(T, P )
+kB ln Z(T, P) + Fe+P Fv
+T
+Gas in a closed tank
+with a piston,
+in contact with a thermostat
+Grand-canonical
+V
+�
+Fe, Fn
+�
+exp
+�
+− Fe(m)−µ Fn(m)
+kB T
+�
+Z(T, µ)
+kB ln Z(T, µ) + Fe−µ F n
+T
+Gas in a porous tank,
+in contact with a thermostat
+Unnamed
+�
+Fe, Fn, Fv
+�
+exp
+�
+− Fe(m)+P Fv(m)−µ Fn(m)
+kB T
+�
+Fe+P Fv−µ Fn
+T
+Gas in a porous tank
+with a piston,
+in contact with a thermostat
+18
+
+7.2. Irreversible conservative macroscopic PHS from a macroscopic dissipative law
+The system (36) models some conservative reversible physics at macroscopic scale. In
+some cases, dissipative phenomena can be observed, for which laws are available only at
+this scale.
+In this part, we assume that such a dissipative phenomenon is described by
+(i) a flow-to-effort mapping law
+zd : fd �→ ed = zd(fd) such that for all fd, zd(fd)⊺fd =: Pd ≥ 0,
+(37)
+(ii) interconnected to the conservative part according to matrix given by
+∇E(x)
+z(w)
+u
+T
+es
+ed
+σext
+eext
+
+
+
+
+˙x
+˙S
+.
+.
+.
+−1
+.
+fs
+.
+Jx
+−K
+.
+−Gx
+w
+fd
+.
+K⊺
+Jw
+.
+−Gw
+y
+Text
+1
+.
+.
+.
+.
+fext
+.
+G⊺
+x
+G⊺
+w
+.
+Jy
+,
+(38)
+(iii) the dissipated power Pd being totally converted into an entropy rate
+σi = Pd/Td ≥ 0 where Td > 0,
+(39)
+denotes the instantaneous macroscopic temperature at which the phenomenon is
+experienced. The positivity of σi reflects the irreversible nature of dissipation.
+From zd, we form the irreversible thermodynamic converter with law
+z : w =
+�
+f ⊺
+d , Td
+�⊺ �−→
+
+zd(fd)⊺, −zd(fd)⊺fd/Td
+�
+��
+�
+−σi
+
+
+⊺
+(40)
+where −σi ≤ 0 accounts for the entropy rate incoming into the converter.
+This law is conservative as z(w)⊺w = 0. Due to irreversibility (σi ≥ 0), it naturally
+fulfills the second principle of thermodynamics.
+Finally, from (iii), the irreversible conservative thermodynamic macroscopic PHS is
+given by
+∇E(x)
+z(w)
+u
+T
+es
+−σi
+ed
+σext
+eext
+
+
+
+
+˙x
+˙S
+.
+.
+−1
+.
+−1
+.
+fs
+.
+Jx
+.
+−K
+.
+−Gx
+w
+Td
+1
+.
+.
+.
+.
+.
+fd
+.
+K⊺
+.
+Jw
+.
+−Gw
+y
+Text
+1
+.
+.
+.
+.
+.
+fext
+.
+G⊺
+x
+.
+G⊺
+w
+.
+Jy
+.
+(41)
+19
+
+8. Conclusion
+In this paper, we revisited equilibrium SP in order to model complex systems with
+numerous degrees of freedom as macroscopic PHS with a reduced number of variables.
+Starting from the choice of a particle’s description and ad hoc characterizing functions,
+we recalled how to derive the probability of a configuration of particles at equilibrium
+based on given experimental conditions. In the end, macroscopic variables are revealed
+to be expectations of the chosen characterizing functions for this probability, and the
+thermodynamic entropy to be a function of these macroscopic variables. Provided that
+the energy has been chosen as a characterizing function from the start, the macroscopic
+energy can in turn be expressed as a function of the thermodynamic entropy and other
+macroscopic variables. Through the PHS formalism, experimental conditions are repre-
+sented as an input flow that acts on the system so that the resulting output is an effort
+shared with the system. With this formulation, the externality of the environment, as
+well as its interactions with the system via exchanges of energy and entropy, are made
+explicit.
+As a result, we proposed two PHS formulations for conservative open systems, a
+reversible one (with no entropy creation), and an irreversible one (with entropy creation).
+An immediate perspective would be to extend this work to non-equilibrium SP [24],
+so that a macroscopic trajectory would not only be a succession of equilibrium states,
+and experimental conditions could change faster.
+20
+
+Appendix A. Proof of Theorem 1
+Proof. To solve Eq. (18), we introduce Lagrange multipliers λ0 and λfree := (λi)i∈Ifree,
+and optimize [32] the Lagrangian L defined by
+L :
+�
+p, λ0, λfree�
+�−→ Sk(p) + λ0
+
+
+
+�
+m∈Ma(θfixed)
+p(m) − 1
+
+
+ +
+�
+i∈Ifree
+λi
+�
+Ep[Fi] − Fi
+�
+.
+(A.1)
+A necessary condition to optimize L is to solve δL
+δp = 0, where δ denotes the functional
+derivative. From Eq. (A.1)-(14)-(10),
+δL
+δp = 0 ⇒
+�
+m∈Ma(θfixed)
+
+−k
+�
+ln p(m) + 1
+�
++ λ0 +
+�
+i∈Ifree
+λi Fi(m)
+
+ = 0.
+This is true in particular if p verifies
+− k
+�
+ln p(m) + 1
+�
++ λ0 +
+�
+i∈Ifree
+λi Fi(m) = 0
+∀m ∈ Ma
+�
+θfixed�
+⇒p(m) = exp
+��
+i∈Ifree λi Fi(m)
+k
+�
+exp
+�λ0
+k − 1
+�
+∀m ∈ Ma
+�
+θfixed�
+.
+A second necessary condition is to solve
+∂L
+∂λ0
+= 0, which, combined to the first
+condition, yields
+∂L
+∂λ0
+= 0 ⇒
+�
+m∈Ma(θfixed)
+p(m) = 1 ⇒
+�
+m∈Ma(θfixed)
+exp
+��
+i∈Ifree λi Fi(m)
+k
+�
+= exp
+�
+1 − λ0
+k
+�
+,
+so that for all m ∈ Ma
+�
+θfixed�
+, p⋆(m) is of the form
+�p⋆
+�
+m | H
+�
+θfree�
+, λfree
+�
+=
+exp
+� �
+i∈Ifree λiFi(m)
+k
+�
+Z
+�
+λfree�
+,
+(A.2)
+with Z
+�
+λfree�
+:=
+�
+m∈Ma(θfixed)
+exp
+��
+i∈Ifree λi Fi(m)
+k
+�
+.
+(A.3)
+A third necessary necessary condition is to solve ∂L
+∂λi
+= 0 for all i ∈ Ifree, which,
+combined with Eq. (A.3), yields
+∂L
+∂λi
+= 0 ⇒Ep[Fi] = Fi
+⇒
+�
+m∈Ma(θfixed) Fi(m) exp
+� �
+i∈Ifree λi Fi(m)
+k
+�
+Z
+�
+λfree�
+= Fi
+⇒ ∂
+∂λi
+k ln Z
+�
+λfree�
+= Fi.
+21
+
+We deduce that the optimal distribution p⋆ is
+p⋆
+�
+m | H
+�
+θfree��
+= �p⋆
+�
+m | H
+�
+θfree�
+, λfree
+�
+,
+with λfree such that
+∂
+∂λi
+k ln Z
+�
+λfree�
+= Fi
+∀i ∈ Ifree.
+Appendix B. Proof of effort equality at the system interface
+Proof. Consider the isolated total system constituted by the system under study and its
+environment. For all i ∈ Ifree, we have
+F
+total
+i
+= F
+sys
+i
++ F
+ext
+i
+.
+The entropy is extensive, therefore,
+Sk
+total
+�
+θfree
+total
+�
+= Sk
+sys
+�
+θfree
+sys
+�
++ Sk
+ext
+�
+θfree
+ext
+�
+.
+(B.1)
+The total system is isolated, therefore the total entropy is maximal with respect to any
+variable, so that for all i ∈ Ifree,
+∂Sk
+total
+∂F
+sys
+i
+= 0
+⇒ ∂Sk
+sys
+∂F
+sys
+i
++ ∂Sk
+ext
+∂F
+sys
+i
+= 0
+⇒ ∂Sk
+sys
+∂F
+sys
+i
+− ∂Sk
+ext
+∂F
+ext
+i
+= 0
+⇒λsys
+i
+− λext
+i
+= 0.
+Moreover, for all i ∈ Ix
+∂E
+∂Fi
+= T λi,
+since
+Sk �
+E(S, Fi), Fi
+�
+= S
+⇒ ∂Sk
+∂Fe
+∂E
+∂Fi
++ ∂Sk
+∂Fi
+= 0
+⇒ 1
+T
+∂E
+∂Fi
+− λi = 0.
+We deduce that ∂Esys
+∂F sys
+i
+= ∂Eext
+∂Fext
+i
+∀i ∈ Ix.
+22
+
+Acknowledgments
+The authors would like to thank Bernhard Maschke for his comments and helpful
+discussion.
+References
+[1] L. Boltzmann, Über die Beziehung zwischen dem zweiten Hauptsatze des mechanischen Wärmethe-
+orie und der Wahrscheinlichkeitsrechnung, respective den Sätzen über das Wärmegleichgewicht, Kk
+Hof-und Staatsdruckerei, 1877.
+[2] P. T. Landsberg, Thermodynamics and statistical mechanics, Courier Corporation, 2014.
+[3] D. Eberard, B. Maschke, Port-Hamiltonian systems extended to irreversible systems: The example
+of the heat conduction, IFAC Proceedings Volumes 37 (13) (2004) 243–248.
+[4] D. Eberard, B. Maschke, A. van der Schaft, An extension of Hamiltonian systems to the thermody-
+namic phase space: Towards a geometry of nonreversible processes, Rep. Math. Phys. 60 (2) (2007)
+175–198.
+[5] H. Ramirez, B. Maschke, D. Sbarbaro, Irreversible port-Hamiltonian systems: A general formulation
+of irreversible processes with application to the CSTR, Chem. Eng. Sci. 89 (2013) 223–234.
+[6] J.-C. Delvenne, H. Sandberg, Finite-time thermodynamics of port-Hamiltonian systems, Physica D
+267 (2014) 123–132.
+[7] H. Ramirez, Y. Le Gorrec, B. Maschke, F. Couenne, On the passivity based control of irreversible
+processes: A port-Hamiltonian approach, Automatica 64 (2016) 105–111.
+[8] A. Van Der Schaft, Classical thermodynamics revisited: A systems and control perspective, IEEE
+Control Systems Magazine 41 (5) (2021) 32–60.
+[9] A. van der Schaft, Liouville geometry of classical thermodynamics, Journal of Geometry and Physics
+170 (2021) 104365.
+[10] A. N. Gorban, N. K. Kazantzis, I. G. Kevrekidis, H. C. Öttinger, C. Theodoropoulos, Model
+reduction and coarse-graining approaches for multiscale phenomena, Springer, 2006.
+[11] H. C. Öttinger, Systematic coarse graining:“Four Lessons and A Caveat” from nonequilibrium sta-
+tistical mechanics, MRS bulletin 32 (11) (2007) 936–940.
+[12] E. Ising, Beitrag zur theorie des ferromagnetismus, Zeitschrift für Physik 31 (1) (1925) 253–258.
+[13] J. Strecka, M. Jascur, A brief account of the Ising and Ising-like models: Mean-field, effective-field
+and exact results, arXiv preprint arXiv:1511.03031 (2015).
+[14] J. E. Hopcroft, R. Motwani, J. D. Ullman, Introduction to automata theory, languages, and com-
+putation, Acm Sigact News 32 (1) (2001) 60–65.
+[15] P. Flajolet, R. Sedgewick, Analytic combinatorics, Cambridge University Press, 2009.
+[16] A. Liechtenstein, M. Katsnelson, V. Gubanov, Exchange interactions and spin-wave stiffness in
+ferromagnetic metals, Journal of Physics F: Metal Physics 14 (7) (1984) L125.
+[17] A. Grothendieck, Récoltes et Semailles (Réflexions et témoignage sur un passé de mathématicien),
+Gallimard (2022), 1986.
+[18] D. Plachky, An ideal theoretic characterization of finite sets, finite algebras, and σ-algebras of
+countably generated type, Mathematica Slovaca 51 (3) (2001) 301–311.
+[19] R. M. Gray, Entropy and information theory, Springer, 2011.
+[20] D. E. Knuth, Dynamic Huffman coding, Journal of algorithms 6 (2) (1985) 163–180.
+[21] A. Patrascioiu, The ergodic-hypothesis: A complicated problem in mathematics and physics, Los
+Alamos Science 15 (1987) 263–279.
+[22] R. K. Zia, E. F. Redish, S. R. McKay, Making sense of the Legendre transform, American Journal
+of Physics 77 (7) (2009) 614–622.
+[23] A. Van der Schaft, B. Maschke, Geometry of thermodynamic processes, Entropy 20 (12) (2018) 925.
+[24] H. C. Öttinger, Beyond Equilibrium Thermodynamics, John Wiley & Sons, 2005.
+[25] V. Duindam, A. Macchelli, S. Stramigioli, H. Bruyninckx, Modeling and control of complex physical
+systems: The port-Hamiltonian approach, Springer Science & Business Media, 2009.
+[26] A. J. van der Schaft, D. Jeltsema, et al., Port-Hamiltonian systems theory:
+An introductory
+overview, Foundations and Trends in Systems and Control 1 (2-3) (2014) 173–378.
+[27] A. Falaize, T. Hélie, Passive guaranteed simulation of analog audio circuits: A port-Hamiltonian
+approach, Applied Sciences 6 (10) (2016) 273.
+[28] R. Müller, Time-continuous power-balanced simulation of nonlinear audio circuits: Realtime pro-
+cessing framework and aliasing rejection, Ph.D. thesis, Sorbonne Université (2021).
+23
+
+[29] T. A. Bigelow, Power and energy in electric circuits, in: Electric Circuits, Systems, and Motors,
+Springer, 2020, pp. 105–121.
+[30] H. Graben, J. R. Ray, Unified treatment of adiabatic ensembles, Phys. Rev. A 43 (8) (1991) 4100.
+[31] J. Najnudel, T. Hélie, D. Roze, R. Müller, From statistical physics to macroscopic port-Hamiltonian
+systems: A roadmap, in: 7th IFAC Workshop on Lagrangian and Hamiltonian Methods for Non-
+linear Control, 2021.
+[32] E. T. Jaynes, On the rationale of maximum-entropy methods, Proceedings of the IEEE 70 (9)
+(1982) 939–952.
+24
+
diff --git a/xdE5T4oBgHgl3EQfMw71/content/tmp_files/load_file.txt b/xdE5T4oBgHgl3EQfMw71/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..bc3ea515ecea43aa2bcde3aa06be1e597db9a73b
--- /dev/null
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf,len=711
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='05485v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='DS]  13 Jan 2023  From Equilibrium Statistical Physics  Under Experimental Constraints  to Macroscopic Port-Hamiltonian Systems  Judy Najnudel,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Thomas Hélie,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' David Roze,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Rémy Müller  S3AM Team,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Laboratory STMS (UMR 9912),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' IRCAM-CNRS-SU,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' 1 Place Igor Stravinsky,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' 75004  Paris,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' France  Abstract  This paper proposes to build a bridge between microscopic descriptions of matter with  internal energy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' composed of many fast interacting particles inside an environment,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' and  their port-Hamiltonian (PH) descriptions at macroscopic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' The environment, as-  sumed to be slow, is modeled through experimental constraints on macroscopic quantities  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' energy, particle number, etc), with a partitioning into two classes: non fluctuating  and fluctuating values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' The method to derive the PH macroscopic laws is detailed in sev-  eral steps and illustrated on two standard cases (ideal gas, Ising ferromagnets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' It revisits  equilibrium statistical physics with a focus on this partitioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' First, the Boltzmann’s  principle is used to provide the statistic law of the matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  It defines a macroscopic  equilibrium characterized by a scalar value, the entropy, together with thermodynamic  quantities emerging from each constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Then, the port-Hamiltonian system is derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  The Hamiltonian (macroscopic energy) is derived as a function of the macroscopic state  (entropy and the macroscopic quantities associated with the fluctuating class).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' The ports  (flows/efforts) are related to the time-derivative of the state and the Hamiltonian gra-  dient in a conservative way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' This open system defines the reversible laws that govern  standard thermodynamic quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Lastly, this paper presents a strategy to extend this  PH system to an irreversible conservative one, given a macroscopic dissipative law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  Keywords:  equilibrium statistical physics, macroscopic port-Hamiltonian Systems,  statistical entropy, experimental conditions  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Introduction  A macroscopic system (of size 10−2 m or bigger) is constituted of matter, that is,  billions of microscopic particles (of size 10−9 m or smaller) which are collectively re-  sponsible for the system’s behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' However, studying a single particle tells nothing  about the macroscopic system, just as following the trajectory of a single person is not  sufficient to predict a crowd movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Yet, solving exhaustive equations with billions  of variables would be all at once much too complex and irrelevant: at a high enough  scale, individual behaviors do not matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Indeed, one is usually not interested in the  particular trajectories of water molecules in one’s glass, but rather in the volume, on  Preprint submitted to Physica D  January 16, 2023  average, that they take.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Likewise, one is not only interested in the day’s weather report,  but rather in the global tendency .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  Averages and tendencies belong to the domain of statistics, which aims to describe  complex systems with a reduced number of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Thus, Statistical Physics (SP)  computes averages on (fast) fluctuations of complex systems in order to derive (slower)  macroscopic quantities, given some experimental conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Statistical arguments for  the description of a system transitioning towards thermodynamic equilibrium were intro-  duced by Ludwig Boltzmann in 1877 [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' This framework allows the prediction of macro-  scopic thermodynamic phenomena such as temperature, entropy creation, and phase  transitions [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  Thermodynamics has been broadly studied in the context of port-Hamiltonian system  (PHS), as well as their modeling and their control (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' [3, 4, 5, 6, 7, 8, 9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' However,  the proper derivation of macroscopic thermodynamic PHS from complex systems with  numerous degrees of freedom is seldom addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' As it happens, the choice of a system  representation for this kind of model reduction is all but inconsequential, and must be  handled with care [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' In this paper, we propose a series of systematical steps in  order to construct a simplified yet physically-based structured macroscopic PHS from a  system that can be described by SP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  Note that in the scope of this work, we limit ourselves to equilibrium SP, in the  sense that average quantities are determined for a system at thermodynamic equilibrium,  given some experimental conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' It is compatible with studying the system dynam-  ics, assuming that thermodynamic relaxation (the process of reaching thermodynamic  equilibrium) is infinitely faster than the rate of change of experimental conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Based  on this assumption, a macroscopic trajectory is to be understood as a succession of ther-  modynamic equilibrium states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  This paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' In Section 2, we formalize the description of  the microscopic configurations of a system through the choice of (i) an ad hoc parti-  cle representation and (ii) a set of characterization functions that evaluate macroscopic  quantities .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Section 3 addresses the experimental conditions at the macroscopic level and  their influence on the system configuration space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' In section 4, we introduce a stochas-  tic description for microscopic configurations .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Then, in Section 5, we determine the  conditional probability distribution according to the Boltzmann principle for a system at  thermodynamic equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' This allows the derivation of relevant macroscopic variables  as expectations for this probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' In Section 6, we introduce the ports and  relates them to those macroscopic variables, leading to the macroscopic PHS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Fi-  nally, section 7 summarizes the practical sequence of these steps to derive the macroscopic  PHS model from the microscopic description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' In addition, it presents how to derive a  conservative irreversible PHS model from an additional macroscopic dissipation law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  All the steps are detailed in the following sections, as recapped in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Microstate of a system  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Particle representation (p ∈ P)  In order to describe a system at a microscopic level, each of its particles must be  described in a relevant way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Depending on the system under study, one may choose to  represent a particle by its position, momentum, charge, magnetic moment, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  2  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Microscopic description - Section 2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' P-valued particle representation  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Particle configuration m ∈ M = P⋆  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Set of characterizing functions F  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Experimental conditions - Section 3  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' F = Ffixed ∪ Ffree  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Fixed values θfixed  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Set of accessible configurations Ma(θfixed)  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Stochastic setting - Section 4  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Probability distribution p  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Surprisal Sb  p(m) (in base b)  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Statistical entropy Sk(p), k = 1/ln b  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Boltzmann principle - Section 5  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Ergodicity Ep  �  Fi ∈ Ffree�  = Fi  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Maximum entropy given  experimental conditions (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=')  → Thermodynamic entropy Sk(Fi)  and Lagrange mult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' λi  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Macroscopic PHS - Section 6  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Entropy to energy representation  Sk(E, Fj) ↔ E(S, Fj  � �� �  x  )  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Connection to ports (effort u, flow y)  Fj ↔ ˙x ↔ u,  λj ↔ ∇E(x) ↔ y  Figure 1: from equilibrium statistical physics to macroscopic port-Hamiltonian systems (PHS): method  recap with the labels of the main mathematical objects introduced in each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  Definition 1 (Particle set P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Given a chosen representation to encode a particle state,  we denote P the set of all its possible values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  Example 1 (Particle represented by its position and momentum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' For a particle chosen  to be represented by its position in space r ∈ R3 and momentum p ∈ R3, such as in a  gas, the particle set is defined as P = R3 × R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  Example 2 (Particle represented by its magnetic moment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' For a particle chosen to be  represented by its magnetic moment s ∈ {−1, 1} such as in the Ising model [12, 13], the  particle set is defined as P = {−1, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Configuration space (m ∈ M)  As an element of P represents the state of one particle, a natural way to represent  a configuration of particles is to concatenate elements of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' By analogy with formal  language theory [14], a particular configuration of particles is chosen to be encoded as a  word over the alphabet P (see remark 1 for other choices) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  Definition 2 (Encoding space W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' We denote W := P⋆ the space of encodable config-  urations (or, for short, the encoding space) , where ⋆ is the Kleene operator defined by  P = {ǫ},  Pi+1 =  �  p1 · p2 | (p1, p2) ∈ Pi × P  �  ∀i ≥ 0,  (1a)  P⋆ =  �  i≥0  Pi,  (1b)  with ǫ the empty configuration and · the concatenation operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  Property 1 (W is a monoid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' By construction, the encoding space W is a monoid  (see Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' (3)) with associative binary operation · (concatenation) and identity element ǫ  (empty configuration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  3  1  2  1  2  1  2  1  2  (a) Examples of microstates for a system of two particles described by their spin s ∈ {−1 (blue) , 1 (red)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  (b) Examples of microstates for a system with three particles described by their position (circle) and momentum  (arrow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  Figure 2: Examples of microstates for different systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  Definition 3 (Monoid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' A set S is a monoid if it is equipped with an associative binary  operation · : S × S �→ S and identity element ǫ, such that for all (s1, s2, s3) ∈ S3, the  following properties hold  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' s1 · (s2 · s3) = (s1 · s2) · s3,  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' ǫ · s1 = s1 · ǫ = s1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  Based on the chosen representation, some configurations may not be physically ad-  missible 1, therefore we introduce the set of microstates as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  Definition 4 (Admissible configuration set M and microstate m ∈ M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' We denote  M ⊆ W the set of encodable configurations that are also physically admissible .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  An  element m ∈ M is called a microstate of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  In the following, we choose M = W (see remark 1(i) for an interpretation and (ii-iii)  for examples with M ̸= W) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Figure 2a shows examples of microstates for a system of  particles described by their spin, and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' 2b shows a system of particles described by  their position and momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  Property 2 (M is measurable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' The pair  �  M, P(M)  �  where P(M) denotes the powerset  of M is a measurable space, that is, it verifies  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' M ∈ P(M),  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' P(M) is closed under complements: M\\P ∈ P(M),  ∀P ∈ P(M),  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' P(M) is closed under countable unions: �∞  i=1 Pi ∈ P(M)  ∀P1, P2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' ∈ P(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  As mentioned above, the set M could be defined on encoding spaces generated with  operators other than the concatenation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Examples are outlined in the following remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  1For instance, if the chosen representation assigns a unique label to each particle, configurations in  which several particles share the same label are not admissible (see remark 1(iii) for more more details) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  4  Remark 1 (Examples of combinatorial structures and interpretations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' The monoid W  provides a simple and natural way to encode microscopic configurations, which consists  in choosing a prioritized and distinguishable representation of particles (see specification  (i) below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' But the configuration encoding can be addressed by using any appropriate  combinatorics of particles, possibly choosing other specifications (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' [15, § I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='2] for  details on operators Seq, Mset, Pset, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=', mentioned below):  (i) Distinguishability with prioritization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' A word pi · pi−1 · · · · · p1 ∈ Pi ⊂ W can be  interpreted as describing the state values pn ∈ P of particles number n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  In this sense, definition 2 encodes a physics with distinguishable particles and with  a priority ordering on involved particles (particle 1 can be encoded alone, particle 2  only if 1 is involved, etc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' From the combinatorics point of view, the encoding space  W = P⋆ corresponds to the sequence construction, also denoted Seq(P) in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  (ii) Undistinguishability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' A natural encoding of a physics described with undistinguish-  able particles is the multiset W(ii) =Mset(P) composed of all the finite sets of P  (the order between elements does not count), in which arbitrary finite repetitions  of elements are allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Examples of admissible configuration sets M(ii) are W(ii),  or the powerset Pset(P) ⊂Mset(P) (no repetition allowed) if the physics under  consideration forbids two particles of a microstate to be excited by the same state  value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  (iii) Distinguishability without priorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' A means other than (i) to encode configu-  rations with distinguishable particles is to add a distinct label to each particle such  as its number n ∈ L ⊆ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' In this case, the encoding space can be the powerset  W(iii) =Pset(Plabeled), where the set of labeled particles Plabeled is the cartesian  product L × P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' An admissible configuration set that forbids the particle ubiquity or  replication is a subset M(iii) of W(iii) for which all the elements make the number  of occurrences of each label n ∈ L be 0 or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  (iv) Complex structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' More generally, the encoding space can be composed of complex  elements, with structures involving sequences, cycles (for e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' aromatic molecules),  trees (etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=') and their combination according to precise combinatorial specifications  (see [15, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Characterizing functions (Fi ∈ F)  In order to characterize the system at a microscopic level, one may choose to equip  M with a finite set of characterizing functions, labeled by i ∈ I, denoted Fi : M → Fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  Definition 5 (Extensivity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' A function Fi is extensive if Fi is a R+-semimodule (see  Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' 6) and if it verifies  m3 = m1 · m2 ⇒ Fi(m3) = Fi(m1) + Fi(m2)  ∀(m1, m2, m3) ∈ M3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  (2)  Definition 6 (R+-semimodule).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' A set S is a R+-semimodule if for all (r1, r2) ∈ R+2  and (s1, s2) ∈ S2, the following properties hold  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' r1 (s1 + s2) = r1 s1 + r2 s2,  5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' (r1 + r2) s1 = r1 s1 + r2 s1,  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' (r1 r2) s1 = r1 (r2 s1),  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' 1 s1 = s1,  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' 0 s1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  Example 1 (Continued).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' For a gas of N identical, non-interacting particles, the function  Fe : M �→ R+ defined as  Fe : m �−→ Fe(m) =  N  �  i=1  ��pi(m)  ��2  2 m  (3)  gives the energy of the system in microstate m, where pi(m) is the momentum of particle  i, and m is the mass of a particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' It fulfills the extensivity property defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  Example 2 (Continued).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' In the Ising model [16] , the function Fe : M �→ R defined as  Fe : m �−→ Fe(m) = −1  2 m⊺ Jex m  (4)  gives the energy of the system in microstate m, where each coefficient Jexi,j is the ex-  change energy between atom i and atom j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' It does not fulfill the extensivity property  defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' The function Fn : M �→ N+ defined as  Fn : m �−→ Fn(m)  (5)  where Fn(m) is the number of particles of the system in microstate m is extensive .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' The function Fr defined as  Fr(m) =  �  ri�  1≤i≤Fn(m) ,  (6)  where ri ∈ R3 is the position of particle i, gives the set of all particle positions for the  system in microstate m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' It is not extensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' The function Fv : M �→ R+ defined as  Fv : m �−→ Fv(m)  (7)  where Fv(m) is the volume occupied by the system in microstate m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' The choice of such  function Fv is hardly unique (see remark 2 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Here, we propose to define Fv(m)  as the minimal bounding volume enclosing all particle positions of microstate m that  accounts for the container geometry and its degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  For instance, for a  cylindrical container of fixed base A closed by a piston moving freely along axis z, we can  define the volume as  Fv(m) = A × h(m),  with h(m) = max{ri  z | ri ∈ Fr(m)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  (8)  This function does not fulfill the extensivity property defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  6  Remark 2 (Volume).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' The mathematical conceptualization of the volume is a challenging  issue2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' In the context of SP, its physical conceptualization is also an issue:  (i) a possible choice could be the volume occupied by the particles, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' that delimited  by the 3D simplicial envelope of all the particle positions,  (ii) an alternative is to consider the volume of a container, in which the particle are  authorized to evolve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  The case (i) allows the definition of a characterizing function Fv, the volume being intrin-  sically related to the microstate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' In (ii), the microstate does not encode the information  of the container volume: its set of particles in contact with the container boundary can  even be empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' This information is an "experimental constraint" (presented in section 3  below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Note also that example 5 corresponds to a hybrid description in between (i) and  (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  In the following, we denote F = {Fi}i∈I the set of characterizing functions on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Experimental conditions and accessible microstates (m ∈ Ma)  Experimental conditions may constrain characterizing functions to take values that  are compatible with these experimental conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Thus, under experimental conditions,  the configuration space becomes restricted to a set of accessible microstates Ma ⊂ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  Definition 7 (Set of accessible microstates Ma).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Denote Ifixed ⊆ I the set of labels of  characterizing functions that are experimentally constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Due to the constraints, a  function Fi, i ∈ Ifixed can only take admissible values in Ffixed  i  ⊆ Fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  Denote θfixed :=  �  Ffixed  i  �  i∈Ifixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' The set of accessible microstates Ma  �  θfixed�  is  Ma  �  θfixed�  =  �  m ∈ M | Fi(m) ∈ Ffixed  i  ∀i ∈ Ifixed�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  (9)  Remark that Ifixed = I defines an isolated system with respect to the chosen charac-  terizing functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  Example 1 (Continued).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Consider a gas of N particles in a closed tank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' The system  cannot exchange particles with the environment, therefore the number of particles Fn(m)  is fixed to N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  Denote Ffixed  n  = {N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' The set of accessible microstates is Ma (N) =  �  m ∈ M | Fn(m) ∈ Ffixed  n  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  Example 1 (Continued).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Consider a gas of N particles in a closed tank occupying a  space Π ⊂ R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Denoting Ffixed  n  = {N} and Ffixed  r  = ΠN, the set of accessible microstates  is Ma (N, Π) =  �  m ∈ M | Fn(m) ∈ Ffixed  n  , Fr(m) ∈ Ffixed  r  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Note that the constraint on  Ffixed  r  corresponds to the case (ii) in remark 2, the container being described by Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  Other examples of experimental conditions are shown on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  2In [17, Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' 3,p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='3], Grothendieck mentions the absence (in most textbooks) of any "serious" defi-  nition of the notion of length (of a curve), of area (of a surface), of volume (of a solid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  7  (a) Fixed number of particles  and fixed volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  (b) Fixed number of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  (c) Fixed volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  Figure 3: Examples of experimental conditions for a gas in a tank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  Intermediate point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' At this step, the physics is described through: M (microscopic rep-  resentation of physical particles, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' microstates), Ma ⊆ M (microstates accessible under  fixed macroscopic experimental constraints), F (links to quantities that can be observed  at the macroscopic scale).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' This is not sufficient to derive an autonomous physical law at  macroscopic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' This issue can be solved by: (i) completing F with a single new func-  tion, namely, the surprisal, which is elaborated from a stochastic description, giving rise  to the entropy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' and (ii) applying the fundamental principle introduced by Boltzmann [1]  to derive a microstate probability that physically makes sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  The following sections revisit this approach, focusing on the propensity of macroscopic  quantities to communicate with their peer in an external environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' To this end, we  assume that all characterizing functions that are not explicitly fixed by experimental  conditions can still depend on microstate m, and we denote Ifree := I\\Ifixed the set of  labels of characterizing functions not fixed by experimental conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Stochastic representation and measure of uncertainty  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Microstate stochastic description  The system fluctuates from one accessible microstate to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' It is considered to  be impossible to predict these fluctuations in a deterministic fashion at the macroscopic  level: SP adopts a stochastic framework that model their random description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  Indeed,  from Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' (2),  �  M, P(M)  �  is measurable, therefore so is  �  Ma, Tr  �  P(M)  �  Ma  �  , where  Tr  �  P(M)  �  Ma denotes the trace of P(M) on Ma [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Assuming that Ma is countable and  that the distribution p is discrete3, we can define a probability distribution p  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' | θfixed� :  Ma  �  θfixed�  �→ [0, 1], denoted p for short below, which assigns to each microstate m ∈  Ma  �  θfixed�  a probability p(m) to be the actual microstate of the system (the Boltzmann  principle in section 5 will provide a tool to determine this probability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  The average of a random quantity F(m) is given by its expectation Ep[F], defined as  Ep[F] =  �  m∈Ma(θfixed)  p(m) F(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  (10)  3Extensions to continuous measurable spaces are available and similar, replacing the sum by an  integral with a Lebesgues measure and using distributions in (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  8  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Statistical entropy  Given some basis of information units b > 1, a microstate m with probability p(m)  has a surprisal Sb  p(m) defined as  Sb  p(m) = logb  1  p(m)  with logb =  1  ln(b) ln .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  (11)  The surprisal, or information content, quantifies how much the occurrence of microstate  m is surprising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' For example, if some microstate m is the state of the system for certain,  it has probability 1 and surprisal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  As the probability of two independent events m1 and m2 verifies  p (m1 · m2) = p(m1) p(m2),  (12)  the surprisal function Sb  p verifies the extensivity property defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  The surprisal allows the definition of a measure of lack of information on average for a  probability distribution p and a basis b, namely, the statistical entropy Sb(p) [19] defined  as  Sb(p) = Ep[Sb  p].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  (13)  The statistical entropy can be interpreted of as “the average number of questions to ask  with b possible answers per question” in order to know the actual microstate for certain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Consider the outcomes of tossing a coin twice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' The coin can come up heads  or tails after each toss, hence 2 × 2 = 4 possible outcomes (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' If all outcomes are  equiprobable, one needs at least two questions with two possible answers each to know the  exact outcome:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Did the coin come up heads or tails after the first toss?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Did the coin come up heads or tails after the second toss?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  As it happens, taking p1 : m �→ p1(m) =  1  4 and b = 2 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' (13) yields Sb(p1) =  − log2  1  4 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  However, if the probability distribution is not uniform, some outcomes are more prob-  able than others, and the uncertainty is lower;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' ditto the entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  For instance, with  a probability distribution p2 assigning  1  2 to outcome (A), 1  4 to outcome (B), and 1  8 to  outcomes (C) and (D), the entropy becomes Sb(p2) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='75 < Sb(p1) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  In information theory, statistical entropy relates to optimal encoding of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  Suppose you repeat the coin toss experiment of Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' (6) for a long period of time, and wish  to record every outcome on a computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' For a sequence of two tosses with distribution  p1, an outcome cannot be encoded in less than two bits;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' while with distribution p2,  outcome (A) can be encoded on one bit, outcome (B) on two, and outcomes (C) and (D)  on three, that is, 1× 1  2 +2× 1  4 +6× 1  8 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='75 bits on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' The most frequent outcome  takes the least encoding space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' conversely, the comparatively large encoding space taken  by outcomes (C) and (D) is compensated by the rarity of their occurrence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  On the  whole, exploiting the knowledge underpinned by distribution p2 reduces the encoding  cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' This principle underlies Morse code (and, more generally, lossless entropy encoding  like Huffman coding [20]): very common letters such as “e” or “i” take much fewer dots  than less common letters like “j” or “q”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  9  (a)  (b)  (c)  (d)  Figure 4: Possible outcomes for a coin tossed twice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  For compacity, the statistical entropy becomes in the following  Sk(p) := Sb=exp(1/k)(p) = −k  �  m∈Ma(θfixed)  p(m) ln p(m),  (14)  for all p defined on Ma(θfixed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  Remark 3 (Random structures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Following remark 1, the cases of microstates involving  combinatorial structures based on elaborated specifications (such as molecules and chem-  ical reaction processes) require elaborate tools addressing random structures (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' [15,  part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' C]) that are out of the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Microstate probability distribution at equilibrium and partition function  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Thermodynamic equilibrium  A system is at thermodynamic equilibrium when its statistics stops evolving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' At this  point, the ergodic hypothesis postulates that over a “sufficiently long” period of time t,  the system explores all its accessible microstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Assuming that a microstate m can  be measured at a time τ through M : R+ �→ Ma, this means that at thermodynamic  equilibrium, the temporal mean of a quantity Fi coincides with its expectation Ep[Fi]:  Fi :=  lim  t→+∞  1  t  � t  0  (Fi ◦ M) (τ) dτ = Ep[Fi].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  (15)  Therefore, for a given set θfree of mean values (Fi)i∈Ifree, the ergodic hypothesis  translates into a set of hypotheses H  �  θfree�  defined as  H  �  θfree�  =  �  Ep[Fi] = Fi  �  i∈Ifree .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  (16)  While still discussed [21] (especially regarding the definition of “sufficiently long”),  this hypothesis is the foundation of equilibrium statistical physics, and we assume its  validity in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Boltzmann principle: maximum entropy at thermodynamic equilibrium (reminder)  By definition, a system at equilibrium does not evolve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Since change is new informa-  tion, the information given by a system at equilibrium is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' As statistical entropy  is a measure of lack of information, it follows that at equilibrium, the entropy is maximal:  this is the Boltzmann principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  10  It follows that the microstate probability distribution at thermodynamic equilibrium  p⋆ is  p⋆ = arg max  p  Sk(p) subject to  �  m∈Ma(θfixed)  p(m) = 1  = arg  p  max  p,λ Sk(p) + λ  \uf8eb  \uf8ec  \uf8ed  �  m∈Ma(θfixed)  p(m) − 1  \uf8f6  \uf8f7  \uf8f8 ,  (17)  where the second line specifies the constraint using a Lagrange multiplier λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Boltzmann principle in a macroscopic external environment  The macroscopic quantities (Fi∈Ifree(m)) whose fluctuation is experimentally allowed  are prone to communicate with the external environment (for example, through exchanges  of particle, energy, etc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Assuming ergodicity and an infinite ratio between macroscopic  and microscopic time scales, this means that the expectation of these fluctuating quanti-  ties (with value Fi∈Ifree) can change over time (at the slow macroscopic scale), while the  thermodynamic equilibrium is satisfied (at the fast microscopic scale) and continuously  updated (at the slow macroscopic scale).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  As a main step of this paper,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' this issue is addressed by deriving the conditional  probability p⋆ �  m | H  �  θfree��  that maximizes entropy (Boltzmann principle) constrained  by the given macroscopic values θfree = (Fi)i∈Ifree,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' namely,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  p⋆ =arg max  p  Sk(p) subject to  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f3  �  m∈Ma(θfixed)  p(m) = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  H  �  θfree�  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  =arg  p  max  p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='λ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='λi∈IfreeSk(p) + λ0  \uf8eb  \uf8ec  \uf8ed  �  m∈Ma(θfixed)  p(m) − 1  \uf8f6  \uf8f7  \uf8f8  +  �  i∈Ifree  λi  �  Ep[Fi] − Fi  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  (18)  where H  �  θfree�  accounts for the experimental conditions (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' (16)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  Note that, by definition, this probability naturally restores that of (17), if the values  Fi∈Ifree are the expectations of Fi∈Ifree computed for probability (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Let θfree :=  �  Fi  �  i∈Ifree ∈ ×  i∈IfreeFi, where ×  i∈IfreeFi denotes the Cartesian  product of the (Fi)i∈Ifree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  Then for all m ∈ Ma  �  θfixed�  ,  p⋆  �  m | H  �  θfree��  =  exp  � �  i∈Ifree λi Fi(m)  k  �  Z  �  λfree�  ,  (19a)  11  where  Z  �  λfree�  :=  �  m∈Ma(θfixed)  exp  ��  i∈Ifree λi Fi(m)  k  �  (19b)  is the partition function of the system, and, for all i ∈ Ifree, λi verifies  ∂  ∂λi  k ln Z  �  λfree�  = Fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  (19c)  The proof is in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  Definition 8 (Thermodynamic entropy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' The thermodynamic entropy Sk �  θfree�  is de-  fined as the statistical entropy for the probability distribution at equilibrium given θfree:  Sk �  θfree�  = Sk  �  p⋆  �  | H  �  θfree���  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  (20)  Property 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' The thermodynamic entropy function Sk is a Legendre transform of k ln Z  and we have  Sk �  θfree�  = k ln Z  �  λfree�  −  �  i∈Ifree  λi Fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  (21)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  Sk �  θfree� (a)  = Sk  �  p⋆  �  | H  �  θfree���  (b)  = −k  �  m∈Ma(θfixed)  p⋆  �  m | H  �  θfree��  ln p⋆  �  m | H  �  θfree��  (c)  = −k  �  m∈Ma(θfixed)  p⋆  �  m | H  �  θfree��  ln  \uf8eb  \uf8ec  \uf8ed  exp  � �  i∈Ifree λi Fi(m)  k  �  Z  �  λfree�  \uf8f6  \uf8f7  \uf8f8  = −k  �  m∈Ma(θfixed)  p⋆  �  m | H  �  θfree�� ��  i∈Ifree λi Fi(m)  k  − ln Z  �  λfree��  (d)  = −  �  i∈Ifree  λi Fi + k ln Z  �  λfree�  ,  using (a) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' (20), (b) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' (14), (c) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' (19a), and (d) Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' (10)-(15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  We deduce that Sk is a Legendre transform of k ln Z (see also [22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  Property 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' It follows from Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' (3) that for all i ∈ Ifree, the Lagrange multiplier λi  is the derivative of the thermodynamic entropy function with respect to average Fi  λi = − ∂Sk  ∂Fi  �  θfree�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  (22)  12  Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' In particular, this defines the system temperature T , chemical potential µ,  and pressure P as  1  T := ∂Sk  ∂Fe  �  θfree�  ,  µ  T := − ∂Sk  ∂Fn  �  θfree�  ,  P  T := ∂Sk  ∂Fv  �  θfree�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  (23)  Adiabatic process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' For a system going through an adiabatic process (no thermal exchange  with the environment), the surprisal is independent of m so that  Sb  p⋆(m) = S  ∀m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  (24)  That implies that for such systems, all microstates have the same probability  p⋆(m) = 1  Ω,  with Ω = card  �  Ma  �  θfixed��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  (25)  From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' (19a), it follows that for such a system, we have  �  i∈Ifree  λi Fi(m) = C,  (26)  where C is independent of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' In particular, a system that can only exchange volume with its environment  verifies Fe(m) + P Fv(m) = C, where C is independent of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Identification of Boltzmann constant  To ensure that the statistical entropy does coincide with the thermodynamic entropy  at equilibrium, the constant k must be chosen as the Boltzmann constant4 kB = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='38 ×  10−23 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='K−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  Indeed, consider an ideal gas of N non-interacting atoms in a box of  volume V at temperature T , represented by their position and momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' The partition  function Z is given by  Z(T | N, V ) = V N  �2 π m k T  h2  �3 N/2  ,  (27)  where here m denotes the mass of an atom, and h is the Planck constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' From Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' (3),  the thermodynamic entropy Sk(Fe, N, V ) is given by  Sk(F e, N, V ) = k ln Z(T | N, V ) + Fe  T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  (28)  Moreover, from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' (23), the pressure P is given by  P = T ∂Sk  ∂V (Fe, N, V ) = N k T  V  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  (29)  Therefore, k must be identified with kB so that the ideal gas law P V = N kB T is  verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  4Note that, from definition (14), choosing the unit USI reference k0 = 1 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='K−1 as a unit information  quantity, this value corresponds to the question number base bB = exp(k0/kB) = exp(1023/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='38) ≈  103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='147e+22 ≈ 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='4736e+21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' This means that explaining +1 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='K−1 requires about 1022 bits for a gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  13  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Final PHS model  The macrosopic description of open system can be achieved by using balanced equa-  tions of variations of entropy, energy, mass, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Port-Hamiltonian systems provide an  adapted framework for such physical descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' This section addresses this issue by  using a standard formulation (recalled in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='1), in which the energy is expressed as  a function of entropy and state variables: this requires to invert S : (E, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' ) �→ S(E, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' )  w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' E, to introduce the hamiltonian H = E : (S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' ) �→ H(S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' ) in a first step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Note  that this inversion is a technical step that could be avoided (to still use the entropy func-  tion) by considering contact forms as in [23] (see also [24] for the alternative GENERIC  formulation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Reminder on port-Hamiltonian systems  The PHS formalism provides a unified formalism for the modeling of multiphysical  systems, in the sense that it recognizes energy as a universal currency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Indeed, any phys-  ical system can be divided into parts that interact with each other via energy exchanges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  Detailed presentations of PHS are available in [25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' In this paper, we rely on a  differential-algebraic formulation adapted to multiphysical systems [27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' This formu-  lation allows the representation of a dynamical system as a network of  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' storage components of state x and energy E(x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' passive memoryless components described by an effort law z : w �→ z(w), such as  the dissipated power Pd = z (w)⊺ w is non-negative for all flows w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' connection ports conveying the outgoing power Pext = u⊺y where u are inputs and  y are outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  The system flows f and efforts e are coupled through a (possibly dependent on x) skew-  symmetric interconnection matrix S = −S⊺, so that  \uf8ee  \uf8f0  ˙x  w  y  \uf8f9  \uf8fb  � �� �  f  = S  \uf8ee  \uf8f0  ∇E(x)  z(w)  u  \uf8f9  \uf8fb  �  ��  �  e  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  (30)  Such systems satisfy the power balance  Ps + Pd + Pext = 0  (31)  where Ps = ∇E(x)⊺ ˙x denotes the stored power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  Ps + Pd + Pext = ∇E(x)⊺ ˙x + z (w)⊺ w + u⊺ y = e⊺f = e⊺Se = (e⊺Se)⊺ = −e⊺Se = 0  due to the skew-symmetry of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  Note that in this paper, we adopt the passive sign convention (also called receiver  convention) for all components, including external sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' This means that a flow is  defined positive when entering the component [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  14  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Macroscopic state and energy  In the previous sections, we derived the thermodynamic entropy as a function of  the energy and other macroscopic variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  In order to obtain a port-Hamiltonian  formulation such as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' (30), we choose to express the energy as a function of the  thermodynamic entropy and other macroscopic variables instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  Denote S := Sk �  θfree�  , and θx := θfree\\Fe the set of macroscopic quantities that are  not the energy, with corresponding set of labels Ix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' We choose to define the (extensive)  macroscopic state x as  x =  �  S, θx�⊺  ,  (32)  so that the flow ˙x accounts for the time variation of extensive quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Assuming  that the entropy function Sk is invertible with respect to Fe, we define the macroscopic  energy function E as  E : x �−→ E  �  Sk(Fe, θx), θx�  = Fe,  (33)  so that the effort ∇E accounts for intensive quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  Otherwise, the macroscopic  energy function can be defined implicitly via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' (21) and contact forms [23, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  Remark: the energy function E should be homogeneous of degree 1, so that it verifies  for all γ  E(γ x) = γ E(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  (34)  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Connection to ports  Ext  Sys  (isolated) {Sys + ext}  fsys  fext  esys = eext  Figure 5:  Flows f and efforts e of a system and its environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  The considered system and its  environment as a whole form an isolated system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  The environment acts on the system flow so that at thermodynamic equilibrium, the  flows are balanced, and the effort is shared at the system interface (Fig 5):  ∂Esys  ∂F  sys  i  = ∂Eext  ∂F  ext  i  ∀i ∈ Ix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  (35)  (see Appendix B for proof).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  Adopting the notations of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' (30), together with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' (22), we obtain the relations  between flows, efforts and external ports in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  15  Table 1: Port variables and their relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  State x  �  S, Fi  �  u + ˙x = 0  Effort ∇E(x)  [T, T λi]  y = ∇E(x)  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Conservative, reversible PHS  Denoting σext the outgoing entropy flow, the conservative PHS interconnection matrix  of an open system is found to be  ∇E(x)  u  T  µ  −P  σext  ˙Next  ˙Vext  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb  ˙S  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  −1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  ˙x  ˙Fn  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  −1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  ˙Fv  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  −1  Text  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  y  µext  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  −Pext  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  (36)  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Summary of the method and generalization route to irreversible systems  This section first summarises the main steps to derive a reversible conservative macro-  scopic PHS from a microscopic description of matter in an experimental context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Second,  a process is proposed to complete this modelling under a macroscopic irreversible con-  servative form, given a dissipation law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Reversible conservative macroscopic PHS  In summary, the macroscopic PHS can be described from the microscopic description  by completing the following steps:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Microstate representation Define P, W = P⋆ and M ⊆ W equipped with char-  acterizing functions F = {Fi : M �→ Fi}i∈I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Experimental conditions and accessible microstates  (a) Partition F = Ffixed ∪ Ffree into the set Ffixed of functions the values of which  are physically constrained by the experiment and its complement Ffree, with  corresponding sets of indices Ifixed and Ifree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  (b) Denote θfixed ⊂ ×i∈IfixedFi the set of experimentally admissible values for  functions in Ffixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  (c) Denote Ma  �  θfixed�  the corresponding set of admissible microstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Stochastic description For all probability distributions p : Ma  �  θfixed�  �−→ [0, 1],  (a) Derive the surprisal Sb  p : m ∈ Ma  �  θfixed�  �−→ logb  1  p(m) ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  (b) Derive the statistical entropy function Sk : p �−→ Ep  �  Sb=exp(1/k)  p  �  ∈  �  0, 1  Ω  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Boltzmann principle for ergodic systems at thermodynamic equilibrium  16  (a) Introduce θfree :=  �  Fi  �  i∈Ifree the values of functions in Ffree observed at a  macroscopic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  (b) Define p⋆ �  m | H  �  θfree��  according to Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  (c) Define the thermodynamic entropy function Sk : θfree �→ Sk  �  p⋆ �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' | H  �  θfree���  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  For common experimental constraints (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=', constraints on F =  �  Fe, Fn, Fv, Sb  p  �  ), we  obtain the results in Table 2 (see also [30]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  Note that if there is no analytic solution for Fe �→ Sk(Fe, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' ) and its inverse (note  that ∂Sk  ∂Fe  is monotonic), approximation strategies can be used (see [31] for an example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  17  Table 2: Statistical ensembles and associated constraints for usual experimen-  tal conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Ω denotes the cardinal of Ma (set of accessible microstates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  Ensemble  θfixed  θfree  p⋆(m)  Entropy  Example  Micro-canonical  (E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' S)  1  Ω  kB ln Ω  Gas in an isolated tank  Isoenthalpic-isobaric  (N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' S)  �  Fe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Fv  �  1  Ω  kB ln Ω  Gas in a closed tank  No  with a piston,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  thermally insulated  thermal contact  Unnamed  (V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' S)  �  Fe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Fn  �  1  Ω  kB ln Ω  Gas in a porous tank,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  thermally insulated  Unnamed  S  �  Fe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Fn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Fv  �  1  Ω  kB ln Ω  Gas in a porous tank  with a piston,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  thermally insulated  Thermal contact  Canonical  (N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' V )  Fe  exp  �  − Fe(m)  kB T  �  Z(T )  kB ln Z(T ) + Fe  T  Gas in a closed tank,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  in contact with a thermostat  Isothermal-isobaric  N  �  Fe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Fv  �  exp  �  − Fe(m)+P Fv(m)  kB T  �  Z(T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' P )  kB ln Z(T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' P) + Fe+P Fv  T  Gas in a closed tank  with a piston,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  in contact with a thermostat  Grand-canonical  V  �  Fe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Fn  �  exp  �  − Fe(m)−µ Fn(m)  kB T  �  Z(T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' µ)  kB ln Z(T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' µ) + Fe−µ F n  T  Gas in a porous tank,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  in contact with a thermostat  Unnamed  �  Fe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Fn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Fv  �  exp  �  − Fe(m)+P Fv(m)−µ Fn(m)  kB T  �  Fe+P Fv−µ Fn  T  Gas in a porous tank  with a piston,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  in contact with a thermostat  18  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Irreversible conservative macroscopic PHS from a macroscopic dissipative law  The system (36) models some conservative reversible physics at macroscopic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' In  some cases, dissipative phenomena can be observed, for which laws are available only at  this scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  In this part, we assume that such a dissipative phenomenon is described by  (i) a flow-to-effort mapping law  zd : fd �→ ed = zd(fd) such that for all fd, zd(fd)⊺fd =: Pd ≥ 0,  (37)  (ii) interconnected to the conservative part according to matrix given by  ∇E(x)  z(w)  u  T  es  ed  σext  eext  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8f0  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fb  ˙x  ˙S  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  −1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  fs  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  Jx  −K  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  −Gx  w  fd  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  K⊺  Jw  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  −Gw  y  Text  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  fext  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  G⊺  x  G⊺  w  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  Jy  ,  (38)  (iii) the dissipated power Pd being totally converted into an entropy rate  σi = Pd/Td ≥ 0 where Td > 0,  (39)  denotes the instantaneous macroscopic temperature at which the phenomenon is  experienced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' The positivity of σi reflects the irreversible nature of dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  From zd, we form the irreversible thermodynamic converter with law  z : w =  �  f ⊺  d , Td  �⊺ �−→  \uf8ee  \uf8ef\uf8f0zd(fd)⊺, −zd(fd)⊺fd/Td  �  ��  �  −σi  \uf8f9  \uf8fa\uf8fb  ⊺  (40)  where −σi ≤ 0 accounts for the entropy rate incoming into the converter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  This law is conservative as z(w)⊺w = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Due to irreversibility (σi ≥ 0), it naturally  fulfills the second principle of thermodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  Finally, from (iii), the irreversible conservative thermodynamic macroscopic PHS is  given by  ∇E(x)  z(w)  u  T  es  −σi  ed  σext  eext  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb  ˙x  ˙S  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  −1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  −1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  fs  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  Jx  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  −K  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  −Gx  w  Td  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  fd  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  K⊺  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  Jw  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  −Gw  y  Text  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  fext  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  G⊺  x  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  G⊺  w  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  Jy  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  (41)  19  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Conclusion  In this paper, we revisited equilibrium SP in order to model complex systems with  numerous degrees of freedom as macroscopic PHS with a reduced number of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  Starting from the choice of a particle’s description and ad hoc characterizing functions,  we recalled how to derive the probability of a configuration of particles at equilibrium  based on given experimental conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' In the end, macroscopic variables are revealed  to be expectations of the chosen characterizing functions for this probability, and the  thermodynamic entropy to be a function of these macroscopic variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Provided that  the energy has been chosen as a characterizing function from the start, the macroscopic  energy can in turn be expressed as a function of the thermodynamic entropy and other  macroscopic variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Through the PHS formalism, experimental conditions are repre-  sented as an input flow that acts on the system so that the resulting output is an effort  shared with the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' With this formulation, the externality of the environment, as  well as its interactions with the system via exchanges of energy and entropy, are made  explicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  As a result, we proposed two PHS formulations for conservative open systems, a  reversible one (with no entropy creation), and an irreversible one (with entropy creation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  An immediate perspective would be to extend this work to non-equilibrium SP [24],  so that a macroscopic trajectory would not only be a succession of equilibrium states,  and experimental conditions could change faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  20  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Proof of Theorem 1  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' To solve Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' (18), we introduce Lagrange multipliers λ0 and λfree := (λi)i∈Ifree,  and optimize [32] the Lagrangian L defined by  L :  �  p, λ0, λfree�  �−→ Sk(p) + λ0  \uf8eb  \uf8ec  \uf8ed  �  m∈Ma(θfixed)  p(m) − 1  \uf8f6  \uf8f7  \uf8f8 +  �  i∈Ifree  λi  �  Ep[Fi] − Fi  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='1)  A necessary condition to optimize L is to solve δL  δp = 0, where δ denotes the functional  derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='1)-(14)-(10),  δL  δp = 0 ⇒  �  m∈Ma(θfixed)  \uf8eb  \uf8ed−k  �  ln p(m) + 1  �  + λ0 +  �  i∈Ifree  λi Fi(m)  \uf8f6  \uf8f8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  This is true in particular if p verifies  − k  �  ln p(m) + 1  �  + λ0 +  �  i∈Ifree  λi Fi(m) = 0  ∀m ∈ Ma  �  θfixed�  ⇒p(m) = exp  ��  i∈Ifree λi Fi(m)  k  �  exp  �λ0  k − 1  �  ∀m ∈ Ma  �  θfixed�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  A second necessary condition is to solve  ∂L  ∂λ0  = 0, which, combined to the first  condition, yields  ∂L  ∂λ0  = 0 ⇒  �  m∈Ma(θfixed)  p(m) = 1 ⇒  �  m∈Ma(θfixed)  exp  ��  i∈Ifree λi Fi(m)  k  �  = exp  �  1 − λ0  k  �  ,  so that for all m ∈ Ma  �  θfixed�  , p⋆(m) is of the form  �p⋆  �  m | H  �  θfree�  , λfree  �  =  exp  � �  i∈Ifree λiFi(m)  k  �  Z  �  λfree�  ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='2)  with Z  �  λfree�  :=  �  m∈Ma(θfixed)  exp  ��  i∈Ifree λi Fi(m)  k  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='3)  A third necessary necessary condition is to solve ∂L  ∂λi  = 0 for all i ∈ Ifree, which,  combined with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='3), yields  ∂L  ∂λi  = 0 ⇒Ep[Fi] = Fi  ⇒  �  m∈Ma(θfixed) Fi(m) exp  � �  i∈Ifree λi Fi(m)  k  �  Z  �  λfree�  = Fi  ⇒ ∂  ∂λi  k ln Z  �  λfree�  = Fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  21  We deduce that the optimal distribution p⋆ is  p⋆  �  m | H  �  θfree��  = �p⋆  �  m | H  �  θfree�  , λfree  �  ,  with λfree such that  ∂  ∂λi  k ln Z  �  λfree�  = Fi  ∀i ∈ Ifree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Proof of effort equality at the system interface  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Consider the isolated total system constituted by the system under study and its  environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' For all i ∈ Ifree, we have  F  total  i  = F  sys  i  + F  ext  i  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  The entropy is extensive, therefore,  Sk  total  �  θfree  total  �  = Sk  sys  �  θfree  sys  �  + Sk  ext  �  θfree  ext  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='1)  The total system is isolated, therefore the total entropy is maximal with respect to any  variable, so that for all i ∈ Ifree,  ∂Sk  total  ∂F  sys  i  = 0  ⇒ ∂Sk  sys  ∂F  sys  i  + ∂Sk  ext  ∂F  sys  i  = 0  ⇒ ∂Sk  sys  ∂F  sys  i  − ∂Sk  ext  ∂F  ext  i  = 0  ⇒λsys  i  − λext  i  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  Moreover, for all i ∈ Ix  ∂E  ∂Fi  = T λi,  since  Sk �  E(S, Fi), Fi  �  = S  ⇒ ∂Sk  ∂Fe  ∂E  ∂Fi  + ∂Sk  ∂Fi  = 0  ⇒ 1  T  ∂E  ∂Fi  − λi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  We deduce that ∂Esys  ∂F sys  i  = ∂Eext  ∂Fext  i  ∀i ∈ Ix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  22  Acknowledgments  The authors would like to thank Bernhard Maschke for his comments and helpful  discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  References  [1] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Boltzmann, Über die Beziehung zwischen dem zweiten Hauptsatze des mechanischen Wärmethe-  orie und der Wahrscheinlichkeitsrechnung, respective den Sätzen über das Wärmegleichgewicht, Kk  Hof-und Staatsdruckerei, 1877.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  [2] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Landsberg, Thermodynamics and statistical mechanics, Courier Corporation, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  [3] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Eberard, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Maschke, Port-Hamiltonian systems extended to irreversible systems: The example  of the heat conduction, IFAC Proceedings Volumes 37 (13) (2004) 243–248.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  [4] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Eberard, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Maschke, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' van der Schaft, An extension of Hamiltonian systems to the thermody-  namic phase space: Towards a geometry of nonreversible processes, Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' 60 (2) (2007)  175–198.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  [5] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Ramirez, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Maschke, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Sbarbaro, Irreversible port-Hamiltonian systems: A general formulation  of irreversible processes with application to the CSTR, Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' 89 (2013) 223–234.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  [6] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Delvenne, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Sandberg, Finite-time thermodynamics of port-Hamiltonian systems, Physica D  267 (2014) 123–132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  [7] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Ramirez, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Le Gorrec, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Maschke, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Couenne, On the passivity based control of irreversible  processes: A port-Hamiltonian approach, Automatica 64 (2016) 105–111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  [8] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Van Der Schaft, Classical thermodynamics revisited: A systems and control perspective, IEEE  Control Systems Magazine 41 (5) (2021) 32–60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  [9] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' van der Schaft, Liouville geometry of classical thermodynamics, Journal of Geometry and Physics  170 (2021) 104365.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  [10] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Gorban, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Kazantzis, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Kevrekidis, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Öttinger, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Theodoropoulos, Model  reduction and coarse-graining approaches for multiscale phenomena, Springer, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  [11] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Öttinger, Systematic coarse graining:“Four Lessons and A Caveat” from nonequilibrium sta-  tistical mechanics, MRS bulletin 32 (11) (2007) 936–940.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  [12] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Ising, Beitrag zur theorie des ferromagnetismus, Zeitschrift für Physik 31 (1) (1925) 253–258.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  [13] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Strecka, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Jascur, A brief account of the Ising and Ising-like models: Mean-field, effective-field  and exact results, arXiv preprint arXiv:1511.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='03031 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  [14] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Hopcroft, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Motwani, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Ullman, Introduction to automata theory, languages, and com-  putation, Acm Sigact News 32 (1) (2001) 60–65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  [15] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Flajolet, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Sedgewick, Analytic combinatorics, Cambridge University Press, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  [16] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Liechtenstein, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Katsnelson, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Gubanov, Exchange interactions and spin-wave stiffness in  ferromagnetic metals, Journal of Physics F: Metal Physics 14 (7) (1984) L125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  [17] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Grothendieck, Récoltes et Semailles (Réflexions et témoignage sur un passé de mathématicien),  Gallimard (2022), 1986.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  [18] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Plachky, An ideal theoretic characterization of finite sets, finite algebras, and σ-algebras of  countably generated type, Mathematica Slovaca 51 (3) (2001) 301–311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  [19] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Gray, Entropy and information theory, Springer, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  [20] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Knuth, Dynamic Huffman coding, Journal of algorithms 6 (2) (1985) 163–180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  [21] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Patrascioiu, The ergodic-hypothesis: A complicated problem in mathematics and physics, Los  Alamos Science 15 (1987) 263–279.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  [22] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Zia, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Redish, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' McKay, Making sense of the Legendre transform, American Journal  of Physics 77 (7) (2009) 614–622.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  [23] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Van der Schaft, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Maschke, Geometry of thermodynamic processes, Entropy 20 (12) (2018) 925.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  [24] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Öttinger, Beyond Equilibrium Thermodynamics, John Wiley & Sons, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  [25] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Duindam, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Macchelli, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Stramigioli, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Bruyninckx, Modeling and control of complex physical  systems: The port-Hamiltonian approach, Springer Science & Business Media, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  [26] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' van der Schaft, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Jeltsema, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=', Port-Hamiltonian systems theory:  An introductory  overview, Foundations and Trends in Systems and Control 1 (2-3) (2014) 173–378.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  [27] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Falaize, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Hélie, Passive guaranteed simulation of analog audio circuits: A port-Hamiltonian  approach, Applied Sciences 6 (10) (2016) 273.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  [28] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Müller, Time-continuous power-balanced simulation of nonlinear audio circuits: Realtime pro-  cessing framework and aliasing rejection, Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' thesis, Sorbonne Université (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  23  [29] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Bigelow, Power and energy in electric circuits, in: Electric Circuits, Systems, and Motors,  Springer, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' 105–121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  [30] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Graben, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Ray, Unified treatment of adiabatic ensembles, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' A 43 (8) (1991) 4100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  [31] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Najnudel, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Hélie, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Roze, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Müller, From statistical physics to macroscopic port-Hamiltonian  systems: A roadmap, in: 7th IFAC Workshop on Lagrangian and Hamiltonian Methods for Non-  linear Control, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  [32] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content=' Jaynes, On the rationale of maximum-entropy methods, Proceedings of the IEEE 70 (9)  (1982) 939–952.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
+page_content='  24' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE5T4oBgHgl3EQfMw71/content/2301.05485v1.pdf'}
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